0 x>ap (vi) If 0 < a < 1 then log a x>p => 0 4 [JEE 2004 (Screening)] "£ 1 [ JEE 2001 Screening, 1 +1 out of 35 ]
EXERCISE-I Q.l
If the roots of the equation [ 1 /(x+p)] + [ 1 /(x+q)] = 1/r are equal in magnitude but opposite in sign, show that p + q = 2r&that the product of the roots is equal to (-l/2)(p 2 + q2).
Q.2
If x2 - x cos (A + B) + 1 is a factor of the expression, 2x4 + 4x3 sin A sin B - x2 (cos 2A + cos 2B) + 4x cos A cos B - 2 . Then find the other factor.
Q.3
a , (3 are the roots of the equation K(x 2 -x)+x+5 = 0 . If K, & K2 are the two values of K for which the roots a , (3 are connected by the relation (a/p) + (p/a) = 4/5 . Find the value of (K(/K2) + (K2/Kj) .
Q.4
If the quadratic equations, x2 + bx + c = 0 and bx2 + cx + 1 = 0 have a common root then prove that either b + c + l = 0 o r b 2 + c 2 + l = b c + b + c. f
2
P P2^ X2 +p(l+q)x + q(q-l) + — = 0 are equal then show that
Q.5
If the roots of the equation Il-q+—
Q.6
p2 = 4q root . of the equation ax2 + bx + c = 0 be the square of the other, prove that If one b3 + a2c + ac2 = 3abc.
Q. 7 Q.8
Q.9
ax2 +2(a + l)x + 9a + 4 Find the range of values of a, such that f (x) = ^ - 8 x + 32
is a l w a y s ne
gative-
Let a, b, c be real. If ax2 + bx + c = 0 has two real roots a & P, where a < - 1 & P > 1 then show that 1 + c/a + | b/a | < 0. 6x 2 - 2 2 x + 21 , , . Find the least value of — ; for all real values of x, using the theory of quadratic 5x -18x + 17 equations.
ii Bansal Classes
Quadratic Equations
[9]
Q.IO
Find the least value of (2p2 + 1 )x2 + 2(4p2 - 1 )x + 4(2p2 + 1) for real values of p and x.
Q.ll
If a be a root of the equation 4x2 + 2x - 1 = 0 then prove that 4a 3 - 3a is the other root.
Q.12
If the equations x 2 +px + q = 0 & x2 + p'x + q' = 0 have a common root, show that it must be equal to (pq'-p'q)/(q-q') or (q-q')/(p'-p) •
Q.13
If a,(3 are the roots of ax 2 +bx + c = 0 & a ' , ~ P are the roots of a'x 2 +b'x + c' = 0, showthat -1 -l "b b' "b b'" 2 + — x +x + + — =0 a , a ' are the roots of a a' c c' —
—
Q.14
If a , P are the roots of x2 - px + 1 = 0 & y, 5 are the roots of x2 + qx + 1 = 0 , show that ( a - y ) ( P - y ) ( a + 5)(P + 8) = q 2 - p 2 .
Q.15
Show that if p , q , r & s are real numbers & pr = 2(q+s), then at least one of the equations x2 + px + q = 0, x 2 + r x + s = 0 has real roots .
Q.16
If a & b are positive numbers, prove that the equation — + —'— + x x-a x+b one between a/3 & 2a/3 and the other between - 2b/3 & - b/3 .
= 0 has two real roots,
If the roots of x2 - ax + b = 0 are real & differ by a quantity which is less than c (c > 0), prove that b lies between (1/4) (a2 - c2) & (l/4)a 2 . At what values of'a' do all the zeroes of the function, f (x) = (a - 2) x2 + 2 a x + a + 3 lie on the interval ( - 2,1)? If one root of the quadratic equation ax2+ bx + c = 0 is equal to the nth power of the other, then show that (acn)1/(n+l) + (a 1 ^) 1 ^ 1 ) + b = 0 .
r3 - 5 ^ ' s4 q4 ? q J - 5 p - 2 ' p-2 q - 2 ' q-2 r-2 r - 2 / and \ s - 2 pqrs = 5(p + q + r + s) + 2 (pqr + qrs + rsp + spq).
V
P
s3-5A s - 2 y are collinear if
Q.21
The quadratic equation x2 + px + q = 0 where p and q are integers has rational roots. Prove that the roots are all integral.
Q.22
If the quadratic equations x 2 +bx+ca = 0 & x 2 +cx+ab = 0 have a common root, prove that the equation containing their other root is x2 + ax + be = 0 .
Q.23
If a , p are the roots of x 2 +px+q = 0 & x 2n +p n x n + qn = 0 where n is an even integer, show that a/p, p/a are the roots of xn +1 + (x + l) n = 0 .
Q.24
If a , p are the roots of the equation x2 - 2x + 3 = 0 obtain the equation whose roots are a 3 - 3 a 2 + 5a - 2 , p 3 - p 2 + p + 5.
Q.25
If each pair of the following three equations x2 + p 1 x+q 1 = 0 ,x 2 +p 2 x + q 2 =0 & x 2 +p 3 x + q3 = 0 has exactly one root common, prove that ; (PI + P2 + P3)2 = 4 [P1P2 + P2P3 + P3P1 ~ ~ % ~ •
Q.26
Show that the function z = 2x 2 +2xy+y 2 -2x + 2y+2 is not smaller than - 3 .
Q.27
If (1/a) + (1/b) + (1/c) = l/(a+b + c) & n is an odd integer, show that ; (l/an) + (l/b n ) + (l/c n ) = l/(an + bn + cn) .
ii Bansal Classes
Quadratic Equations
[9]
Q.28
Find the values o f ' a ' f o r which - 3 < [ ( x 2 + a x - 2 ) / ( x 2 + x + l ) ] < 2 is valid for all real x.
Q.29
b ( 6 11 > -2 X + ~6 1 x°J \ x; Find the minimum value of ™—-—-——r ^' forx>0 / 11 \ 3 1 + X +—rXH xJ V
(X H
0 •30
0
—
Let f (x) = ax2 + bx + c = 0 has an irrational root r. If u = — be any rational number, where a, b, c, p and 1 « q are integer. Prove that — < | f (u) |. q
EXERCISE-II Q.l (a) (c) (d) (f) (h)
Solve the following where x e R . ( x - l ) | x 2 - 4 x + 3| + 2 x 2 + 3 x - 5 = 0 (b) 3 I x2 - 4x + 2 | = 5x - 4 For a < 0, determine all real roots of the equation x 2 - 2 a | x - a j - 3 a 2 = 0. |x 2 + 4 x + 3 | + 2 x + 5 = 0 (e) fx + 3). |x + 2 | + |2x+31 + 1 = 0 | (x + 3) |. (x +1)+12x + 5 | = 0 (g) | x 3 + 1 1 + x2 - x - 2 = 0 2 |x+21 - |2X+1 - 1 | = 2 X + 1 +1
Q.2
Let a, b, c, d be distinct real numbers and a and b are the roots of quadratic equation x2 - 2cx — 5d=0. If c and d are the roots of the quadratic equation x2 - 2ax - 5 6 = 0 then find the numerical value of a + b + c + d.
Q.3
Find the true set of values of p for which the equation p • 2cosZx + p • 2'cos2x - 2 = 0
has real roots.
Q.4
Prove that the minimum value of [(a+x)(b+x)]/(c+x),x>-c is U a - c + 7b-cj .
Q.5
If Xj, x2 be the roots of the equation x2 - 3x + A = 0 & x 3 , x 4 be those of the equation x2 - 12x + B = 0 & X j , x 2 , x 3 , x 4 are in GP. Find A & B .
Q.6
If ax 2 +2bx + c = 0 & a,x2 + 2b 1 x + c1 = 0 have a common root & a/a 1 ,b/bj,c/c, are inAP, show that a t , bj & c t are in GP .
Q.7
If by eleminating x between the equation x 2 +ax+b = 0& xy+/(x+y) + m = 0 , a quadratic in y is formed whose roots are the same as those of the original quadratic in x . Then prove either a = 21 & b = m or b + m = a / . . ,« 2a cos x -2xcosa + l 7 ~ — lies between — and n —r x — 2x cos (3 + 1 , (3 23 sin — cos 2
Q.8
If x be real, prove that
sin
2
2
2
2
2
2
Q.9
Solve the equations, ax +bxy + cy = bx + cxy+ ay = d .
Q.10
Find the values of K so that the quadratic equation x 2 + 2 ( K - l ) x + K + 5 = 0 has atleast one positive root.
Q.ll
Findtiievalues of *b'for which the equation 2 log , (bx + 28) = -log 5 (l2-4x-x 2 jhasonlyonesolution. 25
ii Bansal Classes
Quadratic Equations
[9]
Q.12
Find all th e v aiues of the parameter 'a' for which both roots of the quadratic equation x2 - ax + 2 = 0 belong to the interval ( 0 , 3 ) .
Q.13
Find all the values ofthe parameters c for which the inequality has at least one solution.
Q.14
1 + log2 2x2 + 2x + - > log2 (cx2 + c) . \ 2) " Find the values of K for which the equation x4 + (1 - 2 K) x2 + K2 - 1 = 0 ; (a) has no real solution (b) has one real solution
Q.15
Find the values of p for which the equation 1 + p sin x = p2 - sin2 x has a solution.
Q.16
Solve the equation
-4.3" | x ~ 2 ' - a = 0 for every real numbera.
Q.17 Find the integral values of x & y satisfying the system of inequalities; y - 1 x 2 -2x | + (1/2)> 0 & y+1 x - 1 1 <2 . Q.18
Find all numbers p for each of which the least value of the quadratic trinomial 4x2 - 4px + p2 - 2p + 2 on the interval 0 < x < 2 is equal to 3.
Q.19 If the coefficients of the quadratic equation ax2 + bx + c = 0 are odd integers then prove that the roots of the equation cannot be rational number. Q.20
The equation xn + px2 + qx + r = 0, where n > 5 & r ^ 0 has roots a } , a 2 , a 3 , n
Denoting
an.
a f k by Sk.
il Calculate S7 & deduce that the roots cannot all be real. Prove that Sn + pS2 + qS, +nr = 0 & hence find the value of S n .
(i) (ii)
EXERCISE-III Solve the inequality. Where ever base is not given take it as 10.
Q.3
' x5'2 (log2x)4- l o g l T 201og 2 x+148 < 0. 2 y 2 (log 100 x) + (log 10 x) 2 +log x < 14
Q.5
logx2 . log2x2 . log2 4x > 1.
Q.7
log1/2x + log 3 x>l.
Q.l
Q.2
x 1 / l o 8 x .logx
Q.4
log 1 / 2 (x+l)>log 2 (2-x).
Q.6
log1/5 (2x 2 + 5 x + 1)<0.
Q.8
log x2 (2+x)< 1
4 x 4- S
Q-9 l o g Q 11Q.13
x
Q.10 (log| x+6 |2). log2 ( x 2 - x - 2 ) > 1
^ < - l
Q12.
X tX
log [(x+6)/3] [log 2 {(x-l)/(2+x)}]>0
Find out the values of 'a' for which any solution of the inequality,
~ < 1 is a l s o a
solution of the inequality, x2 + (5 - 2 a) x < 10a. Solve the inequality log
Q.15
Find the set of values of'y' for which the inequality, 2 log0 5 y2 - 3 + 2 x log0 5 y2 - x2 > 0 is valid for atleast one real value of'x'.
ii Bansal Classes
N (x
2
Q.14
-10x + 22) > 0 .
Quadratic Equations
[9]
EXER Q. 1
Prove that the values of the function s i n x
cos
CISE-IV 3 do not lie from -1 & 3 for any real x.
sin3x cosx
3
2
.
[JEE '97,5] [JEE '97,2]
Q.2
The sum of all the real roots of the equation |x - 2| + |x - 2| - 2 = 0 is
Q.3
Let S be a square of unit area. Consider any quadrilateral which has one vertex on each side of S. If a, b, c & d denote the lengths of the sides of the quadrilateral, prove that : 2 < a2 + b2 + c2 + d2 < 4. [JEE '97,5]
Q.4
In a college of 300 students, every student reads 5 news papers & every news paper is read by 60 students. The number of news papers is: (A) atleast 30 (B) atmost 20 (C) exactly 25 (D) none ofthe above [JEE'98,2]
Q.5
If a, p are the roots of the equation x2 - bx + c = 0, then find the equation whose roots are, (a 2 + p2) (a 3 + p3) & a 5 p3 + a 3 p5 - 2a 4 p4. [REE'98,6]
Q.6(i) Let a + ip ; a, p e R, be a root of the equation x3 + qx + r = 0; q, r e R. Find a real cubic equation, independent of a & p, whose one root is 2 a . (ii) Find the values o f a & p , 0 < a , P< n/2, satisfying the following equation, cos a cos p cos (a + p) = - 1 / 8 . Q.7(i) In a triangle PQR, ZR = ~ . If tan ( ^ j & tan ax2 + bx + c = 0 (a*0) then: (A) a + b = c (B) b + c = a
[REE '99,3 + 6]
are the roots of the equation (C)a + c = b
(D)b = c
(ii) If the roots of the equation x2 - 2ax + a2 + a - 3 = 0 are real & less than 3 then (A) a < 2 (B) 2 < a < 3 (C)3
If a, p are the roots of the equation, (x - a) (x - b) + c = 0, find the roots of the equation, (*-
(b)
If a & p (a < P), are the roots ofthe equation, x2 + bx + c = 0, where c < 0 < b, then (A) 0 < a < p (B) a < 0 < p < I a | (C) a < p < 0 (D) a < 0 < | a | < p
(c)
If b > a, then the equation, (x - a) (x - b) - 1 = 0, has : (A) both roots in [a, b] (B) both roots in ( - oo, a) (C) both roots in [b, oo) (D) one root in ( - oo, a) & other in (b, + oo) [JEE 2000 Screening, 1 + 1 + 1 out of 35]
faBansal Classes
Quadratic Equations
[8]
(d)
If a, P are the roots of ax2 + bx + c - 0, (a * 0) and a + 5, P + 5, are the roots of, Ax2 + Bx + C = 0, (A * 0) for some constant 5, then prove that, h 2 — 4a r
3. Q.10
=
R2 — 4 A C
A
2
•
[JEE 2000, Mains, 4 out of 100]
The number of integer values of m, for which the x co-ordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer, is (A) 2 (B) 0 (C)4 (D)l [JEE 2001, Screening, 1 out of 35]
Q. 11 Let a, b, c be real numbers with a * 0 and let a, p be the roots of the equation ax2 + bx + c = 0. Express the roots of a3x2 + abcx + c3 = 0 in terms of a, p. [JEE 2001, Mains, 5 out of 100] 2 Q.12 The set of all real numbers x for which x - |x + 2j + x > 0, is (A) (-a>, -2) U (2, oo)
(B) (-oo, - V 2 ) U (V2 , 00)
(C) (-00,-1) 11 (1, 00)
(D) (V2,00)
[JEE 2002 (screening), 3]
Q. 13 If x2 + (a - b)x + (1 - a - b) = 0 where a, b e R then find the values of 'a' for which equation has unequal real roots for all values of 'b'. [JEE-03, Mains-4 out of 60] [ Based on M. R. test] Q.14 (a) If one root of the equation x2 + px + q = 0 is the square of the other, then (A) p3 + q2 - q(3p + 1) = 0 (B) p3 + q2 + q(l + 3p) = 0 3 2 (C) p + q + q(3p - 1) = 0 (D) p3 + q2 + q(l - 3p) = 0 (b)
If x2 + 2ax + 10 - 3a> 0 for all x e R, then (A) - 5 < a < 2 (B) a < — 5
ii Bansal Classes
(C)a>5
Quadratic Equations
(D)2
[9]
ANSWER KEY EXERCISE-I Q.2
2x 2 + 2x cos (A - B) - 2
Q.IO
minimum value 3 when x = 1 and p = 0
Q.24 x 2 - 3 x + 2 = 0
Q.3 254 *
Q.7
ae
-oo,--
u{2}u(5,6]
Q.18
Q.28 - 2 < a < 1
Q.9 1
Q-29ymin = 6
EXERCISE-II Q.l
(a) x = l (b) x = 2 or 5 (c) x = ( l - f i ) a o r (Ve - l ) a (d) x = - 4 o r - ( 1 + V3) (e) x = ( - 7 - V l 7 ) / 2
Q.3
(f) x = - 2 or - 4 o r - ( l + V ^ ) "4
:,1
(g) x = - l or 1 (h) x > - l or x = - 3
Q.2
30
Q.9
x2 = y2 = d/(a+b+c) ; x/(c - a) = y/(a - b) = K where K2a (a 2 +b 2 +c 2 - ab - be - ca) = d
Q.5 A = 2 or - 1 8 ,
14
Q 12. 2V2 < a <
00
Q 10. K < - 1
Q 11. ( - 0 0 , - 1 4 ) u {4} u
Q.13
Q 14. (a) K < - 1 or K > 5/4 (b) K = - 1
(0,8]
Q 16. x 1 2 = 2 ± l o g 3 ( 2 - V 4 + ^ ) where - 3 < a < 0
Q18.
B = 32or-288
11
Q 1 5 . - 2 < p < - 2 / V ? or 2/VJ < p < 2
Q 17. ( 0 , 0 ) ; ( 1 , 1 ) ; (2,0)
Q 20. S2 = 0, Sn = - nr
a = 1 - V2 or 5 + VlO
EXERCISE-III QL.
X 6
r 11 11 a u (8,16) V 16
Q 4. - 1 < x <
8y
1 - V?
Q 3.
Q 2. (0,1) u (1, 101/10)
1 + V?
Q 5 . 2"V2 < x < 2 - ' ; 1 < x <
or — j — < x < 2
Vio9
Q 6. (-^0,-2.5)u(0,00)
Q 7. 0 < x < 31/1-1083 (where base of log is 2)
Q 8. - 2 < x < - l , - l < x < 0 , 0
Q 10. x < - 7 , - 5 < x < - 2 , x > 4
Qll. x < - | ;
T
< x < 10
Q 9. - < x < 1
Q12. ( - 6 , - 5 ) U ( - 3 , - 2 )
j
Q13. a > | Q.2 4
Q 14. xg(3,5-V3)u(7,oo) Q15. ( - c o , - 2 V 2 ) u ( - ^ , o ) Q.4 C
u(2V2,oo)
EXERCISE-IV
2
Q.5 x - (x, + x 2 ) x + x, x 2 = 0 where x, = (b2 - 2c) (b3 - 3cb); x 2 = c3 (b2 - 4c)
Q.6 (i) x3 + qx - r = 0, (ii) a = p = TT/3, Q.7 (i) A, (ii) A, 2 2 Q.10 A Q.ll y = a p and 6 = a.p or y = ap 2 and 8 = a 2 P Q.14 (a) D ; (b)A
Q.8 (a, b) Q.9 (a) C, (b) B, (c) D Q.12 B Q.13 a > l
m
ii Bansal Classes
Quadratic Equations
[9]
BANSAL CLASSES *
MATHEMATICS TARGETIIT JEE 2007 XI (P, Q,R, S)
SEQUENCE & PROGRESSION
CONTENTS KEY CONCEPTS EXERCISE-I EXERCISE-II EXERCISE-III ANSWER KEY
KEY
CONCEPTS
DEFINITION: A sequence is a set of terms in a definite order with a rule for obtaining the terms, e.g. 1,1/2,1/3, ,1/n, is a sequence. AN ARITHMETIC PROGRESSION (AP): AP is a sequence whose terms increase or decrease by afixednumber. Thisfixednumber is called the common difference. If a is the first term & d the common difference, then AP can be written as a, a + d, a + 2d, a + ( n - l)d, nth term of this AP tn = a + (n - 1 )d, where d = an - a n l . The sum ofthe first n terms of the AP is given by ; Sn =
[2 a+ (n - 1 )d] = ^ [a+/].
where I is the last term. NOTES: (i) If each term of an A.P. is increased, decreased, multiplied or divided by the same non zero number, then the resulting sequence is also an AP. (ii) Three numbers in AP can be taken a s a - d , a, a+d; four numbers in AP can be taken as a - 3 d, a - d , a+d, a+3d; five numbers in APare a - 2 d , a - d , a, a+d, a+2d & six terms in APare a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d etc. (iii)
The common difference can be zero, positive or negative.
(iv)
The sum of the two terms of an AP equidistant from the beginning & end is constant and equal to the sum of first & last terms.
(v)
Any term of an AP (except thefirst)is equal to half the sum of terms which are equidistant from it.
(vi)
tr=Sr-SM
(vii)
If a, b, c are in AP => 2 b = a + c.
GEOMETRIC PROGRESSION (GP): GP is a sequence of numbers whosefirstterm is non zero & each of the succeeding terms is equal to the proceeding terms multiplied by a constant. Thus in a GP the ratio of successive terms is constant. This constant factor is called the COMMON RATIO of the series & is obtained by dividing any term by that which immediately proceeds it. Therefore a, ar, ar2,ar3, ar4, is a GP with a as thefirstterm & r as common ratio. (i) nth term = ar n _ 1 (ii) (iii)
a(r n -l) Sum of the I n terms i.e. Sn = — , if r * 1 . r-1 Sum of an infinite GP when | r | < 1 when n—» oo rn —> 0 if | r | < 1 therefore, st
S ^ f l r K l ) . (iv)
If each term of a GP be multiplied or divided by the same non-zero quantity, the resulting sequence is also a GP.
(v)
Any 3 consecutive terms of a GP can be taken as a/r, a, ar ; any 4 consecutive terms of a GP can be taken as a/r3, a/r, ar, ar3 & so on.
(vi)
If a, b, c are in GP => b2 = ac.
Bansal Classes
Sequence & Progression
[2]
HARMONIC PROGRESSION (HP): A sequence is said to HP if the reciprocals of its terms are in AP. If the sequence a,, a2, a 3 ,...., an is an HP then l/a l5 l/a 2 ,...., l/an is an AP & converse. Here we do not have the formula for the sum of the n terms of an HP. For HP whose first term is a & second term isb, then111 term is tn =
b + (n-l)(a-b)
If a, b, c are in HP => b =
2ac a + c
a
a - b
c
d- c
or — = T
.
MEANS ARITHMETIC MEAN: If three terms are in AP then the middle term is called the AM between the other two, so if a, b, c are in AP, b is AM of a & c . AM for any n positive number a,, a 2 ,..., an is ; A = a ' + a 2 + a ^ +
+a
" .
n-ARITHMETIC MEANS BETWEEN TWO NUMBERS : Ifa,b are any two given numbers & a,A15A2,.... ,An, b are inAP thenA,, A2, ...Anare then AM's between a & b . A1 = a + ^ - , A22 = a + ^ i l n + 1 '
= a + d,
n+ 1
=a+2d ,
,An= a +
,
'
'
n (b - a) n +1 b-a
, A = a + nd, where d = — n
NOTE
n+1
: Sum of n AM's inserted between a & b is equal to n times the single AM between a & b i.e. Xn Ar = nA where A is the single AM between a & b. r= l
GEOMETRIC
MEANS:.
If a, b, c are in GP, b is the GM between a & c. b2 = ac, therefore b = Ja c ; a > 0, c > 0. n-GEOMETRIC MEANS BETWEEN a, b : If a, b are two given numbers & a, G}, G2, , Gn, b are in GP. Then Gj, G2, G 3 ,...., Gn are n GMs between a & b . G, = a(b/a)1/n+1, G2 = a(b/a)2/n+1, = ar , = ar 2 , NOTE
, Gn = a(b/a)n/n+1 = arn, where r = (b/a)1/n+1
: The product of n GMs between a & b is equal to the nth power of the single GM between a & b n
i.e. ^ G r =(G) n where G is the single GM between a & b. HARMONIC MEAN : If a, b, c are in HP, b is the HM between a & c, then b = 2ac/[a+c]. THEOREM: If A, G, H are respectively AM, GM, HM between a & b both being unequal & positive then, (i) G2 = AH (ii) A > G > H (G > 0). Note that A, G, H constitute a GP.
fa B ansa/ Classes
Sequence & Progression
[3]
ARITHMETICO-GEOMETRIC SERIES: A series each term of which is formed by multiplying the corresponding term of an AP & GP is called the Arithmetico-Geometric Series, e.g. 1 + 3x + 5x2 + 7x3 + Here 1,3,5,.... are in AP& l,x,x 2 ,x 3 areinGP. Standart appearance of an Arithmetico-Geometric Series is Let Sn = a + (a + d)r + (a + 2 d) r2 + + [a + ( n - l ) d ] r""1 SUM TO INFINITY : If
|r|
<1
& n —»oo then
Limit r n
= 0
a . S . - — + •
SIGMA
dr 2
(1-r)
.
NOTATIONS
THEOREMS : (i)
Z (ar±br)=£ a r ± £
r=l
(ii)
I
r= 1
ka r • k ±
r=1
(iii)
b,
r=1
r
ar.
r=1
]T k = nk ; where k is a constant. r=1
RESULTS (i)
n r=
X
n (n + 1)
— 9 — (sum of the first n natural nos.) 2
r=l
= n
(ii)
Y
(iii)
£ r3 = T=\
(iv>
z
r=l '
(n+1)
^ s u m 0 f the squares of the first n natural numbers)
( + 1)2
"
4
r= 1
(sum of the cubes of the first n natural numbers)
r4 = - ( n + l)(2n+l)(3n 2 + 3 n - l )
METHOD OF DIFFERENCE : If T j , T 2 , T 3 , , T N are the terms of a sequence then some times the terms T 2 - T j , T 3 - T 2 , constitute an AP/GP. nth term of the series is determined & the sum to n terms of the sequence can easily be obtained. Remember that to find the sum of n terms of a series each term of which is composed of r factors in AP, the first factors of several terms being in the same AP, we "write down the nth term, affix the next factor at the end, divide by the number of factors thus increased and by the common difference and add a constant. Determine the value of the constant by applying the initial conditions".
fa B ansa/ Classes
Sequence & Progression
[3]
EXERCISE-I Q 1.
If the 1 Oth term of an HP is 21 & 2 V* term ofthe same HP is 10, then find the 210th term. logio x + log10 x 1/2 + log,6 X1/4 + 1+3+5+ +(2y-l) 4+7+10+ +(3y+l)
=y 20 71og]0 x
Q 2.
Solve the following equations for x & y,
Q 3.
There are n AM's between 1 & 31 such that 7th mean: (n - 1 )th mean = 5 : 9 , thenfindthe value of n.
Q 4.
Find the sum of the series, 7 + 77 + 777 +
Q 5.
Express the recurring decimal 0.1576 as a rational number using concept of infinite geometric series.
Q 6.
Find the sum of the n terms of the sequence
Q 7.
Thefirstterm of an arithmetic progression is 1 and the sum of thefirstnine terms equal to 369. The first and the ninth term of a geometric progression coincide with thefirstand the ninth term ofthe arithmetic progression. Find the seventh term of the geometric progression.
Q 8. Q 9.
to n terms.
1 2
1 + 1 +1
2 4
2
1+2 +2
3 4
1 + 32 +3 4
r If the pth, qth & rth terms of an AP are in GP. Show that the common ratio of the GP is q~
p-q
If one AM 'a' & two GM's p & q be inserted between any two given numbers then show that p3+ q3 = 2 apq .
Q 10. Thesumofn terms oftwo arithmetic series are in the ratio of (7 n+1): (4 n+27). Find the ratio oftheir nth term. Q l l . If S be the sum, P the product & R the sum of the reciprocals of a GP, find the value of p2
!)•
Q 12. The first and last terms of an A.P. are a and b. There are altogether (2n + 1) terms. A new series is formed by multiplying each of thefirst2n terms by the next term. Show that the sum ofthe new series is (4n2 - l)(a2 + b 2 ) + (4n 2 + 2)ab 6n Q 13. In an AP of which' a' is the 1st term, if the sum of the 1st p terms is equal to zero, show that the sum of the next q terms is - a (p + q) q/(p -1). Q 14. The interior angles of a polygon are in AP. The smallest angle is 120° & the common difference is 5°. Find the number of sides of the polygon, Q 15. An AP & an HP have the samefirstterm, the same last term & the same number of terms; prove that the product of the V th term from the beginning in one series & the rth term from the end in the other is independent of r. Q 16. Find three numbers a, b, c between 2 & 18 such that ; (i) their sum is 25 (ii) the numbers 2, a, bare consecutive terms of an AP & (iii) the numbers b, c, 18 are consecutive terms of a GP . Q 17. Given that ax = by = cz = du & a , b , c , d are in GP,show that x , y , z , u a r e i n H P . Q 18. In a set of four numbers, thefirstthree are in GP & the last three are in AP, with common difference 6. If thefirstnumber is the same as the fourth,findthe four numbers. Q 19. Find the sum of the first n terms of the sequence: l + 2(1+—] +3[1+—] +4(1+—] +
fa B ansa/ Classes
Sequence & Progression
[3]
Q 20. Find the nth term and the sum to n terms of the sequence ; (i) 1 + 5 + 13+29 + 61 + (ii) 6+13 + 22 + 33 + Q 21. The AM of two numbers exceeds their GM by 15 & HM by 27 . Find the numbers. Q 22. The harmonic mean of two numbers is 4. The airthmetic mean A & the geometric mean G satisfy the relation 2 A + G2 = 27. Find the two numbers. Q 23. Sum the following series to n terms and to infinity: (i) w
—5—;+—— + — — + 1.4.7
4.7.10
(ii)
7.10.13
I
r_i
n i © I 7^77 r=1 4r - 1 Q 24. Find the value ofthe sum n
(a) Y v
iti
""""' r=l
n
y
02
rs
r (r+ l)(r+2) (r + 3)
1 1.3 1.3.5 -4+ - 4.6 — + 4.6.8
r s
3 where 5 is zero if r ^ s&8 r is one if r = s. rs
rs
/
(b) i i i i. i=i j=i k=i Q 25. For or 0 < 4> < 7t/2, if : oo
co
x = Z cos2n (j), y = Z n=0
s n2n
i
^ > z = 1l cos2n (j) sin2n (j) then : Prove that
n=0
(i) xyz = xy + z
n=0
(ii) xyz = x + y + z
EXERCISE-II Q 1.
The series of natural numbers is divided into groups (1), (2,3,4), (5,6,7,8,9), that the sum of the numbers in the nth group is (n - 1 )3 + n 3 .
Q 2.
The sum of the squares of three distinct real numbers, which are in GP is S 2 . If their sum is a S, show that a 2 e (1/3 ,1) u (1,3).
Q 3.
If there be m AP's beginning with unity whose common difference is 1,2.3 .... m. Show that the sum of their nth terms is (m/2) (mn - m + n + 1).
Q 4,
If Sn represents the sum to n terms of a GP whose first term & common ratio are a & r respectively, then prove that S,1 + S,3 + S,5 + ^
+ S2 7n ! , = — -
1-r
ar
(1-r) (l + r)
& so on. Show
.
Q 5.
A geometrical & harmonic progression have the same pth, qth & rlh terms a, b, c respectively. Show that a (b - c) log a + b (c - a) log b + c (a - b) log c = 0.
Q.6
A computer solved several problems in succession. The time it took the computer to solve each successive problem was the same number of times smaller than the time it took to solve the preceding problem. How many problems were suggested to the computer if it spent 63.5 min to solve all the problems except for the first, 127 min to solve all the problems except for the last one, and 31.5 min to solve all the problems except for the first two?
Q 7.
If the sum of m terms of an AP is equal to the sum of either the next n terms or the next p terms ofthe same APprove that (m+n)[(l/m)-(l/p)] = (m + p)[(l/m)-(l/n)] (n*p)
Q 8.
If the roots of 1 Ox3 - cx2 - 54x -21 = 0 are in harmonic progression, then find c & all the roots.
fa B ansa/ Classes
Sequence & Progression
[3]
Q 9,(a) Let a,, a2, a3 a
an be an AP . Prove that :
1 1 1 +a a + + a l n 2 n-l a3 an-2
1 n l
+a
a
a +a
l n
1 1 1 1 h + l a2 a3
a
1 +a n
(b) Show that in any arithmetic progression al, a^ a3 a.2 - a22 + a32 - a42 + ...... + a22K _ , - a22K = [K/(2 K - 1)] (&12 - a 2 2K ). Q10.
Let Let
a{, a 2 , , an , a n + i , S, = ax + a2 + a3 + 5 5
2 = an+l + an+2 + 3 = a2n+I + a2n+2 +
be an A.P. + ap + a
2n 3n '
+ S
Prove that the sequence Sj, S 2 , S 3 ,
is an arithmetic progression whose common difference
2
is n times the common difference of the given progression. Q l l . If a, b, c are in HP, b, c, d are in GP & c, d, e are in AP, Show that e = ab2/(2a - b) 2 . Q 12. (i) (ii) (iii)
If a, b, c, d, e be 5 numbers such that a, b, c are in AP ; b, c, d are in GP & c, d, e are in HP then: Prove that a, c, e are in GP . Prove that e = (2 b - a)2/a . If a = 2 & e = 18 , find all possible values of b, c, d .
Q13.
1 1 If A = 1 + - + - + 2 3 b
_£±2 2
[(n + l)n
1 1 +- + and n n+1
n(n-l)
2 + (n-i)(n-2)
+
3-2 j '
then show that A = B. Q 14. If n is a root of the equation x 2 (l -ac)-x(a 2 +c 2 )-(l+ac) = 0&if n HM's are inserted between a & c, show that the difference between the first & the last mean is equal to ac(a - c). Q15.
(a) (b)
The value of x + y + z is 15 if a , x , y , z , b are in APwhile the value of; (l/x)+(l/y)+(l/z) is 5/3 if a, x, y, z, b are in HP . Find a & b . The values of xyz is 15/2 or 18/5 according as the series a, x, y, z, b is an AP or HP. Find the values of a & b assuming them to be positive integer.
Q 16. An AP, aGP &aHP have' a' & 'b' for their first two terms. S how that their (n+2)th terms will be inGP if
h 2 n + 2 - a 2 n + 2 n+1 a n+1 ba(b 2n -a 2n ) n
Q 17. Prove that the sum of the infinite series
13 3.5 5.7 7.9 2 2 2 2
—h——~+T+
°o=23 .
Q 18. If there are n quantities in GP with common ratio r & Sm denotes the sum ofthe first m terms, show that the sum of the products of thesemterms taken two&two together is [r/(r+1)] [Sm] [S m _,]. Q 19. Consider an A.P., with the first term 'a', the common difference'd' and a G.P. with thefirstterm 'a', the common ratio 'r' such that a, d, r >0 and both these progressions have same number of terms as well as the equal extreme terms. Show that the sum of all the terms ofA.P. > the sum of all the terms ofthe G.P.
fa B ansa/ Classes
Sequence & Progression
[3]
Q 20. If n is even & a+(3, a - (3 are the middle pair ofterms, show that the sum of the cubes of an arithmetical progression is n a {a 2 + (n 2 -1) (32}. Q 21. If a, b, c be in GP & logc a, logb c, loga b be in AP, then show that the common difference of the AP must be 3/2. Q 22. If ax = 1 & for n> 1, an = an_, + —1— , then show that 12 < aJ5 < 15. a
Q 23. Sum tonterms:
(i) (ii)
n-l
1 x+1
2x (x + 1) (x + 2)
— - + -———— +
3x 2 (x + 1) (x + 2) (x + 3)
a,
^ + l + »i (1 + a i)( 1 + a2) (l + a i)(l + a 2 )(l + a3) Q 24. In a GPthe ratio of the sum of the first eleven terms to the sum ofthe last eleven terms is 1/8 and the ratio of the sum of all the terms without the first nine to the sum of all the terms without the last nine is 2. Find the number of terms in the GP. Q 25. Given a three digit number whose digits are three successive terms ofa G.P. If we subtract 792 from it, we get a number written by the same digits in the reverse order. Now if we subtract four from the hundred's digit of the initial number and leave the other digits unchanged, we get a number whose digits are successive terms of an A.P. Find the number.
EXERCISE-III Q.l
For any odd integer n > l , n 3 - ( n - 1)3 +
Q.2
x = l+3a + 6a 2 + 10a3+ y = 1+ 4b + 10b2 + 20b3 +
Q.3
The real numbers x p x 2 , x3 satisfying the equation x3 - x2 + p x + y = 0 are in A.P. Find the intervals in which P andy lie . [JEE'96,3]
Q.4
Let p & q be roots of the equation x2 - 2x + A = 0, and let r & s be the roots of the equation x- - 18x + B = 0. If p < q < r < s are in arithmatic progression, then A = , and B = . [JEE'97,2]
Q.5
a, b, c are the first three terms of a geometric series. If the harmonic mean of a & b is 12 and that of b &c is 3 6, find the first five terms ofthe series. [REE'98,6]
Q.6 (a)
Select the correct alternative(s). [ JEE'98,2 + 2 + 8 ] Let Tr be the rth term of an AP, for r = 1, 2, 3,.... If for some positive integers m, n we have 1 1 T = m ~ & T„ = ' Tmn equals : (A)— mn
(b)
(c)
(B) — + — m n
(C) 1 1 + fnx
(B) HP
.
[JEE '96,1]
|a|
If x = 1, y > 1, z > 1 are in GP, then — (A) AP
+ (-l)n~1l3 = _
(D)0
, — - — , —l + £ny
1 + ftiz
(C) GP
are in : (D) none ofthe above
Prove that a triangle ABC is equilateral if & only if tanA+tanB + tanC = 3 •
fa B ansa/ Classes
Sequence & Progression
[3]
Q.7 (a)
The harmonic mean of the roots of the equation (5+Jl j x2 - (4 + V5 j x + 8 + 2%/5 =0 is (A) 2
(B) 4
(C) 6
(D) 8
(b)
Letajjaj,...., a10, be in A.P. & h,,h 2 , (A) 2 (B) 3
Q.8
The sum of an infinite geometric series is 162 and the sum of its first n terms is 160. If the inverse of its common ratio is an integer, find all possible values ofthe common ratio, n and the first terms ofthe series. [REE'99,6]
Q.9 (a)
,h 10 beinH.P. If a1 = h 1 = 2 & a ] 0 = h10 = 3 then a 4 h 7 is: (C) 5 (D) 6 [ JEE '99,2 + 2 out of200 ]
Consider an infinite geometric series with first term 'a' and common ratio r. If the sum is 4 and the second term is 3/4, then: (A) a = ~ , r = | (C)
^ ,
x=
(B) a = 2 , r = |
\
(D) a = 3,
(b)
If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the relation: (A) 0 < M < 1 (B) 1 < M < 2 (C) 2 < M < 3 (D) 3 < M < 4 [ JEE 2000, Screening, 1 + 1 out of 35 ]
(c)
The fourth power of the common difference of an arithmetic progression with integer entries added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer. [ JEE 2000, Mains, 4 out of 100 ]
Q.10
Given that a , y are roots of the equation, A x 2 - 4 x + l = 0 and fi, 5 the roots of the equation, B x2 - 6 x + 1 = 0, find values of A and B, such that a, p, y & 8 are in H.P. [REE 2000, 5.outof 100]
Q.ll
The sum of roots of the equation ax 2 +bx+c = 0 is equal to the sum of squares of their reciprocals. Find whether be2, ca2 and ab2 in A.P., G.P. or H.P.? [ REE 2001, 3 out of 100 ]
Q.12
Solve the following equations for x and y log2x + log4x + log16x + =y 5 + 9 + 13+ 1+ 3 + 5+
Q.13 (a)
(b)
+(4y +1) +(2y-l) =
4l0g
4x
[ R E E 2001
'
5outofl0
°]
Let a, p be the roots of x2 - x + p = 0 and y, 8 be the roots of x2 - 4x + q = 0. If a, P, y, 6 are in G.P., then the integral values of p and q respectively, are (A)-2,-32 (B) - 2 , 3 (C)-6,3 (D)-6,-32 If the sum ofthe first 2n terms of the A.P. 2,5,8, 57,59,61, , then n equals (A) 10
fa Ban sal Classes
(B) 12
is equal to the sum ofthe first n terms of the A.P.
(C)ll
Sequence & Progression
(D) 13
[11]
(c)
Let the positive numbers a, b, c, d be in A.P. Then abc, abd, acd, bed are (A) NOT in A.P./G.P./H.P. (B) in A.P. (C) in G.P. (D)H.P. [JEE 2001, Screening, 1 + 1 + 1 out of 3 5 ]
(d)
Let al5 a2 be positive real numbers in G.P. For each n, let An, Gn, Hn, be respectively, the arithmetic mean, geometric mean and harmonic mean of av a,, a3, an. Find an expression for the G.M. of Gj, G2, Gn in terms of A p A2 An, Hj, H2, Hn. [JEE 2001 (Mains);5]
Q.14 (a)
2
2
3
2
Suppose a, b, c are in A.P. and a , b , c are in G.P. If a < b < c and a + b + c = - , then the value of a is
[JEE 2002 (Screening), 3] (b)
Let a, b be positive real numbers. If a , A, , A7 , b are in A.P. ; a , a! , a2 , b are in G.P. and a, H j , H 2 , b are in H.P., show that GJGJ
H
Q.15
H
-
A!+A 2 H
+ H
-
(2a + b)(a+2b)
[ JEE 2002 , Mains , 5 out of 60 ]
c If a, b, c are in A.P., a 2 , b 2 , c2 are in H.P., then prove that either a = b = c or a, b, - — form a G.P. [JEE-03, Mains-4 out of 60]
Q.16
The first term of an infinite geometric progression is x and its sum is 5. Then ( A ) 0 < x < 10
(B) 0 < x < 10
(C) -10 < x < 0
(D)x>10
Q.17
[JEE 2004 (Screening)] If a, b, c are positive real numbers, then prove that [(1 + a) (1 + b) (1 + c)]7 > 77 a4 b4 c4. [JEE 2004,4 out of 60]
Q.18
If total number of runs scored in n matches is
(n + \)
J " , where 1 < k < n. Find n. V
th
the k match are given by k-2
fa Ban sal Classes
,
(2n+1 - n - 2) where n > 1, and the runs scored in
4
n+1 k
Sequence & Progression
[JEE 2005,2]
[11]
ANSWER KEY
EXERCISE-I Ql. 1 Q 2. x = 105, y = 10 Q3. ji= 14 Q 4. S = (7/81){10n+1 - 9n - 10} Q 5. 35/222 Q6. n(n+l)/2(n 2 + n + 1 ) Q 7. 27 Q 10. (14 n - 6)/(8 n + 23) Q 11. 1 Q 14. 9 Q 16. a = 5 , b = 8 , c = 12 Q 18. ( 8 , - 4 , 2, 8) Q 19. n2 Q20. (i) 2n+1 - 3 ; 2 n + 2 - 4 - 3 n (ii) n 2 + 4 n + l ; ( l / 6 ) n ( n + l ) ( 2 n + 1 3 ) + n Q 21. 120,30 Q 22. 6 , 3 Q 23. (i) sn = (1/24) - [l/{6(3n+ 1) (3n + 4) >] j s ^ l / 2 4 (ii) (l/5)n(n+ l)(n + 2)(n+ 3)(n + 4) (iii) n/(2n+1)
(iv) Sn =
2
1 2
1.3.5 (2n-l)(2n + l) 2.4.6 (2n)(2n + 2)
;" Son = 1
Q 24. (a) (6/5) (6n - 1) (b) [n (n + 1) (n + 2)]/6
EXERCISE-II Q6. 8problems, 127.5 minutes Q.8 C = 9 ; (3,-3/2 ,-3/5) Q 12. (iii) b = 4 , c = 6 , d = 9 OR b = - 2 , c = - 6 , D = - 18 Q 15. (a) a = 1, b = 9 OR b = 1, a = 9 ; (b) a = 1 ; b = 3 or vice versa
Q 23. (a) 1 -
(x + 1) (x + 2)
Q 24. n = 38
(x + n)
(b) 1 - (l + a i ) ( l + a 2 )
(l + a n )
Q 25. 931
EXERCISE-III Ql.
|(2n-l)(n+l)2
Q 2. S =
Where a = 1 - x"1/3 & b = 1 - y"1/4
Q 4. - 3 , 7 7 Q 6. (a) C (b) B
Q3.
p < (1/3) ; y > -(1/27)
Q 5. 8,24,72,216,648 Q 7. (a) B (b) D
Q 8. r = ± 1/9 ; n = 2 ; a = 144/180 OR r = ± 1/3 ; n = 4 ; a = 108 OR r = 1/81 ; n = 1 ; a = 160 Q9. (a) D
(b) A
Q 10. A = 3 ; B = 8
Q l l . A.P.
Q 12. x = 2V2 andy = 3
Q13. (a) A, (b)C, (c) D, (d)[(A,, A 2 , Q14.
(a) D
fa Ban sal Classes
Q.16
B
A n ) (Hl5 H2, Q.18
H n )]
n=7
Sequence & Progression
[11]
BANSAL CLASSES MATHEMATICS I TARGETIITJEE
2007
XI (P> Q, R, S)
r
*
i I
-yfi
iInn cLae csi
CONTENTS KEY- CONCEPTS EXERCISE-I EXERCISE-II EXERCISE-III ANSWER-KEY
KEY STANDARD
1.
CONCEPTS
RESULTS:
EQUATION OF A CIRCLE IN VARIOUS FORM : (a) The circle with centre (h, k) & radius 'r' has the equation; (x-h) 2 + ( y - k ) 2 =r 2 . (b) The general equation of a circle is x2 + y2 + 2gx + 2fy + c = 0 with centre as : (-g, -f) & radius = ^ g 2 + f 2 - c . Remember that every second degree equation in x & y in which coefficient of x2 = coefficient of y2 & there is no xy term always represents a circle. If g2 + f 2 - c > 0 => real circle. 2 2 g + f - c = 0 => point circle. 2 2 g + f - c < 0 => imaginary circle. Note that the general equation of a circle contains three arbitrary constant s, g, f & c which corresponds to the fact that a unique circle passes through three non collinear points. (c) The equation of circle with (x,, y}) & , y2) as its diameter is : ( X - X l ) ( x - x ^ + ( y - y i ) ( y - y 2 ) = 0.
Note that this will be the circle of least radius passing through 2.
, yj) & (xj, y2).
INTERCEPTS MADE BY A CIRCLE ON THE AXES : The intercepts made by the circle x2 + y2 + 2gx + 2fy + c = 0 on the co-ordinate axes are 2 vg2 - c & 2 ^ / f ^ c respectively. NOTE : circle cuts the x axis at two distinct points. => If g2 - c > 0 2 circle touches the x-axis. If g =c 2 => circle lies completely above or below the x-axis. If g
3.
POSITION OF A POINT w.r.t. A CIRCLE : The point (x t , yj) is inside, on or outside the circle x2 + y2 + 2gx + 2fy+ c~ U. according as Xj2 + y 2 + 2gx, + 2fv, + c O 0 . Note : The greatest & the least distance of a point Afrom a circle with centre C & radius r is AC + r & AC - r respectively.
(Xi,yi) p\
4.
LINE & A CIRCLE: Let L = 0 be a line & S = 0 be a circle. If r is the radius of the circle & p is the length of the perpendicularfrom,the centre on the line, then: (i) p > r <=> the line does not meet the circle i. e. passes out side the circle. (ii) p = r o the line touches the circle. (iii) p < r o the line is a secant of the circle. (iv) p = 0 => the line is a diameter of the circle.
5.
PARAME TRIC EQUATIONS OF A CIRCLE: The parametric equations of (x - h)2 + (y - k)2 = r2 are: x = h + rcos9 ; y = k + rsin9 ; -TC < 0 < TC where (h, k) is the centre, r is the radius & 9 is a parameter. Note that equation of a straight line joining two point a & (3 on the circle x2 + y2 = a2 is a+B , . a+B a-B x cos —— + y sin = a cos — - .
(!%Bansal
Classes
Circles
[12]
6. (a)
TANGENT & NORMAL: The equation of the tangent to the circle x 2 +y 2 = a2 at its point (x t , y t ) is, xxj + y y, = a2. Hence equation of a tangent at (a cos a, a sina) is; x cos a + y sin a = a. The point of intersection of the tangents at the points P(a) and Q(f3) is a+ff
acos —2—
.
a-fl
(b) (c)
a-p
COS — —x—- vcos uo-2 The equation of the tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 at its point (x t , yj) is XX, + yyj + g (x + X j ) + f (y + yj) + c = 0. y = mx + c is always a tangent to the circle x 2 +y 2 =a 2 if c2 = a2 (1 + m2) and the point of contact ( a 2m a 2\ is c
(d)
a+6
asrn^-11
cJ
If a line is normal/orthogonal to a circle then it must pass through the centre of the circle. Using this fact normal to the circle x2 + y2 + 2gx + 2fy + c = 0 at(x ; , y,) is
y_y = Zi±£ (X-Xl). x,+g
7. (a)
A FAMILY OF CIRCLES : The equation of the family of circles passing through the points of intersection of two circles St = 0 & S2 = 0is : S ! + K S 2 = 0 (K*-l).
(b)
The equation of the family of circles passing through the point of intersection of a circle S = 0 & a line L = 0 is given by S+KL = 0.
(c)
The equation of a family of circles passing through two given points (x}, y t ) & in the form:
, y2) can be written
y i (x-x 1 )(x-x 2 ) + ( y - y 1 ) ( y - y 2 ) + K Yi 1 = 0 where K is a parameter. x2 y2 i (d)
(e)
The equation of a family of circles touching afixedline y - yL = m (x - x}) at thefixedpoint (xL, yj) is (x - x t ) 2 + (y - yx)2 + K [y - yj - m (x - Xj)] = 0, where K is a parameter. In case the line through (xj, yj) is parallel to y - axis the equation of the family of circles touching it at (Xj, y t ) becomes (x - x,)2 + (y - y^ 2 + K (x - Xj) = 0. Also if line is parallel to x - axis the equation of the family of circles touching it at (xi»Yi) becomes ( x - X j ) 2 + ( y - y ^ 2 + K ( y - y i ) = 0. Equation of circle circumscribing a triangle whose sides are given by Lj = 0 ; L2 = 0 & L3 = 0 is given by; LjL2 + A. L2L3 + \x L3Lj = 0 provided co-efficient of xy = 0 & co-efficient of x2 = co-efficient of y2.
(f)
Equation of circle circumscribing a quadrilateral whose side in order are represented by the lines Lj = 0, L2 = 0, L 3 = 0 & L4 = 0 is L,L 3 + A L 2 L 4 = 0 provided co-efficient of x2 = co-efficient of y2 and co-efficient of xy=0.
8.
LENGTH OF A TANGENT AND POWER OF A POINT : The length of a tangent from an external point (x t , y^ to the circle S = x2 + y2 + 2gx + 2fy + c = 0 is given by L= Jx 1 2 +y 1 2 +2gx,+2f 1 y+c = Js^. Square of length of the tangent from the point P is also called THE POWER OF POINT w.r.t. a circle. Power of a point remains constant w.r.t. a circle. Note that : power of a point Pis positive, negative or zero according as the point 'P'is outside, inside or on the circle respectively.
(!%Bansal
Classes
Circles
[12]
9.
DIRECTOR CIRCLE
:
The locus ofthe point of intersection of two perpendicular tangents is called the DIRECTOR CIRCLE ofthe given circle. The director circle of a circle is the concentric circle having radius equal to V2 times the original circle. 10.
EQUATION OF THE CHORD WITH A GIVEN MIDDLE
POINT:
The equation of the chord of the circle S = x2 + y2 + 2gx + 2fy + c = 0 in terms of its mid point Xi
2
M(xj, yj) is y - yj = - - 1 —- (x - Xj). This on simplication can be put in the form Yj+r xxj + yyj + g (x + Xj) + f (y + yj) + c = Xj2 + Y!2 + 2gx} + 2fyj + c which is designated by T = S,. Note that : the shortest chord of a circle passing through a point 'M' inside the circle, is one chord whose middle point is M. 11.
CHORD O F C O N T A C T : If two tangents PTj & PT2 are drawn from the point P (x ]; y t ) to the circle S = x2 + y2 + 2gx + 2fy + c = 0, then the equation of the chord of contact T t T 2 is: xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0.
REMEMBER : (a) Chord of contact exists only if the point 'P' is not inside. 2LR (b)
Length of chord of contact T, T2 =
•
(c)
RL3 Area ofthe triangle formed by the pair of the tangents & its chord of contact = |> 2+T2 Where R is the radius of the circle & L is the length of the tangent from (x1; y}) on S = 0.
(d)
Angle between the pair of tangentsfrom(xt, yj) = tan 1
' 2RL ^
vL2-*2/
(f)
where R=radius ; L = length of tangent. Equation of the circle circumscribing the triangle PTj T2 is: (x-Xj) (x + g) + ( y - y i ) (y + f) = 0. The joint equation of a pair of tangents drawn from the point A(xj,yj)to the circle x2 + y2 + 2gx + 2fy + c = 0 is : S S ^ T 2 . Where S s x2 + y2 +2gx + 2 f y + c ; Sj =Xj2 + y2 + 2gXj + 2fyj + c T= xxj + yyl + g(x + XJ) + f(y + y^ + c.
12.
POLE &
(i)
If through a point P in the plane of the circle, there be drawn any straight line to meet the circle in Q and R, the locus of the point of intersection of the tangents at Q & R is called the POLAR O F THE POINT P ; also P is called the POLE O F THE POLAR. The equation to the polar of a point P (xj, y,) w.r.t. the circle x2 + y2 = a2 is given by xxj + yy t =s a 2 , & if the circle is general then the equation of the polar becomes xx1 + yy,+g(x + Xj) + f (y + y^ + c = 0. Note that if the point (x t , yj) be on the circle then the chord of contact, tangent & polar will be represented by the same equation.
(e)
(ii)
(iii)
POLAR:
2
2
2
Pole of a given line Ax + By + C = 0 w.r.t. any circle x + y = a is
(!%Bansal
Classes
Circles
Aa2
Ba 2 ^
[12]
(iv) (v)
If the polar of a point P pass through a point Q, then the polar of Q passes through P. Two lines L, & L2 are conjugate of each other if Pole of Lj lies on L2 & vice versa Similarly two points P & Q are said to be conjugate of each other if the polar of P passes through Q & vice-versa.
13. (i)
COMMON TANGENTS TO TWO CIRCLES: Where the two circles neither intersect nor touch each other, there are FOUR common tangents, two of them are transverse & the others are direct common tangents. When they intersect there are two common tangents, both of them being direct. When they touch each other: (a) EXTERNALLY : there are three common tangents, two direct and one is the tangent at the point of contact. (b) INTERNALLY: only one common tangent possible at their point of contact. Length of an external common tangent & internal common tangent to the two circles is given by:
(ii) (iii)
(iv)
L e W
(v)
14.
(a) (b) (c) (d) (e) (f) (g) (h) 15.
d 2
- (
r
. -
r
2 )
2
&
L
int=
A
/d2-(r
1 +
r2)
2
.
Where d = distance between the centres of the two circles. ^ & r2 are the radii of the two circles. The direct common tangents meet at a point which divides the line joining centre of circles externally in the ratio of their radii. Transverse common tangents meet at a point which divides the line joining centre of circles internally in the ratio of their radii. RADICAL AXIS & RADICAL CENTRE : The radical axis of two circles is the locus of points whose powers w.r.t. the two circles are equal. The equation of radical axis of the two circles S j = 0 & S2 = 0 is given; S 1 - S 2 = 0 i.e. 2 ( g 1 - g 2 ) x + 2 ( f 1 - f 2 ) y + (c 1 -c 2 ) = 0. NOTE THAT: If two circles intersect, then the radical axis is the common chord of the two circles. If two circles touch each other then the radical axis is the common tangent of the two circles at the common point of contact. Radical axis is always perpendicular to the line joining the centres of the two circles. Radical axis need not always pass through the mid point of the line joining the centres of the two circles. Radical axis bisects a common tangent between the two circles. The common point of intersection of the radical axes of three circles taken two at a time is called the radical centre of three circles. A system of circles, every two which have the same radical axis, is called a coaxal system. Pairs of circles which do not have radical axis are concentric. ORTHOGONALITY OFTWO CIRCLES: Two circles St = 0 & S 2 =0 are said to be orthogonal or said to intersect orthogonally if the tangents at their point of intersection include a right angle. The condition for two circles to be orthogonal is : 2 g[ g2 + 2 f j f 2 = Ci + C2 .
Note : (a) Locus of the centre of a variable circle orthogonal to twofixedcircles is the radical axis between the twofixedcircles. (b) . If two circles are orthogonal, then the polar of a point 'P' onfirstcircle w.r.t. the second circle passes through the point Q which is the other end of the diameter through P . Hence locus of a point which moves such that its polars w.r.t. the circles S j = 0, S2 = 0 & S3 = 0 are concurrent in a circle which is orthogonal to all the three circles.
(!%Bansal
Classes
Circles
[12]
EXERCISE-I Q 1.
Find the equation of the circle circumscribing the triangle formed by the lines ; x + y = 6,2x + y = 4& x + 2y = 5, without finding the vertices of the triangle.
Q 2.
If the lines a, x + b, y + Cj = 0 & a2X + b2y + c 2 =0 cut the coordinate axes in concyclic points. Prove that aj bj b2.
Q 3.
One of the diameters of the circle circumscribing the rectangle ABCD is4y=x + 7. If A& B are the points (-3,4) & (5,4) respectively. Then find the area of the rectangle.
Q 4.
Lines 5x + 12y - 10 = 0 & 5 x - 12y-40 = 0 touch a circle Cj of diameter 6. If the centre of Cj lies in the first quadrant, find the equation of the circle C2 which is concentric with C} & cuts interceptes of length 8 on these lines.
Q 5.
Find the equation of the circle inscribed in a triangle formed by the lines 3x + 4y = 12; 5x + 12y = 4 & 8y = 15x + 10 withoutfindingthe vertices of the triangle.
Q 6.
Find the equation of the circles passing through the point (2,8), touching the lines 4x - 3y - 24 = 0 & 4x + 3y - 42 = 0 & having x coordinate of the centre of the circle less than or equal to 8.
Q 7.
Find the equation of a circle which is co-axial with circles 2x2 + 2y2 - 2x + 6y - 3 = 0 & x2 + y2 + 4x + 2y + 1 = 0. It is given that the centre of the circle to be determined lies on the radical axis of these two circles.
Q 8.
Let A be the centre of the circle x2 + y2 - 2x - 4y - 20 = 0. Suppose that the tangents at the points B(1,7) & D(4,-2) on the circle meet at the point C. Find the area of the quadrilateral ABCD.
Q 9.
The radical axis of the circles x2 + y2 + 2gx + 2fy + c = 0 and 2x2 + 2y2 + 3x + 8y + 2c = 0 touches the circle x2 + y2 + 2x - 2y + 1 = 0. Show that either g = 3/4 or f = 2 .
Q. 10 Find the equation of the circle through the points of intersection of circles x 2 +y 2 -4x-6y-12=0 and x2 + y2 + 6x + 4y - 12 = 0 & cutting the circle x2 + y2 - 2x - 4 = 0 orthogonally. Q 11. Consider a curve ax 2 +2 hxy + by2 = 1 and a point P not on the curve. Aline is drawnfromthe point P intersects the curve at points Q & R. If the product PQ. PR is independent of the slope of the line, then show that the curve is a circle. Q 12, Find the equations of the circles which have the radius Vl3 & which touch the line 2x-3y+1 = 0at(l, 1). Q 13. A circle is described to pass through the origin and to touch the lines x = 1, x + y = 2. Prove that the radius of the circle is a root of the equation ^3 - 2-j2j t 2 - 2 j 2 t + 2 = 0. Q 14. The centre of the circle S = 0 lie on the line 2x-2y + 9 = 0 & S = 0 cuts orthogonally the circle x2 + y2 = 4. Show that circle S = 0 passes through twofixedpoints &findtheir coordinates. Q 15. Show that the equation x2 + y2 - 2x - 2 Ay - 8 = 0 represents, for different values of A, a system of circles passing through twofixedpoints A B on the x - axis, andfindthe equation of that circle of the system the tangents to which at A & B meet on the line x + 2y + 5 = 0.
(!%Bansal
Classes
Circles
[12]
Q 16. Find the equation ofthe circle which passes through the point (1, 1) & which touches the circle x2 + y2 + 4x - 6y - 3 = 0 at the point (2, 3) on it. Q 17. Find the equation of a circle which touches the lines 7x2 - 18xy + 7y2 = 0 and the circle x2 + y2 - 8x- 8y = 0 and is contained in the given circle. Q 18. Find the equation of the circle which cuts the circle x2 + y2 -14x - 8y + 64 = 0 and the co-ordinate axes orthogonally. Q 19. Obtain the equations ofthe straight lines passing through the point A(2,0)&making 45° angle with the tangent at A to the circle (x + 2)2 + (y - 3)2 = 25. Find the equations of the circles each of radius 3 whose centres are on these straight lines at a distance of 5 V2 from A. Q 20. Find the equations of the circles whose centre lie on the line 4x + 3y - 2 = 0 & to which the lines x + y + 4 = 0 & 7 x - y + 4 = 0 are tangents. Q 21. Find the equations to the four common tangents to the circles x2 + y2 = 25 and (x-12) 2 + y2 = 9. Q 22. If 4/2 - 5m2 + 6/ + 1 = 0. Prove that /x + my + 1 = 0 touches a definite circle. Find the centre & radius of the circle. Q 23. Find the condition such that the four points in which the circle x2 + y2 + ax + by + c = 0 and x2 + y2 + a'x + b'y + c' = 0 are intercepted by the straight lines Ax + By + C = 0 & A'x + B'y + C' = 0 respectively, lie on another circle. Q 24. Show that the equation of a straight line meeting the circle x2 + y2 = a2 in two points at equal distances d2 'd' from a point (x}, yj) on its circumference is xxt + yy} - a + —• = 0. 2
Q 25. If the equations of the two circles whose radii are a & a' be respectively S = 0 & S - 0, then prove that S S' the circles — + — = 0 will cut each other orthogonally, a a Q 26. Let a circle be given by 2x (x - a) + y (2y - b) = 0, (a * 0, b ^ 0). Find the condition on a & b if two chords, each bisected by the x-axis, can be drawn to the circle from
v ^j
Q 27. Prove that the length of the common chord of the two circles x2 + y2 = a2 and (x - c)2 + y2 = b2 is -7(a+b+c)(a-b+c)(a+b-c)(-a+b+c) . c Q 28. Find the equation of the circle passing through the points A (4,3) & B (2, 5) & touching the axis of y. Also find the point P on the y-axis such that the angle APB has largest magnitude. Q 29. Find the equations of straight lines which pass through the intersection of the lines x - 2y - 5 = 0, 7x + y = 50 & divide the circumference of the circle x2 + y2 = 100 into two arcs whose lengths are in the ratio 2:1. Q 30. Find the equation of the circle which cuts each of the circles x2 + y2 = 4, x2 + y 2 - 6 x - 8 y + 10=0 & x2 + y2 + 2 x - 4 y - 2 = 0 at the extremities of a diameter.
(!%Bansal Classes
Circles
[12]
EXER CISE-II Q 1.
A point moves such that the sum of the squares of its distancefromthe sides of a square of side unity is equal to 9. Show that the locus is a circle whose centre coincides with centre of the square. Find also its radius.
Q 2.
A triangle has two of its sides along the coordinate axes, its third side touches the circle x2 + y2 - 2ax - 2ay + a2 = 0. Prove that the locus of the circumcentre of the triangle is : a2 - 2a (x + y) + 2xy = 0.
Q 3.
A variable circle passes through the point A (a, b) & touches the x-axis; show that the locus ofthe other end of the diameter through A is (x - a)2 = 4by.
Q 4.
(a)
Find the locus of the middle point of the chord of the circle, x2 + y2 + 2gx + 2fy + c = 0 which subtends a right angle at the point (a, b). Show that locus is a circle.
(b)
Let S= x 2 +y 2 + 2gx+2fy+c=0 be a given circle. Find the locus of the foot of the perpendicular drawnfromthe origin upon any chord of S which sustends a right angle at the origin.
Q 5.
A variable straight line moves so that the product ofthe perpendiculars on itfromthe twofixedpoints (a, 0) & (- a, 0) is a constant equal to c2 . Prove that the locus ofthe feet of the perpendiculars from each of these points upon the straight line is a circle, the same for each.
Q 6.
Showthat the locus of the centres of a circle which cuts two given circles orthogonally is a straight line & hence deduce the locus of the centers of the circles which cut the circles x2 + y2 + 4x - 6y + 9 = 0 & x2 + y2 - 5x + 4y + 2 = 0 orthogonally .
Q 7.
Afixed circle is cut by afamilyofcirclespassingthroughtwofixedpointsA(x1,y1)andB(x2,y2). Show that the chord ofintersection ofthefixedcircle with any one ofthe circles offamily passes through afixedpoint.
Q 8.
The sides of a variable triangle touch the circle x2 + y2 = a2 and two of the vertices are on the line y2 - b2 = 0 (b > a > 0) . Show that the locus of the third vertex is; (a2 - b2) x2 + (a2 + b2) y2 = (a(a2 + b2))2.
Q 9.
Show that the locus of the point the tangentsfromwhich to the circle x2 + y2 - a2 = 0 include a constant angle a is (x2 + y2 - 2a2)2 tan2 a = 4a2 (x2 + y2 - a 2 ).
Q 10. ' O' is afixedpoint & P a point which moves along afixedstraight line not passing through O; Q is taken on OP such that OP. OQ=K(constant) . Prove that the locus of Q is a circle. Explain how the locus of Q can still be regarded as a circle even if thefixedstraight line passes through 'O'. Q 11. P is a variable point on the circle with centre at C. CA & CB are perpendiculars from C on x-axis & y-axis respectively. Show that the locus of the centroid of the triangle PAB is a circle with centre at the centroid of the triangle CAB & radius equal to one third of the radius ofthe given circle. Q 12. A(-a, 0) ; B(a,0) arefixedpoints. C is a point which divides AB in a constant ratio tana. If AC & CB subtend equal angles at P, prove that the equation ofthe locus of P is x2 + y2 + 2ax sec2a + a2 = 0.
Bansal Classes
Circles
Q 13. The circle x2 + y2 +2ax- c2 = 0 and x2 + y2 + 2bx- c2 = 0 intersect at Aand B. Aline through Ameets one circle at P and a parallel line through B meets the other circle at Q. Show that the locus of the mid point of PQ is a circle. Q 14. Find the locus of a point which is at a least distance from x2 + y2 = b2 & this least distance is equal to its distance from the straight line x = a. Q 15. The base of a triangle is fixed. Find the locus of the vertex when one base angle is double the other. Assume the base of the triangle as x-axis with mid point as origin & the length ofthe base as 2a. Q 16. An isosceles right angled triangle whose sides are 1, 1, V2 lies entirely in thefirstquadrant with the ends of the hypotenuse on the coordinate axes. If it slides prove that the locus of its centroid is (3x-y) 2 + ( x - 3 y ) 2 = - 3 ^ . Q 17. The circle x2 + y2 - 4x - 4y + 4 = 0 is inscribed in a triangle which has two of its sides along the coordinate axes. The locus of the circumcentre ofthe triangle is: x + y - xy + K ^ / P + y 1 = 0. Find K. *
Q 18. Find the locus of the point ofintersection of two perpendicular straight lines each of which touches one of the two circles (x - a)2 + y2 = b 2 , (x + a)2 + y2 = c2 and prove that the bisectors of the angles between the straight lines always touch one or the other of two otherfixedcircles. Q.19 Find the locus ofthe mid point of the chord of a circle x2 + y2 = 4 such that the segment intercepted by the chord on the curve x2 - 2x - 2y = 0 subtends a right angle at the origin. Q 20. TheendsAB ofafixedstraightlineoflength'a'&endsA'&B'ofanotherfixedstraightlineoflength 'b' slide upon the axis ofx&the axis ofy (one end on axis of x& the other on axis of y). Find the locus of the centre of the circle passing through A, B, A' & B'. Q 21. The foot of the perpendicularfromthe origin to a variable tangent of the circle x 2 +y 2 - 2x = 0 is N. Find the equation ofthe locus ofN. Q 22, Find the locus of the mid point of all chords of the circle x 2 +y 2 - 2x - 2y = 0 such that the pair of lines joining (0,0) & the point of intersection of the chords with the circles make equal angle with axis of x. Q 23. P (a) & Q (p) are the two points on the circle having origin as its centre & radius 'a' & AB is the diameter along the axis of x. If a - p = 2 y, then prove that the locus of intersection of AP & BQ is x2 + y2 - 2 ay tany = a2. Q 24. Show that the locus of the harmonic conjugate of a given point P (xl5 y t ) w.r.t. the two points in which any line through P cuts the circle x2 + y2 = a2 is xxj + yy} = a2. Q 25. Find the equation of the circle which passes through the origin, meets the x-axis orthogonally & cuts the circle x2 + y2 = a2 at an angle of 45°.
(!%Bansal
Classes
Circles
[12]
EXERCISE-III Q.l
(a)
The intercept on the line y=x bythe circle x 2 + y 2 - 2 x = 0 isAB . Equation ofthe circle with AB as a diameter is .
(b)
The angle between a pair of tangents drawn from a point P to the circle x2 + y2 + 4x - 6y + 9 sin2a + 13 cos2a = 0 is 2 a. The equation of the locus of the point P is : (A) x2 + y2 + 4 x - 6 y + 4 = 0 (B) x2 + y2 + 4x - 6y - 9 = 0 2 2 (C) x + y + 4 x - 6 y - 4 = 0 (D) x2 + y2 + 4 x - 6y + 9 = 0
(c)
Find the intervals of values of a for which the line y + x = 0 bisects two chords drawn from a ' l + V2a 1-V2a 1 rpoint — — — , — - — to the circle; 2x 2 +2y 2 - (1+V2 a) x - (1 - V2 a) y = 0. V
2
J
[JEE'96, 1+1+5] Q.2
Atangent drawnfromthe point (4,0)tothecircle x2+y2 = 8 touches it at apointAin the first quadrant. Find the coordinates of the another point B on the circle such that AB = 4. [ REE '96, 6 ]
Q.3
(a)
The chords of contact of the pair of tangents drawn from each point on the line 2x + y = 4 to the circle x2 + y2 = 1 pass through the point .
(b)
Let C be any circle with centre (o, 42 ) • Prove that at the most two rational point can be there on C. (A rational point is a point both of whose co-ordinate are rational numbers). [JEE '97, 2+5] 2 2 2 2 The number of common tangents to the circle x + y = 4 & x + y - 6x - 8y = 24 is : (A) 0 (B) 1 (C) 3 (D) 4 Cj & C2 are two concentric circles, the radius of C2 being twice that of Cj. From a point P on C2, tangents PA & PB are drawn to C}. Prove that the centroid of the triangle PAB lies on C j. [ JEE '98, 2 + 8 ]
Q.4
(a) (b)
Q. 5
Find the equation of a circle which touches the line x + y = 5 at the point (-2, 7) and cuts the circle x2 + y2 + 4x - 6y + 9 = 0 orthogonally. [ REE '98, 6 ]
Q.6
(a)
If two distinct chords, drawn from the point (p, q) on the circle x2 + y2 = px + qy (where pq q) are bisected by the x-axis, then: (A) p2 = q2 (B) p2 = 8q2 (C) p2 < 8q2 (D)p2>8q2
(b)
Let Lj be a straight line through the origin and L2 be the straight line x+y = 1. If the intercepts made by the circle x2 + y2 - x + 3y = 0 on L} & L2 are equal, then which of the following equations can represent Lj? (A) x + y = 0 (B)x-y = 0 ( C ) x + 7y = 0 (D)x-7y = 0
(c)
Let Tj, T2 be two tangents drawnfrom( - 2,0) onto the circle C: x2 + y2 = 1. Determine the circles touching C and having T t , T2 as their pair of tangents. Further,findthe equations of all possible common tangents to these circles, when taken two at a time. [ JEE '99, 2 + 3 + 10 (out of200) ] 2 2 The triangle PQR is inscribed in the circle, x + y = 25. IfQ and Rhave co-ordinates (3,4) & ( - 4,3) respectively, then Z QPR is equal to :
Q.7
(a)
(A) § (b)
(B) f
(C) f
(D) f
If the circles, x2 + y2 + 2x + 2ky + 6 = 0 & x2 + y2 + 2 ky + k = 0 intersect orthogonally, then 'k' is: (A) 2 or - |
(B) - 2 or - |
(C) 2 or |
(D) - 2 or \
[ JEE '2000 (Screening) 1 + 1 ]
^Bansal Classes
Circles
[10]
Q.8
Q.9
(a)
Extremities of a diagonal of a rectangle are (0,0) & (4,3). Find the equation ofthe tangents to the circumcircle of a rectangle which are parallel to this diagonal.
(b)
A circle of radius 2 units rolls on the outerside of the circle, x2 + y2 + 4 x = 0, touching it externally. Find the locus ofthe centre ofthis outer circle. Alsofindthe equations of the common tangents of the two circles when the line joining the centres ofthe two circles makes on angle of 60° with x-axis. [REE '2000 (Mains) 3 + 5]
(a)
Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. IfPS and RQ intersect at a point X on the circumference of the circle then 2r equals
[JEE'2001 (Screening) 1 out of 35] 2
2
(b)
Let 2x + y - 3xy = 0 be the equation of a pair of tangents drawn from the origin 'O' to a circle of radius 3 with centre in thefirstquadrant. IfAis one ofthe points of contact,findthe length of OA [JEE '2001 (Mains) 5 out of 100]
Q. 10 (a)
Find the equation of the circle which passes through the points of intersection of circles x2 + y2 - 2x - 6y + 6 = 0 and x2 + y2 + 2x - 6y + 6 = 0 and intersects the circle x2 + y2 + 4x + 6y + 4 — 0 orthogonally. [ REE '2001 (Mains) 3 out of 100 ]
(b)
Q. 11 (a)
(b)
Tangents TP and TQ are drawnfroma point T to the circle x2 + y2 = a2. If the point T lies on the line px + qy = r,findthe locus of centre of the circumcircle of triangle TPQ. [ REE '2001 (Mains) 5 out of 100 ] If the tangent at the point P on the circle x2 + y2 + 6x + 6y = 2 meets the straight line 5x - 2y + 6 = 0 at a point Q on the y-axis, then the length of PQ is (A) 4 (B)2V5 (C)5 (D)3V5 If a > 2b > 0 then the positive value of m for which y = mx-b-Jl + m is a common tangent to x2 + y2 = b2 and (x - a)2 + y2 = b2 is v'a2 - 4 b 2
2b
2b
b
(C)
W - z T -
T-Tib [ JEE '2002 (Scr)3 + 3 out of270]
Q .12 The radius of the circle, having centre at (2, 1), whose one of the chord is a diameter of the circle x2 + y2 — 2x — 6y + 6 = 0 (A)l
(B)2
(C)3
(D)V3
[JEE'2004 (Scr)] Q.13 Line2x + 3y+ 1 = 0 is a tangent to a circle at (1,-1). This circle is orthogonal to a circle which is drawn having diameter as a line segment with end points (0,-1) and (- 2,3). Find equation of circle. [JEE'2004, 4 out of 60] 2 2 Q.14 A circle is given by x + (y -1) = 1, another circle C touches it externally and also the x-axis, then the locus of its centre is (A){(x,y):x 2 = 4y}u{(x,y):y = 0} (B) {(x, y) : x2 + (y - 1)2 = 4} u {x, y): y = 0} (C) {(x, y): x2 = y} VJ {(0, y): y = 0} (D*) {(x, y): x2 = 4y} u {(0, y): y = 0} [JEE '2005 (Scr)]
(!%Bansal
Classes
Circles [12]
ANSWER
KEY
EXERCISE-I 2
2
Q 4. x2 + y2 - lOx-4y + 4 = 0
Q 1. x + y - 17x - 19y + 50 = 0
Q 3. 32 sq. unit
Q 5. x2 + y2 - 2x - 2y + 1 - 0
Q 6. centre (2 ,3), r = 5 ; centre
Q 7. 4x2 + 4y2 + 6x + lOy- 1 = 0
Q 8. 75 sq.units
f
182
"
2 0 5
Q 10. x2 + y2 + 16x+ 14y- 12 = 0
Q 12. x2 + y2 - 6x + 4y = 0 OR x2 + y2 + 2 x - 8 y + 4 = 0
Q 14. ( - 4, 4) ; f _ 1 V
I
2' 2 J
Q 15. x2 + y2 - 2x - 6y - 8 = 0 Q 16. x2 + y2 + x - 6y + 3 = 0 2 2 Q 17. x + y - 12x -12y + 64 = 0 Q18. x2 + y2 = 64 Q 19. x - 7y = 2, 7x + y = 14 ; (x - l) 2 + (y - 7)2 = 32 ; ( x - 3 ) 2 + (y + 7)2 = 32 ; (x - 9)2 + (y - l) 2 = 32 ; (x + 5)2 + (y + I)2 = 32 Q 20. x2 + y2 - 4x + 4y = 0 ; x2 + y2 + 8x - 12y + 34 = 0 Q 21. 2 x - V 5 y - 1 5 = 0, 2 x + V 5 y - 1 5 = 0 , x - ^ I J y - 3 0 = 0, x + ^/35 y - 3 0 = 0 a-a' b-b' Q 23. A B C A' B'
Q 22. Centre == (3, 0), 1 (radius) = S 2
2
2
c-c' C
Q 26. (a2 > 2b2)
2
Q 28. x + y - 4x - 6y + 9 = 0 OR x + y - 2 0 x - 2 2 y + 1 2 1 =0, P(O,3),0 = 45° Q 29. 4x - 3y - 25 = 0 OR 3x + 4 y - 2 5 = 0 Q 30. x2 + y 2 - 4 x - 6 y - 4 = 0
EXER CISE-II 2
2
Ql. r = 2 Q 4. (b) 2(x + y ) + 2gx + 2fy + c = 0 Q 6. 9 x - 10y + 7 = 0 Q 10. aline 2 2 2 2 Q 14. y = (b + a) (b + a - 2x) OR y = (b - a) (b - a + 2x) Q 15. 3 x - y ± 2 a x - a 2 = 0 Q 17. K = 1 Q 18. {± cy + b(x + a)}2 + { - by ± c(x - a)}2 = (a2 - x2 - y2)2 Q 19. x2 + y2 - 2x - 2y = 0 Q 20. (2ax - 2by)2 + (2bx - 2ay)2 = (a2 - b2)2 Q 21. (x2 + y2 - x ) 2 = x2 + y2
Q 22. x + y = 2
Q 25. x2 + y2 ± aV2 x = 0
EXER CISE-II I ( n2 f if 1 Q.l (a) x— + y—I = - , (b) D, (c) ( - 00, -2) u (2,00) Q.2 (2, -2) or (-2, 2) Q.3 (a) (1/2, 1/4) 2J 2 V 2 y V 2 4- „2 2 Q.4 (a) B Q.5 x + y2 +. 7 x - l l y +38 = 0 Q.6
(a) D
(b) B, C
(c)
Cl
: ( x - 4 ) 2 + y2 = 9 ; c2 :
+
+y2= I
common tangent between c & Cj : Tj = 0; T2 = 0 and x - 1 = 0 ; common tangent between c & c2 : T t = 0; T2 = 0 and x + 1 = 0 ; common tangent between c, & c2 : T, = 0 ; T, = 0 and y = ± -jL= ^x + where ^ : x - V 3 y + 2 = 0 and T2 : x + v / 3 y + 2 = 0 Q.7 Q.8
(a) C (b) A (a) 6 x - 8 y + 25 = 0 & 6 x - 8 y - 2 5 = 0 (b) x2 + y2 + 4 x - 12= 0, T. : V3x-y + 2^3 +4 = 0, T2 : V 3 x - y + 2V3-4 = 0(D.C.T.) T 3 : x + V 3 y - 2 = 0, T4 : x + V3y + 3 = 0 (T.C.T.)
Q.9 (a) A ; (b).OA=3(3 +V10) Q.ll (a) C ; (b). A Q.12 C
(!%Bansal Classes
Q.IO (a) x2 + y2 + 14x-6y + 6 = 0 ; (b) 2px + 2qy = r Q.13 2x2 + 2 y 2 - 1 0 x - 5 y + 1 = 0
Circles
[12]
BANSAL CLASSES MATHEMATICS TARGET IIT JEE 2007 XI(PQRS)
PERMUTATION AND COMBINATION
CONTENTS KEY- CONCEPTS EXERCISE-I EXERCISE-II EXERCISE-III ANSWER-KEY
KEY
CONCEPTS
DEFINITIONS: 1. PERMUTATION
: Each ofthe arrangements in a definite order which can be made by taking some or all ofa number of things is called a P E R M U T A T I O N .
2.
Each of the groups or selections which can be made by taking some or all of a number ofthings without reference to the order ofthe things in each group is called a C O M B I N A T I O N . COMBINATION:
FUNDAMENTAL PRINCIPLE OF COUNTING:
If an event can occur in'm' different ways, following which another event can occur in'/?' different ways, then the total number of different ways of simultaneous occurrence of both events in a definite order is m x n. This can be extended to any number of events. RESULTS:
(i)
A Useful Notation :n! = n ( n - l ) ( n - 2 ) 3. 2. 1 ; n! =n. ( n - 1) ! n 0! = 1! = 1 ; (2n)! = 2 . n ! [1. 3. 5. 7...(2n- 1)] Note that factorials of negative integers are not defined.
(ii)
If nPr denotes the number of permutations of n different things, taking r at a time, then n! n Pr = n (n - 1) (n - 2) ( n - r + 1)= ( n _ r ) j Note that, nPn = n !.
(iii)
If nCr denotes the number of combinations of n different things taken r at a time, then n n! p n Cr = . = L where r < n ; n e N and r e W . r!(n-r)! rj The number ofways in which (m+n) different things can be divided into two groups containing m & n
(iv)
things respectively is : ( m + n ) - if m=n, the groups are equal & in this case the number of subdivision m!n! is
(v)
; for in any one way it is possible to interchange the two groups without obtaining a new n!n!2! distribution. However, if 2n things are to be divided equally between two persons then the number of (2n)! ways = n!n! Number ofways in which (m + n + p) different things can be divided into three groups containing m, n & p things respectively is
m! n!p!
5
m ^ n ^ p. /A
If m = n = p then the number of groups^
(vi)
(vii)
\ I
n!n!n!3!'
(3n)! However, if 3 n things are to be divided equally among three people then the number of ways = . (n!) The number ofpermutations ofn things taken all at a time whenp of them are similar & of one type, q of them are similar & of another type, r of them are similar & of a third type & the remaining I n - (p + q + r) are all different is: -——. p!q!r! The number of circular permutations ofn different things taken all at a time is; (n-1)!. Ifclockwise& anti-clockwise circular permutations are considered to be same, then it is • Note : Number of circular permutations ofn things when p alike and the rest different taken all at a time distinguishing clockwise and anticlockwise arrangement is^——. p!
(!iBansalClasses
Permutation and Combination
[7]
(viii)
Given n different objects, the number of ways of selecting atleast one of them is , C[ + nC2 + nC3 + + nCn = 2n - 1. This can also be stated as the total number of combinations of n distinct things.
n
(ix)
Total number of ways in which it is possible to make a selection by taking some or all out of p+q+r+ things, where p are alike of one kind, q alike of a second kind, r alike of third kind & so on is given by: (p+ l ) ( q + l ) ( r + 1) -1.
(x)
Number of ways in which it is possible to make a selection ofm + n + p = N things, where p are alike of one kind, m alike of second kind & n alike of third kind taken r at a time is given by coefficient of xr in the expansion of (1 + X + X 2 +
+ X ? ) ( 1 + X + X2+
+ X m ) (1 + X + X2 +
+xn).
Note : Remember that coefficient ofx r in (1 -x) _n = n+r_1 C r (n e N). For example the number ofways in which a selection of four letters can be madefromthe letters of the word PROPORTION is given by coefficient of x4 in (1 + x + x2 + x3) (1 + x + x2) (1 + x + x2) (1 + x) (1 + x) (1 + x). (xi)
Number ofways in which n distinct things can be distributed to p persons if there is no restriction to the number of things received by men = pn.
(xii)
Number of ways in which n identical things may be distributed among p persons if each person may receive none, one or more things is; n+p_1 Cn.
(xiii)
a.
n
C r = n C n _ r ; n C 0 = nCn = 1
c.
n
c r + nCr_! = n+1Cr
;
b.
n
Cx = nCy =>x = y orx + y = n
(xiv)
n
(xv)
Let N = p8- qb- r°- where p, q, r. are distinct primes & a, b, c are natural numbers then: (a) The total numbers of divisors ofN including 1 & N is = (a + 1 )(b + 1 )(c + 1)
Cr is maximum if: (a) r = y if n is even, (b) r = o r
- y - if n is odd.
(b)
The sum ofthese divisors is - (p° + p1 + p2+.... +p a )(q°+q 1 + q2+.... + qb) (r° +r 1 + r 2 +....+r°)....
(c)
Number of ways in which N can be resolved as a product of two „ . factors is
4(a + l)(b + l)(c +1).... if N is not a perfect square j [(a + l)(b + l)(c +1).... +1] if N is a perfect square
(d)
(xvi)
Number of ways in which a composite number N can be resolved into two factors which are relatively prime (or coprime) to each other is equal to 2 n_I where n is the number of different prime factors inN. [ Refer Q.No.28 of Ex-I ] Grid Problems and tree diagrams.
DEARRANGEMENT: Number of ways in which n letters can be placed in n directed letters so that no letter goes into its own envelope is = n!
1
2!
1
1
+ —+ 3! 4!
1
/ IV. +(-1) — v ' n!
(xvii) S ome times studentsfindit difficult to decide whether a problem is on permutation or combination or both. Based on certain words / phrases occuring in the problem we can fairly decide its nature as per the following table: PROBLEMS OF COMBINATIONS PROBLEMS OF PERMUTATIONS
• • • •
Selections, choose Distributed group is formed Committee Geometrical problems
(!i Bansal Classes
• Arrangements • Standing in a line seated in a row IB problems on digits • Problems on letters from a word
Permutation and Combination
[7]
EXERCISE-I Q.l (a) (b) (c)
In how many ways 8 persons can be seated on a round table If two of them (say Aand B) must not sit in adjacent seats. If 4 of the persons are men and 4 ladies and if no two men are to be in adjacent seats. If 8 persons constitute 4 married couples and if no husband and wife, as well as no two men, are to be in adjacent seats?
Q.2
A box contains 2 white balls, 3 black balls & 4 red balls. In how many ways can 3 balls be selected from the box if atleast 1 black is to be included in the draw ?
Q.3
How manyfivedigits numbers divisible by 3 can be formed using the digits 0, l,2,3,4,7and8 ifeach digit is to be used atmost once.
Q.4
During a draw of lottery, tickets bearing numbers 1, 2, 3, , 40, 6 tickets are drawn out & then arranged in the descending order of their numbers. In how many ways, it is possible to have 4th ticket bearing number 25.
Q.5
In how many ways can a team of 6 horses be selected out of a stud of 16, so that there shall always be 3 out of AB C A' B' C ' , but never A A ' , B B' or C C' together.
Q.6
5 boys & 4 girls sit in a straight line. Find the number ofways in which they can be seated if 2 girls are together & the other 2 are also together but separatefromthefirst2.
Q.7
In how many ways can you divide a pack of 52 cards equally among 4 players. In how many ways the cards can be divided in 4 sets, 3 of them having 17 cards each & the 4th with 1 card.
Q. 8
Find the number ofways in which 2 identical kings can be placed on an 8 x 8 board so that the kings are not in adjacent squares. How many on n x m chessboard?
Q.9
The Indian cricket team with eleven players, the team manager, the physiotherapist and two umpires are to travelfromthe hotel where they are staying to the stadium where the test match is to be played. Four of them residing in the same town own cars, each a four seater which they will drive themselves. The bus which was to pick them up failed to arrive in time after leaving the opposite team at the stadium. In howmany ways can they be seated in the cars ? In how many ways can they travel by these cars so as to reach in time, if the seating arrangement in each car is immaterial and all the cars reach the stadium by the same route.
Q.IO How many 4 digit numbers are there which contains not more than 2 different digits? Q.ll
An examination paper consists of 12 questions divided into parts A & B. Part-A contains 7 questions & Part - B contains 5 questions. A candidate is required to attempt 8 questions selecting atleast 3fromeach part. In how many maximum ways can the candidate select the questions ?
Q.12
A crew of an eight oar boat has to be chosen out of 11 menfiveof whom can row on stroke side only, four on the bow side only, and the remaining two on either side. How many different selections can be made?
faBansal Classes
Permutation and Combination
[4]
Q.13
There are p intermediate stations on a railway line from one terminus to another. In how many ways can a train stop at 3 of these intermediate stations if no 2 of these stopping stations are to be consecutive ?
Q.14
The straight lines l x , l2 & /3 are parallel & lie in the same plane. A total of m points are taken on the line /j, n points on l2 & k points on /3. How many maximum number oftriangles are there whose vertices are at these points?
Q. 15 Prove that if each of m points in one straight line be joined to each of n in another by straight lines terminated by the points, then excluding the given points, the lines will intersect — mn(m - l)(n -1) times. 4 Q.16 Afirmof Chartered Accountants in Bombay has to send 10 clerks to 5 different companies, two clerks in each. Two of the companies are in Bombay and the others are outside. Two of the clerks prefer to work in Bombay while three others prefer to work outside. In how many ways can the assignment be made if the preferences are to be satisfied. Q.17 Find the number of words each consisting of 3 consonants & 3 vowels that can be formed from the letters of the word "Circumference". In how many of these c's will be together. Q.18
There are n straight lines in a plane, no 2 of which parallel, & no 3 pass through the same point. Their point of intersection are joined. Show that the number of fresh lines thus introduced is
n(n-l)(n-2)(n-3) 8 Q. 19 Find the number of distinct throws which can be thrown with 'n' six faced normal dice which are indistinguishable among themselves. Q . 20 There are 2 women participating in a chess tournament. Every participant played 2 games with the other participants. The number of games that the men played between themselves exceeded by 66 as compared to the number of games that the men played with the women. Find the number of participants & the total numbers of games played in the tournament. Q.21 Find the number of ways 10 apples, 5 oranges & 5 mangoes can be distributed among 3 persons, each receiving none, one or more. Assume that the fruits ofthe same species are ail alike. Q.22 All the 7 digit numbers containing each of the digits 1,2,3,4, 5, 6,7 exactly once, and not divisible by 5 are arranged in the increasing order. Find the (2004)th number in this list. Q. 23 (a) (b) (c)
How many divisors are there of the number x = 21600. Find also the sum of these divisors. In how many ways the number 7056 can be resolved as a product of 2 factors. Find the number of ways in which the number 300300 can be split into 2 factors which are relatively prime.
Q. 24 There are 5 white, 4 yellow, 3 green, 2 blue & 1 red ball. The balls are all identical except for colour. These are to be arranged in a line in 5 places. Find the number of distinct arrangements.
(!i Bansal Classes
Permutation and Combination
[7]
Q.25
(1) 00 (iii) (iv) (v)
Prove that: nPr = n"1Pr + r. n"1Pr_1 If 20 C r+2 = 20 C 2r _ 3 find 12 C r Find the ratio 20Cr to 25Cr when each of them has the greatest value possible. Prove that n_1 C3 + C4 > nC3 if n > 7. Find r if 15C3r = 15Cr+3
Q. 26 In a certain town the streets are arranged like the lines of a chess board. There are 6 streets running north & south and 10 running east & west. Find the number ofways in which a man can gofromthe north-west corner to the south-east corner covering the shortest possible distance in each case. Q.27 A train goingfromCambridge to London stops at nine intermediate stations. 6 persons enter the train during the journey with 6 different tickets of the same class. How many different sets ofticket may they have had? Q.28 How many arrangements each consisting of 2 vowels & 2 consonants can be made out of the letters of the word4 DEVASTATION' ? Q. 29 0
(ii)
If'ri things are arranged in circular order, then show that the number ofways of selecting four of the things no two ofwhich are consecutive is n(n - 5) (n - 6) (n - 7) 4! If the 'ri things are arranged in a row, then show that the number of such sets of four is (n-3)(n-4)(n-5)(n-6) 4!
Q. 3 0 There are 20 books on Algebra & Calculus in our library. Prove that the greatest number of selections each ofwhich consists of 5 books on each topic is possible only when there are 10 books on each topic in the library.
EXERCISE-II Q. 1
There are 5 balls of different colours & 5 boxes of colours same as those of the balls. The number of ways in which the balls, 1 in each box could be placed such that a ball does not go to the box ofits o^/n colour.
Q.2
How many integral solutions are there for the equation ;x + y + z + w = 29 when x > 0, y > 1, z > 2 & w>0.
Q. 3
There are counters available in 7 different colours. Counters are all alike except for the colour and they are atleast ten of each colour. Find the number ofways in which an arrangement of 10 counters can be made. How many of these will have counters of each colour.
Q.4
A man has 7 relatives, 4 of them are ladies & 3 gentlemen; his wife has also 7 relatives, 3 of them are ladies & 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies & 3 gentlemen so that there are 3 of the man's relative & 3 of the wife's relatives?
Q. 5
Find the number of 7 lettered words each consisting of 3 vowels and 4 consonants which can be formed using the letters ofthe word "DIFFERENTIATION".
(!iBansalClasses
Permutation and Combination
[7]
Q.6
A shop sells 6 different flavours of ice-cream. In how many ways can a customer choose 4 ice-cream cones if (1) they are all of different flavours (ii) they are non necessarily of different flavours (iii) they contain only 3 different flavours (iv) they contain only 2 or 3 different flavours?
Q.7
6 white & 6 black balls of the same size are distributed among 10 different urns. Balls are alike except for the colour & each urn can hold any number of balls. Find the number of different distribution ofthe balls so that there is atleast 1 ball in each urn.
Q. 8
There are 2n guests at a dinner party. Supposing that the master an d mistress of the house have fixed seats opposite one another, and that there are two specified guests who must not be placed next to one another. Show that the number of ways in which the company can be placed is (2n - 2)! ,(4n2 - 6n+4).
Q.9
Eachof3 committees has 1 vacancy which is to befilledfroma group of 6 people. Find the number of ways the 3 vacancies can befilledi f ; (l) Each person can serve on atmost 1 committee. (ii) There is no restriction on the number of committees on which a person can serve. (iii) Each person can serve on atmost 2 committees.
Q . 10 A party of 10 consists of 2 Americans, 2 Britishmen, 2 Chinese & 4 men of other nationalities (all different). Find the number of ways in which they can stand in a row so that no two men ofthe same nationality are next to one another. Find also the number of ways in which they can sit at a round table, Q.ll
5 balls are to be placed in 3 boxes. Each box can hold all 5 balls. In how many different ways can we place the balls so that no box remains empty if, (i) balls & boxes are different (ii) balls are identical but boxes are different (iii) balls are different but boxes are identical (iv) balls as well as boxes are identical (v) balls as well as boxes are identical but boxes are kept in a row.
Q.12 (i) (ii) (iii)
In how many other ways can the letters of the word MULTIPLE be arranged; without changing the order of the vowels keeping the position of each vowelfixed& without changing the relative order/position ofvowels & consonants.
Q.13 Find the number of ways in which the number 3 0 can be partitioned into three unequal parts, each part being a natural number. What this number would be if equal parts are also included. Q. 14 In an election for the managing committee of a reputed club, the number of candidates contesting elections exceeds the number of members to be elected by r (r > 0). If a voter can vote in 967 different ways to elect the managing committee by voting atleast 1 ofthem & can vote in 55 different ways to elect (r - 1 ) candidates by voting in the same manner. Find the number of candidates contesting the elections & the number of candidates losing the elections. Q.15 Find the number of three digits numbersfrom100 to 999 inclusive which have any one digit that is the average ofthe other two.
(!i Bansal Classes
Permutation and Combination
[7]
Q.16 Prove by combinatorial argument that: n+I (a) C r = »Cr + »C r _ 1 /T-A n + nif> —c Jif . mr 4. n/» . m^ c W 0 r 1 r- 1 r
4. 2
C.
r-2
>
C•
r
C.
0'
Q.17
A man has 3 friends. In how many ways he can invite one friend everyday for dinner on 6 successive nights so that nofriendis invited more than 3 times.
Q.18
12 persons are to be seated at a square table, three on each side. 2 persons wish to sit on the north side and two wish to sit on the east side. One other person insists on occupying the middle seat (which may be on any side). Find the number of ways they can be seated.
Q.19
There are 15 rowing clubs; two of the clubs have each 3 boats on the river;fiveothers have each 2 and the remaining eight have each 1;findthe number ofways in which a list can be formed ofthe order ofthe 24 boats, observing that the second boat of a club cannot be above the first and the third above the second. How many ways are there in which a boat of the club having single boat on the river is at the third place in the list formed above?
Q.20
25 passengers arrive at a railway station & proceed to the neighbouring village. At the station there are 2 coaches accommodating 4 each & 3 carts accommodating 3 each, Find the number ofways in which they can proceed to the village assuming that the conveyances are always fully occupied & that the conveyances are all distinguishablefromeach other.
Q.21
An 8 oared boat is to be manned by a crew chosen from 14 men ofwhich 4 can only steer but can not row & the rest can row but cannot steer. Of those who can row, 2 can row on the bow side. In how many ways can the crew be arranged.
Q. 22 How many 6 digits odd numbers greater than 60,0000 can be formed from the digits 5, 6, 7, 8,9,0 if (i) repetitions are not allowed (ii) repetitions are allowed. Q. 23 Find the sum of all numbers greater than 10000 formed by using the digits 0 1 , 2 , 4 , 5 no digit being repeated in any number. Q. 24 The members of a chess club took part in a round robin competition in which each plays every one else once. All members scored the same number of points, except four juniors whose total score were 17. J. How many members were there in the club? Assume that for each win a player scores 1 point, for di uw 1/2 point and zero for losing. Q.25
In Indo-Pak one day International cricket match at Shaijah, India needs 14 runs to win just before the start ofthefinalover. Find the number ofways in which India just manages to win the match (i.e. scores exactly 14 runs), assuming that all the runs are made off the bat & the batsman can not score more than 4 runs off any ball.
Q.26 A man goes in for an examination in which there are 4 papers with a maximum of m marks for each paper; show that the number of ways of getting 2m marks on the whole is I (m+ l)(2m 2 + 4m + 3). Q.27
The number of ways in which 2n things of 1 sort, 2n of another sort & 2n of a 3rd sort can be divided between 2 persons so that each may have 3 n things is 3 n 2 +3 n + I.
(!iBansalClasses
Permutation and Combination
[7]
Q. 28
Six faces of an ordinary cubical die marked with alphabets A, B, C, D, E and F is thrownntimes and the list of n alphabets showing up are noted. Find the total number ofways in which among the alphabets A, B, C, D, E and F only three of them appear in the list.
Q.29 Find the number of integer betwen 1 and 10000 with at least one 8 and atleast one 9 as digits. Q.30
The number of combinations n together of 3n letters of which n are 'a' and n are 'b' and the rest unlike is (n + 2). 2"- 1 .
EXERCISE-III Q.l
Let n & k be positive integers such that n > kfr+1). The number of solutions (xj.xj,.... , x k ) , x j > 1, X j > 2,... , x k > k , all integers, satisfying Xj + X2+.... +x k =n, is . [ JEE '96,2 ]
Q. 2
Find the total numb er of ways of selectingfivelettersfromthe letters of the word INDEPENDENT. [REE'97, 6]
Q.3 (l)
Select the correct alternative(s). Number of divisors of the form 4n + 2 ( n > 0) of the integer 240 is (A) 4 (B) 8 (C)10
[ JEE '98, 2 + 2 ] (D)3
(ii)
An n-digit number is a positive number with exactly 'n' digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5&7. The smallest value ofn for which this is possible is : (A) 6 (B)7 (C)8 (D)9
Q.4
How many different nine digit numbers can be formedfromthe number 2233 55888 by rearranging its digits so that the odd digits occupy even positions ? [JEE '2000, (Scr)] (A) 16 (B) 36 (C) 60 (D) 180
Q. 5
Let Tn denote the number of triangles which can be formed using the vertices of a regular polygon of ' n' sides. If T n + 1 - Tn = 21, then V equals: [ JEE '2001, (Scr) ] (A) 5 (B)7 (C)6 (D)4
Q.6
The number of arrangements of the letters of the word BANANAin which the two N's do not appear adj acently is [JEE 2002 (Screening), 3 ] (A) 40 (B) 60 (C) 80 (D) 100
Q.7
Number of points with integral co-ordinates that lie inside a triangle whose co-ordinates are (0, 0), (0, 21) and (21,0) [JEE 2003 (Screening), 3] (A) 210 (B) 190 (C) 220 (D)None
Q. 8 Q.9
(n 2 ) ! Using permutation or otherwise, prove that . ..„ is an integer, where n is a positive integer. (n!) [JEE 2004, 2 out of 60] A rectangle with sides 2m - 1 and 2n - 1 is divided into squares ofunit length by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is (A) (m + n+ l) 2 (B) 4m + n ~ 1 (C) m2n2 (D) mn(m + l)(n + 1) [JEE 2005 (Screening), 3]
(!iBansalClasses
Permutation and Combination
[7]
ANSWER KEY EXERCISE-I Q.l
(a) 5-(6!), (b) 3! • 4!, (c) 12
Q.2
6 if the balls of the same colour are alike & 64 if the balls of the same colour are different
Q.3
744
Q J
Q.4
24
960
Q.6
Q 8
n[*C2-(m-l) + m[»C2-(n-l)]
Q.9
Q.ll
420
'
C2 . 15C3
Q.5
43200 111.4! 12!;(3!)42|
Q.12
145
Q.13 P~2C.
Q.16
5400
Q,17
22100,52
Q.21
29106
Q.22
4316527
Q.23 (a) 72 ; 78120 ; (b) 23 ; (c) 32
Q.24
2111
Q.25
(ii) 792 ; (iii) ^
Q.26
(14)! 5!9!
Q.27
45
Q.28
1638
Q.4
485
Q.7
26250
Q.14
10,3
Q.IO
576
Q.14
m+n+k
Q.19
n+5
C 3 - (mC3 + nC3 + k C 3 )
C,
Q.20
13 , 156
; (v) r = 3
Cfi
EXERCISE-II 49 710 ; | — | 10
Q.l
44
Q.2
2600
Q.5
532770
Q.6
(i) 15, (ii) 126, (iii) 60, (iv) 105
Q.9
120, 216, 210
Q. 10
(i) linear: (47) 8! ; (ii) circular: (244). 6!
Q.ll
(i) 150 ; (ii) 6 ; (iii) 25; (iv)2; (v) 6
Q.12
(i) 3359 ; (ii) 59; (iii) 359
(!iBansalClasses
Q.3
Q.13
61,75
Permutation and Combination
[7]
Q.15
121
Q.17
510
Q.22 240,15552
Q.23 3119976
Q.25
Q.28
1506
Q.29 974
6
C3[3n - ^ ^
Q.18
Q.24
2! 3! 8!
27
- 2) - 3C2]
EXERCISE-III
Q.l
m
Ck_, where m = (1/2) (2n - k2 + k - 2)
Q.2
72
Q.3
(i) A; (ii) B
Q.4
C
Q.5
B
Q.6
A
Q.7
B
Q.9
C
(!i Bansal Classes
Permutation and Combination
[7]
ft
BANSAL CLASSES MATHEMATICS TARGET IIT JEE 2007 XI (P. Q, R, S)
BINOMIAL
CONTENTS KEY- CONCEPTS EXERCISE - 1(A) EXERCISE - 1(B) EXERCISE-II EXERCISE - 111(A) EXERCISE - 111(B) EXERCISE-IV ANSWER-KEY
KEY
CONCEPTS
BINOMIAL EXPONENTIAL & LOGARITHMIC SERIES 1.
BINOMIAL THEOREM : The formula by which any positive integral power of a binomial expression can be expanded in the form of a series is known as BINOMIAL THEOREM . If x,y e R and n e N , then ; n
n
n
n
n
1
n
n 2 2
n
(x + y) = C0 x + Cj x"- y + C2 x " y +
n r r
+ Crx " y +
n
n
+ Cny = X n C r x n " r y r .
r=0
This theorem can be proved by Induction . OBSERVATIONS : (i) The number of terms in the expansion is (n + 1) i.e. one or more than the index. (ii) The sum of the indices of x & y in each term is n . (iii) The binomial coefficients of the terms n C 0 , nC j.... equid istant from the beginning and the end are equal. 2. (i) (iii)
IMPORTANT TERMS IN THE BINOMIAL EXPANSION ARE : General term (ii) Middle term Term independent of x & (iv) Numerically greatest term
(i)
The general term or the(r+ l) th term in the expansion of (x + y)n is given by; Tr+i = nCr x n - r . yr The middle term(s) is the expansion of (x + y)n is (are) : (a) If n is even, there is only one middle term which is given by ;
(ii)
T
(b)
(n+2)/2
= nf1
Yn/2
x
n/2
„n/2
'l
If n is odd, there are two middle terms which are : T(n+l)/2
&
T[(n+l)/2]+l
(iii)
Term independent of x contains no x; Hencefindthe value of r for which the exponent of x is zero.
(iv)
To find the Numerically greatest term is the expansion of (1 + x) n , n e N find n T . C1x r n-r+ 1 —^ = —r = x . Put the absolute value of x &findthe value ofr Consistent with the T Cf_jX
T.j inequality - y > 1. Note that the Numerically greatest term in the expansion of (1 - x) n , x > 0, n e N is the same as the greatest term in (1 +x) n . If (VA + b)"= I + f> where I & n are positive integers, n being odd and 0 < f < l , then
3.
(I + f). f = Kn where A - B 2 = K > 0 & V A - B < 1 . If n is an even integer, then (I + f) (1 - f) = Kn. 4.
BINOMIAL COEFFICIENTS :
(i)
C 0 + CJ + C 2 +
+ CN = 2*
(ii)
C0 + C2 + C4 +
= CJ + C 3 + C 5 +
(iii)
C 0 2 + C J 2 + C 2 2 + .... + C N 2 =
(iv)
C0.Cr + Cj.C^j + C2.Cr+2 + ... + C n _ r .C n -
?fe\Bansal
Classes
2N
CN =
= 211-1 ^ ( ! (n+ r^ _r).
Binomial
[6]
REMEMBER : (i) (2n)!=2 n .n! [1.3.5 5.
(2n-l)]
BINOMIAL THEOREM FOR NEGATIVE OR FRACTIONAL INDICES : . , n ( n - l ) 2 n ( n - l ) ( n - 2 )J 3s I f n e Q , then (1 +x)n= l + nx + —^—-x — ^ -x +
.. ,, G . O Provided | x | < 1.
Note : (i) When the index n is a positive integer the number of terms in the expansion of (1 +x) n is finite i.e. (n+ 1) & the coefficient of successive terms are : np np np np np M> 2' 3 n (ii) When the index is other than a positive integer such as negative integer or fraction, the number of terms in the expansion of (1 +x) n is infinite and the symbol nCr cannot be used to denote the Coefficient of the general term. (iii)
Following expansion should be remembered (| x | < 1). (a) (1 +X)"1 = 1 - x + x 2 - x 3 + x 4 -.... oo (b) (1 - x ) - 1 ^ 1 + x + x2 + x3 + x 4 +.... oo (c) (1 + x)~2 = 1 - 2 X + 3 X 2 - 4 X 3 + . . . . oo (d) (1 -x)~ 2 = 1 + 2 X + 3X 2 +4X 3 + oo
(iv)
The expansions in ascending powers of x are only valid if x is 'small'. If x is large i.e. | x | > 1 then
6.
we may find it convinient to expand in powers of —, X which then will be small. APPROXIMATIONS : (1 +x) n = 1 +nx+
- 1 ) ( n - 2 ) x3 1.2 1.2.3 If x < 1, the terms of the above expansion go on decreasing and if x be very small, a stage may be reached when we may neglect the terms containing higher powers of x in the expansion. Thus, if x be so small that its squares and higher powers may be neglected then (1 + x)n = 1 + nx, approximately. This is an approximate value of (1 +x)n. x* +
7.
EXPONENTIAL SERIES:
(i)
x x2 x3 ex=l + — + — + — + 2
(ii)
n(n
( lV oo ; where x may be any real or complex & e = ^u^t \ \ + — J 3
ax = 1 + —In a + — / n 2 a + — / n 3 a + 1! 2! 3!
oo where a > 0
Note: (a)X
/
(b) (c)
1
1
1
e = l1 +4 . — + — + — + oo 1! 2! 3! e is an irrational number lying between 2.7 & 2.8. Its value correct upto 10 places of decimal is 2.7182818284. 1
e + e" = 2
' 1,+— 1 +1 1—+ —+ V
2! 4! 6!
N
0.0
y
N
(d) (e)
r i i i 1 00 e - e"-1=0 = 2 l+ _ + _ + _ + ^ 3! 5! 7! J Logarithms to the base 'e' are known as the Napierian system, so named after Napier, their inventor. They are also called Natural Logarithm.
?fe\Bansal
Classes
Binomial
[6]
8.
LOGARITHMIC
(i)
/n(l+x) = x - ~
(ii)
In v(1- x) = - x
SERIES:
2
1
2
(ho
3
4 3
2
f X+
x3
+
b
oo w h e r e - l < x < l
4
— + 4
3
3
V
REMEMBER :
4
2
^
(1 + x)
3
x5
+
00
5
oo where - 1 < x < 1 X <1
1 1 1
1 - - + - - - +... oo = /n2 2 3 4 /n2 = 0.693
(a)
(c)
EXERCISE-I Q. 1
(b)
e/n x - x
(d)
/n!0 = 2.303
(A)
Nil
a +• i v ' Find the coefficients : (i) x in ' ax 7
V
(ii) x
bx
7
in ax--
bx'
(iii) Find the relation between a & b, so that these coefficients are equal. Q.2
If the coefficients of (2r + 4) th , (r - 2)th terms in the expansion of (1 + x)18 are equal,findr.
Q3
If the coefficients of the rth, (r + l) th & (r + 2)th terms in the expansion of (1 +x) 14 are inAP, find r. /x 3
V3 2x'
Q.4
Find the term independent of x in the expansion of (a)
Q.5
1 3r V 15r t h—T- + —j- + Find the sum of the series ^ ( - l ) r . n C r —r + ——
r=0
nd
rd
2
2
2
2
10 1/3 +x~1/5 (b) —x 2
up to m terms
& 4th terms in the expansion of (1 +x) 2n are in AP, show that
Q.6
If the coefficients of 2 , 3 2n2 - 9n + 7 = 0.
Q.7
Given that (1 + x + x2)n = a0 + ajX + s^x2 + ....+ a^x 2 ", find the values of : (i)a0 + aj + a2 + + 8^ ; (ii) a 0 - a 1 + a 2 - a 3 + a2n ; (iii) a 0 2 - a t 2 + a2-ag2 +
Q.8
If a, b, c & d are the coefficients of any four consecutive terms in the expansion of (1 + x)n, n e N, prove that ^
Q. 9
a
+
c
+ a22
2b
Find the value ofx for which the fourth term in the expansion,
^|log5 V 4x+44+
1 +7
is 336.
Q. 10 Prove that : n"1Cr + n~2Cr + n"3Cr + .... + rCr = nCr+1. Q.ll
(a)Which is larger : (99 S0 + 10050) or (101)50. v (b) ' Show that
te Bonsai Classes
4n - C n— ,z+ 2.2n"2CBn~i. + 2n~2Cnn> n 71 , n e N , n>2
2n 2
Binomial
[4]
vn Q.12 In the expansion of \1 + x + - J find the term not containing x.
Q.13
Show that coefficient of x5 in the expansion of (1 +x 2 ) 5 . (1 + x) 4 is 60.
Q.14 Find the coefficient of x4 in the expansion o f : (i) ( l + x + x 2 + x 3 )
n
(ii) ( 2 - X + 3 X 2 ) 6
Q.15 Find numerically the greatest term in the expansion of : (i) (2 + 3x)9 when x = Q.16 Given s n = l + q + q2 + prove that
a+1
(ii) (3 - 5x)15 when x = j + qn &
Sn = 1 + ^
+
+ .... + ( ^ j , q * 1,
C t + n+1C2.S! + n+1C3.s2 +....+ n+1Cn+1.sn = 2 n . S n .
Q.17 Prove that the ratio of the coefficient of x10 in (1 -x 2 ) 1 0 & the term independent of x in x-|j
is 1 : 32 .
Q.18 Find the term independent of x in the expansion of (1 + x + 2x3)
3x 2 2
3 xj
" (l + rlog 10) Q. 19 Prove that for n e N X (~l) r nCr 7 = 0. r=0 (l+log e 10 n ) Q.20 Prove the identity
+
=
. Use it to prove £
=
Q. 21 If the coefficient of a r_1 , a r , afTl in the expansion of (1 + a)n are in arithmetic progression, prove that n2 - n (4r + 1) + 4r2 - 2 = 0. (1 - x n )(1 •- x n_1 )(1 - x n " 2 ) Q22
If nJ = r
"iMO^Xi-x3)
(1 - x n_r+1 )
...,.(i-xr)~~'provethatnJ- = nJ-
n Q.23 Prove that ^ n C K sinKx. cos(n - K)x = 2 n_1 sin nx. K=0
Q.24 The expressions 1 + x, 1+x + x2, 1 + x + x2 + x3, 1 + x + x2 + + xn are multiplied together and the terms ofthe product thus obtained are arranged in increasing powers ofx in the form of a0 + ajX + a^x2 + , then, (a) how many terms are there in the product. (b) show that the coefficients of the terms in the product, equidistantfromthe beginning and end are equal. (c)
(n + 1)! show that the sum of the odd coefficients = the sum of the even coefficients = 2
Q.25 Find the coeff. of
2n
2n
x 2 r= b
(a) (b) (c)
x6 in the expansion of (ax2 + bx+c) 9 . x2 y3 z4 in the expansion of (ax - by + cz)9 . a2 b3 c4 d in the expansion of (a - b - c + d)10.
x3r
Q.26 If 2>r( - ) S r( - ) & a k = 1 for all k>n, then show that bn = 2n+1Cn+1. r=0 r=0
?fe\Bansal
Classes
Binomial [6]
/=k-l /. ^ n Q.27 If P k (x)= 2 x1' then prove that, £ n C k P k (x) = 2""1-Pn' /=o \ ^j k=1 Q.28 Find the coefficient of xr in the expression of : (x + 3)n_1 + (x + 3)n"2 (x + 2) + (x + 3)n"3 (x + 2)2 +
+ (x + 2)n~l
fx 2V Q.29(a) Find the index n of the binomial I •- + -1 if the 9th term of the expansion has numerically the greatest coefficient (n e N). (b) For which positive values of x is the fourth term in the expansion of (5 + 3x)10 is the greatest. Q.30 Prove that
(72)1 (36!
- 1 is divisible by 73.
f
Q. 31 If the 3rd, 4th, 5th & 6th terms in the expansion of (x+y) n be respectively a, b, c & d then prove that b 2 -ac 5a c 2 ~bd 3c' Q.32 Find x for which the (k+ l) th term of the expansion of (x + y)n is the greatest if x + y = 1 andx>0, y>0. *Q,33 If x is so small that its square and higher powers may be neglected, prove that:
(l - 3x)y2+(i ~ x)5/a *Q.34 w(a) (b)
=
Mr+Mr5
j
If x = - + — + HJL + _L11L +
oo then Fprove that x2 + 2 x - 2 = 0.
tfy=!
then find the value of y2 + 2y.
3
3.6
3 . 6 . 9 3.6.9.12
+~if)
+
*Q.35 If p = q nearly and n >1, show that (n-l)p + (n + l)q (*)
= 1 + (Ji)x+,±)x or
P U
Only for CBSE. Not in the syllabus of HT JEE.
EXERCISE-I (B) Q.l
Show that the integral part in each of the following is odd. n e N ( B ) (s + 3 V7) n
( A ) (5 + 2 Ve)"
Q.2
Show that the integral part in each of the following is even, n e N (B) (5V5 + ll)2n+1
(A) (3V3 + 5 p ' Q.3
( C ) (e + V3?) n
If (7 + 4-^3
=
p+P where n & p are positive integers and P is a proper fraction show that
(1-P)(p + P)=l. Q.4
If x denotes (2 + V3) , n e N & [x] the integral part of x then find the value of : x - x 2 + x[x].
?fe\Bansal Classes
Binomial
[6]
Q.5
If P = (s + 3V7) and f = P - [P], where [ ] denotes greatest integer function. Prove that: P (1 - f) = 1 (n e N)
Q.6
If (6V6 + 14)2n+1 = N & F be the fractional part of N, prove that NF = 202n+1 (n e N)
Q. 7
Prove that if p is a prime number greater than 2, then the difference (2 + V5)P - 2p+1 is divisible by p, where [ ] denotes greatest integer.
Q.8
Prove that the integer next above (v'3 + lj contains 2n+1 as factor (n e N)
Q.9
Let I denotes the integral part & F the proper fractional part of (3 +
where n e N and if p
denotes the rational part and 0 the irrational part of the same, show that p=|(I+l)and
a =
(I + 2 F - 1).
2n
C
Q.IO Prove that
is an integer, V n e N .
n+1
EXERCISE-II (NOT IN THE SYLLABUS OF I IT-JEE) PROBLEMS ON EXPONENTIAL & LOGARITHMIC SERIES For Q.l TO Q.15, Prove That: Q.l Q2
v
>2 . , 1 1 1 1 1 1 ,1 + — + — + -— ,+ 1+ +— 2!
e-1
4!
6!
[ 1
1 1 + — +— +, V2! 4! 6!
e + 1
3!
— + — +. 5! 7!
1
1 1
—+ — + — +, .1! 3! 5! 1 1 + — + — +1 — + ,
Q3
e 2 - 1 = ( _1 + _1 + _1 + , — e + 1 U 3! 5!
~A Q.4
1.1+2 1+2+3 1+2+3+4 ! + _ _ + — + + 2! 3! 4!
_L
Q5
1
2!
+
2
3
3
4
4!
— +— +— +
Q.ll
1 +1 — 1 + — + ... = 1- +1 1 +1 —
Q.12
1.2 1
2
3.4
2.2
?fe\Bansal
°° = \2J
Q8.
5e
Q.9
4.5
6!
6
1 + 2 + 22 + 23 2 +. =e -e
3
1 + 2! — + 3!— +4!— + 2.3
4!
1.2.3.4.5.7
, 1 + 2 1 + 2 + 22 1 +— + 3! + 2! 3
1
1
1.3 + 1 . 2 . 3 . 5
Q.6 0V 7
1
1
00 = 1 - log 2
6.7
5.6 1
3.2
&e
2
1 4.2
Classes
+
1.2.3
3.4.5
- ln3 - ln2
+
2
3
6
11
18
1!
2!
3!
4!
5!
— +— + — + — + —+ ^
Q 10.
5.6.7
1 1 +2 — + 4t 3.2
5.2
+
,\
=3 (e-1) v '
1
loge3
r +• 7.2 6
+ .... = .In_2 1 1
1
nn13. - + — - + —^ + — 1 T1+ Q 3
Binomial
3.3
5.3
7.3
=
In 2 [6]
Q.14
2 \2
V
4 V2
6V2 3
3 /
Q.15 If y = x - +
+
3
where|x|< 1, then prove that x = y +
+ 2L +
+
EXERCISE-HI (A) If C 0 , CJ , C 2 , , Cn are the combinatorial coefficients in the expansion of (1 +x) n , n s N , then prove the following :
(2n)!
C - + CV + C - +
Q.2
C0 q + Cj C2 + C2 C3 +....+Cn_j Cn =
Q.3
Cj + 2C2 + 3C3 +
+ n . Cn = n . 2n_1
Q.4
C0 + 2Cj + 3C2 +
+ (n+l)Cn = (n+2)2n~1
Q.5
C0 + 3Cj + 5C2 +
+ (2n+l)Cn = (n+1) 2n
Q.6
(C0+C1)(C1+C2)(C2+C3)
Q.7 v
V
C0
'
+
C,
+
0
C2
+ c
2
Q.l
+
2
+
n
=-^
(Cn_j+Cn) C a _,
3
Qv 8
2
4
n+1
Q. 10 C0Cr + CjCr+1 + C2Cr+2 + .... + C n . r Cn = Q.ll
+
2
+ (-Dn
3
C
" "
n+1
2
+
^
3
+
+
n+1
n+1
n+1 ( n
_r^
+ r) ,
1
n+1
Q.12 C 0 - C j + C 2 - C 3 + .... + ( - l ) r . Cr< Q.13
0
(-1X _(n_- 1)! (n r 1)s
r|
C0 - 2Cj + 3C2 - 4C3 + .... + (-l) n (n+l) Cn = 0
Q. 14 C 0 2 -C, 2 + C 2 2 -C 3 2 +
+ (-l) n C n 2 = 0 or (-l) a / 2 Ca/2 according as n is odd or even.
Q.15 If n is an integer greater than 1, show that ; a - n Cj(a-l) + nC2(a-2) + (~l) n (a - n) = 0 Q. 16 (n-1) 2 . Cj + (n-3) 2 . C3 + (n-5) 2 . C5 + Q. 17 V
1 . C0 2 + 3 . C,1 2 + 5 . C2 2 +
= n (n + l)2n~3
+ V(2n+l) Cn n2 = -(n + '
l)
n!n!
Q.18 If a 0 , a;, a,, be the coefficients in the expansion of ( l + x + x2)n in ascending powers of x, then prove that : (1) a0 a, - aj % + % % - .... = 0 . + (ii) a^-aja3 + a ^ ^n - 2 *2n = an +1 o r an-in-! (iii) Ej = E2 = E 3 = 3 ; where E t = a0 + % + a6 + ; E2 = a1 + a 4 +a 7 +
E^ a2 + a^ + ag +
1-2 Q.19 Prove that : Z("x c r • ^+2) = r=o
n
'
&
(2n)! ( n - 2 ) ! (n+2)!
2
Q.20 If (l+x) = C0 + CjX + C2x + .... + C n x n , then show that the sum of the products of the C .' s taken two at a time, represented by
ItBansat Classes
2 X C Cj . '.
0 < i < j < n
Binomial
is equal to 22n_1 n
2n! 2(n!) 2
[8]
Q.21
J c ^ + yfc7 +
Q.22
V ^
+ Jc~n<2n-l
+
n-1
+
2 -il/2
+ V c T + A / c T + •••••• + V c 7 ^ [ n ( 2
n
-l)]
forn>2.
EXERCISE-III (B) Q.l
If ( l + x y ^ C o + Cj. x + C2. x 2 +.... + C15. x15, then find the value o f : C
Q.2
2
+ 2C3 + 3 C
4
+ ....
+14C
If(1 + x + x 2 + . . . + x P ) a,+2a2
n
1 5
= a0 + a1x + a2x2+..,+anp.xnP
,
then
find
the value of :
t- 3 a 3 + . . . . + n p . a n p
Q.3
l 2 . C 0 + 2 2 . C t + 3 2 . C 2 + 4 2 . C 3 + .... + ( n + 1 ) 2 C
Q.4
^ r 2 . C r = n ( n + l)2n-2
Q.5
Given
n
= 2n~2(n+1) (n+4) .
t=0
p + q = l
j>>2.nCr.p1 .q'w = n p [ ( n -
, showthat
l)p+l]
i=0
II Q.6
S h o w that
2
^C
1=0
r
= n.2n
(2r-n)
w h e r e Cf denotes the combinatorial coeff. in the expansion
+X)n.
(1
Q .7 v
r
Q.8
P r o v e t h a t , 2 . C n + — . c 1, + — . C2, +
v
Q.9
0
+
2
3
+
4
0
If(l+x)
n
=
Z c
r=0
22.C0
r
QI.O
+
. x '
;
24,C2
|
3.4
aQ.12 i ^ -Ca Ci + — O9+ — Ci + 2 -+ — 2 3 4 5
Cn n+2
13
Q.13 Si_SL+Sl_Si + 2
3
4
Q. 15
2
3
% ( l - x )
5
?fe\Bansal
— —
11
31""2 - 2 n - 5
+ %
Classes
4n.n!
1.5.9.13
(4n-3)(4n+l)
l+n.2 (n+1) (n+2) n+2
1
(n+1) (n+2)
+ (_!)»-i .Si = I + I + i + A + v
4 2
=
+ (_!)" . . Cn _
I f ( l + x ) n = C 0 + CjX + C 2 x 2 +
q o - x ) -
. C10, 0 =
2n+2.Cn
4n+l
Q.14 Si _ Si + Si _ Si + 1
11
2n n+1
+
9
x n+ ) ' (n + l ) x
(n+1) (n+2) ~ (n+1) ( n + 2 )
( i)"
5
+
+ —
|
Q 11 Si _Sl + Si _ Si + 1
(1
t h e n p r o v e that ;
2.3
+
=
n + 1
3
2
23.Ct
|
1.2
v
of
y
n + Cnx
2 3 4 n
, then show
(1-x)3 -....+ (-l)n-l I ( l _
x )
n =
( 1
+I
n that : _
Binomial [6]
x ) +
I
( 1
_
X
2
) +
I (1—x3) +
+ ^(1-X«)
Q. 16 Prove that , \ - C r f »C2+ j »C3Q.17
n
n
x+1
x+2
+
^
. »Cn= - 1
n+1
n+1
n
C.L -i C +•
If n e N ; show that
+^ + (-l)n
C
x+n
=
n! x ( x + l) (x + 2) .... ( x + n )
Q.18 Prove that, ( ^ C ^ 2 . (2nC2)2 + 3 . (2nC3)2 + ... + 2n. (2nC2n) Q.19
(4n - 1)! [(2n-l)!]2
2n If(l+x+x ) = X a r x r , n s N , then prove that 2 n
r=0
(r+l)ar+1 =(n-r)ar + (2n-r+l)ar_j. Q.20 Prove that the sum to (n + 1) terms of -
n(n+1)
(0
(n+1) (n+2)
equals
(n+2) (n+3)
J xn_1. (l-x) n + 1 . dx&evaluate the integral. 0
EXERCISE-I V Q. 1
Let (1+x2)2. (l+x) n = 2 X . xK . If aj, % & % are in AP, find n.
Q.2
In the expansion ofthe expression (x+a) 15 , if the eleventh term is the geometric mean of the eighth and twelfth terms, which term in the expansion is the greatest ? [REE '96,6]
Q.3
The sum ofthe rational terms in the expansion of (4l +31/5)10 is
Q4
n
n
r=0
r=0
1 If a n = I — - - , then
Q. 6
.
[JEE'97,2] [JEE'98,2]
equals
<-r
(B) na n
(A) (n-l)a„
Q.5
r
[REE '96, 6]
(D) None of these
(C) n a „ / 2
3 5 9 15 23 Find the sum ofthe series Y[ + 2! + 3 j + 4 j + ' 5 7 + '
[REE '98,6]
.oo
If in the expansion of (1 + x)m (1 - x)n, the co-efficients of x and x2 are 3 and - 6 respectively, th n m is : [JEE '99, 2 (Out of200)] (A) 6 (B) 9 (C) 12 (D) 24
Q.7(i) For 2 < r < n, (A)
Vrj
'n + O vr-1/
+2 (B)2
n r- 1 n +1 r + 1,
'
n ^
\ r - 2/
(C) 2
n +2 r J
(D)
'n + 2 V
r
(ii) In the binomial expansion of (a-b) n , n > 5, the sum ofthe 5th and 6th terms is zero. Then — equals: b [ JEE '2000 (Screening), 1 + 1 ] n- 4 n- 5 (C) n - 4 (B) (A)
?fe\Bansal
Classes
Binomial
[6]
Q.8
For any positive integers m, n (with n > m), let VmJ
+
'n-r
+
HI J
( n-1 n 1\ \
m
vm;
/ +
+
m
= "Cm . Prove that
\
n +1 m +1
ImJ
Hence or otherwise prove that. 'n-lN ' nN n_2 l + +3 f +2 K in J ni J V m;
/ m\ + ( n - m + 1) VrnJ
f n + 2l + 2,
[JEE'2000 (Mains), 6]
Find the largest co-efficient in the expansion of (1 + x) n , given that the sum of co-efficients ofthe terms in its expansion is 4096 . [ REE '2000 (Mains) ] a Q.10 In the binomial expansion of (a - b)n, n > 5, the sum of the 5th and 6th terms is zero. Then — equals Q.9
(A)
n-5
(B)
n-4
(C)
v»—5
n-4
[JEE'2001 (Screening), 3] Q.ll
49
[REE'2001 (Mains), 3]
Find the coefficient of x in the polynomial \ / ^ \ ' C,N ' 2 r 2 C x 2 • — x 3 • — x — -—C2y v C0y v C„
Q. 12 The sum
Q. 13(a) Coefficient of t24 (A) 12C6 + 2
2
cA
x - 50 • —^
V
C49y
where C = 5 0 C .
(where ( n = 0 i f P < q ) is maximum when m is
i=0
(A) 5
'
(D) 20 [JEE'2002(Screening), 3] in the expansion of (1+t2)12 (1+t 12 ) (1+t 24 ) is (B) 12 C 6 + 1 (C) 12 C 6 (D) none [JEE 2003, Screening 3 out of 60] (C)15
(B)10
iK (-1) K A O K j[JEE 2003, Mains-2 out of 60]
nYn 2 (b) P r o v e « h a t : 2 K . l S ) ( ^ - 2 - ( i ' ) ( r - l ) + 2 K - 22' K -" 2
Q.14
n_1
Cr = (K2 - 3).nCr+1, if KG
( A ) [ - V 3 , V3]
Q.15 The value of [ q ] ( j q
wfs:
(!%Bansal Classes
(C)(2,oo)
(B)(-oo,-2)
(B)f^
+
301(30
12
(o CSS"
Binomial
(D)(V3,2] [JEE 2004 (Screening)] + 20 30 is, where
l = nCr
[JEE 2005 (Screening)]
[11]
ANSWER KEY EXERCISE-I Ql.
(i) n C 5 p - (u)
(HQ a b = l Q2. r = 6 Q3. r = 5 o r 9
Q 7. (i) 3n (ii) 1, (iii) an
Q5. ( J ^ L )
(A)
Q9. x = O o r l
Q 4. (a) ^ (b) T6 =7 Q10.x = 0 o r 2
Q 11. (a)10150(Prove that 10150 - 9950= 1005° + some+ive qty) Q 12. 1 + £ u C 2 k . 2kCk 7k k = l1 Q14. (i) 990 (ii)3660 Q.24 (a)
Q15.(i)T?=-^
(ii) (ii) 455x3^2 455x3 12
Q18<1Z Q18.
n 2 +n + 2 2
Q 25. (a) 84b6c3 + 630ab4c4 + 756a2b2c5 + 84a3c6
; (b) -1260 . a2b3c4
; (c) -12600
Q 28. nCr (3n-r - 2n~r) Q 29. (a) n = 12 (b) | < x < Q.32 Q 34. (a) Hint: Add 1 to both sides & compare theRHS series with the expansion (l+y)n to get n & y (b) 4 ~
EXERCISE-I(B) Q.4
1
EXERCISE-III
(B)
15
Q 1. divide expansion of (1+x) both sides by x& diff. w.r.t.x, putx= 1 to get 212993 Q 2. Differentiate the given expn. & put x = 1 to get the result Q 9.
(p+l) n
Integrate the expn. of (1 + x)n. Determine the value of constant of integration by putting x=0. Integrate the result again between 0 & 2 to get the result.
Q 10. Consider j [(l+x)n + (l-x) n ] = C0 + C2x2 + C4x4 +
Integrate between 0 & 1.
Q 12. Multiply both sides by x the expn. (l+x) n . Integrate both sides between 0 & 1. Q 14. Note that
(1 x)
~
= - Cj+ C2x - C3x2 +....+ Cn. x11"1. Integrate between 1 & 0
(n-1)! (n+1)! (2n +1)!
EXER CISE-IV Q.l n = 2 or3 or4 Q.6 C Q.ll -22100
?fe\Bansal
Classes
Q.2 Tg Q.3 41 Q.7 (i) D (ii) B Q.12 C Q.13 (a) A
Binomial
Q.4 C Q.9 12C6 Q.14 D
Q.5 4 e - 3 Q.10 B Q.15 A
[6]
BANSAL CLASSES M
II MATH€MAfiC5 TARGET IIT JEE 2007 XI (P.Q.R.S)
FUNCTIONS &
INVERSE TRIGONOMETRIC FUNCTIONS Trigonometry Phase - IV
. ir»W : HMT I ;:: fS:1=CMWT• FUNCTIONS KEY CONCEPT. EXERCISE-I EXERCISE-II EXERCISE-III
Page Page Page Page
-2 -10 -12 -13
INVERSE TRIGONOMETRIC FUNCTIONS KEY CONCEPT. EXERCISE-I. EXERCISE-II EXERCISE-III
Page Page Page Page
-16 -21 -23 -25
ANSWER KEY
Page -26
KEY CONCEPTS (FUNCTIONS) THINGS T O REMEMBER 1.
:
GENERAL DEFINITION
:
If to every value (Considered as real unless other-wise stated) of a variable x, which belongs to some collection (Set) E, there corresponds one and only onefinitevalue ofthe quantity y, then y is said to be a function (Single valued) of x or a dependent variable defined on the set E ; x is the argument or independent variable. If to every value of x belonging to some set E there corresponds one or several values of the variable y, then y is called a multiple valued function of x defined on E.Conventionally the word " F U N C T I O N " is used only as the meaning of a single valued function, if not otherwise stated. { } f(x)=y Pictorially: x > f >, y is called the image of x & x is the pre-image of y under f. input
output
Everyfonctionfrom A—» B satisfies the following conditions. (i) f c AxB (ii) V a e A=> (a, f(a)) s f (iii) (a, b) e f & (a, c) s f => b = c 2.
DOMAIN,
CO-DOMAIN
&
RANGE
OF A
and
FUNCTION:
Let f: A-»B,then the set Ais known as the domain of f&the set B is known as co-domain off. The set of all f images of elements of A is known as the range of f . Thus : Domain of f = {a | a e A, (a, f(a)) e f} Range of f = (f(a) | a e A, f(a) e B} It should be noted that range is a subset of co-domain. If only the rule offunction is given then the domain of the function is the set of those real numbers, where function is defined. For a continuous function, the interval from minimum to maximum value of a function gives the range. 3.
IMPORTANT TYPES OF FUNCTIONS
(i)
POLYNOMIAL FUNCTION :
:
If a function f is defined by f (x) = aQxn+ a, xn_1+a,xn_2+... + an l x + an where n is a non negative integer and a0, a p a2,..., an are real numbers and a0 ^ 0, then f is called a polynomial function of degree n. NOTE : (a) A polynomial of degree one with no constant term is called an odd linear function, i.e. f(x) - ax, a^O (b)
(ii)
There are two polynomial functions, satisfying the relation; f^x).f(l/x) = f(x) + f(l/x). They are : (i) f(x) = xn + 1 & (ii) f(x) = 1 — xn , where n is a positive integer.
ALGEBRAIC FUNCTION
:
y is an algebraic function ofx, ifit is a function that satisfies an algebraic equation ofthe form P0 (x) yn + Pj (x) yn_1 + + Pr l (x) y + Pn (x) = 0 Where n is a positive integer and P0 (x), P; (x) are Polynomials in x. e.g. y = | x | is an algebraic function, since it satisfies the equation y2 - x 2 =0. Note that all polynomial functions are Algebraic but not the converse. A function that is not algebraic is called TRANSCEDENTAL FUNCTION . (iii)
FRACTIONAL RATIONAL FUNCTION
:
A rational function is a function of the form, y = f (x)
g(x) h(x)
g (x) & h (x) are polynomials & h (x) * 0.
feBansal Classes
Functions & Trig.-fl-IV
where
(iv)
EXPONENTIAL FUNCTION : x xtaa
A function f(x) = a = e (a > 0, a * 1, x e R) is called an exponential function. The inverse of the exponential function is called the logarithmic function. i.e. g(x) = logax. Note that f(x) & g(x) are inverse of each other & their graphs are as shown.
g(x) = l o g t x
(V)
ABSOLUTE VALUE FUNCTION
A function y=f (x) = |x| is called the absolute value function or Modulus function. It is defined as : y= |x|
(Vi)
:
"x if x > 0 - x if x < 0
SIGNUM FUNCTION
:
A function y= f (x) = Sgn (x) is defined as follows: y = f(x)
y y = 1 if x>0
" 1 for x > 0 0 for x = 0 - 1 for x < 0
0 y = -1 if x < 0
It is also written as Sgn x = |x|/ x ; x * 0 ; f(0) = 0 (Vii)
>
y= Sgnx
GREATEST INTEGER O R STEP U P FUNCTION :
The function y = f(x) = [x] is called the greatest integer function where [x] denotes the greatest integer less than or equal to x. Note that for: - 1
(viii)
-• - 3
FRACTIONAL PART FUNCTION :
It is defined as : g(x)={x}=x-[x], e.g. the fractional part of the no. 2.1 is 2.1-2 = 0.1 and the fractional part of - 3.7 is 0.3. The period ofthis function is 1 and graph ofthis function is as shown.
^Bansal Classes
- 2
Functions &
Trig.-IV
graph ofy= {x}
[3]
4.
DOMAINS AND RANGES OF COMMON FUNCTION : Function (y = f(x))
Domain (i. e. values taken by x)
Range (i. e. values taken by f (x))
R = (set of real numbers)
R, if n is odd + R u { 0 } , ifniseven
R-{0}
R - {0} , if nis odd
Algebraic Functions
C O
Xn
, (n e N)
1 " , (n e N) x
R+, (iii)
(iv)
B.
x
1/n
if n is even
, (n e N)
R, if n is odd R + u { 0 } , ifniseven
R, ifnisodd R + u { 0 } , ifniseven
,(n eN)
R - {0}, ifnisodd
R - (0},
ifnisodd
R+,
R+,
if n is even
xl/n
if n is even
Trigonometric Functions © (ii)
cosx
R R
(iii)
tanx
R-(2k+ 1) - ,kel
(iv)
sec x
R-(2k+l) - ,kel
sinx
[-1. + 1] [-1. + 1]
7C
TC
R-bc,k eI (v) cosec x R-kn;, k e I (vi) cotx Inverse Circular Functions (Refer after Inverse is taught) (i) (ii)
sin 1 x 1
cos x tan 1 x
]u[l ,00) (-00,-1 ]u[ 1,00 ) R n%
[-1. + 1] [-1. + 1]
2' 2
[0, 7t] 71 It
R
'v 2 2/ j
Tt Tt
(iv)
cosec _1x
(- O O , - 1 ] U [ 1 , 00 ) r
(v)
sec-1 x
(vi)
cotx
(- C O , - 1 ] u [ 1 , 00 ) [0,7C] - { f } (0,71) R
i&Bansal Classes
Functions & Trig.-- IV
2'2
{0}
[4]
Function (y = f (x)) D.
Range (i. e. values taken by f (x) )
Exponential Functions (i) Cn) (iii)
(iv)
E.
Domain (i. e. values taken by x)
ex e!/x ax, a > 0
a1/x, a > 0
R R- { 0 } R
R+ R+- { 1 } R+
R
R+-{1)
R -{ 0}
Logarithmic Functions
(i)
logx,(a>0)(a * 1)
R+
(ii)
logxa=]^
R+-{1}
R-{0}
(a > 0 ) (a * 1) F.
Integral Part Functions Functions (i)
[x]
R
®
^
R-[0,1)
I
g,nSI-{0}j
Fractional Part Functions
0)
(*) 1 to
H.
I.
R
[0,1)
R-I
(1,QO)
Modulus Functions (i)
|x|
(ii)
j^j
R+w{0}
R
1
R+
R-{ 0 }
Signum Function sgn(x)=^,x*0
R
{-1,0,1}
R
{c}
A.
=0,x=0 J.
Constant Function say f (x) = c
^Bansal Classes
Functions &
Trig.-IV
[5]
5. (i) (ii) (iii)
6.
EQUAL OR IDENTICAL FUNCTION : Two functions f & g are said to be equal if: The domain of f = the domain of g. The range of f = the range of g and f(x) = g(x) , for every x belonging to their common domain, eg. 1 x f(x) = — & g(x) = —2 are identical functions . x x CLASSIFICATION OF FUNCTIONS : One-One Function (Injective mapping): A function f: A-^Bis said to be a one-one function or injective mapping if different elements of A have different f images in B . Thus for x p x2 e A& f(Xj), f(x2) e B, f(x,) = f(x2) <=> x, = x, or x, * x2 o f(x,) * f(x2). Diagramatically an injective mapping can be shown as A
B
A
B
Note : (i)
Any function which is entirely increasing or decreasing in whole domain, then f(x) is one-one. (ii) If any line parallel to x-axis cuts the graph of the function atmost at one point, then the function is one-one. Many-one function: Afunction f: A—>B is said to be a many one function iftwo or more elements ofA have the same f image in B . Thus f: A-» B is many one if for ; x p x 2 e A, f(Xj) = f(x2) but x] ^ x2 . Diagramatically a many one mapping can be shown as A
Note : (i) (ii)
B
A
B
Any continuous function which has atleast one local maximum or local minimum, then f(x) is many-one. In other words, if a line parallel to x-axis cuts the graph ofthe function atleast at two points, then f is many-one. If a function is one-one, it cannot be many-one and vice versa.
Onto function (Surjective mapping): If the function f: A B is such that each element in B (co-domain) is the f image of atleast one element inA then we say that fis a function ofA'onto'B . Thus f : A ^ B i s suijective iff V b e B, 3 some a e A such that f (a) = b . Diagramatically surjective mapping can be shown as A
B
A
Functions &
Trig.-IV
B
[6]
Into function: If f: A -» B is such that there exists atleast one element in co-domain which is not the image of any element in domain, then f(x) is into. Diagramatically into function can be shown as B
B OR
Note that: If a function is onto, it cannot be into and vice versa. A polynomial of degree even will always be into. Thus a function can be one of these four types : (a)
one-one onto (inj ective & suij ective)
(b)
one-one into (injective but not surjective)
(c)
many-one onto (suij ective but not inj ective)
(d)
many-one into (neither suij ective nor injective)
Note : (i) (ii)
If f is both injective & surjective, then it is called a Bijective mapping. The bijective functions are also named as invertible, non singular or biuniform functions, If a set A contains n distinct elements then the number of different functions defined from A —»A is nn & out of it n! are one one.
Identity function: The function f: A -» A defined by f(x) = x V x e Ais called the identity ofA and is denoted by IA. It is easy to observe that identity function is a bijection. Constant function: A function f: A—> B is said to be a constant function if every element ofA has the same f image inB . Thus f: A—» B ; f(x) = c, V x e A, c e B is a constant function. Note that the range of a constant function is a singleton and a constant function may be one-one or many-one, onto or into . 7.
ALGEBRAIC OPERATIONS ON FUNCTIONS : If f & g are real valued functions of x with domain set A, B respectively, then both f & g are defined in A n B. Now we define f+ g, f - g, (f. g) & (f/g) as follows : (i) (f±g)(x) = f(x)±g(x) domain in each case is A n B (f.g)(x) = f(x).g(x) (Hi)
(-) VgJ
i&Bansal Classes
g(x)
domain is {x | x e A n B s.t g(x) * 0}
Functions &Trig.--IV
[63]
8.
COMPOSITE OF UNIFORMLY & NON-UNIFORMLY DEFINED FUNCTIONS :
Let f : A —» B & g : B —» C be two functions . Then the function gof: A C defined by (gof) (x) = g (f(x)) V x e A is called the composite of the two functions f & g. f(x) x Diagramatically + g(f(x)). Thus the image of every x e A under the function gof is the g-image of the f-image of x. Note that gof is defined only if V x e A , f(x) is an element of the domain of g so that we can take its g-image. Hence for the product gof oftwo functions f & g, the range of f must be a subset ofthe domain ofg. PROPERTIES O F COMPOSITE FUNCTIONS :
(i) (ii) (iii) 9.
The composite of functions is not commutative i. e. gof ^ fog. The composite offunctions is associative i.e. if f, g, h are three functions such that fo (goh) & (fog) oh are defined, then fo(goh) = (fog) oh. The composite of two bijections is a bijection i.e. if f & g are two bijections such that gof is defined, then gof is also a bijection.
HOMOGENEOUS
FUNCTIONS:
A function is said to be homogeneous with respect to any set of variables when each of its terms is of the same degree with respect to those variables . For example 5 x2 + 3 y2 - xy is homogeneous in x & y. Symbolically if, f(tx, ty) = t n . f(x, y) then f(x, y) is homogeneous function of degree n. 10.
BOUNDED FUNCTION:
A function is said to be bounded if | f(x) | < M, where M is afinitequantity. 11.
IMPLICIT & EXPLICIT FUNCTION :
A function defined by an equation not solved for the dependent variable is called an 3 3 IMPLICIT FUNCTION . For eg. the equation x + y = 1 defines y as an implicit function. If y has been expressed in terms of x alone then it is called an EXPLICIT FUNCTION. 12.
INVERSE OF A FUNCTION :
Let f: A-»B be a one-one & onto function, then their exists a unique function g: B A such that f(x) = y o g(y)=x, V x e A & y e B . Then g is said to be inverse of f. Thus g = f-i; B ->A= {(f(x), x) | (x, f(x)) € f} . PROPERTIES O F LNVERSE FUNCTION :
(i) (ii)
The inverse of a bij ection is unique. If f: A B is a bijection & g : B -> Ais the inverse of f, then fog = IB and gof = I A , where I & IB are identity functions on the sets A & B respectively. Note that the graphs of f & g are the mirror images of each other in the line y = x. As shown in thefiguregiven below a point (x ',y') corresponding to y = x2 (x >0) changes to (y ',x') corresponding to y=+Vx , the changed form of x = 1/y .
fig. l
(iii) (iv)
fig. 2
fig. 3
The inverse of a bijection is also a bijection. If f & g are two bijections f : A B, g : B - ^ C then the inverse of gof exists and (gof)-1 = f o g " 1 . "
^Bansal Classes
Functions & Trig.-
[8]
13.
ODD & EVEN FUNCTIONS :
If f (-x) = f (x) for all x in the domain o f f then f is said to be an even function, e.g. f (x) = cos x ; g (x) = x2 + 3 . If f (-x) = - f (x) for all x in the domain o f f then f is said to be an odd function, e.g. f (x) = sin x ; g (x) = x 3 +x . NOTE
: (a) f (x) - f (-x) = 0 => f (x) is even & f (x) + f (-x) = 0 = > f (x) is odd . (b) A function may neither be odd nor even. (c) Inverse of an even function is not defined . (d) Every even function is symmetric about the y-axis & every odd function is symmetric about the origin. (e)
Every function can be expressed as the sum of an even & an odd function.
e^f(x)=f(x)+f(-x)
+
f(x)
EVEN
(f) (g) 14.
-f(-x)
ODD
The only function which is defined on the entire number line & is even and odd at the same time is f(x) = 0. If f and g both are even or both are odd then the function f.g will be even but if any one of them is odd then f.g will be odd.
PERIODIC FUNCTION:
A function f(x)is called periodic if there exists a positive number T(T>0) called the period ofthe function such that f(x+T) = f(x), for all values of x within the domain of x. e.g. The function sin x & cos x both are periodic over 2% & tan x is periodic over n. NOTE : (a) f (T) = f (0) = f (-T), where'T' is the period. (b) Inverse of a periodic function does not exist. (c) Every constant function is always periodic, with no fundamental period. (d) If f(x) has a period T & g(x) also has a period T then it does not mean that f(x)+g(x) must have a period T. e.g. f(x) = [ sinx j + | cosx |. (e) (f) 15.
If f(x) has a period p, then -j— and i/f(x) also has a period p . t (x) if f(x) has a period T then f(ax + b) has a period T/a (a > 0) .
GENERAL:
If x, y are independent variables, then: (i) f(xy) = f(x) + f(y) f(x) = k In x or f(x) = 0 . (ii) f(xy) = f(x). f(y) f(x) = x n , n e R (iii) f(x+y) = f(x) . fly) => f(x) = a"*. (iv) f(x + y) = f(x) + f(y) => f(x) = kx, where k is a constant.
^Bansal Classes
Functions &Trig.-
[9]
EXERCISE-I Q. 1
Find the domains of definitions of the following functions : (Read the symbols [*] and {*} as greatest integers andfractionalpart functions respectively.)
(i) f (x) = Vcos2x + V16 - x2
(ii) f(x) = log7log5log3log: (2x3 + 5x2 - 14x)
(iii) f(x) = In j^Vx2 - 5 x - 24 - x - 2
(iv) f(x)'
1-5* 7~ x - 7 f
(v) y = logiosin(x-3) + Vl6-x 2
(vi) f(x)= log100;
21og10 x + 1
\
(vii) f(x) =
—+lnx(x 2 -1) V^x -"1
(via) f(x)=
2
~
x
j
J o g , —
(x) f (x) = .v/(x2 - 3x -10). In2 (x - 3)
(ix) f(x)=Jx 2 -|x| +-j=L=
cosx —-
:
(xi) f(x) = ^/logx(cos27tx)
(xii) f(x)
(xiii) f(x)= V logj/3 (log4 ([x] 2 - 5) )
(xiv) f(x)= ~^ + log(2{x}_5)(x2 -3x + 10)+-~
•\/6 + 35x-6x 2
(xv) f(x) = logxsinx \\
(xvi) f(x) = log2 -k>gi/2 1 + + Vlo8io 0°§iox) - togio ( 4 - log!0x) - log10 3 3in (iw) JJ V v. (XVii) f ( x ) - ^
+ lo g l _ { x } (x 2 -3x + 10)+ - 7 = 2
(xviii)f(x) = j(5x-6-x )
[{/n{x}}] +
+
A/(7x-5-2x
2
Vn_1 ^ (l /n — x v v2 yy
)
(xix) If f(x)= J x 2 - 5x + 4 & g(x) = x+3, thenfindthe domain of - (x). g Q. 2 Find the domain & range of the following functions. (Read the symbols [*] and {*} as greatest integers andfractionalpari functions respectively.) (i) y=log^
sinx-cos x) + sj
00 y
2x 1+x
y=^2-x
(iv)f(x)=^|x|
(iii) f(x) =
x2 - 3x + 2 x2 +x-6
+
2
(vi) f (x) = l°g(cosecx.1)(2- [sinx] - [sinx] ) Q. 3
Draw graphs of the following function, where [ ] denotes the greatest integer function. (i) f(x) = x + [x] (ii) y = (x)M where x=[x] + (x)& x > 0 & x < 3 (iii) y = sgn [x] (iv) sgn (x-! x |)
tlBansal Classes
Functions & Trig.-
[10]
Q. 4
Classify the following functions f(x) definzed in R —» R as inj ective, suij ective, both or none. (a) f(x) = x ' 2+ 4 x + 3 0 (b) f(x) = x 3 -6x 2 + l l x - 6 x - 8x + 18
Q.5
(c) f(x) = (x 2 +x+5)(x 2 +x-3)
Let f(x) = ------. Let f2(x) denote f[f(x)J and f3(x) denote f[f{f(x)}]. Find f3n(x) where nis a natural number. Also state the domain ofthis composite function.
Q.6
Function f & g are defined by f(x) = sin x, xeR ; g(x) = tan x , xeR K e I .Find
Q. 7
Q.8 Q.9
(i) periods of fog & gof.
k where
(ii) range of the function fog & gof.
Which of the following pairs of functions are identical ? (A) f(x) = log e, g(x) = —— logex
(B) sgn (x2 + 1); g (x) = sin2x + cos2x
(C) f(x) = sec2x - tan2x ; g(x) = cosec2x - cot2x
(D) f(x) =
; g(x) = Vx"2" x The function f(x) is defined on the interval [0,1], Find the domain of definition ofthe functions, (a) f(sinx) (b) f(2x+3) Find whether the following functions are even or odd or none :
2 (a) f(x) = log fx+Vl+x v J)
(d) f(x) = x 2 - | x |
x a +1 (b) f(x) =_ -A( x * ) a -1 (e) f(x) = x Sin2x - x3
A
_i_
(c) f(x) = sin x + cos x (f) f(x) = K, where K is constant
0x \
(g) f(x)= sinx-cosx (h) f(x) = L _ i _
(i) f(x)=
+\ + 1
1/3
(j) f(x) = [(x+l)2]1/3 + [(x-l) 2 ] Q.IO Find the peri od for each of the following functions : (a) f(x)= sin^x + cos'bc (b) f(x) = | cosx!
(c) fix)= | sinx I +1 cosx |
3 2 (d) f(x)= cos - x - sin - x. Q.ll
Prove that the functions; (c) f(x) = x + sinx
(a) f(x) = cosVx (d) f(x) = cosx2
(b) f(x) = sin Vx are not periodic.
Q.12 Write explicitly, functions of y defined by the following equations and alsofindthe domains of definition ofthe given implicit functions: (a) 10X +10y = 10 (b) x + | y | = 2y Q.13
Find out for what integral values of n the number f(x) = cos nx. sin (5/n)x.
Q.14
Compute the inverse of the functions: (a) f(x) = In (x + Vx 2 +l)
is a period of the function:
(b) f(x) = 2 ^
(c) y =
10 x -10" x 10x+ 10^
Q.15
Show if f(x) = n/a - xn , x > 0 n > 2 , n e N , then (fof) (x) = x. Find also the inverse of f(x).
Q.16
(a)
Represent the function f(x) = 3x as the sum of an even & an odd function,
(b)
For what values of p e z, the function f(x) =
^Bansal Classes
, n e N is even.
Functions & Trig.-0- IV
[U]
Q.17
1 A function f: r 2
Q.18
A function f defined for all real numbers is defined as follows for x > 0 : f (x) = [ *'x°> j*"1
j^
j
-00 .4
How is f defined for x < 0 if: (a) f is even Q.19
(b) f is odd ?
If f(x) = max ( x ' ~ ) for x > 0 where max (a, b) denotes the greater of the two real numbers a and b. Define the function g(x) = fj[x). f f ~ j and plot its graph.
Q.20
Find two distinct linear functions which map the interval [ - 1 , 1 ] onto [ 0 , 2 ] ,
EXERCISE-II Q.l
Let f be a one-one function with domain {x,y,z} and range {1,2,3}. It is given that exactly one of the following statements is true and the remaining two are false. f(x) = 1 ; f(y) * 1 ; f(z) * 2 . Determine f1'(1)
Q.2
Let/(x) be a polynomial with real coefficient. If/(x)=/(x 2 + x + 1) for all x e R, show that/ (x) is an even degree polynomial.
Q.3
Prove that if the function f(x) = sinx + cospx is periodic, then p is a rational number.
Q.4 —J
Prove that the inverse of the linear fractional function f(x) =
\ (ad-be * 0) is also a linear cx + d fractional function. Under what condition f(x) coincide with its inverse.
Q.5
A function f: R—»R satisfies the condition, x2 f (x) + f (1 - x) = 2x - x 4 . Find f (x) and its domain and range.
Q.6
Let f be a real valued function with domain R. Now if for some positive constant p, the equation f(x + p) = l+(2 - 3 f(x) + 3 f(x) - f5 (x))1/3 holds good for x e R , then prove that f(x) is a periodic function. e~J\hi{x}\
Q. 7
_
{x}
3X +
l \j 1 /n{x}
whereeyer
it exigts
Prove that the function defined as , f (x) = . {x}
,^
otherwise,then
f (x) is odd as well as even. (where {x} denotes the fractional part function) Q.8
Q.9
f 7tx n = 4 cos2—— + x cos In a function 2f(x) + xfl - ) - 2f V2 sin 7t X + — 2 x v I 4// Prove that (i) f(2) + f(l/2) = 1 and (ii) f(2) + f(l) = 0 A function f, defined for all x, y e R is such that f ( l ) = 2 ; f(2) = 8 & f(x+y)-kxy=f(x)+2y 2 ,wherekis some constant. Find f(x) & show that : f ( x + y ) f P ^ ] =kfor x+y*0.
Q.IO
Let 'f be a real valued function defined for all real numbers x such that for some positive constant' a' the equation f(x+a)=^+-Jf(x)-(f(x))2 holds forall x.Prove that the function f is periodic.
feBansal Classes
Functions & Trig.-
[12]
Q.ll
If
f(x) = - l + i x - 2 | , 0 < x < 4 g(x)= 2 - | x | , - 1 < x < 3 Then find fog(x) & gof(x) .Draw rough sketch of the graphs of fog (x) & gof(x).
Q.12 Find the domain of definition of the implicit function defined by the implicit equation, 2 24x
3y + 2x* =
-1
Q. 13 Let {x} &[x] denote thefractionaland integral part ofa real number x respectively. Solve 4{x}=x+ [x] Q.14 (a) (b)
2tanx Iff 1 + tan2x.
(cos2x + 1) (sec2x + 2tanxj -,finddomain & range of f(x).
71 If f (x) = tanx; g(f(x)) = fI x ~ ~ I and f(x), g(x) are real valued function. For all permissible values of x, then prove that ®^
Q.15
If f(x) = log
=
3g(x) +1 "S^T
] V
+ 2
-x
2x
'®
' f 2x +1 •l ^ ^ l J J
= 1
> ^
71\ = f l 4/ T s( 2x > • I ^ O T T .
; g(x) = {x} where {x} denotes the fractional part of x. If the function
(fog) (x) exists, thenfindthe range of g(x). Q.16 Find a formula for a function g (x) satisfying the following conditions (a) domain of g is ( - oo, oo) (b) range of g is [-2, 8] (c) g has a period tz and (d) g(2) = 3 2x (sinx 4* tsnx) 311 Q.17 Prove that f(x) = — ^ ->,i is °dd function, where [] denotes greatest integer function. [x + 2711 2 n - 3 Q.18 Find the set of real x for which the function f(x) =
1 is not defined, where [x] [|x-l|] + [ | l 2 - x | ] - l l
denotes the greatest integer function. Q.19 Ais a point on the circumference ofa circle. ChordsAB and AC divide the area ofthe circle into three equal parts. If the angle B AC is the root of the equation, f(x) = 0 then find f (x). Q.20
Iffor all real values of u&v, 2f(u) cosv = f(u + v) + f(u-v), prove that, for all real values of x (i) f(x) + f(-x) = 2a cosx (ii) f(7i - x) + f(-x) = 0 (iii) f(7i - x) + f(x) = - 2b sinx. Deduce that f(x) = a cosx - b sinx, a, b are arbitrary constants.
EXERCISE-III Q.l(a) Fill in the blanks. (i) If f is an even function defined on the interval (-5, 5), then 4 real values of x satisfying the equation x+1 f(x) = f x + 2. are (ii)
&
If f(x) = sin2x+sin2(x+~-J + cosxcos(^x+^j and t
=1 , then (gof) (x) = [IIT'96,1+2]
^Bansal Classes
Functions &Trig.-
[13]
Q.2
Let f: {x,y,z}->{a,b,c}be a one-one function. It is known that only one ofthe following statements is true: (i) f(x) * b (ii) f (y) = b (iii)f(z)*a Find the function f. [REE '96, 6]
Q.3
If the functions f, g, h are defined from the set of real numbers R to R such that ; r 0, if x < 0 f(x)=x 2 -l,g(x)=^~Ti ,h(x)= ; thenfindthe composite function ho(fog) & determine L x, if x > 0 whether the function (fog) is invertible & the function h is the identity function.
[REE '97, 6]
Q.4(a) If g (f(x)) = | sinx | & f (g(x» = (sin Vx)*, then : (A) f(x) = sin2 x, g(x) = Vx
(B) f(x) = sinx, g(x) = i x |
(C) f(x) = x2, g(x) = sin Vx
(D) f & g cannot be determined
-1
(b) If f(x) = 3x-5,thenf (x) 1 3x - 5 (C) does not exist because f is not one-one (A) is given by
x+5 (B) is given by —— 3 (D) does not exist because f is not onto [JEE!98, 2 + 2]
Q.5
If the functions f & g are defined from the set of real numbers R to R such that f(x) = e\ g(x) = 3x-2, then find functions fog & gof. Alsofindthe domains of functions (fog)-' & (gof)-1. [REE'98, 6]
Q.6
If the function f: [1, » ) - > [1, oo) is defined by f(x) = 2x(x-1>, then f-1(x) is: [JEE'99,2] / .y(x-I) . ^ (A) ( - J (B) - (l + ^l + 41og2I) (C) - (l - Vl + 41og2x) (D) not defined
Q.7
The domain of definition of the function, y (x) given by the equation, 2X + 2y = 2 is: (A) 0 < x < 1
(B) 0 < x < 1
(C) - o o < x < 0
(D)
- oo < x < 1
[ JEE 2000 (Screening), 1 out of 35 ] Q.8
Given x = {1,2, 3, 4}, find all one-one, onto mappings, f: X —» X such that, f(l) = I, f(2) * 2 and f(4) * 4 . [ REE - 2000, 3 out of 100 ]
Q.9
[JEE 2001 (Screening) 5 x 1 = 5 j -1
(a)
L ©t g(x) = 1 + x - [x] & f(x) = <{ 0 1
(A)x (b)
, x = 0 , Then for all x, f (g (x)) is equal to ,
x>0
(B) 1
(C)f(x)
(D)g(x)
If f: [1, oo) -» [2, oo) is given by, f(x) = x + — , then f - 1 (x) equals : X (A) l l J U l 2
(c) W
, x <0
(B)
The domain of definition of f(x) W = (A) R \ {-1, - 2 }
^B ansaI Classes
(C) I z E I l 2
1 + x" + 2
x + 3x + 2
(B) (-2, oo)
(D) 1 -
is : (C) R\{-1, - 2 , - 3 }
Functions & Trig.-fl- IV
(D) (-3, QO) \ {-1, - 2 }
[14]
(d) ;'
Let E = {1, 2, 3,4} & F = {1, 2}. Then the number of onto functions from E to F is (A) 14 (B) 16 (C) 12 (D) 8
(e)
a x Let f(x) = — j - , x * - 1 . Then for what value of a is f (f (x)) = x ? (A) ,J2
(B)-V2
(C) 1
(D) - 1 .
Q. 10(a) Suppose f(x) = (x + l)2 for x > -1. If g(x) is the function whose graph is the reflection ofthe graph of f(x) with respect to the line y=X, then g(x) equals (A)-Vx - L , x > 0
(B)^-^r,x>-L
(C) A/^+T,X>-1
(D)V^-L,x>0
(b) Let function / : R --> R be defined by / (x) = 2x + sinx for x e R . Then / is (A) one to one and onto (B) one to one but NOT onto (C) onto but NOT one to one (D) neither one to one nor onto [JEE 2002 (Screening), 3+3] x 2 +x + 2 Q. 11 (a) Range of the function f (x) = —~ x +x + l
(B)[l,co)
(A) [1,2]
is
(D)
(C)
v1'!
defined from (0, 00) —> [ 0,00) then by f (x) is 1+x (A) one- one but not onto (B) one- one and onto (C) Many one but not onto (D) Many one and onto
(b) Let f(x)
Q.12
[JEE 2003 (Screening), 3+3] Letf(x) = sinx + cosx, g(x) = x - 1 . Thusg(f(x))isinvertibleforx e 2
(A)
•f.O
7T
(B) ~~2'71
(C)
71
TC
4' 4
(D) 0 . * [JEE 2004 (Screening)]
Q. 13 (a) If the functions/ (x) and g (x) are defined on R 0. f(*)= ,
x,
x e rational . . , , g(x) = x e irrational
then ( f - g)(x) is (A) one-one and onto (C) one-one but not onto
R such that x e irrational
X,
x e rational
(B) neither one-one nor onto (D) onto but not one-one
(b j) X and Y are two sets and f: X —» Y. If {f (c) = y; c
(B) f _ 1 (f(a)) = a
(C) f (f - 1 (b))=b, b e y
(D) f~ ! (f(a)) = a , a c x [JEE 2005 (Screening)]
^Bansal Classes
Functions & Trig.-
[15]
KEY CONCEPTS (INVERSE TRIGONOMETRY FUNCTION) G E N E R A L DEFEVITION(S):
sin-1 x, cos -1 x, tan -1 x etc. denote angles or real numbers whose sine is x, whose cosine is x and whose tangent is x , provided that the answers given are numerically smallest available. These are also written as arc sinx, arc cosx etc.
1.
If there are two angles one positive & the other negative having same numerical value, then positive angle should be taken . 2.
PRINCIPAL VALUES AND DOMAINS OF INVERSE CIRCULAR FUNCTIONS :
(i)
y = sin -1 x where - 1 < X < 1
(ii)
y = cos - 1 x where - 1 < X < 1 ; 0 < y < 7 t and cosy = x .
(iii)
y = tan - 1 x
(iv)
y = cosec-1 x where x < - 1 or x > 1 ; ~ < ; y s , y ^ O and cosecy = x .
(v) (vi)
y = sec-1 x where x < - 1 or x > 1 ; 0 < y < 7 t ; y = cot -1 x where x e R , 0 < y < 7t and coty = x .
NOTE THAT
;
siny = x .
~ < Y < a n d
where x e R ; ~ y < x < f "
an(
*
tan
^
= x
•
and secy = x .
: (a) (b)
1st quadrant is common to all the inverse functions . 3rd quadrant is not used in inverse functions .
(c)
4th quadrant is used in the CLOCKWISE DIRECTION i.e.
< y <, 0 .
3.
PROPERTIES OF INVERSE CIRCULAR FUNCTIONS :
P-l
(i) sin (sin-1 x) = x , - 1 < x < 1
(ii) cos (cos -1 x) = x , - 1 < x < 1
(iii) tan (tan-1 x) = x , x e R
(iv) sin-1 (sin x) = x ,
(v) cos -1 (cosx) = x ; 0 < x < 7i
(vi) tan -1 (tanx) = x ; - j < x < ~
P-2
(i) cosec-1 x = sin-1 -
;
x<-l,x>l
(ii) sec-1 x = cos-1 — X
;
x<-l , x>l
(iii) cot -1 x = tan -1 -X
; x>0
x
= 7t + tan -1 — ; x < 0 X
P-3
(i) (ii) (iii) (iv)
P-4
sin-1 (-x) = - sin"1 x , - 1 < x < 1 tan -1 (-x) = - tan -1 x , x e R cos -1 (-x) = % - cos -1 x , - 1 < x < 1 cot -1 (-x) = it - cot -1 x , x e R
(i) sin-1 x + cos -1 x = j
-1 < x < 1
(iii) cosec-1 x + sec-1 x = —
^Bansal Classes
(ii) tan -1 x + cot -1 x = y
xeR
I x | >1
Functions & Trig.-
[16]
P-5
tan -1 x + tan"1 y = tan -1 t- 2 --^
where x > 0 , y > 0 & x y < l
X+V = k + tan -1 -—— where x > 0 , y > 0 & x y > l
tan 1 x-tan~'yJ = tan -1 ——— where x > 0 , y > 0 1 + xy P-6
(i)
sin-1 x + sin-1 y = sin-1 x -Jl - y 2 + y -Jl
71
Note that: x2 + y2 < 1 . = (ii)
where x > 0 , y > 0 & x2 + y 2 > l
71 — < sin-1 x + sin-1 y < 7t
=>
(iii)
sin _1 x-sin -1 y = sin -1 [x^/l-y 2 - y V l - x 2 J
wherex > 0 , y > 0
(iv)
cos -1 x ± cos -1 y = cos"1 jxy + yjl-x2 yj\-y2 j
where x > 0, y > 0
i x + y + z-xyz If tan *x + tan ! y + tan 1lz"7 =— tan ton 1' - xy - yz - zx
(ii)
if, x > 0, y > 0, z > 0 & xy + yz + zx < 1
If tan -1 x + tan -1 y + tan -1 z = n then x + y + z - x y z
Note : (i)
P-8
0 < sin 1 x + sin 1 y < —
sin-1 x + sin-1 y = % - sin-1 x J l - y 2 + y J l - x 2 Note that: x2 + y2 >1
P-7
where x > 0 , y > 0 & (x2 + y2) < 1
If tan -1 x + tan -1 y + tan"1 z = ~ then xy + yz + zx = 1
2 tan"1 x = sin"
2x l + x1
:
cos'
1—X 2x = tan"1 • 1 + x' 1-x2
Note very carefully t h a t : 2x sin 1 + x2
tan
2x 1-x2
2 tan"1 x it - 2tan_1 x -(7c+2 tan"1 x) 2tan _1 x
if if if
jx|
if
|x|
7i+2tan x
if
x<-l
-(7T-2tan_1x)
if
x>l
_1
cos
1-x2 1 + x2
2tan"1 x if x > 0 -2tan"1x if x < 0
REMEMBER THAT :
3 7t
(i)
sin 1 x + sin 1 y + sin 1 z =
(ii)
cos"1 x + cos -1 y + cos -1 z — 3%
(iii)
tan"1 1 + tan"1 2 + tan"1 3 = 71
^Bansal Classes
=> x = y = z = 1 => and
x =y= z= -l tan"1 1 + tan"1 J + tan"1 j
Functions & Trig.-
= j
[17]
INVERSE TRIGONOMETRIC
FUNCTIONS
SOME USEFUL GRAPHS
1.
y= s k r ' x , | x | < 1, y
71 7C
e
2.
2 ' 2
y
=
cos-1 x , | x | < l , y e [0,7t]
y= arc cos x y=stnx
x y=sinx v= cos X
y=arc sinx
3.
y = tan~ 1 x,xeR, y
4.
2 ' 2
y = cot - 1 1, x e R , y e (0 , it)
y=tanx
v= arc cot x y= arc tan x
y=arc cot x • x
TT/2
y= arc tan x
5.
6. y = cosec"'x, | x | > 1, y e
y=sec _ 1 x, |x| > 1, y
2
, 00
TT/2
^Bansal Classes
v
71/2
7t
o
;
y
y1
-lj
2
1
0 1
X
—7t/2
Functions & Trig.-
[18]
y = sin-1 (sinx),x e R, y e K 7t
7. (a)
7.(b)
~2 '~2
=x
Periodic with period 2n
x e [-1,1], y e [-1,1] , y is aperiodic
n/21i y
xV
y = sin (sin 1 x),
1
f y \ \ \V V+y-Ic/2 /)45°i 3tt/2 // V -2n -2n/2 ~it\ ! 0 /Tt/2 7t\ /2x
-1
-TT/2 8.(a)
8. (b)
y = cos ( c o s x ) , =x x e [-1,1], y e [-1,1], y is aperiodic
y = cos "'(cosx), x e R , y e [0, TC], periodic with period 2 it
=x
>y
V
/yy y -2%
1
71
•tiHY
—tt/2 0
9. (a)
it/2
O 7t
1
-1
y = tan (tan~'x) , j e e R , i y e R , y i s aperiodic =x
9. (b)
y = tair ! (tanx), =x xeR periodic with period n
%/i•y -in/
•X
O
—3k 2
f / —71 V 2
/-Tt
/
/ o
/71
n 2
3k 2
/2k
•x
-Tt/2
10. (a) y = cot"1 (cotx), =x x e R - {n7i} , y e (0, TT) , periodic with it
Tt A
J y y -2%
/ -K
^Bansal Classes
10, (b) y = cot (cot_Ix), =x x e R, y e R , y is aperiodic
•y
>y
.y/
^
/ y O
/ Tt
2n
•x
/
Functions &Trig.-
/
0
[19]
11. (a) y = cosec-1 (cosec*),
11. (b)
y = cosec (cosecx), =x
= x
xsR-{n7i,nsI},ye
|*| >1, lyl >1, yisaperiodic
y is periodic with period 2n
12. (a) y= sec-1 (sec*), =* y is periodic with period 2n;
12. (b) y = sec (sec-1*), =* |*| > 1; l.vi >l],y is aperiodic
j
^Bansal
Classes
Functions &Trig.-
[120]
EXERCISE-I Q. 1
Find the following 1 1 (i) tan cos - + tan
(iv) tan 11 tan Q.2
(ii) sin — - sin
s.
2tc
(v) cos tan
(iv) cos
4Tt cos-
1% 6
r . -x3 i3 (vi) tan sm — + cot -
3
5
'-VT (ii) cos cos v 2 ,
2
(vi). tan-1
3sin2a 5 + 3cos2a
% +—
6
3 it (iii) tan - tan-
-i3 (v) sin cos
3
+ tan -1
tana 4
J
where - — < a < — 2
2
Prove that : _3
(a) 2 cos-1 -4=
Vl3
-i — L =% + cot -1 — + - cos-1 63
(c) cot 1 9 + cosec-1 Q.4
(iii) cos-^cos
Find the following : Tt . ' - V T (i) sin -— sin 2 V2 ,
Q.3
1
2
25
=—
(b) tan -1 2 + tan -1 3 = (d) arc cos
3%
- arc cos
76 + 1 _ TC 2V3 ~ 6
Find the domain of definition the following functions. (Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.) (i) f(x) = arc cos
2x 1 +x
(ii) f(x)=-+2 arcsmx + 1 •- •' x yfx-2
(iii) 7cos(sinx) +sin (v) f(x) =
(iv) f(x) — sin"
12 sm x z
log 5 (l-4x )
(vi) f (x) =
x + cos-11 • J
l2
+ log6 (2|x| - 3) + sin-1 (log2 x)
'+tan -1
x
2 + sin 97IX
+ in( V ^ W )
(ix) f(x) = v;sm(cosx) + /n ( - 2 cos 2 x + 3 cosx+ 1) + e°
^Bansal Classes
-log 10 (4-x)
+ cos-1 (1 - {x}) , where {x} is the fractional part of x.
(vii) f(x) = logjg (l - log7 (x 2 - 5 x+13)) + cos-1 (viii) f(x)=e
3
Functions &
-1 2 sinx + 1 2«j2si sinx
Trig.-IV
[21]
Q. 5
Find the domain and range of the following functions. (Read the symbols [*] and {*} as greatest integers andfractionalpart functions respectively.) (i) f (x) = cot -1 (2x-x 2 ) f i—r2 \ _1 ^2x + 1 (iii) f(x) = cos
(ii) f(x) = sec 1 (log3 tanx+ log tanx 3) (iv) f (x) = tan"1 log4 (5x2 - 8x + 4 ) V 5
Q.6
Find the solution set ofthe equation, 3 cos"1 x = sin
Q.7
Prove that : 71 (a) sin 1 cos (sin 1 x) + cos 1 sin (cos 1 x) = —,
2mn
+ tan
/ ~ \ 2pq
(x * 0)
2MN where M = mp - nq, N = np+mq, vM2 -N 2 .
tan
VP - q
f - J l - x 2 (4x 2 - l) j .
| x | <1
(b) 2 tan"1 (cosec tan_1x - tan cot-1x) = tan -1 x (c) tan
1
n N q <1
Find the simplest value of, arc cosx + arc cos [ ~ + ^ V3 ~ 3x2 j , x e
Q.9
—l If cos 1 — + cos 1 — = a then prove that a
x -d
b
2.xy y . , — cosa H—- = sin a . ab b
Q.IO If arc sinx + arc siny + arc sinz = n then prove that : (a)
X^ 1 - x 2
,1
(x, y, z > 0)
-y2 + z-^l - z2 = 2xyz
(b) x4 + y4 + z4 + 4 x2y2z2 = 2 (x2 y2 + y2 z2 + z2x2) QJ1
Find the greatest and the least values of the function, f(x) = (sin-1 x)3 + (cos-1 x)3
Q.12
Solve the following equations/system of equations : (b) tan"1—1— + tan"1 — — = tan 1 -72
(a) sin lx + sin 1 2x - —
1 + 2x
1 + 4x
x
(c) tan -1 (x-l) + tan-1(x) + tan -1 (x+l) = tan-1(3x) i 1 i TC (d) sin - 1 ^j +cos - 1 x= —
_i
2x
(e) cos 1 ~ — - + tan 1 2 x -1 x +1
_
2n
(f) sin lx + sin ! y = — & cos *x - cos 'y= — 3
3
2
1-a 1-b (g) 2 t a n - 1 x - c o s - i Y 7 ^ - cos 'Y^tf
a>0, b>0
Q.13
If tan-1x, tan-1y, tan -1 z are in A. P., then prove that, y2 (x + z) + 2y (1 -xz) = x + z where y e (0, 1) ; x z < 1 & x > 0 , z >0.
Q.14
Find the value of sin-1 (sin5) + cos-1(coslO) + tan-1 [tan (-6)] + cot -1 [cot (-10)].
Q.15
• if • 3370 i I 467t^ f 137t | Showthat: sm I sin——J + cos |^cos—— | + tan | - t a n — | + cot
^Bcmsal Classes
Functions &
Trig.-IV
cot -
1971
1 3tc
[22] *
Q.16
In a A ABC if ZA = 90°, then prove that tan"1 — ^
+ tan"1 — V = T
c+a
a+b
Q.17 Prove that: sin cot-1 tan cos-1 x = sin cosec-1 cot tan_1x = x Q.18
4
where x e (0,1]
If sin2x + sin2y < 1 for all x, y e Rthen prove that sin-1 (tanx. tany) e ' 2 ' 2
Q.19 Find all the positive integral solutions of, tan Jx + cos 1
y
V!
:
sin
+y /
VTo'
%
Q. 20 Let f (x) = cor (x + 4x+a - a) be a function defined R -> 0 thenfindthe complete set of real v ' 2 values of a for which f (x) is onto. 1
2
2
EXERCISE-II Q. 1
Prove that : K 1 K 1 —i a + — cos a + tan cos b 4 2 b (a) tan 4 2 —
w
c o s -l
—
—
cosx + cosy = 1 + cosx cosy
—
,
tan X
v
V1 + X2 - VL - X2
=
2b a
* 2
2J
a- b x . tan— a+b 2
w
cos
b + a cosx a + bcosx
Q.2
If y = tan-
Q.3
If u = cot -1 i/cos20 - tan -1 V c o s 2 e then prove that sin u = tan2 9.
Q.4
. f 1 - x2 for 0 < x < 1, then prove that a + (3 = If a = 2 arc tan 1 + X1 &„ pn = arc sin 1 + x2 J-x.
2
1 + x +Vl-x
2
prove that x2 = sin 2y.
what the
value of a + P will be if x > 1. Q.5
1
Ifx e
2
^en express the function f (x) = sin-1 (3x - 4x 3 ) + cos -1 (4x3 - 3x) in the form of
a cos -1 x + b7t, where a and b are rational numbers. Find the sum of the series :
1 , . , 4l - 1 , —f= + sm —1=— t V2 V6
. Vn - Jn - 1 + sm 1 — i —LVn(n + 1)
(a)
sm
(b)
tan-1 ^ + tan -1 - +
(c)
cot"17 + cot"113 + cot-121 + cot"131 +
(d) w
l tan -1 —22— + tan -1 + tan-l. 2 • + tan"1 •2 to n terms. x + X+ 1 x' + 3x + 3 x + 7x + 13 x + 5x + 7
(e)
tan-1 - + tan - 1 1 + tan -1 — + tan"1 ^ + , 2 8 18 32
^Bansal Classes
+ tan - 1 1 + ^
+
QO
oo to n terms .
oo
Functions & Trig.-
[23]
Q.7
Solve the following : (a) cor'x + cot"1 ( n 2 - x + 1) = cot"1 ( n - 1 ) (b) sec-1 — - sec-1 ~ - sec_1b - sec_1a a > 1 ; b > 1 , a ^ b . (c) tan'-i
a x- 1 x +1
b
2x — 1
23
+ tan~' 2X7T = tan ~ 36
Q.8
33 1 1 1 _i a a P — +— F sec^ -tan — Express — cosec22 -tan" 2 a 2 .2 P
Q9
Find the integral values of K for which the system of equations; arccosx + (arcsiny)
Q.IO Express the equation cot
KTI2
=
(arcsiny) 2 . (arccosx)
as an integral polynomial in a & p.
4
possesses solutions & find those solutions.
= —
1
VT
2
2
x - y
= 2 tan
3 - 4x 4X
r I3 4 x 2 tan"1 V—X— as a rational integral ~
equation is x & y. Q.ll
If X= cosec . tan -1 . cos . cot -1 . sec . sin-1 a & Y= sec cot -1 sin tan -1 cosec cos"1 a ; where0 < a < 1 . Find the relation between X & Y . Express them in terms of 'a'.
Q.12
If A = ~ c o r 1 Q j + | c o r ^ | j H-^cof'f-M ; B = 1 cot -1 (l) + 2cot"1(2) + scores), then find the value of (A2 + B2 - 2 AB)1/2.
Q.13 Q.14 Q.15 Q.16
Q.17
1 7 Prove that the equation ,(sin_1x)3 + (cos_1x)3 = a n3 has no roots f o r a < — anda> — Solve the following inequalities : (c) tan2 (arc sinx)> 1 (a) arc cot 2 x - 5 arc cotx + 6 > 0 (b) arc sinx > arc cosx Solve the following system of inequations 4 arc tan2x - 8arc tanx + 3 < 0 & 4 arc cotx - arc cot2 x - 3 > 0 cos ! x
/
\
• -1
sm
x
\
=0 v y J v y ; The sets Xj, Xj e [-1, 1] ; Yj, Y2 c I - {0} are such that Xj : the solution set of equation (i) X2 : the solution set of equation (ii) Yj : the set of all integral values of y for which equation (i) possess a solution Y2 : the set of all integral values of y for which equation (ii) possess a solution Let: Cj be the correspondence : Xl Yj such that xC, y for x e X p y e Yj & (x, y) satisfy (i). C2 be the correspondence : X2 -> Y2 such that x C2 y for x e X 2 , y e Y2 & (x, y) satisfy (ii). State with reasons if C j & C2 are functions ? If yes, state whether they are bijjective or into ? Consider the two equations in x;
(i) sin
=1
cos-1 (sin(x + 3-)) f 4 — 1 Given the functions f(x)= e " , g(x) = cosec"1 I — —
(ii) cos
2C0SX^
J & the function h(x) = f(x)
defined only for those values of x, which are common to the domains of the functions f(x) & g(x). Calculate the range of the function h(x). Q.18
(a)
2x. 1 x^ are identical functions, then compute If the functions f(x) = sin"1 -——z & g(x) = cos-1 2
(b)
their domain & range. If the functions f(x) = sin"1 (3x~4x3) & g(x) = 3 sin-1 x are equal functions, then compute the maximum range ofx.
1+x
^Bansal Classes
~"
1+ x
Functions & Trig.-
[24]
Q.19
Convert the trigonometric function sin (2 cos 1 (cot (2tan 1x))) into an algebraic function f(x). Then from the algebraic function,findall the values of x for which f(x) is zero. Express the values of x in the form a ± Vb where a & b are rational numbers. f
Q.20
Solve for x:
(»2x 2 +4„Y\ sin < TC — 3. sin" 1 + x' J) V V 1
EXERCISE-III Q. 1
i
The number of real solutions of tair ! Jx (x+1) + sin"5 jx 2 + x + 1 = — is: (A) zero
Q.2
(B) one
(C) two
[JEE'99,2 (out of200)]
(D) infinite
Using the principal values, express the following as a single angle: [REE'99, 6]
3 tan- ( ! ) + 2 * * *
Q.3 Q. 4
bx ax sin-1 — + sin-1 — = sin"1 x c c
Solve,
where a2 + b2 = c 2 , c * 0. [REE2000(Mains), 3 out of 100]
Solve the equation: cos_1(V6x) + cos_I(3V3x2) = -
Q.5
x2 x3 If sin"1 | x ~ y + ~
[REE2001 (Mains), 3 out of 100] f
2 ;
+ cos"
4
X
2
6
X + —
for 0 < | x | <
4
thenx equals to
[JEE 2001 (screening)] (A) 1/2
(B) 1
(C) - 1 / 2
Q, 6
Prove that cos tan"1 sin cot _1 x =
Q.7
1 Domain of f(x) : ^|sin- (2x)+ 6
%
(A)
Q8
( 1 1 2'2. v
(B)
(A)
1
x2 + l x+2
(B)|
1
[JEE 2002 (mains) 5]
is
" 1 3^
If sin (cot 1 (x +1)) - cos(tan
(D)-l
(C)
i r 4' 4_
(D)
i r
4'2_ [JEE 2003 (Screening) 3]
x), then x = (C)0 .
^Bansal Classes
Functions &Trig.-
[25]
ANSWER KEY
FUNCTIONS EXERCISE-I Q l . (i)
571 -371
u
( i v ) ( - o o , - l ) u [ 0 , oo)
7C 7t u 4'4
1
-4,--
U ( 2 , 0 0 ) (iii) ( - 0 0 , - 3 ]
(v) ( 3 - 2 T C < X < 3 - T C ) U ( 3 < X < 4 )
(vii) (-1 < x < -1/2) U (x > 1) (x) { 4 } ^ [ 5 , o o )
3 7C 5 7t (ii) T ' T
1-V?
(viii)
u
i +VJ
1 1 '100 J l IOO'VIO u
00 (ix) (-3, - 1 ] U {0} U [ 1,3 )
(xi) ( 0 , 1 / 4 ) U ( 3 / 4 , 1 ) U { x : x e N , x > 2 }
(xiii) [ - 3 , - 2 ) u [3,4)
(vi)
(xii)
(xiv) (j>
A 1 u 5TC , T'6 6' 3
(xv) 2KTC < x < ( 2 K + L)7i but x ^ L where K is non-negative integer (xvi) {x | 1000 < x < 10000} (xvii) (-2, -1) U (-1, 0) U (1, 2)
(xviii) (1, 2) u
V
(xix) (-oo, -3) u (-3,1] u [4, oo) Q2. (i) D : x e R
R: [0,2]
D: { x l x e R ; x * - 3 ; x * 2 }
(iv)
D : R ; R : (-1, 1)
(vi)
D : x s (2nn, (2n+l)7t) - |2n7i+-|, 2n7i+-|, 2n7t+^f, n e l | and
Q 4.
L
J
(ii) D = R ; range [ - 1 , 1 ]
(iii)
R : log 2; a e (0, oo) - {1}
H)
R : (f(x)|f(x) e R , f ( x ) * 1/5 ; f(x)* 1} (v)D:-l
R : [V3 , V? ]
Range is (-oo, oo)- {0} (b) surjective but not injective
(a) neither suijective nor injective (c) neither injective nor suijective
Q5. f3n(x) = x ; Domain = R - {0,1} Q6. (i) period of fog is 7r, period of gof is 2% ; (ii) range of fog is [-1,1], range ofgof is [-tanl, tanl] Q 7. A, B, D
Q9.
Q 8. (a) 2KTC < x < 2KTC + TC where K e l
(a) odd (b) even (c) neither odd nor even (d) even (e) odd (g) neither odd nor even (h) even (i) even (j) even
(b) [ - 3 / 2 , - 1 ]
(f) even
Q 10. (a) TC/2 (b) % (c) TC/2 (d) 70 TC Q 12. (a) y = l o g ( 1 0 - 1 0 x ) , - o o < x < 1 (b) y = x/3 when - o o < x < 0 & y = x w h e n 0 < x < + oo Q 13. ±1, ±3, ±5, ±15
Q 14. (a)
e
"
e
2
(b)
3 + 3"x
3X - 3"
2
2
X
toga* l0g2x - 1
M
1 ,
1+X
(c) - log^ 1-x
Q 15. f !(x) = (a-x n ) 1/n
Q 16. (a)
Q 17. x = 1
Q 18. (a) f(x) = 1 for x < - 1 & - x for - 1 < x < 0 (b) f(x) = - 1 for x < - 1 and x for - 1 < x < 0
^Bansal Classes
(b) p = 2k, k e l
Functions &Trig.-
[26]
Q19.
x
g(x)=
if 0
Q20.
f(x) = 1 + x or 1 - x
- x 2 ifx>l
EXERCISE-II ]
Q i . f- (i) = y
Q 4.
a+D=0
Q5. f ( x ) = 1 - x 2 , D = x e R ; range = ( - QO, 1]
Q 9. f(x) = 2x2
x+1 , 0 < x < 1 Q 11. f o g ( xJ ) = - ( 1 + X ) x—1
- x -0 • , 0
'
5 f ( xW) =
3-x , 1 < x ^ 2 x-1 , 2 < x < 3
'
5-x , 3 < x < 4 x
fof(x) = 4 _ x f
Q 12.
. ., , n0 < x < 1
3
V3 + 1 1 - VT) V2 ' V2 J
u
;
-x
, -1 < x < 0
x
'
°
4-x
,
2
(VI - 1 V3 + 1 { V2 ' 4i
Q.13
x — 0 or 5/3
Q.14 Domain [-1,1], Range [0, 2] Q 15. (0, 10"2) u (10-2, 10"1)
Q 16.
Q 18. (0,1) u {1, 2,
Q 19. f(x) = sinx + x -
, 12} u (12, 13)
g (x) = 3 + 5 sin(mr + 2x - 4), n e I -
EXERCISE-III n t t \ r\ - l + VI —1 —A/5 -3 + VS -3-45
Q.1 ( « ) ( , ) _
.... , (11) 1
Q.2 {(x,b),(y,a),(z,c)} Q.3 (hofog)(x) = h(x2) = x2 for XGR, Hence h is not an identity function, fog is not invertible Q.4 (a) A, (b) B Q.5 (fog)(x) = e3x~2; (gof) (x) = 3 e x - 2 ; D : x s R+ and x e (-2, QO) Q.6 B Q.7 D Q.8 {(1, 1), (2, 3), (3, 4), (4, 2)} ; {(1, 1), (2, 4), (3, 2), (4, 3)} and {(1, 1), (2, 4), (3, 3), (4, 2)} Q.9 (a) B, (b) A, (c) D, (d)A, (e) D Q.IO (a) D ; (b)A Q.ll (a) D, (b)A Q.12 C, Q.13 (a) A; (b) D
INVERSE TRIGONOMETRY EXERCISE-I Q 1. (i) - L , (ii) 1, (iii) 5-f, (iv) - f , (v) | , (vi) j Q 4.
FUNCTIONS
Q 2. (i) | , (ii) -1, (iii)
, (iv) f , (v) | , (vi) a
(i) (v)
-1/3 < x < 1 (ii) (j) (not defined for any real x) x €(-1/2, 1/2), x * 0 (vi) (3/2,2]
(iii) {1,-1}
(vii)
{7/3,25/9}
{x|x = 2n7r + - , n e 1}
(!%Bansal Classes
(viii)
(-2, 2) - {-1,0,1}
Functions &
(ix)
Trig.-IV
(iv)l
6
[27]
Q5.
Q 6. Q12.
(i)
D : x s R R : [7c/4,7c)
(ii)
D:XG
(iii)
D:xe R
7t mz, mi + - - j x | x = n7t + -^l n e l
;
R:
V
s (a)x=iJ|
R:
(iv)
o,
7C
D:xgR
n 7i 2'4
R:
Q8.f
TT3 Q 11. ll^L when x = - l & — when x = 4 = 8 32 V2
(b) x = 3
(e)x = 0 , I , - I
1 (e) x = 2-73 or V3 (f) x = - , y = 1 Q 14. 8tc - 21
2ti" I ' 3 _ 71
(d)x
a -b (g)x = 1 + ab
1 + Vl7
Q.20
Q 19. x = 1; y = 2 & x = 2 ; y = 7
Vio
EXERCISE-II 9tt 9 Q5. 6 c o s 2 x - y , so a = 6, b = - —
Q 4. -7i (b)|
7C (d) arc tan (x + n) - arc tan x (e) —
(a) f
Q 7.
(a) x = n 2 - n + 1 or x = n (b) x = ab (c) x = 2
(c) arc cot
2n + 5
Q6.
2
Q 9. K = 2; cos—,1 & c o s — , - 1 4 4 Q12. — + | c o r 1 ( 3 ) 24 6 Q15. ( t a n i , cotl Q17. [e®6, e*]
Q 8. (a 2 + P2) (a + p) 2
Q 10. y2 = — (9 - 8x2)2 27
Q 14. (a) (cot2, oo)u(-oo, cot3) (b)
Q 11. X = Y = ^
V2
/
(c)
v 2
,1 u ,
Q16. C, is abijective function, C2 is many to many correspondence, hence it is not a function
1 1 Q 18.(a) D : [0, 1], R : [0, n/2] (b) - - < x < - (c) D : [ - 1 , 1 ] , R : [0, 2] 2
Q.19 x = 0 ± 7 i ; x = 1 ± V 2 ; x = - 1 ± V 2
Q.20
2
x e (-1, 1)
EXERCISE-III Q.l C
Q.2 Tt
^Bansal Classes
Q.3 x e { - l , 0, 1}
1 Q.4 x = -
Q.5B
Functions & Trig.-
Q.7 D
Q.8 A
[28]
BANSAL CLASSES M ATH€M ATI C5 TARGETIIT JEE 2007
TRANSIT DPP For Class XI to XII moving students
FUNCTIONS AND INVERSE TRIGONOMETRY FUNCTIONS Time Limit: 5 Sitting Each of 60 Minutes duration approx. This D P P will be discussed during commencement of class- XII
There are 95 questions in this question bank. Only one alternative is correct. Q. i Let f be a real valued function such that f(x)+
2f
/ 2002^
v
x
J
= 3x
for all x > 0. Find f (2). (A)1000
(B)2000
(D)4000
(C)3000
Q.2
Solution set ofthe equation, cos-1 x - sin"1 x - cos" ] (x y3 ) (B) consists of two elements (A) is a unit set (D) is a void set (C) consists of three elements
Q.3
If f(x) = 2 tan 3x + 5-^1 - cos 6x ;g(x) is a function having the same time period as that off(x), then which ofthe following can be g(x). (A) (sec23x + cosec23x)tan23x cos2 3x + cosec3x
(C) Q.4
(B) 2 sin3x + 3cos3x (D) 3 cosec3x + 2 tan3x
Which one of the following depicts the graph of an odd function? ,y
10
10-
o
(A)
(B)
-»x
10
0 -10
(C) -10
10
/ -10
-10
10-
y
10-
,y
0
(D)
10
-10
o
J
7*
-10
Q. 5
The sum of the infinite terms ofthe series
cot"1 [ l 2 + £ j + cot"1 [ 22 (A) tan"1 (1) Q6
(B) tan"1 (2)
Domain of definition of the function f (x) = log (A) [0,1]
Q.7
+ cot"1
(B) [1,2]
j+ (C) tan"1 (3)
is equal to (D) tan"' (4)
03x~2 - 9X_1 - 1 + ^ c o s ' 0 - x ) is (C)(0,2)
(D)(0,1)
The value of tan"1 Q tan 2A\ + tan "' (cot A)+tan (cot3 A) for 0 < A < (k/4) is (A) 4 tan"1 (1)
(B) 2 tan"1 (2)
(Q0
(D)none
fa B ansa! Classes Transit Dpp on Functions & Inverse trigonometry functions
[11]
Q.8
Let
f(x) =
max.{sint: 0
g(x) -
min. {sint: 0
and h(x) =
[f(x)-g(x)]
where [ ] denotes greatest integer function, then the range of h(x) is (A) {0,1} (B){1,2} (C) {0,1,2} (D) { - 3 , - 2 , - 1 , 0 , 1,2,3} Q. 9
Q.IO
'TCX^
The value of
sec sin
1071
(A)secQ.ll
9
f f . 50tc> -sin + cos"1 COS I 9J (B) sec
7T
Given f(x)
If x = tan 1 1 - cos 1 [ (A) x = Tty
Q.15
and
3x 2 - 7 x - f -
n
1 + x2
g(x)=
(D) ( - 2 , 5 ] then:
;
(C) tan a = tan P f (sin x)
where [ *] denotes the
+
f (cosx)
(D) tan 06 = - t a n P theng(x)iis
(B) periodic with period Tt (D) aperiodic
. . ,1 (1 j + sin 1 ^ ; y = cos - cos
(B) y = Ttx
iOJ 1
then
(C) tanx = -(4/3)y
(D) tanx = (4/3)y
In the square ABCD with side AB = 2, two points M & N are onthe adjacent sides ofthe square such that MN is parallel to the diagonal BD. If x is the distance of MN from the vertex A and f (x) = Area (A AMN), then range of f (x) is: (A) (0,V2]
Q.16
(D)-l
a = sin -1 jcos(sin -1 xjj and P = cos - 1 [sin(cos-1 x)j (B) t a n a = - c o t p
(D) 30
is equal to
(C) [0,1]
(A) periodic with period 7t/2 (C) periodic with period 2n Q.14
31TIn] 9 J
The domain of definition of the function, f (x) = arc cos
(A) tan a = cot P Q.13
—
(C)l
9
greatest integer function, is: (A) (l,o) (B) [0, 6) Q.12
( 7TX '
The period of the function fix) — sin 2 rex + sin + s i n ^ ~ | is \3 J (A) 2 (B)6 (C)15
cos| cos
(A) 1
1
(B) ( 0 , 2 ]
cos^y-j +tan
1
tan]
(B)-l
(C) (o,2V2]
(D) (0,2V3
j j has the value equal to 7t (C) cos-
(D)0
fa B ansa! Classes Transit Dpp on Functions & Inverse trigonometry functions
[11]
Q.17
+ [log10(6-x)]~ is:
The domain of the definition of the function f(x) = sin"
(B) (-7, - 3 ) u ( - 3 , 7) (D) (-3, 3) u (5, 6)
(A) (7, 7) (C) [ - 7 , - 3 ] u [3, 5) u (5, 6) tc 1 . . f a^ f« + - sin — + tani Q .18 The value of tan< — 4 2 VbJ [4 b
»il
(B)
value of/(40)is (A) 15
(C)
2b
Q.19 Let/be a function satisfying /(xy) : (B) 20
1 . J a sin — 2 Kb
f(x)
, where (0 < a < b), is
vV 2b
(D)
Vb^j 2a
for all positive real numbers x and y. If /(30) = 20, then the (C) 40
(D) 60
Q.20 Number ofreal value ofx satisfying the equation, arc tan ^x(x +1) + arc sin^x(x + l) + l = ~ is (A) 0
(B) 1
(C) 2
(D) more than 2
Q.21 Let f (x) = sin2x + cos4x + 2 and g (x) = cos (cos x) + cos (sin x) also let period of f (x) and g (x) be Tj and T2 respectively then (A)T,=2T 2 ' (B) 2Tj = T2 (C)Tj-T2 (D)T } =4T 2 Q.22 Number of solutions of the equation (A) 0 (B) 1 Q.23
2 cor 1 2 + cos-1 (3/5) = cosec"1 x is (C) 2 (D) more than 2
The domain of definition of the function : f(x) = /n(Vx 2 - 5 x - 2 4 - x - 2 ) is (A) ( - « , - 3 ]
(B) (-QO, -3 ] U [ 8 , oo) (C) -00, v
28 9J
(D)none
Q. 24 The period of the function f(x) = sin [ c o s ~ + cos(sinx) equal (A) Q.25
Tt
(B) 2n
(D) 4;t
(C)7t
If x = cos"1 (cos4) ; y = sin"1 (sin3) then which ofthe following holds ? (A) x - y = 1 (B) x + y + 1 = 0 (C)x + 2 y - 2 (D) tan (x + y) = - tan7
Q.26 Let f (x) — e {eWsgnx} andg(x)= e [ e ' s g n x ] , x e R where {x} and [ ] denotes the fractional part and integral part functions respectively. Also h (x) = In (f (x)) + In (g(x)) then for all real x, h (x) is (A) an odd function (B) an even function (C) neither an odd nor an even function (D) both odd as well as even function Q. 27 The number of solutions of the equation tan"1 V3y + tan" (A) 3
(B) 2
(C)l
J
= tan"1 x
is
(D)0
fa B ansa! Classes Transit Dpp on Functions & Inverse trigonometry functions
[11]
f Q.28 Which of the following is the solution set of the equation 2 cos-1 x = cot _1 (B)(-l, 1)- {0}
(A) (0,1) Q.29
(D)[-l,l]
( 9 (-1,0)
]_
2
1 The value of tan — - tan v2 (A)
Q.31
2x VT
Suppose that / is a periodic function with period — and that / (2) = 5 and f(9/4) = 2 then f (-3) - f (l/4) has the value equal to (A) 2 (B)3
Q.30
2x2 - 1 ^
K
(B)
(C)5
CD) 7
- 2V6 is equal: 1 +V6
71
(Of
(D)none
Given f(x) = (x+1>C(2x_8); g(x) = ( 2x ~ 8 )C (x+1) and h (x) = f(x). g (x), then which of the following holds? (A) The domain of 'h' is (j) (B) The range of 'h* is { - 1 } (C) The domain of 'h' is {x / x > 3 or x < - 3 ; x e I (D) The range of *h' is {1 >
Q.32 The sum ]T
t a n -1
n = 1
,1 ,2 (A) tan"1 - + tan"1 -
4n is equal to: n - 2n + 2 4
(B) 4 tan -J 1
(D) sec"1 (-V2)
(C)
Q 33 Range ofthe function f(x) = tan"> J [ x ] +[-x] + p-\x\
+-- is x"
where [*] is the greatest integer function. (A)
(B)!^!2'00)
(D)
(C){^2|
Q. 3 4 Let [x] denote the greatest integer in x. Then in the interval [0,3] the number of solutions of the equation, x 2 - 3 x + [x] = 0 is: (A) 6
(B) 4
(C) 2
(D) 0
Q. 3 5 The range of values of p for which the equation, sin cos-1 (cosOan1 xyj = p has a solution is: (A)
1
_L
V2'V2.
Q.36 Let f (x) =
(B) [0, 1)
r 0
if x is rational
L
if x is irrational
X
Then the function ( f - g) x is (A) odd (C) neither odd nor even
(C) and g(x) =
1
^
(D)(-U) if x is irrational if x is rational
(B)even (D) odd as well as even
fa B ansa! Classes Transit Dpp on Functions & Inverse trigonometry functions
[11]
Q. 3 7 Number of value of x satisfying the equation sin (A) 0 Q.38
vx; (C)2
(B) 1
Consider a real valued function f(x) such that a +b is satisfied are .1 + ab (A) a e (-oo, 1); b e R (C) a e (-1, 1); b e [-1, 1]
+ sin-1
1 - ef(x) 1 + ef(x)
Tt
v
x
y
=
iiS (D) more than 2
x. The values of 'a' and V for which
f(a) + f(b) = f
( B ) a s ( - o o , 1); b e (-l,oo) ( D ) a e ( - l , l ) ; b s ( - l , 1)
1 Q. 3 9 The value of tan - c o r (3) equals 2
(Aj^-Vio)"1 Q.40
(B) (lO + V3 )_1
(D)(lO+V3)
The period of the function cos V2 x + cos 2x is : (A) n
Q.41
(C) (3 + V10)
(B) K4~2
(C) 2tt
(D) none of these
f f4Ni 2^ fx +— 1 1 xThe real values of x satisfying tan -tan" -tan" = 0 are V xj V xj —
(A)±7J
(B)±V2
(C)± 4V2
—
(D)±2
Q.42
Which of the following is true for a real valued function y = f (x), defined on [ - a, a]? (A) f (x) can be expressed as a sum or a difference of two even functions (B) f (x) can be expressed as a sum or a difference of two odd functions (C) f (x) can be expressed as a sum or a difference of an odd and an even function (D) f (x) can never be expressed as a sum or a difference of an odd and an even function
Q.43
cos 1^2 tan "
lV, Y
(A)sin(4cor 1 3) Q. 44
(C) cos(3cor ! 4)
(D) cos(4cor13)
(B) {4, 5}
(C) [4, 5]
(D) [4, 5)
The range ofthe function, f(x) = cot-1 log05(x4 - 2x2 + 3) is: (A) (0,7i)
Q. 46
(B) sin(3cor'4)
Let fix) = sin *J[a] x (where [ ] denotes the greatest integer function). If f is periodic with fundamental period %, then a belongs to : (A) [2,3)
Q.45
equals
7C
3?c"
(B)
4 _
(C)
T""J
(D)
371
2'T
Which of the following is the solution set of the equation sin"!x=cos"1x + sin"1 (3x - 2)1
(A)
(B)
(C)
I 3'
(D)i^
fa B ansa! Classes Transit Dpp on Functions & Inverse trigonometry functions
[11]
Q. 47 Which ofthe following functions are not homogeneous ? x + vcosx ' ysinx + y
xy
(A) x + ycos;
(B) x+y"
( D ^ J x V y J * y \x/ x vy;
v(C)
Q. 48 Which of the following is the solution set ofthe equation 3 cos_1x = cos-1 (4x3 - 3x)? (B)
(A) [-1,1]
Q.49
1 l"
.
(C)
3'3.
The function f : R - > R , defined as f(x) = II— 6 x
r1
J
+
is:
3x - 3 - x 2
(A) injective but not suijective (C) injective as well as suijective
(D)
(B) suijective but not injective (D) neither injective nor suijective
Q. 50 The solution of the equation 2cos_1x = sin-1 (2xVl - x 2 ) (A) [-1,0]
(B) [0,1]
(C)[-l,l]
(D) V2'
Q. 51 The period ofthe function/(x) = sin(x + 3 - [x+3 ]), where [ ] denotes the greatest integer function is (A) 2TI+ 3 (B) 27t (C)l (D) 3 Q. 52 If tan-'x + tan"1 2x + tan~'l3x = TC, then (A) x = 0 (B) x = 1
(C)x = -1
(D)xe|
Q. 53 If f(x + ay, x - ay) = axy then f(x, y) is equal to : 2 X2 - V2 x2 + v (A) (B) (C) 4xy
(D) none iz
2 Q. 54 The set of values ofx for which the equation cos_1x + cos-1 —+ —V3--3X v2 2
(A) [0,1] Q.55
(B)
The range ofthe function y : (A) ( -
°°) -
(C)
1
0
(D) {-1,0,1}
8 9-x 2
1S
3} (B)
(C)
(D)(-®,0)u
V 9y
Q. 5 6 The domain of definition of the function f (x) =
\ +2 X cot ,2cosec x + 5.
oo
r
f
+
, 2x a tan J l o 8 i ,3sec"x + 5y is V 2
7T
(A) R - {mc, n e 1}
(B) R - {(2n+ 1)—, n e 1} 71
(D)none
( C ) R - { n i c , ( 2 n + l ) - , n e 1} Q. 57 The solution set of the equation sin-1 *J\- X 2 (A) [-1, 1] - {0}
holds good is
(B) (0, 1] U {-1}
+
cos_1x = cot-
VT -x V
(C) [-1, 0) U {1}
X
,
-sin 'x
(D) [-1,1]
fa B ansa! Classes Transit Dpp on Functions & Inverse trigonometry functions
[11]
Q. 58
Given the graphs of the two functions, y = f(x) & y = g(x). In the adjacent figure from point A on the graph of the function y = f(x) corresponding to the given value ofthe independent variable (say Xq), a straight line is drawn parallel to the X-axis to intersect the bisector of thefirstand the third quadrants at point B. From the point B a straight line parallel to the Y-axis is drawn to intersect the graph ofthe function y - g(x) at C. Again a straight line is drawnfromthe point C parallel to the X-axis, to intersect the line NN' at D . If the straight line NN' is parallel to Y-axis, then the co-ordinates of the point D are (A)fi:x0),g(f(x0)) (B)x0,g(x0) (C)x0,g(f(x0)) (D)f(x0),f(g(x0))
Q. 59 The value of sin"1 (sin(2cor1 (V2 -1))) is (A)-
7t <
B)
7t 4
(C)
3%
(D)
In
Q. 60
The function f: [2, QO) - » Y defined by f(x) - x2 - 4x + 5 is both one-one and onto if: (A) Y = R (B) Y= [1, oo) (C) Y= [4, o o ) (D) [5, oo)
Q. 61
If f(x) = cosec"1 (cosecx) and cosec(cosec"1x) are equal functions then maximum range of values ofx is (A)
71
KJ
71
(C) (-oo,-l]u[l,oo) Q.62
7C
(D)[-1,0)u[0,1)
1-x2
(B) " 5x
(C)
5x
(D)
1-x4
n Sum of the roots of the equation, arc cot x - arc cot (x + 2) = — (A)V3
Q. 64
0,
If 2 f(x2) + 3 f(l/x 2 ) = x2 - 1 (x * 0) then f(x2) is : 1-x4 (A) 5x2
Q. 63
LJ
(B)
(B)2
Range of the function f (x) :
(C)-2 {x}
2x 4 + x 2 - 3
5x2
is (D)-V3
where {x} denotes thefractionalpart function is
l + {x} (A) [0,1) Q. 65
(B) 0. 1
(C)
-3
(D)
Range of the function sgn [ In (x2 - x + 1) ] is (A) {-1,0,1} (B) {-1,0} (C)-{1>
Q. 66 Number of solution(s) of the equation cos 1 (1 - x) - 2cos (A) 3
(B)2
(C)l
••I
(D){-1,1} 7t — is (D)0
Q. 67 Let /(x) and g (x) be functions which take integers as arguments. Let / ( x + y) =/(x) + g (y) + 8 for all integer x andy. Let / ( x ) = x for all negative integers x, and let g(8) = 17. The value of /(0) is (A) 17 (B) 9 (C) 25 (D) - 17
fa B ansa! Classes Transit Dpp on Functions & Inverse trigonometry functions
[11]
Q. 68
There exists a positive real number x satisfying cos(tan-1x)=x. The value of cos-1 (A)
Q.69
7t 10
_
2tz 2TZ (C)y
Tt
(B) 5
v_/
(D)
v2 y
is
4n
The domain ofthe function, f(x) = (x + 0.5)log°5+x 4Xx2+-4x-3 is 1 (A) l - i . o o 1
(B) [1,3] 3
tr\ f ^ l!>^ - f. c o )j (C) /
Q.70
cos (A)
cos
771 . 27t sin — is equal to 5 5 j
23tc 20
(B)
1371
(C)
20
1771
33TC
(D) 20
~20~
Q. 71 Let/(x) be a function with two properties (a) for any two real number x and y. f(x + y) = x+/(y) and (b) f(0) = 2. The value of/(100), is (A) 2 (B) 98 (C) 102 Q.72
(D) 100
Let/be a function such that f(3)= 1 and / (3x) = x +/(3x ~ 3) for all x. Then the value of /(300)is (A)5050 (B)4950 (C)5151 (D)none
Q. 73 I f / (x) is an invertible function, and g (x) = 2/(x) + 5, then the value ofg~l (x), is (A) 2/~'(x) - 5
1
(B) 2/- 1 (x) + 5
(C)\r\x)
+5
Q. 74 If /(2x + 1) = 4x2 + 14x, then the sum ofthe roots of /(x) = 0, is (A) 9/4 (B) 5 (C) - 9/4
(D)/-'
x-5
(D)-5
Q. 75 If y - / ( x ) is a one-one function and (5,1) is a point on its graph, which one of the following statements is correct? (A) (1, 5) is a point on the graph of the inverse function y =/ _ 1 (x) (B)/(5) = f ( l ) (C) the graph of the inverse function y =/"'l(x) will be symmetric about the y-axis (D)/(/_1(5)) = l Q. 76 Domain of definition of the function f (x); (A) (-oo,0] (C) ( - oo, -1) u [0, 4) Q. 77
3X-4S
is x -3x-4 (B) [0, oo) (D)(-oo,l)u(l,4)
Suppo se/ and g are both linear functions, with/(x) = - 2x + 1 and / (g(x)) = x. The sum ofthe slope and the y-intercept ofg, is (A)-2 (B) - 1 (C)0 (D) 1
fa B ansa! Classes Transit Dpp on Functions & Inverse trigonometry functions
[11]
Q. 78
The range of the function/ (x) =
4
(A)
Q.79
I i (B) 0 , 1 u 6 ' 3
(C) ( - QO, 0) ^ (0, QO) (D) (0, oo)
I f / ( x , y) - (max(x, y)) mm(x ' y) and g (x, y) = max(x, y) - min(x, y), then
/
f r g
,g(-4,-1.75)
equals
vV
(D) 1.5
(C)l
(B)0.5
(A)-0.5 Q. 80
Vx+4-3 :— is x-5
The domain and range of the function f(x) = cosec""1 / l o § 3-4s«x 2 are respectively V
(A)R;
Tt (B)R+;[0,-
Tt Tt 2'2
r
1 - 2 sec x
r
%
(C) 2mt —2 ',2mt + -2 J -{2mt}; Or
r Tt - Tt (D) 2nn—2 ',2n7i+—2) -{2MC}; 2 ' 2 -{0} %
More than one alternatives are correct. Q.81
11 + sinx The values of x in [-2k, 2k], for which the graph of the function y = J T — : — - secx and ' 1 - sinx I I - sinx + secx, coincide are 1 + sinx
y = .
(C)
371 (B) V 2
37C^l , ,(3% _
It
(A) (
Tt Xs
71
U
Tt 3% 12'~2
r % 37ri (D) [-271,2*]- \ ± ~ , ±~
Q. 82
sin4(sin3) + sin"1 (sin4) + sin"1 (sin5) when simplified reduces to (A) an irrational number (B) a rational number (C) an even prime (D) a negati ve integer
Q. 8 3
The graphs of which of the following pairs differ. (A) y — ,
sinx
cosx -2 - + + tan x -Jl + cot2x
y = sin2x
(B) y = tanx cotx; v = sinx cosecx (C) y = | cos x | + | sin x i ; y
|secx| + |cosecx| |secx cosecxj
(D) none of these Q.84
1
1
If f(x) = cos -71"2" x + sin 2
(A) f (0) = 1
—
(B)f
2
^
V3+1
(C) f
(D) f(7t) — 0
faB ansa! Classes Transit Dpp on Functions & Inverse trigonometry functions
[11]
Q.85
14tcV The value of cos — cos cos 2 v V 5 J /AN cos (i - ^—71 (A)
(B) sin
(C) cos
10J
Q 86 The functions which are aperiodic are: (A) y = [x + 1 ] (B) y = sin x2 where [x] denotes greatest integer function Q.87
IS :
271 vT
(D) - cos
(C) y = sin2x
(D) y = sin ' x
tan 1 x, tan 1 y, tan ! z are in A.P. and x, y, z are also in A.P. (y 0, 1 , - 1 ) then (A) x, y, z are in G.P. (B) x, y, z are in H.P. (C) x = y = z (D) (x-y) 2 + ( y - z ) 2 + ( z - x ) 2 = 0
Q.88 Which of the following function(s) is/are periodic with period Tt. (A) f(x) - | sinx | (B) f(x) = [x + Tt] (C) f(x) = cos (sinx) (where [. ] denotes the greatest integer function) Q.89
m K 5J
(D) f(x) = cos2x
For the equation 2x = tan(2tan_1 a) + 2tan(tan_1 a + tan"1 a3), which ofthe following is invalid? (A) a2x + 2a = x (B) a2 + 2ax + 1 = 0 ( C ) a ^ 0 (D)a*-1, 1
Q.90 Which ofthe functions defined below are one-one function(s) ? (A) f(x) = (x+ 1), ( x > - l ) (B) g(x) = x+(l/x) ( x > 0 ) 2 (C) h(x) = x + 4x - 5, (x > 0) (D) f(x) = e "x, ( x > 0) Q.91
If cos_1x + cos ' y + cos 'z = Tt, then (A) x2 + y2 + z2 + 2xyz - 1 (B) 2(sin~1x + sin_1y + sin'z) = cos_1x + cos_1y + cos_1z (C) xy + yz + zx = x + y + z - l (D) x + X-
Q.92
' y + -11 + f z + - 0 >6 v yJ v zJ ~
Which of the following homogeneous functions are of degree zero ? (A) — /n—+— In— x y y
x
(x-y)
' y(x+y)
(C)
^
(D) x sin — - y cos
2 x +y 2
x
f 71 xsina x - cosa tan-l is, for a e 0,— ; x ^1-xcosa J sma J v 2J (A) independent of x (B) independent of a
Q. 93 The value of tan-l Tt
(C) - - a Q.94
GR
+
, is
(D) none of these
D = [-1,1 ] is the domain of the following functions, state which of them has the inverse. (A) f(x) = x 2
(B) g(x) = x 3
(C) h(x) = sin 2x
(D) k(x)= sin (TTX/2)
Q.95 Which ofthe following fimction(s) have no domain? (A) f(x) = logx_ j (2 - [x] - [x]2) where [x] denotes the greatest integer function. (B) g(x) = cos_1(2-{x}) where {x} denotes thefractionalpart function. (C) h(x) = In /n(cosx) (D)f(x):
sec ' ( s g n ( e - ) )
fa Bansa! Classes Transit Dpp on Functions & Inverse trigonometry functions
[11]
[ZlJ
suoijDuti/iQjdwouoSuj
3Sd3AU]
3p suoijounj uo dd(j
j i s u v u j
s s s s v j j
jvsuvgf^
Q.l
B
Q.2
C
Q.3
A
Q.4
D
Q.5
Q.6
C
Q.7
A
Q.8
C
Q.9
D
Q.IO D
Q.ll
A
Q.12
A
Q.13
A
Q.14
C
Q.15 B
Q.16
B
Q.17
C
Q.18
C
Q.19
A
Q.20
Q.21
C
Q.22
A
Q.23 A
Q.24
D
Q.25 D
Q.26
A
Q.27
A
Q.28 A
Q.29
B
Q.30 A
Q.31
D
Q.32
D
Q.33
C
Q.34
C
Q.35 B
Q.36
A
Q.37
B
Q.38 D
Q.39
A
Q.40 D
Q.41
B
Q.42
C
Q.43 A
Q.44
D
Q.45 C
Q.46 A
Q.47
B
Q.48
D
Q.49
D
Q.50 D
Q.51
C
Q.52
B
Q.53 A
Q.54
B
Q.55 D
Q.56
C
Q.57
C
Q.58
C
Q.59
B
Q.60
Q.61
A
Q.62
D
Q.63
C
Q.64
C
Q.65 A
Q.66
C
Q.67
A
Q.68
c
Q.69
D
Q.70 D
Q.71
C
Q.72
A
Q.73
D
Q.74
D
Q.75 A
Q.76
c
Q.77
C
Q.78
B
Q.79
D
Q.80
Q.81 A,C
Q.82
B, D
Q.83 A,B,C
Q.84
A,B,C
Q.85 B,C,D
Q.86 AB,D
Q.87
A,B,C,D
Q.88 A,C,D
Q.89
B,C
Q.90 A,C,D
Q.91
Q.92
A,B,C
Q.93 A,C
Q.94
B,D
Q.95 A,B,C,D
AB
A3H
V3MSNV
B
C
B
C
BANSAL CLASSES MATHEMATICS TARGET IIT JEE 2007 XI (PQRS)
SOLUTIONS OF TRIANGLE Trigonometry Phase-Ill
CONTENTS KEY-CONCEPTS EXERCISE-I EXERCISE-II EXERCISE-III ANSWER KEY
KEY
I.
SINE FORMULA
n.
COSINE FORMULA
:
CONCEPTS
In any triangle :
(i) cosA =
ABC,
=
sinA
b 2 + c2 - a 2 2bc
sinB
=
or a2 = b2 + c2 - 2bc. cos A
2 2 b2 c' +• a~ (ii) cosB = 2ca
m.
PROJECTION
(iii) cosC : v
a + b - c2
'
(i) a = b cosC + c cosB
FORMULA:
sinC
2ab
(ii) b = c cosA + a cosC
(iii) c = a cosB + b cos A IV.
VI.
B-C 2
b - c A cot— b +c 2
(I) tan—— = v
.... '
(m) tan——— = v
C-A 2
(11) tan——— = v v.
... '
NAPIER'S ANALOGY - TANGENT RULE :
c-a B cot— c+a 2
A-B 2
'
a-b C cot— a +b 2
TRIGONOMETRIC FUNCTIONS O F HALF ANGLES :
0)
. A j(s-b) (s-c) . B sin— = J 7 ; s m2 V be
(ii)
cos-
(iii)
t a n
(iv)
Area of triangle = Js(s-a) (s-b) (s-c) .
s (s-a) B — r — ; cos— be ' 2
A
A T
M - N RULE
(s-b) (s-c) V s(s-a)
=
s (s-b) ca
A 1(^1)
=
. c
(s-c) ( s - a ) ca
Where
sin
i
C cos
|(s-a) (s-b) =
s (s-c) =
i
a +b + c . & 2
S =
i
. . A =
arCa
°
f tnangle
, '
: In any triangle,
(m+n) cot6 = m cota - n cotp = ncotB-mcotC VII.
~ ab sinC= ^ be sin A = ^ ca sin B = area of triangle ABC sinA
sinB
Note that Vin.
sinC
R =
4 A
; Where Ris the radius of circumcircle & A is area of triangle
Radius of the incircle 'r' is given by :
..
A
,
(a) r = — where s = (c)r =
abc
2R
A
A+B+C
a sin I2 sin-?- „
...
——- & soon
. A
IX.
Radius of the Ex-circles r j , r 2 & r 3 are given by :
(a)
r
=
A
A
s-a
; 12
^Bansal Classes
r 1 s- b ; 3
C
. B
. C
(d) r = 4Rsin-j sin— sin—
COS I
i
B
(b) r = (s-a)tan— = ( s - b ) t a n - = (s-c)tan— Z Z ^
=
A
s-c
(b)
A B C r} = s tan— ; r2 = s t a n - ; r3 = s t a n -
Trig.-t-III
[2]
acosl-cos5-
(C)
COSy
„
—- & so on
(d)
A B C f j = 4 R sin— . cos— . cos—
A C AT> • B r,2 = 4 R sin— . cos— . cos—
2
X.
2
A JY
2
LENGTH O F ANGLE BISECTOR &
•
C
A
2
2
B
r,=4Rsin— . cos — . cos— 3 2
MEDIANS :
If ma and (3a are the lengths of a median and an angle bisector from the angle A then, 1 I 2^0084 2 2 m. = - y2b + 2c - a and B = — 2 b +c Note that m2 + m2 + m2 = - (a2 + b2 + c2)
XL
ORTHOCENTRE A N D P E D A L TRIANGLE :
The triangle KLM which is formed by joining the feet ofthe altitudes is called the pedal triangle. the distances of the orthocentre from the angular points of the A ABC are 2 R cosA, 2 R cosB and 2 R cosC the distances of P from sides are 2 R cosB cosC, 2 R cosC cosA and 2 R cosA cosB the sides ofthe pedal triangle are a cosA (= R sin 2A), b cosB (= R sin 2B) and c cosC (= R sin 2C) and its angles are it — 2A, % — 2B and TC - 2C. circumradii of the triangles PBC, PCA, PAB and ABC are equal.
xn
EXCENTRAL TRIANGLE :
The triangle formed by joining the three excentres I1? I2 and I3 of A ABC is called the excentral or excentric triangle. Note that: Incentre I of A ABC is the orthocentre ofthe excentral A Ijl 2 I 3 . A ABC is the pedal triangle of the AIM the sides ofthe excentral triangle are A B C 4Rcos— , 4Rcos— and 4 R cos— 2 '
2
2
and its angles are — - — 2
2
7C
B
2
2
,
and
2
2
A B C IIj = 4 R s i n - j ; II 2 = 4 R s i n y ; II 3 = 4 R s i n y . Xin.
T H E DISTANCES B E T W E E N T H E S P E C I A L POINTS :
(a)
The distance between circumcentre and orthocentre is = R. J 1 - 8 cos A cosB cosC
(b)
The distance between circumcentre and incentre is = VR2 - 2Rr
(c)
The distance between incentre and orthocentre is ^2r 2 - 4R2 cos A cosB cosC
faBansal Classes
Trig.-0-III
[132]
XIV.
Perimeter (P) and area (A) of a regular polygon of n sides inscribed in a circle of radius r are given by
XV.
71 1 <> • . 271 P = 2nr sin— and A = — n r sin— n 2 n Perimeter and area of a regular polygon of n sides circumscribed about a given circle ofradius r is given by n . tc P = 2nrtan— and A = n r tan— n n In many kinds oftrignometric calculation, as in the solution oftriangles, we often require the logarithms of trignometrical ratios. To avoid the trouble and inconvenience ofprinting the proper sign to the logarithms ofthe trignometric functions, the logarithms as tabulated are not the true logarithms, but the true logarithms increased by 10. The symbol L is used to denote these "tabular logarithms". Thus: L sin 15° 25' = 10 + log10 sin 15° 25' and L tan 48° 23'= 10 + log 10 tan48° 23' •
,
EXERCISE-I Q. 1
With usual notation, if in a A ABC, ^ = ^ = — ; then prove that, 11 12 13 7
Q.2
For any triangle ABC , if B = 3 C, show that cosC = J ^ t l & sin^ = -. y 4c 2 2c
Q.3
•v3 Tt In a triangle ABC, BDisamedian. If /(BD) = —-/(AB) and Z DBC = ~ . Determine the ZABC.
=
^ = . 19 25
»
Q.4
ABCD is a trapezium such that AB,DC are parallel & BC is perpendicular to them. If angle ADB = 9 , BC = p & CD = q, show that AB =
Q.5
+ q2) sin9
pcosB + qsinG
.
Let 1 < m < 3 . In a triangle ABC, if 2b = (m+1) a & cos A= |
Prove
that there
are two values to the third side, one of which is m times the other. Q.6
If sides a, b, c of the triangle ABC are in A.P, then prove that sin2 — cosec2A;
sin2 — cosec2B ; sin2 — cosec2C areinH.P.
2
Q. 7 Q.8 Q.9 Q.10
2
2
Find the angles of a triangle in which the altitude and a median drawnfromthe same vertex divide the angle at that vertex into 3 equal parts. A B C Inatriangle ABC, if tan—, tan— , tan— areinAP. Showthat cos A, cosB, cosC are in AP. Show that in any triangle ABC; a 3 cosBcosC + b 3 cosC. cosA+c 3 cosAcosB = a b c ( l - 2 cos A cos B cos C). 3R A point'0'is situated on a circle ofradius Rand with centre O, another circle of radius — isdescribed. Inside the crescent shaped area intercepted between these circles, a circle ofradius R/8 is placed. If the same circle moves in contact with the original circle of radius R, thenfindthe length ofthe arc described by its centre in moving from one extreme position to the other.
Q.ll
ABC is a triangle. D is the middle point of BC. If AD is perpendicular to AC, then prove that cos A. cos C :
3 ac
Q. 12 Ifin a triangle ABC,
faBansal Classes
cos A + 2cosC cosA + 2cosB
sinB sinC
=
, prove that the triangle ABC is either isosceles cr right angled.
Trig.-0-III
[4]
Q.13
In a A ABC, (i) —— = —— v(ii) 2 sinAcosB = sinC cosA cosB A A C (iii) tan2 — + 2 tan — tan — - 1 = 0, prove that (i) => (ii) => (iii) (i). Ai jL JL
Q.14
Sides a, b, c of the triangle ABC are in H.P., then prove that cosec A (cosec A + cot A); cosec B (cosec B +cot B) & cosec C (cosec C + cot C) are in A.P.
Q.15
In a triangle the angles A, B, C are in AP. Show that 2 cos
A— C
a+c ac + c2
Q. 16 If pj, p2, p3 are the altitudes of a triangle from the vertices A, B, C & A denotes the area of the ,
.
l
l
l
p,
p2
p3
triangle, prove that — +
2ab
c
(a + b + c) A
2
= ——-—-— cos"2 —.
Q.17 Let ABCD is a rhombus. The radii of circumcircle of AABD and AABC are Rj and Rj respectively then show that the area of rhombus is TTz D 2 x2 . (Kj + K 2 ; Q.18 In a AABC, GA, GB, GC makes angles ot, P, y with each other where G is the centroid ot the AABC then Show that, cot A + cot B + cot C + cot a + cot p + cot y = 0. A +B Q.19 If atanA+btanB = (a+b)tan—-— , prove that triangle ABC is isosceles.
Q.20
The two adjacent sides of a cyclic quadrilateral are 2 & 5 and the angle between them is 60°. If the area of the quadrilateral is 4 fi, find the remaining two sides.
Q.21
The triangleABC (with side lengths a, b, c as usual) satisfies log a2 = log b2 + log c2 - log (2bc cosA). What can you say about this triangle?
2 Q.22 Ifthe bisector of angle C oftriangle ABC meets AB in D & the circumcircle in E prove that, CE _ (a+b) 2
DE "
Q.23
c
In a triangleABC, the median to the side BC is of length , 1 & it divides the angle A into y n - 6V3 angles of 30° & 45°. Find the length ofthe side BC.
Q.24
Given the sides a, b, c of a triangle ABC in a G.P. (a, b, c * 1) . Then prove that; x = rb2_c2) X
tanB + tanC
tanB - tanC
• '
v = Y
( c 2 . a * tanC + tanA (C 4 > tan'C - tanA
. '
Z
(a
b)
tanA + tanB tanA-tanB
are also in G.P. Further, if a2 = logxe ; b j = l o ^ e & Cj=log z e are the sides of the triangle A B C Aj Bj Cj, then prove that : sin2-^- , sin2 - - , sin2-^- are in H.P. Q.25 With reference to a given circle, Aj and B, are the areas of the inscribed and circumscribed regular polygons of n sides, A^ and B2 are corresponding quantities for regular polygons of 2n sides. Prove that (1) A2 is a geometric mean between Aj and B j (2) B2 is a harmonic mean between A^ and B j.
EXERCISE-II Q.l
I +± +i = i n r2 r3 r
^
b-c r.
Q.4A
+
c-a r„
+
faBansal Classes
a-b r„
Q . 3 — ^ — r +^ — ^ — A -3 (s-b)(s-c) (s-c)(s-a) (s-a)(s-b) r
Q.2 rj + r2 + r3 - r = 4R „
=0
„
r
i ~ r +. a
Q.5c —~
r
2 ~
o
r
_ c - 7
r3
T rig
Q.6
.-(f> - III
abc s
A 2
B 2
C' 2
cos— cos— cos— = A
[5]
Q.7
a cosB cosC +b cosC cosA+ c cos AcosB = —
Q.8
(ri+r2)tan| =(r3-r)cot|=c
R
Q.9
Q.10 4R sin A sinB sinC = a cos A + b cosB + c cosC B-C
, ,
,
.
C-A , , ,
.
a cot A+ b cotB + c cotC = 2 (R+r)
Q.ll
(r 1 -r)(r 2 -r)(r 3 -r) = 4 R ^
A-B
Q.12 (r + fj) t a n — + (r + r2) t a n - — + (r + r3) t a n — = 0 a„ 2 +. bv 2+ ,c„2
Q.13 _L _L _L J_ 2
+
+
2
2
A
r
1
Q. 15
—
Q.17
1 Vr
Q. 19
-
be
+
—
r
1
ca
+
—
1
ab
+
—
Tj
+
—
r2
3
1 =
2Rr
1 v Vr +
Q. 14 (r3+ fj) (r3+ r2) sin C = 2 r3 Jr2r3 + r ^ + r ^
2
r
i V vr
1 1 1 1
r
+
2
—
r3;
r
3/
4R r2 s2
Q.16
T, be
Q.18
bc - r2 r3 _ ca - r3 rt _ a b - rt r2 — r
|
I; ca
|
r3 _ 1 ab r
2R
4I I 1 1 + —+— r vr,i % r3;
Q.20 In acute angled triangle ABC, a semicircle with radius ra is constructed with its base onBC and tangent to the other two sides. rb and rc are defined similarly. Ifr is the radius ofthe incircle of triangle ABC then 2 1 1 1 prove that, — = — + — + — r ra r,b rc Q.21
If I be the in-centre ofthe triangle ABC and x, y, z be the circum radii ofthe triangles IBC, ICA& IAB, showthat 4 R 3 - R ( x 2 + y2 + z 2 )-xyz = 0.
Q.22 IfAq denotes the area of the triangle formed by joining the points of contact ofthe inscribed circle ofthe triangle ABC and the sides of the triangle; Aj, Aj and A3 are the corresponding areas for the triangles thus formed with the escribed circles of A ABC. Prove that A1 + A^ + A3 = 2A + Aq where A is the area of the triangle ABC. Q.23 Consider a A DEF, the pedal triangle ofthe A ABC such that A-F-B and B-D-C are collinear. If H is the incentre of A DEF and Rj, R^ R3 are the circumradii of the quadrilaterals • AFHE; • BDHF and • CEHD respectively, then prove that I R p R + r where R is the circumradius and r is the inradius of A ABC. A
B
C
Q.24 Prove that in a triangle, 8 r R(cos2— + cos2— + cos2—) = 2bc + 2ca + 2ab - a2 - b2 - c2. Q.25 Prove that in a triangle — + — + — = 2R Q.26
b
aJ
vc
by
va
c.
The triangle ABC is a right angled triangle, right angle at A. The ratio of the radius of the circle circumscribed to the radius of the circle escribed to the hypotenuse is, -Jl: (S+-Ii). Find the acute angles B & C. Also find the ratio ofthe two sides ofthe triangle other than the hypotenuse.
Q.27 Let the points PJ, P2, , PN_1 divide the side BC of the triangle ABC into n parts. Let ij, i^ i3, in be the radii of the inscribed circle ; el5 e2, e3, , e n bethe radii of the escribed circles corresponding to the vertex A for the triangles A B P J , A P j P 2 , A P 2 P 3 , , A P N _ 1 C respectively, then show that r + r, = - , where R t , I^, , Rn are the radii ofthe circumcircle (i) n i L = £ & ( i i ) t h R R, of triangles ABP j, AP j P2, , AP n _jC &R is the circumradius, r is the inradius & r t is the exradius as usual of A ABC. Trig.-0-III faBansal Classes [6]
Q.28 In a plane of the given triangleABC with sides a, b, c the points A', B', C' are taken so that the A A'BC, A AB'C and A ABC' are equilateral triangles with their circum radii Ra, Rb, Rc ; in-radii ra, rb, rc & ex-radii r a ', r b ' & rc' respectively. Prove that ; (ii) r,r 2 r 3 = £ ( 3R * + 6 r * + 2 r *')] U t m A 648 V 3 2 Q.29 In a scalene triangle ABC the altitudes AD & CF are dropped from the vertices A& C to the sides BC & AB. The area of A ABC is known to be equal to 18, the area of triangle BDF is equal to 2 and length of segment DF is equal to 2V2 • Find the radius of the circle circumscribed. Q.30 Consider a triangleABC with Aj, Bj, Cj, as the centres ofthe excirlces opposite to the verticles A B, C respectively. Ar.(AA1BC) + Ar.(AAB 1C)+Ar.(AABC1)_ 1 Showthat S(R?+R 2 +R 2 ) ~ 2R (i) n r a : IT R : IT r ' = 1: 8 :27
&
Where R, Rj, R^, R3 are the circum radii of AABC, AAjBC, AABjC and AABCj respectively and S is the semiperimeter of AABC.
EXERCISE-III Q.l
In a AABC, Z C = 60° & z A= 75°. If D is a point on AC such that the area of the A BAD is S times the area of the A BCD, find the Z ABD.
[REE'96,6]
Q.2
In a A ABC, a : b : c = 4:5:6. The ratio of the radius of the circumcircle to that ofthe incircle is . [JEE '96,1]
Q.3
If in a A ABC, a = 6, b = 3 and cos(A- B) = 4/5 thenfindits area.
[REE'97,6]
Q.4
If in a triangle PQR, sin P, sin Q, sin R are in A.P., then (A) the altitudes are in A.P. (B) the altitudes are in H.P. (C) the medians are in G.P. (D) the medians are in A.P.
[JEE '98,2]
Q. 5
Two sides of a triangle are of lengths ^6 and 4 and the angle opposite to smaller side is 3 0°. How many such triangles are possible ? Find the length of their third side and area. [REE '98,6]
Q. 6
Let ABC be a triangle having 'O' and T as its circumcentre and incentre respectively. If R and r are the circumradius and the inradius respectively, then prove that, (IO)2 = R2 - 2 Rr. Further show that the triangle BIO is a right triangle if and only ifb is the arithmetic mean of a and c. [JEE'99,10 (out of200)]
Q.7
The radii r } , r2 , r3 of escribed circles of a triangle ABC are in harmonic progression. If its area is 24 sq. cm and its perimeter is 24 cm,findthe lengths of its sides. [REE '99,6] n Q.8(a) In a triangleABC, Let Z C = —. If 'r* is the inradius and ' R' is the circumradius of the triangle, then 2 (r + R) is equal to : (A) a + b
(B) b + c
(C) c + a
(D) a + b + c
(b)
In a triangleABC, (A) a2 + b2 - c2 2 ac(B)sinc2~+(aA2 -- Bb2+ C) = (C) b2 - c2 - a2
Q.9
LetABC be a triangle with incentre T and inradius 'r'.Let D,E,F be the feet ofthe perpendiculars from I to the sides BC, CA & AB respectively. If rj, r2 & r3 are the radii of circles inscribed in the quadrilaterals AFIE, BDIF & CEID respectively, prove that —
faBansal Classes
+—
+—
=
, J1 r2 \ , t . ( r - r,) ( r - r 2 ) (r - r3)
7
Trig.-0-III
(D) c2 - a2 - b2 [JEE '2000 (Screening) 1 + 1]
[JEE '2000,7]
[7]
Q. 10 If A is the area of a triangle with side lengths a, b, c, then show that : A < — ^(a + b + c)abc Also show that equality occurs in the above inequality if and only if a = b = c.
[JEE' 2001 ]
Q. 11 Which of the following pieces of data does NOT uniquely determine an acute-angled triangle ABC (R being the radius of the circumcircle)? (A) a, sinA, sinB (B)a,b, c (C)a, sinB,R (D)a,sinA,R [ JEE' 2002 (Scr), 3 ] Q. 12 If I n is the area of n sided regular polygon inscribed in a circle of unit radius and On be the area of the polygon circumscribing the given circle, prove that On
1 + , 1-
v «
[JEE 2003, Mains, 4 out of 60]
J
Q. 13 The ratio ofthe sides of a triangle ABC is 1: ^3 :2. The ratio A: B : C is (A) 3 : 5 : 2
(B) 1 :
:2
(D) 1 :2 : 3 [JEE 2004 (Screening)] Q. 14 In AABC, a, b, c are the lengths of its sides and A, B, C are the angles of triangle ABC. The correct relation is f\ \ B-C B-C^j = asm = a cos (B) (b-c)cos (A) (b-c)sin 2 V^J V 2 ,
J
(C) (b + c) sin
B + C> 2
J
(C) 3 : 2 : 1
—
= a cos
fA> U ;
' B + C^ v 2 y [JEE 2005 (Screening)]
= 2a sin
(D) (b-c)cos
—
ANSWER KEY EXERCISE-I Q.3
120°
Q.7
Q.10
TT/6, tc/3, n/2
7tiR 12
Q20. 3 cms & 2 cms Q 23. a = 2
EXERCISE-II Q.26B=f
Q.29 - units
c
2
EXERCISE-III Q.l angleABD = 30°
Q.2
Q.4 B
Q.3 9 sq. unit
Q.5 2, (2V3-V2) , (2V3+V2) , (2V3-V2) & (2V3+V2) sq. units Q.8 (a) A, (b) B
Q.ll D
Q.13
D
Q.14
Q.7 6, 8, 10 cms
D
o
faBansal Classes
Trig.-^-III
15]
ft
BANSAL CLASSES MATHEMATICS TARGET IIT JEE 2007 XI (P, Q, R, S)
TRIGONOMETRIC EQUATIONS AND INEQUATIONS Trigonometry Phase-H
CONTENTS KEYCONCEPTS EXERCISE-I EXERCISE-II EXERCISE-III ANSWER KEY
KEY
CONCEPTS
THINGS TO REMEMBER : 1. 2. 3.
it n ,ne I . 2'2 If cos0 = cosa => 9 = 2n% ± a where a e [0, Tt], n e I
If sinG = sina => G = n7t + (~l) n a where a e
4.
If tanG = tana => G = nTt + a where a e -— — , n e I . I 2'2J If sin2G = sin 2 a=> G = nTt ± a .
5.
cos2G = cos 2 a => G = nTt ± a .
6. 7.
tan2G = tan 2 a => 9 = nTt ± a . [Note: a is called the principal angle] TYPES OF TRIGONOMETRIC EQUATIONS : (a)
Solutions of equations by factorising. Consider the equation ; (2 sinx - cosx) (1 + cosx) = sin2x ; cotx - cosx = 1 - cotx cosx
(b)
Solutions of equations reducible to quadratic equations . Consider the equation : 3 cos 2 x- 10 cosx+ 3 = 0 and 2 shfo + -Js sinx + 1 = 0
(c) (d)
(e)
(f)
Solving equations by introducing an Auxilliary argument. Consider the equation : sinx + cosx = ^[2 ; V3 cosx + sinx = 2 ; secx- 1 =(V2 - l)tanx Solving equations by Transforming a sum of Trigonometric functions into a product. Consider the example : cos 3 x + sin 2 x - sin 4 x = 0 ; sin2x + sin22x + sin23x + sin24x = 2; sinx + sin5x = sin2x + sin4x Solving equations by transforming a product of trigonometric functions into a sum. Consider the equation : sin 6x . . . sin5x. cos3x= sin6x.cos2x ; 8cosxcos2xcos4x = —: : sin3G = 4sinG sin2G sin4G sin x Solving equations by a change of variable : (i) Equations of the form of a. sinx + b. cosx + d = 0, where a, b & d are real numbers & a, b ^ 0 can be solved by changing sinx& cos x into their corresponding tangent of half the angle. Consider the equation 3 cosx + 4 sinx = 5. (ii)
Many equations can be solved by introducing a new variable. eg. the equation sin4 2x + cos4 2x = sin 2x. cos 2x changes to 2(y+l)
(g)
=0
by substituting, sin2x. cos2x = y.
Solving equations with the use of the Boundness of the functions sinx& cosx or by making two perfect squares. Consider the equations: \
/•
„
\
1+sin—-2cosx .cosx = 0 ; 4 4 11 "7= tanx - sinx + — = 0 sin2x + 2tan2x + ~~r= V3 12 TRIGONOMETRIC INEQUALITIES: There is no general rule to solve a Trigonometric inequations and the same rules of algebra are valid except the domain and range of trigonometric functions should be kept in mind. sinx | cos— - 2sinx + 4
<
xV
• f
Consider the examples: log2 sin — < - t1 : sin x cos x + 2 V
faBansal Classes
Trig.-II
O < 0 ; V5-2sin2x > 6 s i n x - l [2]
EXERCISE-I Q 1.
Solve : 2 + 7 tan2 0 = 3.25 sec2 6 (0° < 6 < 360°).
Q 2.
Solve : tanQ + secQ = S
Q 3.
Find all the values of 0 satisfying the equation ; sin9 + sin59 = sin3 9 such that0< 9 < TC.
Q 4.
Solve the inequality: tan2 x - (V3 + l)tan x + V3 < 0
Q.5
cos2x + cos2 2x + cos2 3x= 1.
Q 6.
Find all value of 9, between 0 & it, which satisfy the equation ; cos 9 . cos2 9 . cos 3 9 = 1/4.
Q 7.
Find the general solution of the trigonometric equation
for values of 9 between 0° & 360°.
•^/l6cos 4 x-8cos 2 x + l + ^/l6cos 4 x-24cos 2 x + 9 = 2. Q 8.
Solve for x , the equation j 13 - 18tanx = 6 tan x - 3 , where -2it
Q 9.
If a & p are two distinct roots of the equation, a tan 9 + b sec 9 = c then prove that : tan(a + P ) = - ^
T
a - c
.
.
Q 10. Find the principal solution of the trigonometric equation '
Jcot3x + sin2 x-— + Jv J 3 cosx + s i n x - 2 = sin — - — V 4 2 2
Q l l . Determine the smallest positive value of x which satisfy the equation, v 1 + Sin2x - V2 cos3x = 0 .
Q 12. 2 sin |^3x + ^J =
+ 8 sin2x . cos2 2x
Q 13. Given that A,B are positive acute angle, solve : Js sin2A= sin2B & S sin2A+ sin2B =
2
Q 14. Solve : sin5x = cos2x for all values of x between 0°& 180° . Q 15. Find all values of 9 between 0°& 180° satisfying the equation ; cos 69 + cos 49 + cos 29 + 1 = 0 . Q 16. If a & P satisfy the equation, acos29 + bsin29 = c then prove that : cos2 a + cos2 P = d - + 2 dt ' i ^ b . & 4" b Q 17. Find all values of 9 lying between 0&2TC satisfying the equations : r sin9 =
& r + 4 sin9 = 2 ( S +1).
Trig.-4-II
[3]
Q 18. Find the value of 9 , which satisfy 3 - 2 c o s 6 - 4 sin 6 - c o s 26 + sin2 6 = 0. Q 19. Solve the inequality: sin3x < sinx. Q 20. Solve for x, ( - TC
Find the general values of8 for which the quadratic function ' cos6 + sin8 . „ (sin6)x2 + (2cos0)x + — is the square of a linear function.
Q 23. If sin A= sinB & cosA= cosB, find the values of A in terms of B. Q 24. Solve the equation : (1 - tanG) (1 + sin26) = 1 + tanG. Q25. Find the general solution of sec48-sec28 = 2.
EXERCISE-II Q 1,
V3 Solve the equation — sinx - c o s x = cos2x.
Q 2.
cos3x. cos3x + sin3x. sin3x = 0.
Q3.
Find all the solutions of, 4cos 2 x sinx-2sin 2 x = 3 sinx.
Q4,
If a & p are the roots ofthe equation, a cos 6 + bsin8 = c then prove that : 2b c a +b
(i)
sina + sin P = —
(w111)
tan — + tan - = 2
2
(ii) (iv)
w
a+c
sin a . sin P = '
c2 — a2 a2+b2
tan— . t a n - = 2
2
Q 5.
Find the least positive angle measured in degrees satisfying the equation sin3x + sin32x + sin33x = (sinx + sin2x + sin3x)3.
Q 6.
Find the general solution of the following equation: 2 (sinx - cos 2x) - sin 2x (1 + 2 sinx) + 2 cos x = 0.
Q 7.
Solve the inequality sin2x > 42 sin2x + (2 - V2 ) cos2x.
Q 8.
Find the values of x, between 0 & s a t i s f y i n g the equation; 3x
x
2
2
c
+a
cos3x + cos2x= sin-— +skiT-. Q 9.
Solve for x : sin3a = 4 sin a sin (x + a ) s i n ( x - a ) where a is a constant * nu.
Q 10. Solve : cos (TC . 3*) - 2 cos2 (TC . 3X) + 2 cos (4 TC . 3X) - cos (7 TC . 3X) = sin(7t. 3*) + 2 sin2(TC . 3X) - 2 sin(4TC. 3*) + 2 sin(Tt. 3* +! ) - sin(7TC. 3X)
faBansal Classes
Trig.-(f>-II
[4]
Q 11. Find the set of values of'a' for which the equation, sin 4 x+cos 4 x + sin2:x + a = 0 possesses solutions. Alsofindthe general solution for these values of 'a'. Q 12. Solve : tan2 2x + cot2 2x + 2 tan 2x + 2 cot 2x = 6. Q 13. Solve the equation : 1 +2cosecx=
•
3 Q 14. sin4x + cos 4 x-2sin 2 x + - sin22x = 0. Q 15. Solve : tan2 x . tan2 3x . tan 4x = tan2 x - tan2 3x + tan Ax. 29
Q 16. Solve : sin10* + cos10* = — cos4 2x. 16
Q 17. Find the set of values of x satisfying the equality sin
f I
m\ X
—
4J
3 -cos fX + —
^
4J
1 and the inequality
2cos7x cos3+sin3
> z
eos2x
Q 18. Find the sum of all the roots of the equation, sin Vx = - 1 , which are less than 100 n 2 . Also Find the sum of the square roots of these roots. Now, can we conclude that all the roots cos Vx = 0 are also the roots of sinVx = - 1 ? Justify your answer. Q 19. Solve : sin
' V R V
+ COS /
v2j
= V2 sinVx.
. , 1 1 • n, • . 2x +1 . 2x +1 „ 7, 2x +1 Q 20. Find the general solution of the equation, sin—-— + sin—;:— — = 0. 3x - 3 cos —3x Q 21. Let S be the set of all those solutions of the equation, (1 + k)cos x cos (2x - a) = (1 + k cos 2x) cos(x - a) which are independent of k & a. Let H be the set of all such solutions which are dependent on k & a. Find the condition on k & a such that H is a nonempty set, state S. If a subset of H is (0, tz) in which k = 0, then find all the permissible values of a. Q 22. Solve the equation ; sin5x= 16 sin5x. xcos3y + 3xcosy sin2y = 14 Q23. Solve for x & y , . , 2 - 1 , xsin3 y + 3xcos y siny = 13 Q 24. Solve the equation : c o t x - 2 sin2x= 1. Q 25. Find all values of'a' for which every root of the equation, acos2x+ | a | cos4x + cos6x= 1 is also a root of the equation, sinx cos2x = sin2x cos3x
sin5x, and conversely, every root of the second equation is also
a root of the first equation.
Bansal Classes
Trig.-j-II
[5]
EXERCISE-III Q. 1
Q.2
Q. 3
Find all values of 8 in the interval f - - | ,
satisfying the equation ;
(1 - t a n 8 ) ( l + tan 8) sec2 8 + 2tenS 6 = 0.
[JEE'96,2]
The number of values of x in the interval [0, 5K] satisfying the equation 3 sin2x - 7 sinx + 2 = 0 is (A) 0 (B) 5 (C) 6 (D) 10
[JEE'98'2]
Find the general values ofx and y satisfying the equations 5 sinx cosy = 1, 4 tanx = tany
[REE '98,6]
Q.4
Find real values of x for which, 27 cos2x . 8 l sin2x is minimum. Alsofindthis minimum value. [REE 2000,3]
Q.5
Solve the following system of equations for x and y [REE'2001(mains), 3] 2 2 5 (cosec x-3sec y) _ 2 (2cosecx + V3jsecyj) _ 6 4
Q.6
The number of integral values of k for which the equation 7cosx + 5sinx = 2k + 1 has a solution is (A) 4 (B) 8 (C)10 (D) 12 [JEE 2002 (Screening), 3]
Q.7 . cos(a-{3) = 1 and cos(a + (B) = 1/e, where a, (3 e [-11,11], numbers of pairs of a, P which satisfy both the equations is (A) 0 (B) 1 (C) 2 (D)4 [JEE 2005 (Screening)]
faBansal Classes
Trig.-j-II
ANSWERKEY EXERCISE-I Q 2. 0 = 30°
Q 1. 30°,150°,210°,330° - - 71 71 271 57t
7t TC 0 4. n7i + — < x < n 7 i + — ; n s l 4 3
D
Q 5. x = (2n + l ) ^ ; x = (2n + l ) - | ,x = nn± •-.• , n e l y
71 7T 37t 5n 2% 1% . i W T ' T T 271 57t n7i + — , n7t + — n e l 3 6
71 7C Q 7. x e n7t+— , n7t + — 6 3
Q 8. a - 2 7t ; a - 7t, A, a + TC , where tan a = ~ Q 10. x = TC/6 only
Q l l . x = 7t/16
Q 12. x= 2n7t +— or 2n7t 12
90° 450° 810° 7 7 7 _ jc it 2 it 5% Q 1 7 -6'7'T'T Q 14. — , 3 0 ° ,
12 15QO
;nel
Q 13. A= 15°,B = 30°
1170°
Q 15. 30°,45°,90°,135°,150° Q 18. 0 = 2n7t or 2 n 7 t + - ; n e l
z' _ f 71 371 \ Q 19. x e 2n7t+—,2n7t + — u 2tm —, 2n7i 4 V 4 4
u
Q21. -V2 £y
71 Q 22. 2n7t+- or (2n+l)7t - tan-*2, ns I
Q 23. A=2n7t + B
Q 24. n7t or nit —
571 2n7r + 7t,2n7T + ;ne I
„ _ _ „ 2n7l 71 _ 7t Q25. 0 = — ± — or 2n7t± - n e l
EXER CISE-II Q 1. x = 2n7t±7t or2n7t + - n e l
3
Q 3. nn ; tin + (-l) n ^
or rnt + (-l) n (-—)
Q6. x = 2n7t or x = n7t + (-l) n 7t % 0v 7. n7t+—
feBansal Classes
Q 5. 72c
or x = n7t + (-l) n ^
„ „ 7t 5x 719it 13 "T 7 > 7— > 7> 7> Z
V
Trig.-
Q 9. nic±
it
m
Q 10. x = l o g 3 ( f 4 j , k e N ; x = log 3 (j) , n e N ; x = l o g 3 Q + | j Q 11. j [nir + (-l) n sin"1 (l - V2a+I) where n e l and a e TC
N%
,meNu{0}
3 1 2 ' 2
n+1 — Q 12. x = — + (~l) n — -8 or — — 4 + (-l)
Q 13. x = 2n7t -
Q 14. n7c± \ cos"1 (2 - V5)
Q 15. <2n + *> n , kn , where n, k e I 4 3tc Q 17. x = 2mt + — , n e l
^ n7t k T Q 16. x= — + — , n e I 4 8
Q 18.
765
4
2
. All the roots of cos Vx = 0 are not the same as those of sinVx = -1
4m7i 71V 7t\ 2 +Q 19. x - | 4n7t + —^ or x = 3 2 Q 20. x =
6nn + 3.7c - 4
or
Q 21. (i) | k sin a | < 1
where m, n € W.
3n7t + 3(-l) n sin-1 J - 2
where n e l
(ii) S = n 71, n e I (iii) a e ( - m7i , 2 7t - m 7i) m e I
Q 22. x = nTt or x = nn ± — 6
Q 23. x = ± 5^5 & y = nn + tan-1 | Q24. x = | +
or X = ^ + KTC
Kel
Q 25. a = 0 or a < - 1
EXERCISE-III Q.2 C
Q.l
f
Q.3
n
m
n
i \
m
y = (n-m)— + (-l) — -(-l) — ;x = (m+n)-+ ( - l ) - + ( - l ) - wherea = sin"V -V , m, nel 2 4 2 2 4 2
Q.4 Min. value = 3 ~5 for x=(4n+1)^ - ^tan 1 ^-,neI;max.value=3 5 forx=(4n-l)^ - ^tan ^ n e l ] Q.5
% 71 x = mr + (-1 )n — and y = mrc + — where m & n are integers.
Q.7
D
6
feBansal Classes
6
Trig.-
Q.6B
BANSAL CLASSES TARGET IIT JEE 2007
MATHEMATICS XII (ABCD)
COMPLEX NUMBERS
CONTENTS KEY- CONCEPTS EXERCISE-I EXERCISE-II EXERCISE-III ANSWER-KEY
KEY 1.
CONCEPTS
DEFINITION:
Complex numbers are defmited as expressions of the form a + ib where a, b e R & i = /+L. It is denoted by z i.e. z = a + ib. 'a' is called as real part of z (Re z) and 'b' is called as imaginary part of z (Im z). EVERY COMPLEX NUMBER CAN B E REGARDED A S
Purely real if b = 0
Purely imaginary if a = 0
Imaginary ifb*0
Note : (a) The set R of real numbers is a proper subset ofthe Complex Numbers. Hence the Complete Number system is N c W c I c Q c R c C. (b)
Zero is both purely real as well as purely imaginary but not imaginary.
(c)
i= ^/-l is called the imaginary unit. Also i2 = - l ; i3 = - i ; i 4 = l etc.
(d)
Va Vb = ^rab only if atleast one of either a or b is non-negative.
2.
CONJUGATE COMPLEX: If z = a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted by z . i.e. z = a - ib. Note that : (i) z + z = 2Re(z) (ii) z - z = 2ilm(z) (iii) z z = a 2 + b2 which is real (iv) If z lies in the 1st quadrant then z lies in the 4th quadrant and - z lies in the 2nd quadrant. 3.
ALGEBRAIC OPERATIONS : The algebraic operations on complex numbers are similiar to those on real numbers treating i as a polynomial. Inequalities in complex numbers are not defined. There is no validity if we say that complex number is positive or negative, e.g. z > 0, 4 + 2i < 2 + 4 i are meaningless . However in real numbers if a2 + b2 = 0 then a = 0 = b but in complex numbers, Z;2 + z22 = 0 does not imply z, = z2 = 0.
4.
EQUALITY IN COMPLEX NUMBER: Two complex numbers ^a, + ib, &. z7 = a 2 +ib 2 are equal if and only if their real & imaginary pails coincide.
5.
REPRESENTATION OF A COMPLEX NUMBER IN VARIOUS FORMS :
(a)
Cartesian Form (Geometric Representation): Every complex number z = x + i y can be represented by a point on the cartesian plane known as complex plane (Argand diagram ) by the ordered pair(x,y). length OP is called modulus ofthe complex number denoted by | z | & 9 is called the argument or amplitude. eg. | z | = Vx^Ty5" & 9 = tan t' ?— (angle made by OP with positive x-axis)
faBansal Classes
Complex Numbers
[2]
NOTE z
z is always non negative. Unlike real numbers z [ J &
(i)
if z > 0
-z
if
z<0
is not correct
(ii)
Argument of a complex number is a many valued function. If 9 is the argument of a complex number then 2 rnt + 9 ; n e l will also be the argument of that complex number. Any two arguments of a complex number differ by 2nn.
(iii)
The unique value of 9 such that - n < 9 < n is called the principal value of the argument.
(iv)
Unless otherwise stated, amp z implies principal value of the argument.
(v)
By specifying the modulus & argument a complex number is defined completely. For the complex number 0 + 0 i the argument is not defined and this is the only complex number which is given by its modulus.
(vi)
There exists a one-one correspondence between the points of the plane and the members of the set of complex numbers.
(b)
Trignometric / Polar Representation : z = r (cos 9 + i sin 9) where | z | = r ; a r g z = 9 ; Note: cos 9 + i sin 9 is also written as CiS 9. „ix
Also cos x = (c)
.
e +e
-ix
,Jx
. e & sin x =
-e
z = r (cos 9 - i sin 9)
-ix
are known as Euler's identities.
Exponential Representation: z = re,ie .. J z | = r ; arg z = 9 ; z = re
10
6.
IMPORTANT PROPERTIES OF CONJUGATE/MODULI/AMPLITUDE : If z , Zj, z2 e C then ;
(a)
z+z=2Re(z)
; z-z=2ilm(z)
;
(z) = z r
z
(b)
i ~z2
-
z
i ~ Z2
'
Z Z
12
-
z
i • z2
| z | > 0 ; | z | > Re (z) ; | z | > Im (z); | z, z2!
h Z \ 2j
Zj |. I z2
;
z x +z 2 = z,+ z 2 ;
\ = zr- ;
\z
| - z I; zz = |z , Z2*0,
|zn| = |z|n
;
z1 + z 2 | 2 + ! z 1 - z 2 p = 2 [ j z i | 2 + | z 2 | 2 ] z, (c)
(i) (ii) (iii)
(7)
z J < I z, + z J < z, ! + z. amp (Zj. z2) = amp Zj + amp z2 + 2 krc.
[TRIANGLE INEQUALITY] kel
: O amp z, - amp z2 + 2 kn ; k e l VZ; amp(zn) = n amp(z) + 2k7t. where proper value of k must be chosen so that RHS lies in (— u, tc ].
amp
VECTORIAL REPRESENTATION OF A COMPLEX: Every complex number can be considered as if it is the position vector of that point. If the point P represents the complex number z then, OP = z & | OP | = | z |.
Complex Numbers
[3]
NOTE : (i)
If OP = z = re i e then OQ = z1 = re i < 9+ « = z.e i ( Uf OP and OQ are of unequal magnitude then QQ = OP e^
(ii)
If A, B, C & D are four points representing the complex numbers z p z 2 ,z 3 & z4 then AB I I CD if
is purely real;
Z
4~ Z 3 is purely imaginary ] 2 Z1 If Zp z2, z3 are the vertices of an equilateral triangle where zQ is its circumcentre then
AB_LCD if (iii)
(a) z] + z 2 + z 2 - z l z 2 - z 2 z 3 - z 3 z ] = 0
(b) z 2 + z 2 + z 2 = 3 z 2
8.
DEMOIVRE'S THEOREM: Statement: cos n9 + i sin n 9 is the value or one of the values of (cos 9 + i sin 9)n ¥ n e Q. The theorem is very useful in determining the roots of any complex quantity Note : Continued product of the roots of a complex quantity should be determined using theory of equations.
9.
CUBE ROOT OF UNITY:
(i)
The cube roots of unity are
(ii)
If w is one of the imaginary cube roots of unity then 1 + w + w2 0. In general 1 +w r + w 2 r =0 ; where r € I but is not the multiple of 3. In polar form the cube roots of unity are : 271 In 471 471 cos 0 +1 sin 0 ; cos— +1 sin—, cos— + i sin— 3 3 3 3 The three cube roots ofunity when plotted on the argand plane constitute the verties ofan equilateral triangle. The following factorisation should be remembered: (a, b, c g R & co is the cube root of unity) 3 3 a - b = (a - b) (a - cob) (a - co2b) ; x2 + x + 1 = (x - co) (x - co2) ; a3 + b3 = (a + b) (a + cob) (a + co2b) ; a3 + b3 + c3 - 3abc = (a + b + c) (a + cob + co2c) (a + co2b + coc) nth ROOTS OF UNITY: If 1 , ctj, a 2 , a 3 a n _ j are the n, nth root of unity then : (i) They are in G.P. with common ratio el(2lt/n) &
(iii)
(iv) (v)
10.
- l + iV3
-l-iV3
2
2
l p + af + a£ + ....+a£_, = 0 if p is not an integral multiple of n = n if p is an integral multiple of n (iii) (l-a,)(l-a2) (l-an_,) = n & (1 + a,) (1 + a 2 ) (1 + a n _,) = 0 if n is even and 1 if n is odd. a = (iv) 1 . ocj. a 2 . a 3 1 or - 1 according as n is odd or even. n-1 THE SUM OF THE FOLLOWING SERIES SHOULD BE REMEMBERED : sin(nQ/2) n+1 cos 9 + cos 2 9 + cos 39 + + cos n 9 = 9. sin (9/2) cos (ii)
11. (i)
(ii)
sin in(n8/2) n+1 sin 9. sin(9/2) Note : If 9 = (2rc/n) then the sum of the above series vanishes. sin 9 + sin 2 9 + sin 3 9 +
+ sin n9 =
Complex Numbers
[4]
12. (A)
STRAIGHT LINES & CIRCLES IN TERMS OF COMPLEX NUMBERS : nZj + mz7 If Zj & z2 are two complex numbers then the complex number z = ——- divides the joins of z, & z, in the ratio m: n.
Note: (i)I f a , b , c are three real numbers such that azj + bz2 + cz3 = 0 ; where a + b + c = 0 and a,b,c are not all simultaneously zero, then the complex numbers z,, z? & z3 are collinear. (ii)
If the vertices A, B, C of a A represent the complex nos. z p z2, z3 respectively, then :
(E) (F)
(b)
Orthocentre ofthe AABC =
(c)
(a sec A)ZJ + (b sec B)Z 2 + (c sec C)z3 ZJ tan A + z 2 tan B + z 3 tan C asecA + bsecB + csecC tanA + tanB + tanC Incentre of the A ABC = (az, + bz2 + cz3) + (a + b + c).
Z
(H)
Circumcentre ofthe AABC= : (Zj sin 2A + Z2 sin 2B + Z3 sin 2C) + (sin 2A + sin 2B + sin 2C).
amp(z) = 8 is a ray emanating from the origin inclined at an angle 0 to the x - axis. | z — a i — ! z — b | is the perpendicular bisector of the line joining a to b. The equation of a line joining z} & z2 is given by; z = z, +1 (Zj - z2) where t is a perameter. z = z, (1 + it) where t is a real parameter is a line through the point z, & perpendicular to oz,. The equation of a line passing through z, & z2 can be expressed in the determinant form as z z,
(G)
Z^
Centroid ofthe AABC =
(d) (B) (C) (D)
Zj I
(a)
2
z 1 z, 1 Z
2
:
0. This is also the condition for three complex numbers to be collinear.
^
Complex equation of a straight line through two given points Zj & z2 can be written as z (zj - z2 ) - z (zj - z2 )+ (z,z2 - ZjZ2 )= 0, which on manipulating takes the form a s a z + a z + r = 0 where r is real and a is a non zero complex constant. The equation of circle having centre zQ & radius p is : I z — z0 | = p or z z - z 0 z - z Q z + z 0 z 0 - p 2 = 0 which is of the form zz+az+az+r = 0 , r is real centre - a & radius
(I)
Circle will be real if a a - r > 0. The equation of the circle described on the line segment joining Zj & z2 as diameter is : z-z, (i)arg—=
(J)
aa-r .
%
or ( z - z j ) ( z - z 2 ) + ( z - z 2 ) ( z - z , ) = 0
Condition for four given points z;, z 2 , z3 & z4 to be concyclic is, the number Z 3 Z1 Z4 Z2 is real. Hence the equation of a circle through 3 non collinear points z,, z & z can be 2 3 Z 3 Z2 Z4 Z1 (z-z2)(z3-z,) . (z-z 2 )(z 3 -z 1 ) (z-z2xz3-zi) taken as i yj \ is real => i v? \ - jz _ \/_ \ (z-z1J(z3-z2) ( z - z 1 j ( z 3 - z 2 j [z - Zj )(z3 — z 2 j
Classes
Complex Numbers
[5]
13.(a) Reflection points for a straight line: Two given points P & Q are the reflection points for a given straight line if the given line is the right bisector of the segment PQ. Note that the two points denoted by the complex numbers z, & z, will be the reflection points for the straight line a z + a z + r = 0if and only if; a z j + a z 2 + r = 0, where r is real and a is non zero complex constant, (b) Inverse points w.r.t. a circle : Two points P & Q are said to be inverse w.r.t. a circle with centre 'O' and radius p, if : (i) the point O, P, Q are collinear and on the same side of O. (ii) OP . OQ = p2. Note that the two points Zj & z2 will be the inverse points w.r.t. the circle zz+az+az+r=0 ifandonlyif z,z 2 +az 1 +az 2 +r=0. 14.
PTOLEMY'S THEOREM: It states that the product of the lengths of the diagonals of a convex quadrilateral inscribed in a circle is equal to the sum of the lengths of the two pairs of its opposite sides. Z + z, - z i.e. I Zj z2 2 4 I 2 Z3 Z
15. (i)
Z
LOGARITHM OF A COMPLEX QUANTITY: qA 1 ' -1 Log (a + i p) 2 Loge (a 2 + p2) + i 2nrc + tan — where n e l . a) v 2nn+~
(ii)
i1 represents a set of positive real numbers given by e
Q.l
VERY E L E M E N T A R Y EXERCISE ON COMPLEX NUMBER Simplify and express the result in the form of a + bi (a)
Q.2 Ml Q.3 Q.4
'1 + 202
2+i
(b) - i (9 + 6 i) (2 ~ i)-1 (c)
f 4i A-3- l^ 2i + l
nel.
2 ... 3 + 2i 3 — 2i , .(2 + i) (d) — — + (e) 2 — 5i 2 + 5i 2-i
(2-i) 2 2+i
Given that x , y e R, solve : (a) (x + 2y) + i (2x - 3y) = 5 - 4i (b) (x + iy) + (7 - 5i) = 9 + 4i (c) x 2 - y 2 - i ( 2 x + y) = 2i (d) (2 + 3 i ) x 2 - ( 3 - 2 i ) y = 2 x - 3y + 5i + IV (e) 4x2 + 3xy + (2xy - 3x2)i = 4y2 - (x2/2) + (3xy - 2y2)i .PfTC ^slcSS' I J J S Find the square root of : (a) 9 + 40 i (c) 50 i (b) —11 — 60 i i Ir+Mi) If f (x) = x4 + 9x3 + 35x2 - x + 4, find f ( - 5 + 4i) (a) % 4 3 2 If g(x) = x - x + x + 3 x - 5 , find g(2 + 3i) (b) iH-J
Q.5
Among the complex numbers z satisfying the condition z + 3 - S i = S, find the number having the least positive argument.
Q.6
Solve the following equations over C and express the result in the form a + ib, a, b e R. (a) ix2 - 3x - 2i = 0 (b) 2 (1 + i) x2 - 4 (2 - i)x - 5 - 3 i = 0
Q.7
Locate the points representing the complex number z on the Argand plane: (a) | z + l - 2 i | = V7 ; (b) |z - l|2 + |z + l|2 = 4 ; (c)
z + 3
3 ; (d) I z - 3 ! - I z - 6 I
Q.8
If a & b are real numbers between 0 & 1 such that the points Zj = a + i, z 2 = 1 + bi & z3 - 0 form an equilateral triangle, then find the values of'a' and 'b'.
Q.9
For what real values of x & y are the numbers - 3 + ix2 y & x2 + y + 4i conjugate complex?
Q.10
Find the modulus, argument and the principal argument of the complex numbers. 2+ i (i) 6 (cos 310°-i sin 310°) (ii) - 2 (cos 30° + i sin 30°) (iii) 4i + (1 + i)2
Complex Numbers
[6]
Q.ll
If (x + iy)1/3 = a + bi; prove that 4 (a2 - b2) = - + f a b
. _ a + ib 2 a2 + b2 Q. 12(a) If — — = p + qi, prove that rpz + q1 - 2 2 c + id c +d (b) Let Zj, z2, z3 be the complex numbers such that Zj + Z2 + Z3 = Z]Z2 + Z2Z3 + Z3Zj = 0. Prove that | z} | = | z2 | = | z31. 1 + z + z2 Q.13 Let z be a complex number such that z e c\R and j e R, then prove that | z I = 1. 1-z +z Q.14 Prove the identity, 11 - z,z 2 |2 -1 z, - z 2 |2 = (l-1 z, | 2 ) (l-1 z 2 | 2 ) A
Q.15
For any two complex numbers, prove that |Zi + ^ +
- z2| = 2 |Zj| + |z2| . Also give the
•
Q.16
geometrical interpretation of this identity. (a) Find all non-zero complex numbers Z satisfying z =iZ 2 . (b) If the complex numbers z,, z2, zn lie on the unit circle \z\ = 1 then show that |Zj + z2 + +Zn| = | z f z 2 - ' + +zn-1| .
Q.17 Find the Cartesian equation of the locus of 'z' in the complex plane satisfying, | z - 4 | + | z + 4 | = 16. Q.18
If oo is an imaginary cube root of unity then prove that : (a) (1 + co - co2)3 - (1- © + co2)3 = 0 (b) (1 - co + co2)5 + (1+ oo - ©2)5 = 32 (c) If co is the cube root of unity, Find the value of, (1 + 5co2 + oo4) (1 + 5co4 + co2) (5co3 + oo + oo2).
Q.19 If co is a cube root of unity, prove that; (i) (1 + © - co2)3 - (1 - © + ©2)3 ' = «2 (iii) (1 - ©) (1 - ©2) (1 - ©4) (1 - ©8) = 9 c + aco + bco" If x = a + b ; y = aco + b©2 ; z = a©2 + b©, show that (i) xyz = a3 + b3 (ii) x2 + y2 + z2 = 6ab (iii) x3 + y3 + z3 = 3 (a3 + b3) (ii)
Q.20
Q.21
a + bm + cco
1 If(w* 1) is a cube root of unity then 1 - i -i (A)0 (B)l
1 + i + w2 -1 - i + w -1 (C)i
w2 w2-l -1
(D)w Q.22(a) (1 + w) = A+ Bw where w is the imaginary cube root of a unity and A, B e R , find the ordered pair (A,B). (b) The value of the expression ; 1. (2 - w) (2 - w2) + 2. (3 - w) (3 - w2) + + (n - 1). (n - w) (n - w2), where w is an imaginary cube root of unity is . 7
Q.23
If n e N, prove that (1 + i)n + (1 - i)n =
Q.24
Show that the sum ^ k=i
+1
. cos ™ .
(sin ^ ^ - i c o s ^ ^ i simplifies to a pure imaginary number. 2n + l 2n + l7
V
Q.25
If x = cos 9 + i sin 9 & 1 + -^/l - a2 = na, prove that 1 + a cos 9 =
Q.26
The number t is real and not an integral multiple of 7t/2. The complex number Xj and x2 are the roots of the equation, tan2(t) • x2 + tan (t) • x + 1 = 0
(1 + nx) (1 + — \.
/ 2mi ^ Showthat (x,)n + (x2)n = 2 c o s - ^ ] cotn(t).
fa Bans al Classes
Complex Numbers
m
EXERCISE-I Q. 1
Simplify and express the result in the form of a + bi : f ,.3 .\ 2 (b) 4 i - l 2i + l
1
(a) - i (9 + 6 i) (2 -
,(c), 3 + 2i+ 3 —2i r 2 - 5 i 2 + 5i
(2 + i)2 (2-i) 2 . (e) Vi+V-i 2-i 2+i Find the modulus, argument and the principal argument of the complex numbers. (d)
Q.2
rioTt^ poTt^ (i) z = 1 +cos + i sin 19 J 19 J (iii)z=V5TT2i+y5^l2i
i-1 271 . 271 1 — cos + sin5
(iv)
V5 + 12i—>/5-12i Q.3
(ii) (tanl -i) 2
Given that x, y e R, solve : (a) (x + 2y) + i (2x - 3y) = 5 - 4i
(b) v y
+
l + 2i 3 + 2i 8 i - l (c) x - y - i (2x + y) = 2i (d) (2 + 3i) x2 - (3 - 2i) y = 2x - 3y + 5i (e) 4x2 + 3xy + (2xy - 3x2)i = 4y2 - (x2/2) + (3xy - 2y2)i 2
2
Q.4(a) Let Z is complex satisfying the equation, z2 - (3 + i)z + m + 2i = 0, where m e R. Suppose the equation has a real root, then find the value of m. (b) a, b, c are real numbers in the polynomial, P(Z) = 2Z4 + aZ3 + bZ2 + cZ + 3 If two roots of the equation P(Z) = 0 are 2 and i, then find the value of'a'. Q.5(a) Find the real values of x & y for which z, = 9y2 - 4 - 10 i x and z2 = 8y2 - 20 i are conjugate complex of each other. (b) Find the value of x4 - x3 + x2 + 3x - 5 if x = 2 + 3i Q.6
Solve the following for z : (a) z2 - (3 - 2 i)z = (5i - 5)
(b) | z | + z = 2 + i
Q.7(a) If i Z3 + Z2 - Z + i = 0, then show that | Z | = 1. (b) Let z, and z2 be two complex numbers such that
Z
1 - 2 Z 2 = 1 and J z J * 1, find i z, I. 2-ZjZ 2
z - z 1 • Tt (c) Let z, = 10 + 6i & z 2 =4 + 6i. If z is any complex number such that the argument of, is —, then z-z,. prove that | z - 7 - 9i | = 3V2 , Q. 8
Show that the product, ri+n 1
fi+i> i+ 1+ 12 J _ 12 J -
faBansal Classes
1
fi+n 1+ 12 J
22 "
fi+n 1+ 12 J
2" ~ is equal to
Complex Numbers
V
2 J
(1+i) where n > 2 .
[8]
Q. 9
Let a & b be complex numbers (which may be real) and let, Z = z3 + (a + b + 3i) z2 + (ab + 3 ia + 2 ib - 2) z + 2 abi - 2a . (i) Show that Z is divisible by, z + b + i (ii) Find all complex numbers z for which Z = 0 (iii) Find all purely imaginary numbers a & b when z = 1 + i and Z is a real number.
Q.10
Interpret the following locii in z e C. (a)
1 < | z - 2i | < 3
(b)
(c)
Re iz + 2 <4 (z * 2i) Arg (z - a) = n/3 where a = 3 + 4i.
Arg (z + i) - Arg (z - i) = n/2 (d) t o act fU dw > Q. 11 Prove that the complex numbers Zj and z2 and the origin form an isosceles triangle with vertical angle 2TI/3 if z\ + z\ + z, z 2 = 0.
•
1 $1\
0
Q. 12 Pisa point on the Aragand diagram. On the circle with OP as diameter two points Q & R are taken such that Z POQ = Z QOR = 0. If 'O' is the origin & P, Q & R are represented by the complex numbers Z 1 , Z2 & Z3 respectively, show that: Z 2 2 . cos 2 9 = Z,. Z3 cos29. Q. 13 Let z,, z2, z3 are three pair wise distinct complex numbers and t p t2, t3 are non-negative real numbers such that t, +12 +13 = 1. Prove that the complex number z = t,z, + t2z2 + t3z3 lies inside a triangle with vertices z,, z2, z3 or on its boundry. Q.14 I f a C i S a , bCiSp,cCiSy represent three distinct collinear points in an Argand's plane, then prove the following: (i) Z ab sin (a - (3) = 0. (ii)
(a CiS a)
v
/
b 2 + c 2 - 2 b c c o s ( P - y ) ± 0> CiS p) ^ / a 2 + c 2 - 2 a c c o s ( a - y )
+ (c CiS y) A / a 2 + b 2 - 2 a b c o s ( a - P ) = 0. Q. 15 Find all real values of the parameter a for which the equation (a - 1 )z4 - 4z2 + a + 2 = 0 has only pure imaginary roots. Q. 16 Let A=z,; B = z2; C s z3 are three complex numbers denoting the vertices of an acute angled triangle. If the origin 'O' is the orthocentre of the triangle, then prove that z , z 2 + Z, Z 2 = Z 2 Z 3 + Z 2 Z 3 = Z 3 Z, + Z 3 ZJ
hence showthat the AABC is a right angled triangle <=>ZjZ2 + z, z2 = z 2 z 3 + z,z 3 = z 3 z, + z3z, = 0 Q.l7
If the complex number P(w) lies on the standard unit circle in an Argand's plane and z = (aw+ b)(w - c)"1 then, find the locus of z and interpret it. Given a, b, c are real.
Q. 18(a) Without expanding the determinant at any stage, find KG R such that 4i 8 + i 4 + 3i -8 +i 16i i has purely imaginary value, - 4 + Ki i 8i (b) If A, B and C are the angles of a triangle e
D=
-2iA
elC eiB
£
iC
e _]B eiA
£
iB
elA e _2iC
where i = JZ\
thenfindthe value of D.
Complex Numbers
[9]
Q.19
If w is an imaginary cube root of unity then prove that : (a) (1 - w + w2) (1 - w2 + w4) (1 - w4 + w8) to 2n factors = 2 2n . (b) If w is a complex cube root of unity, find the value of (1 + w) (1 + w2) (1 + w4) (1 + w8) to n factors .
Q.20
Prove that '
. Tt
1 + sin 9 + i cos 0 v + sin9-icos9 J .
Tt^
'
:
cos
71
.
nTt
-n9 + i sin
- n 9 . Hence deduce that
7T^ 5
+ i 1 + sin — icos— 0 + sin —+ icos — 1 v 5 5, 5 5 y v If cos (a - P) + cos (P - y) + cos (y - a) = - 3/2 then prove that : (a) £ cos 2a = 0 = E sin 2a (b) E sin (a + P) = 0 = E cos (a + p) 2 2 (c) £ sin a = £ cos a = 3/2 (d) £ sin 3a = 3 sin (a + P + y) (e) £ cos 3a = 3 cos (a + P + y) (f) cos3 (9 + a) + cos3 (9 + p) + cos3 (9+y) = 3 cos (9 + a ) . cos (9 + p). cos (9+y) where 9 e R. '
Q.21
n7t
/
Q.22
Resolve Z5 + 1 into linear & quadratic factors with real coefficients. Deduce that: 4-sin-p- -cos y = 1.
Q.23
I f x = l + i V 3 ; y= 1 -iV3 & z = 2, then prove that xp + yp = zp for every prime p > 3.
Q.24
Prove that for all complex numbers z with | z | = 1 <|l-z| + |l+z2|<4
Q.25(a) Let z = x + iy be a complex number, where x mdy are real numbers. Let Aand B be the sets defined by A = { z | | z | <2} and B = {z | (1 -z')z + (l +/) z >4}. Find the area ofthe region A n B. (b) For all real numbers x, let the mapping f (x) =
;, where i = ^/ZJ. If there exist real number X I a, b,c and d for which/(a),/(b),/(c) and/(d) form a square on the complex plane. Find the area of the square.
EXERCISE-II P
Q.l
q
r
If q r p = 0 ; where p, q, r are the moduli of non-zero complex numbers u, v, w respectively, r
p
q
/ w - u A2 w prove that, arg — = arg V
v —U /
3n
Q.2
Prove that X* -3 )' ' 3"C2r-i = 0, where k= — &n is an even positive integer.
Q. 3
Show that the locus formed by z in the equation z3 + iz = 1 never crosses the co-ordinate axes in the
r= 1
2
-Im(z) Argand's plane. Further show that |z|= ^ 2 R e ( z ) I m ( z ) Q.4 Q.5
+ 1
4,
WW*" 1 If co is the fifth root of 2 and x = co + co2, prove that x5 = 1 Ox2 + 1 Ox +•«6 A N ^ ^ T / Prove that, with regard to the quadratic equation z2 + (p + ip') z + q + iq' = 0 where p , p', q, q' are all real. (i) if the equation has one real root then q ' 2 - pp' q' + qp ' 2 = 0 . (ii) if the equation has two equal roots then p 2 - p ' 2 = 4 q & p p ' = 2q'. State whether these equal roots are real or complex.
Classes
Complex Numbers
[10]
Q.6
if the e q u a t i o n (z + i
)7 + z7 = 0 has roots z,, z2,.... z7, find the value of
7
(a)
X
7
Re
r=l
Z
( r)
311(1
ZIm(Zr)
^
r=l
n
Q.7
Find the roots of the equation Z = (Z + 1 )n and show that the points which represent them are collinear on the complex plane. Hence show that these roots are also the roots of the equation \2 / \2 . n u t 2 Z + 1 = 0. 2 sin Z + 2 sin n n v
Q.8
Dividing f(z) by z - i , we get the remainder i and dividing it by z + i, we get the remainder 1 + i. Find the remainder upon the division of f(z) by z2 + 1.
Q. 9
Let z j & z2 be any two arbitrary complex numbers then prove that: ZL +
Z2| > I ( | z , | + | z 2 | )
z, I |z 2
Q. 10 If Zr, r = 1,2,3,
2m, m e N are the roots of the equation 2m 1 Z2m + Z2m-1 + Z2m-2 + + Z + 1 = Q then prove that I ^ _ j = -m
Q.ll
If (1 +x) n = c 0 + C , X + C 2 X 2
.... + c n x n (n e N), prove that:
n7t 2"-' + 2"n cos — (b)C 1 + C5 + C9 + ....= i 1
(a) C0 + C4 + C R + ....= (c) C2 + C6 + C ] 0 +
+
=
2
-2 n / 2 0
(e) C0 + C3 + C6 + C9 +
n
n
COS —
2" + 2 cos
(d)C 3 + C7 + C n + ....= I
+2
sm — 4
2 - 1 _ 2 n/2 sin —
n7t
Q. 12 Let z,, z 2 , z 3 , z4 be the vertices A, B , C , D respectively of a square on the Argand diagram taken in anticlockwise direction then prove that : (i) 2Z2 = (1 + i) z, + (1- i)z3 & (ii) 2z4 = (1- i) Zj + (1 + i) z3 Q. 13 Show that all the roots of the equation Q.14
1 + ix |
1 + ia
1 - ixy
1 - ia
a e R are real and distinct.
Prove that: (a) cos x + nC, cos 2x + nC2 cos 3x + (b) sin x + n Cj sin 2x + nC2 sin 3x + (c) cos
271 2 n + 1,
+ COS
471 2n + 1
+ nCn cos (n + 1) x = 2" . cos" | . cos + nCn sin (n + 1) x = 2n . cos"
+ cos
671 2n + 1
+
Q. 15 Show that all roots of the equation a0zn + aj zn " 1 +
+ cos
2n7i. 2n + 1
. sin
n + 2
n + 2
x
= - - When n e N. 2
+ an _, z + an = n,
n-1 where | a-1 < 1, i = 0,1,2,...., n lie outside the circle with centre at the origin and radius
Complex Numbers
[11]
Q.16
The points A, B, C depict the complex numbers z,, z 2 , z3 respectively on a complex plane & the angle B & C of the triangle ABC are each equal to ^-(rc - a) . Show that (z2 - z3)2 = 4 (z3 - z,) (z, - z2) sin2 | .
Q.17
A2 A2 Show that the equation — 1 — + —-— + x
aj
x
a,, a 2 , a3.... an & A,, A2, A3
a2
A2 + —-— = k has no imaginary root, given that: ^
An, k are all real numbers.
Q.18
If z ] 2 + z22 + z 3 2 -z 1 z 2 -z 2 z 3 -z 3 z 1 — 0, prove that all the three z,,z 2 &z 3 simultaneously need not be equal, where z,, z2 & z3 are complex numbers.
Q.19
Let a, P be fixed complex numbers and z is a variable complex number such that, | z - a | 2 + | z - p | 2 =k. Find out the limits for 'k' such that the locus of z is a circle. Find also the centre and radius of the circle.
Q.20
C is the complex number, f: C —» R is defined by f (z) = | z3 - z + 2|. What is the maximum value of f on the unit circle | z | = 1 ?
Q.21
Let f (x) = l°g cos3x (cos 2 i x) if
x
* 0 and f (0) = K (where i = V^T ) is continuous at x = 0 then find
the value of K. Use of L Hospital's rule or series expansion not allowed. Q.22
If z,, z2 are the roots of the equation az2 + bz + c = 0, with a, b, c > 0 ; 2b2 > 4ac > b2 ; z, e third quadrant; z2 e second quadrant in the argand's plane then, show that f z, \ arg VZ2
Q.23
f u2\ 1/2 :
J
2cos
4ac
Find the set of points on the argand plane for which the real part of the complex number (1 + i) z2 is positive where z = x + iy,x, y e R and i - - / - l .
Q.24
If a and b are positive integer such that N = (a + zb)3 -107 i is a positive integer. Find N.
Q.25
If the biquadratic x4 + ax3 + bx2 + cx + d = 0 (a, b, c, d e R) has 4 non real roots, two with sum 3 + 4i and the other two with product 13 + i. Find the value of'b'.
EXERCISE-III Q. 1
flQ / ' f ] (3 p + 2) f f s i n — - • cos^H)
Evaluate:
p=l
\q=l
.
[REE '97,6]
11
Q.2(a) Let Zj and z2 be roots of the equation z2 + pz + q = 0 , where the co-efficients p and q may be complex numbers. Let A and B represent z, and z2 in the complex plane. If ZAOB = a ^ 0 and OA = OB, where O is the origin. Prove that p2 = 4 q cos2 (b) Prove that
n-1 k=l
( n - k ) cos
Classes
2k7t
n
n
2
\.
where n > 3 is an integer.
Complex Numbers
[JEE '97 , 5] [JEE'97,5]
[12]
Q.3 (a) If © is an imaginary cube root of unity, then (1 +00 — co2)7 equals (A) 128© (B) - 128co (C) 128co2 13 ,
.
(b) The value of the sum ]Mi n +i n + 1 ) , where i =
(D) - 128co2
, equals
n=l
(A) i Q.4
(B) i - 1
(C) - i
(D) 0
[JEE' 98, 2 + 2 ]
Find all the roots of the equation (3z - 1 )4 + (z - 2)4 = 0 in the simplified form of a + ib. [REE'98,6]
Q.5(a) If i = p ,
then 4 + 5
(A)l-iV3
. r\334
1
—+ 2
i V3 | 2
,
f rf 1 i V3 + 3 — + -—\
2
2 .
(B) - 1 + iV3
is equal to :
(C) iV3
(D)-iV3
(b) For complex numbers z & co, prove that, |z|2 co - |co|2 z = z - co if and only if, z — co or z co = 1 [JEE'99,2+10(outof200)] Q.6
2ni 20 7 and f(x)=A 0 + X Ak xk, then find the value of, k=1 f(x) + f(a x) + + f(a 6 x) independent of a .
If a = e
[REE '99,6]
Q.7(a) If Zj, z 2 , z3 are complex numbers such that | z, | = | z2 | = | z 3 | = I Zj + z2 + z31 is : (A) equal to 1
(B) less than 1 (C) greater than 3
1 —
1 +
—
1 +
—
1, then
(D) equal to 3
(b) If arg (z) < 0, then arg ( - z) - arg (z) = (A) 71
(B)-TT
( O - f
(D)
~
[ JEE 2000 (Screening) 1 + 1 out of 35 ] Q.8
Given, z = cos
2
71
2n +
a = z + z3 +
_
^ + i sin
+ z2n _ 1
2 2n +
&
71
j , 'n' a positive integer, find the equation whose roots are,
|3 = z2 + z4 +
+ z2n. [ REE 2000 (Mains) 3 out of 100 ]
z, - z 3 l-iV3 Q.9(a) The complex numbers z,, z2 and z3 satisfymg — - — = — - — are the vertices of atriangle which is Z2 ^ (A) of area zero (B) right-angled isosceles (C) equilateral (D) obtuse - angled isosceles (b) Let z j and z2 be nth roots of unity which subtend a right angle at the origin. Then n must be of the form (A) 4k + 1 (B) 4k + 2 (C)4k + 3 (D)4k [JEE 2001 (Scr) 1 + 1 out of 35] 12 6 Q. 10 Find all those roots of the equation z - 56z -512 = 0 whose imaginary part is positive. [ REE 2000, 3 out of 100 ]
Complex Numbers
[13]
1 1 1 1 . V3 2 2 Q.ll (a) Let co = — + 1 — . Then the value of the determinant 1 — 1 — co CO is 2 2 ,1 co2 co4 (A) 3co
(C) 3co2
(B) 3co (co - 1)
(D) 3co(l -©)
(b) For all complex numbers zp z2 satisfying \zx\ = 12 and |z2 — 3 — 4i| = 5, the minimum value of |Zj - z 2 | is (A) 0 (B)2 (C) 7 (D) 17 [JEE 2002 (Scr) 3+3] (c) Let a complex number a , a * 1, be a root of the equation '* ' where p,q are distinct primes. z p+q_ z p_ z q+1 =0 Show that either l + a + a 2 + + aP-1 = 0 or l + a + a 2 + + a ^ 1 = 0 , but not both together. / otse [JEE 2002, (5) ] 1 - z , z2 Q. 12(a) If z, and z2 are two complex numbers such that j z , j < 1 < ] z2 j then prove that Zj z 2
<1.
z r =1 where | ar | < 2.
(b) Prove that there exists no complex number z such that | z j < — and r=l
[JEE-03,2 + 2 out of 60] Q. 13(a) co is an imaginary cube root of unity. If (1 + co ) = (1 + co ) , then least positive integral value of m is (A) 6 (B)5 (C)4 (D)3 [JEE 2004 (Scr)] 2m
4 m
(b) Find centre and radius ofthe circle determined by all complex numbers z=x+i y satisfying (z- -a) = k, (z--P) where a = 04 + ia 2 , p = P, +i{32 a r e fixed complex and k 1.
[JEE 2004,2 out of 60]
Q. 14(a) The locus of z which lies in shaded region is best represented by (A) z : |z + 11 > 2, |arg(z + 1)| < 7t/4 (B) z : jz -1| > 2, |arg(z-l)|<7t/4 (C) z : |z + lj < 2, |arg(z + 1)| < rc/2 (D) z : |z - 11 < 2, |arg(z - 1)| < TC/2
p(V2-l,V2)
(-i,o)\
1TM
Q(V2-l,-V2f
(b) If a, b, c are integers not all equal and w is a cube root of unity (w * 1), then the minimum value of ia + bw + cw2| is V3
(C)T
(B)l
(A)0
(D)
1
[JEE 2005 (Scr), 3 + 3] (c) If one of the vertices of the square circumscribing the circle |z - 1 j = i s 2 + V3 i • Find the other vertices of square. [JEE 2005 (Mains), 4] Q.15
If w = a + ip where P * 0 and z * 1, satisfies the condition that values ofz is (A) {z : | z | = 1} If-
Classes
(B){z:z=z)
w-wz is purely real, then the set of 1-z
(C) {z: z ^ 1} J ^
a
Complex Numbers
(D) {z : | z | = 1, z * 1} [JEE 2006,3]
[14] (1-
(ck\\)l t b^
A N S W E R KEY VERY ELEMENTARY EXERCISE ON COMPLEX NUMBER Q.l
7 (a) —
Q.2
( a ) x = l , y = 2 (b) (2 , 9) (c) ( - 2 , 2 ) or
Q.3
(a) ± (5
Q.5
2
Q.7
H
24 i
21 12 ; (b) y - y i
(b) ± (5
+ 4i);
8
(c) ± 5(1
- 6i)
2
-, y j
5K (e) x = K , y = — K e R
(d) (1 ,1)1
Q.4
+ i)
Q.6
1
2
?? + 0i ; (e) y i
; (c) 3 + 4 i ; (d)
( a ) - 1 6 0 ; (b) - ( 7 7 + 1 0 8 i)
(a) - i , -
3 — 5i
(b) —
2i
2
or
+ I
2
(a) on a circle of radius V 7 with centre ( - 1 , 2 ) ; (b) on a unit circle with centre at origin (c) on a circle with centre ( - 1 5 / 4 , 0 ) & radius 9/4 ; (d) a straight line
Q.8
a = b = 2 -V3
;
Q.9
Q.10
(i) M o d u l u s = 6 , A r g = 2 k n + — (K e I ) , Principal A r g = — ( K e I)
x = 1, y
18
Q.16
(a)
V3
i
2
2
Q.22" (a) (1,1); (b)
= -4
-— 6
, A r g = 2 k n - t a n ' 1 2 (K e I ) , Principal A r g = - tan~'2 z.
V3
or x = - 1, y
18
(ii) M o d u l u s = 2 , A r g = 2 k 7 i + -— , Principal A r g = 6 (iii) M o d u l u s = — 6
=-4
— 2 , i ; vQ.17
£
v 6— 4 +4 8^ - = l ; Q . 1 8
v
(c) ' 64 ;
Q.21 A
n(n + l)
EXERCISE-I Q . l (a) ^ - - - ^ i
j
—
(b)
3 + 4 i (c) - A + 0 i
(d) ^ i
29
4tc Q.2 (i) Principal A r g z = - —— ;
(iii) Principal value of A g r z = - — & \ z \ = y 71
1
a
^
Principal A r g = (2 -
x = 1, y = 2;
(b) x
Q.4
(a) 2, ( b ) - 1 1 / 2
Q.6
(a)
Q.7
(b)2
Q.9
(ii)
=
1
& y= 2
; (c)
Q.5
(a)
z = (2 + i) or ( l - 3 i ) ;
z = - (b + i ) ; - 2 i , - a
(b)
11 TT,
(iii)
n)
( - 2 , 2 ) or
. . . .
Principal A r g =
1 lrc 20
2 2 3 . -- jJ ; (d) (1,1) I 0, (b) - (77 + 1 0 8
[(-2, 2); ( - 2 , - 2 ) ]
z =
0±V2i
; Principal v a l u e of A r g z = — & | z | =
(iv) M o d u l u s = - 7 = c o s e c — , A r g z = 2ri7i + 6 V2 5 20
Q.3(a)
or
4 7t ; A r g z = 2 k TT - —— k e I
471 | z | = 2 cos——
(ii) M o d u l u s = s e c 2 l , A r g = 2 n 7 t + ( 2 - TT),
/•xx,,,
(e) ± V 2 + 0 i
r?
; (e) x
=K,
3K
y = — Ke R
i)
3+ 4i
2ti 3t + 5
, ti
where
Complex Numbers
t e R
[15]
Q.IO (a) The region between the co encentric circles with centre at (0,2) & radii 1 & 3 units 1 1 (b) region outside or on the circle with centre — + 2i and radius —. (c) semi circle (in the 1 st & 4th quadrant) x2 + y2 = 1 (d) a ray emanating from the point Q.15
(3 +4i) directed away from the origin & having equation Vs x - y + 4 - 3/3 = 0 [-3,-2] Q.17 (1 - c2) | z |2 - 2(a + be) (Re z) + a2 - b2 = 0
Q.18 Q.22 Q.25
(a) K = 3 , (b) - 4 Q.19 (b) one if n is even; - w2 if n is odd (Z+ l ) ( Z 2 - 2 Z c o s 3 6 ° + l ) ( Z 2 - 2 Z c o s 108°+ 1) (a) 7t - 2 ; (b) 1/2
EXERCISE-II Q.8
(a) - — , (b) zero
Q.20
| f (z) | is maximum when z = co, where co is the cube root unity and | f (z) | = ^13
2
—+- +i
2
2
Q.19
k > ^ !«• ~ P|2
Q.6
2
4 Q.21 K = - ~ ' • 9 Q.23 required set is constituted by the angles without their boundaries, whose sides are the straight lines y = (V2-l) xandy+(V2 + 1) x = 0 containing the x - axis Q.24 198 Q.25 51
EXERCISE-III Q.l 48(1 - i) ^ Q.4
^ L
Q.6
7A
0
Q.3 (a) D
(b) B
(29+20V2) + i(±l5 + 25\/2) 82 + 7A?X
7
+ 7AI4X
14
Q.9 (a) C, (b) D
Q.7 (a)
A
,
(29-20V2) + i(±15-25V2) 82 (b)
Q.IO ± l + i V 3 ,
A
Q.8 z2 + z+
sin2 n 0
+
V2
^ *, , „
Q.5 (a)
=
0,
where
Q.H
(a)B
C
2 7t 0
2n + 1
; (b) B
2
, Radius = — j — V l a - k 2 p | 2 -(k 2 .| (312 -1 a | 2 ) ( k 2 - l ) 1 (k — 1)
Q.13
(a) D ; (b) Centre =
Q.14
(a) A, (b) B, (c) z2 = - V 3 i ; z 3 = ( l - V s ) + i ; z 4 = ( l + V3)-i
Classes
k
Complex Numbers
Q.15
D
[16]
BANSAL CLASSES TARGETIIT JEE 2007
XII (ABCD)
CONIC SECTION (PARABOLA, ELLIPSE & HYPERBOLA)
CONTENTS PARABOLA KEY CONCEPT EXERCISE-I EXERCISE-II EXERCISE-III
Page -2 Page -5 Page -7 Page -8
ELLIPSE KEY CONCEPT EXERCISE-I EXERCISE-II . EXERCISE-III
Page -10 Page -13 Page -14 Page -16
HYPERBOLA KEY CONCEPT EXERCISE-I EXERCISE-II EXERCISE-III
Page -18 Page -22 Page -24 Page -25
ANSWER KEY
Page-27
PARABOLA KEY
CONCEPTS
1.
CONIC SECTIONS:
^
A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line. The fixed point is called the Focus. Thefixedstraight line is called the DIRECTRIX. The constant ratio is called the ECCENTRICITY denoted by e. The line passing through the focus & perpendicular to the directrix is called the Axis. A point of intersection of a conic with its axis is called a VERTEX.
2.
GENERAL EQUATION OF A CONIC: FOCAL DIRECTRIX
PROPERTY:
The general equation of a conic with focus (p, q) & directrix Ix + my + n = 0 is : (I2 + m2) [(x - p)2 + (y - q)2] = e2 (Ix + my + n)2 = ax2 + 2hxy +"by2 + 2gx + 2fy + c = 0 3.
DISTINGUISHING BETWEEN THE CONIC :
The nature of the conic section depends upon the position of the focus S w.r.t. the directrix & also upon the value of the eccentricity e. Two different cases arise. CASE ( I ) : W H E N T H E Focus L I E S O N T H E DIRECTRIX. In this case D = abc + 2fgh - af2 - bg2 - ch2 - 0 & the general equation of a conic represents a pair of straight lines if: e > 1 the lines will be real & distinct intersecting at S. e = 1 the lines will coincident, e < 1 the lines will be imaginary. CASE ( I I ) : W H E N T H E
a parabola e=l;D*0, h2 = ab 4.
Focus D O E S N O T L I E an ellipse 0
O N DIRECTRIX.
a hyperbola rectangular hyperbola e>l;D*0; e>l;D*0 2 h > ab h2 > ab ; a + b = 0
PARABOLA : DEFINITION:
A parabola is the locus of a point which moves in a plane, such that its distancefromafixedpoint (focus) is equal to its perpendicular distancefromafixedstraight line (directrix). Standard equation of a parabola is y2 = 4ax. For this parabola: (i) Vertex is (0,0)
(ii) focus is (a, 0)
(iii)Axisisy=0 (iv) Directrix is x + a = 0
FOCAL DISTANCE:
The distance of a point on the parabola from the focus is called the FOCAL DISTANCE O F T H E FOCAL CHORD
:
A chord of the parabola, which passes through the focus is called a FOCAL DOUBLE
CHORD.
ORDINATE:
A chord of the parabola perpendicular to the axis of the symmetry is called a DOUBLE LATUS
POINT.
ORDINATE.
RECTUM:
A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola is called the LATUS RECTUM. For y2 = 4ax. • Length of the latus rectum = 4a. • ends of the latus rectum are L(a, 2a) & L' (a, - 2a). Note that: (i) Perpendicular distancefromfocus on directrix = half the latus rectum. (ii) Vertex is middle point of the focus & the point of intersection of directrix & axis. (iii) Two parabolas are laid to be equal if they have the same latus rectum. Four standard forms the parabola are y2 = Ellipse, 4ax ; y2 =Hyperbola) - 4ax ; x2 = 4ay; x2 = - 4ay [17] ^B ansaIClasses ConicofSection (Parabola,
5.
POSITION OF A POINT RELATIVE TO A PARABOLA: The point (x( y t ) lies outside, on or inside the parabola y2 = 4ax according as the expression y,2 - 4ax, is positive, zero or negative.
6.
LINE & A PARABOLA: The line y=mx + c meets the parabola y 2 =4ax in two points real, coincident or imaginary according as a a > c m => condition of tangency is, c = —.
7.
Length of the chord intercepted by the parabola on the line y = m x + c is :
f
A
\ 2
-ja(l + m 2 ) ( a - m c ) .
vm ; Note: length of the focal chord making an angle a with the x - axis is 4aCosec2 a.
8.
PARAMETRIC REPRESENTATION: The simplest & the best form of representing the co-ordinates of a point on the parabola is (at2,2at). The equations x - at2 & y = 2at together represents the parabola y2 = 4ax, t being the parameter. The equation of a chord joining t, & t2 is 2x - (t[ +12) y + 2 at} t2 = 0. Note: If the chord joining t]512 & t3, t4 pass through a point (c 0) on the axis, then t,t2 = t3t4 = - c/a.
9.
TANGENTS TO THE PARABOLA y2 = 4ax:
(0
yy1 = 2a(x + Xj)atthepoint(x,,y 1 ) ;
(iii)
t y = x + a t2 at (at2,2at). Note : Point of intersection of the tangents at the point t, & t2 is [ at, t2 a(t, +12) ].
10.
NORMALS TO THE PARABOLA y2 - 4ax :
(i)
y - y i =-h. ( x - x , ) a t ( x , y,) ; 2a y + tx = 2at + at3 at (at2' 2at).
(iii)
(ii)
(ii)
y = mx - 2am - am3 at (am2, - 2am)
Note : Point of intersection of normals at t, & t2 are, a (t2 + 11. (a)
(b)
+ t,t 2 + 2); - at, t2 (t, +12).
THREE VERY IMPORTANT RESULTS: If t, & t2 are the ends of a focal chord of the parabola y2 = 4ax then t,t 2 = -1. Hence the co-ordinates fa 2a) at the extremities of a focal chord can be taken as (at2' 2at) & - j > . vt tJ 2 If the normals to the parabola y = 4ax at the point t, meets the parabola again at the point t2, then 2^ 7l \ ,J v If the normals to the parabola y2 = 4ax at the points t, & ^ intersect again on the parabola at the point 't3' then tj t2 = 2; t3 = - (tj +12) and the line joining t, & t2 passes through a fixed point (-2a, 0).
h= ' (c)
' a 2a ^ y = mx + — ( m ^ O ) a t V m2 ' m m
'
t l +
General Note : (i) Length of subtangent at any point P(x, y) on the parabola y2 = 4ax equals twice the abscissa of the point P. Note that the subtangent is bisected at the vertex. (ii) Length of subnormal is constant for all points on the parabola & is equal to the semi latus rectum. (iii) If a family of straight lines can be represented by an equation A,2P + XQ + R = 0 where X is a parameter and P, Q, R are linear functions of x and y then the family of lines will be tangent to the curve Q2 = 4 PR. 12.
The equation to the pair of tangents which can be drawn from any point (x, y,) to the parabola y2 = 4ax is given by : SS, = T2 where : S = = y - 4 a x . S1=y,2-4ax1 ; T = yy, ~2a(x + x,). 2
^B ansaIClasses
Conic Section (Parabola, Ellipse, Hyperbola)
[17]
13.
DIRECTOR
CIRCLE:
Locus of the point of intersection of the perpendicular tangents to the parabola y2 = 4ax is called the DIRECTOR CIRCLE. It's equation is x + a = 0 which is parabola's own directrix. 14.
CHORD OF CONTACT :
Equation to the chord of contact of tangents drawn from a point P(x, y,)is yy, =2a(x + x,). Remember that the area of the triangle formed by the tangents from the point (x, y,) & the chord of contact is (y,2 - 4ax1)3/2 2a. Also note that the chord of contact exists only if the point P is not inside. 15.
POLAR & POLE:
Equation of the Polar of the point P(x, y,) w.r.t. the parabola y2 = 4ax is' yy 1 =2a(x + x1) ' , W 2 (ii) The pole of the line /x + my + n = 0 w.r.t. the parabola y = 4ax is —, 1 Note: vl , (i) The polar ofthe focus ofthe parabola is the directrix. (ii) When the point (x, y,) lies without the parabola the equation to its polar is the same as the equation to the chord of contact of tangents drawn from (x,, y,) when (x,, y,) is on the parabola the polar is the same as the tangent at the point. (iii) If the polar of a point P passes through the point Q, then the polar of Q goes through P. (iv) Two straight lines are said to be conjugated to each other w.r.t. a parabola when the pole of one lies on the other. (v) Polar of a given point P w.r.t. any Conic is the locus of the harmonic conjugate of P w.r.t. the two points is which any line through P cuts the conic. (i)
16.
CHORD WITH A GIVEN MIDDLE POINT
:
Equation of the chord of the parabola y2 = 4ax whose middle point is 2a (x, y,) i s y - y j = — ( x - x,). This reduced to T = Sj where T = y y, - 2a (x + x,) & S, =y, 2 -4ax s . 17.
DIAMETER:
The locus ofthe middle points of a system of parallel chords of a Parabola is called a DIAMETER. Equation to the diameter of a parabola is y = 2a/m, where m = slope of parallel chords. Note: (i) The tangent at the extremity of a diameter of a parabola is parallel to the system of chords it bisects. (ii) The tangent at the ends of any chords of a parabola meet on the diameter which bisects the chord. (iii) A line segmentfroma point P on the parabola and parallel to the system of parallel chords is called the ordinate to the diameter bisecting the system of parallel chords and the chords are called its double ordinate. 18.
IMPORTANT HIGHLIGHTS
(a)
If the tangent & normal at any point 'P' of the parabola intersect the axis at T & G then ST = SG = SP where 'S' is the focus. In other words the tangent and the normal at a point P on the parabola are the bisectors of the angle between the focal radius SP & the perpendicular from P on the directrix. From this we conclude that all rays emanating from S will become parallel to the axis of the parabola after reflection. The portion of a tangent to a parabola cut off between the directrix & the curve subtends a right angle at the focus. The tangents at the extremities of a focal chord intersect at right angles on the directrix, and hence a circle on any focal chord as diameter touches the directrix. Also a circle on any focal radii ofa point P (at2,2at) as diameter touches the tangent at the vertex and intercepts a chord of length a^i + t2 on a normal at the point P.
(b) (c)
^B ansaIClasses
:
Conic Section (Parabola, Ellipse, Hyperbola)
[17]
(d) (e) (f) (g)
Any tangent to a parabola & the perpendicular on it from the focus meet on the tangtent at the vertex. If the tangents at P and Q meet in T, then: • TP and TQ subtend equal angles at the focus S. • ST2 = SP. SQ & • The triangles SPT and STQ are similar. Tangents and Normals at the extremities of the latus rectum of a parabola y2 = 4ax constitute a square, their points of intersection being ( - a 0) & (3 a> 0). Semi latus rectum of the parabola y2=4ax, is the harmonic mean between segments of any focal chord L 1 is • ; 2a o = of the parabola
(h) (i) (j) (k)
2bc
•i.e. —+—=—. 1 1 1 b+c b c a The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus. The orthocentre of any triangle formed by three tangents to a parabola y2 = 4ax lies on the directrix & has the co-ordinates - a, a (tj +1, +13 + t,t2t3). The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points, If normal drawn to a parabola passes through a point P(h, k) then k = mh - 2am - am3 i.e. am3 + m(2a - h) + k = 0. ™ Then gives m, + m, + m3 = «0
;
mjm 2 + m 2 m 3 + m 3 m, = J
2a
~h
k . ; m, m 2 m3 = —
a a where m, m2 &m 3 are the slopes of the three concurrent normals. Note that the algebraic sum of the: • slopes of the three concurrent normals is zero. • ordinates of the three conormal points on the parabola is zero. • Centroid of the A formed by three co-normal points lies on the x-axis. (/) A circle circumscribing the triangle formed by three co-normal points passes through the vertex of the parabola and its equation is, 2(x2 + y2) - 2(h + 2a)x - ky = 0 Suggested problems from Loney: Exercise-25 (Q.5,10,13,14,18,21), Exercise-26 (Important) (Q.4, 6,7,16,17,20,22,26,27,28,34,38), Exercise-27 (Q.4,7), Exercise-28 (Q.2,7,11,14,17,23), Exercise-29 (Q.7, 8, 10, 19, 21, 24, 26, 27), Exercise-30 (2, 3, 13, 18, 20, 21, 22, 25, 26, 30) Note: Refer to the figure on Pg. 175 if necessary.
EXERCISE-I — Q. 1
Show that the normals at the points (4a, 4a) & at the upper end of the latus ractum of the parabola y2 = 4ax intersect on the same parabola.
Q. 2
Prove that the locus of the middle point of portion of a normal to y2 = 4ax intercepted between the curve & the axis is another parabola. Find the vertex & the latus rectum of the second parabola.
Q.3
Find the equations Qf the tangents to the parabola y2 = 16x, which are parallel & perpendicular respectively 1 to the line 2 x - y + 5 = 0. Find also the coordinates of their points of contact, n ' i ^ /
Q.4
A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of a parabola y2 = 4ax. Prove that the common chord of the circle and parabola bisects the distance between the vertex and the focus.
Q. 5
Find the equations of the tangents of the parabola y2 = 12x, which passes through the point (2,5).
Q.6
Through the vertex O of a parabola y 2 =4x, chords OP & OQ are drawn at right angles to one another. Show that for all positions of P, PQ cuts the axis of the parabola at afixedpoint. Alsofindthe locus of the middle point of PQ.
Q. 7
Let S is the focus of the parabola y2 = 4ax and X the foot of the directrix, PP' is a double ordinate of the curve and PX meets the curve again in Q. Prove that P'Q passes through focus.
^B ansaIClasses
Conic Section (Parabola, Ellipse, Hyperbola)
[17]
t
Q.8
Three normals to y2 = 4x pass through the point (15,12). Show that one of the normals is given by y = x - 3 & find the equations of the others.
Q. 9
Find the equations of the chords of the parabola y2 = 4ax which pass through the point (-6a, 0) and which subtends an angle of 45° at the vertex.
Q.IO Through the vertex O of the parabola y 2 =4ax, a perpendicular is drawn to any tangent meeting it at P & the parabola at Q. Show that OP • OQ = constant. Q.ll
'O' is the vertex of the parabola y2 = 4ax & L is the upper end of the latus rectum. If LH is drawn perpendicular to OL meeting OX in H, prove that the length of the double ordinate through H is 4a •
Q.12
The normal at a point P to the parabola y2 = 4ax meets its axis at G. Q is another point on the parabola such that QG is perpendicular to the axis of the parabola. Prove that QG2 - PG2 = constant. If the normal at P( 18,12) to the parabola y2= 8x cuts it again at Q, show that 9PQ = 80 VlO
Q.13 Q.14
Prove that, the normal to y2 = 12x at (3' 6) meets the parabola again in (27,-18) & circle on this normal chord as diameter is x2 + y2 - 30x + 12y - 27 = 0.
Q.15
Find the equation of the circle which passes through the focus of the parabola x2 = 4y & touches it at the point (6,9).
Q.16
P & Q are the points of contact o f t h e tangents drawn from the point T to the parabola y2 = 4ax. If PQ be the normal to the parabola at P- prove that TP is bisected by the directrix.
Q.17
Prove that the locus of the middle points of the normal chords of the parabola y2 = 4ax is y2 4a 3 — + —2T- = x - 2 a • 2a yFrom the point (-1,2) tangent lines are drawn to the parabola y2 = 4x. Find the equation of the chord of contact. Also find the area of the triangle formed by the chord of contact & the tangents. Show that the locus ofa point that divides a chord of slope 2 of the parabola y 2 =4x internally in the ratio 1 :2 is a parabola. Find the vertex of this parabola.
Q.18 Q.19 Q.20
From a point A common tangents are drawn to the circle x2 + y2 = a2/2 & parabola y2 = 4ax. Find the area of the quadrilateral formed by the common tangents, the chord of contact of the circle and the chord of contact of the parabola.
Q.21
Prove that on the axis of any parabola y2 = 4ax there is a certain point K which has the property that, if a chord PQ of the parabola be drawn through it, then — — j + — — j is same for all positions of the chord. Find also the coordinates of the point K. '^ ^ (Q^)
Q.22
Q.23
Prove that the two parabolas y2 = 4ax & y2 = 4c (x - b) cannot have a common normal, other than the b axis, unless 7 ~ > 2. (a-c) Find the condition o n ' a ' & ' b ' s o that the two tangents drawn to the parabola y2 = 4ax fromapoint are normals to the parabola x2 = 4by.
Q.24
Prove that the locus ofthe middle points of all tangents drawnfrompoints on the directrix to the parabola y2 = 4ax is y2(2x + a) - a(3x + a)2.
Q.25
Show that the locus of a point, such that two of the three normals drawn from it to the parabola y2 = 4ax are perpendicular is y2 = a(x - 3a).
^B ansaIClasses
Conic Section (Parabola, Ellipse, Hyperbola)
[17]
EXERCISE-II Q.l
In the parabola y2=4ax, the tangent at the point P, whose abscissa is equal to the latus ractum meets the axis in T & the normal at P cuts the parabola again in Q. Prove that PT: PQ = 4:5.
Q.2
Two tangents to the parabola y2= 8x meet the tangent at its vertex in the points P & Q. If PQ = 4 units, prove that the locus of the point of the intersection of the two tangents is y2 = 8 (x + 2).
Q.3
A variable chord tj t, of the parabola y2 = 4ax subtends a right angle at a fixed point tQ of the curve. Show that it passes through a fixed point. Alsofindthe co-ordinates of thefixedpoint.
Q.4
Two perpendicular straight lines through the focus of the parabola y2 = 4ax meet its directrix in T & T' respectively. Show that the tangents to the parabola parallel to the perpendicular lines intersect in the mid point of T T
Q.5
Two straight lines one being a tangent to y2 = 4ax and the other to x2 = 4by are right angles. Find the locus of their point of intersection.
Q.6
A variable chord PQ of the parabola y2 = 4x is drawn parallel to the line y = x. If the parameters of the points P & Q on the parabola are p & q respectively, show that p + q = 2. Also show that the locus of the point of intersection of the normals a t P & Q i s 2 x - y = 1 2 .
Q. 7
Show that an infinite number of triangles can be inscribed in either of the parabolas y2 = 4ax & x2 - 4by whose sides touch the other.
Q.8
If (Xj, yj), (x2, y2) and (x3, y3) be three points on the parabola y2 = 4ax and the normals at these points . , Xi X, X7 Xi X-i XiL meet ma point then prove that —1 +— +— = 0. ys yi yi
Q. 9
Show that the normals at two suitable distinct real points on the parabola y2 = 4ax intersect at a point on the parabola whose abscissa > 8a.
Q. 10 The equation y = x2 + 2ax + a represents a parabola for all real values of a. (a) Prove thaj>eacn of these parabolas pass through a common point and determine the coordinates of this point, s (b) The vertices of the parabolas lie on a curve. Prove that this curve is a parabola andfindits equation. Q. 11 The normals at P and Q on the parabola y2 = 4ax intersect at the point R (x,, y,) on the parabola and the tangents at P and Q intersect at the point T. Show that, /(TP) • /(TQ) = | (x, - 8a) / y f + 4 a ^ Also show that, if R moves on the parabola, the mid point of PQ lie on the parabola y2 = 2a(x + 2a). Q. 12 If Q (Xj, y,) is an arbitrary point in the plane of a parabola y2 = 4ax, show that there are three points on the parabola at which OQ subtends a right angle, where O is the origin. Show furhter that the normal at these three points are concurrent at a point Redetermine the coordinates of R in terms of those of Q. Q. 13 PC is the normal at P to the parabola y2 = 4ax, C being on the axis. CP is produced outwards to Q so that PQ = CP; show that the locus of Q is a parabola, & that the locus of the intersection of the tangents at P & Q to the parabola on which they lie is y2 (x + 4a) + 16 a3 = 0. Q. 14 Show that the locus of the middle points of a variable chord of the parabola y2 = 4ax such that the focal distances of its extremities are in the ratio 2 :1, is 9(y2 - 2ax)2 = 4a2(2x - a)(4x + a). Q. 15 A quadrilateral is inscribed in a parabola y 2 =4ax and three of its sides pass throughfixedpoints on the axis. Show that the fourth side also passes throughfixedpoint on the axis of the parabola. Q. 16 Prove that the parabola y2 = 16x & the circle x2 + y2 - 40x - 16y - 48 = 0 meet at the point P(36,24) & one other point Q. Prove that PQ is a diameter of the circle. Find Q.
^Bansal Classes
Conic Section (Parabola, Ellipse, Hyperbola)
[13]
Q.17
A variable tangent to the parabola y2 = 4ax meets the circle x2 + y2 = r2 at P & Q. Prove that the locus of the mid point of PQ is x(x2 + y2) + ay2 = 0.
Q. 18 Find the locus of the foot of the perpendicular from the origin to chord of the parabola y2 = 4ax subtending an angle of 45° at the vertex. Q.19
Show that the locus of the centroids of equilateral triangles inscribed in the parabola y2 = 4ax is the parabola 9y2 - 4ax + 32 a2 = 0.
Q.20 The normals at P, Q, R on the parabola y2 = 4ax meet in a point on the line y = k. Prove that the sides of the triangle PQR touch the parabola x2 - 2ky - 0. Q.21
Afixedparabola y2 = 4 ax touches a variable parabola. Find the equation to the locus of the vertex of the variable parabola. Assume that the two parabolas are equal and the axis of the variable parabola remains parallel to the x-axis.
Q.22
Show that the circle through three points the normals at which to the parabola y 2 =4ax are concurrent at the point (h, k) is 2(x2 + y2) - 2(h + 2a) x - ky = 0.
Q.23
Prove that the locus of the centre of the circle, which passes through the vertex of the parabola y2 = 4ax & through its intersection with a normal chord is 2y2 = ax - a2.
Q.24 The sides of a triangle touch y2 = 4ax and two of its angular points lie on y2 = 4b(x + c). Show that the locus of the third angular point is a2y2 = 4(2b - a)2.(ax + 4bc) Q.25
Three normals are drawn to the parabola y2 = 4ax cos a from any point lying on the straight line y = b sin a. Prove that the locus of the orthocentre of the traingles formed by the corresponding tangents x2 y 2 is the ellipse — + ~ = 1, the angle a being variable, a b
EXERCISE-III Q. 1
Find the locus of the point of intersection of those normals to the parabola x2 = 8 ^ which are at right angles to each other. [REE '97,6]
Q. 2
The angle between a pair oftangents drawn from a point P to the parabola y2 = 4ax is 45°. Show that the locus of the point P is a hyperbola. [ JEE '98,8]
Q. 3
The ordinates of points P and Q on the parabola y2 = 12x are in the ratio 1:2. Find the locus of the point of intersection of the normals to the parabola at P and Q. [ REE '98,6]
Q. 4
Find the equations of the common tangents of the circle x2 + y2 - 6y + 4 _= 0 and the parabola y2 = x. [ REE '99,6 ]
Q.5(a) If the line x - 1 = 0 is the directrix of the parabola y 2 - k x + 8 = 0 then one of the values of' k' is (A) 1/8 (B) 8 (C) 4 ' (D) 1/4 (b) If x + y = k is normal to y2 = 12 x, then' k' is : (A) 3 (B) 9 (C)-9 Q.6
[JEE'2000 (Scr), 1+1] (D) - 3
Find the locus of the points of intersection of tangents-drawn at the ends of all normal chords ofthe parabola y2 = 8(x - 1). [REE '2001,3]
Q.7(a) The equation of the common tangent touching the circle (x - 3)2 + y2 = 9 and the parabola y 2 =4x above the x-axis is ( A ) ^ y = 3x+1 (B) V^y = -(x + 3) (C) ^ y = x + 3 (D) ^ y = -(3x + 1)
^B ansaIClasses
Conic Section (Parabola, Ellipse, Hyperbola)
[17]
\
(b) The equation of the directrix of the parabola, y2 + 4y + 4x + 2 = 0 is (A) x = - 1
(B) x = 1
(C)X =
~|
( D ) X =
|
[JEE'2001(Scr), 1+1] Q.8
The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y2 = 4ax is another parabola with directrix [ JEE'2002 (Scr.), 3 ] (A) x = - a
Q.9
(B)x = - |
(C) x = 0
(D)x=|
The equation of the common tangent to the curves y2 = 8x and xy=-1 is (A) 3y = 9x + 2 (B)y = 2x+1 (C) 2y = x + 8 (D) y = x + 2
[JEE'2002 (Scr), 3]
Q.l0(a) The slope of the focal chords of the parabola y2 = 16x which are tangents to the circle (x - 6)2 + y2 = 2 are (A) ± 2 (B)-1/2,2 (C)±l (D)-2,1/2 [JEE'2003, (Scr.)] (b) Normals are drawn from the point 'P' with slopes m,, m2, m3 to the parabola y 2 =4x. If locus of P with m s m2 = a is a part of the parabola itself then find a. [JEE 2003,4 out of 60] Q. 11 The angle between the tangents drawn from the point (1,4) to the parabola y2 = 4x is (A) 7T/2 (B) 7T/3 (C) 7I/4 (D) TC/6 [JEE 2004, (Scr.)] Q. 12 Let P be a point on the parabola y2 - 2y - 4x + 5 = 0, such that the tangent on the parabola at P intersects the directrix at point Q. Let R be the point that divides the line segment QP externally in the ratio 1:1. Find the locus of R.
[JEE 2004,4 out of 60]
Q. 13(a) The axis of parabola is along the line y=x and the distance of vertex from origin is J 2 and that from its focus is 2 v2 • If vertex and focus both lie in the 1st quadrant, then the equation of the parabola is (A) (x + y)2 = (x - y - 2) (C) (x - y)2 = 4(x + y - 2)
(B)(x-y) 2 = (x + y - 2 ) (D) (x - y)2 = 8(x + y - 2)
[JEE 2006,3] (b) The equations of common tangents to the parabola y = x2 and y = - (x - 2)2 is/are (A)y = 4(x-1) (B) y = 0 (C)y = - 4 ( x - l ) (D)y = - 3 0 x - 5 0 [JEE 2006,5] (c) Match the following Normals are drawn at points P, Q and R lying on the parabola y2 = 4x which intersect at (3,0). Then (i) Area of APQR (A) 2 (ii) Radius ofcircumcircle of APQR (B)5/2 (iii) Centroid of APQR (C) (5/2,0) (iv) Circumcentre of APQR (D) (2/3,0) [JEE 2006,6]
^B ansaIClasses
Conic Section (Parabola, Ellipse, Hyperbola)
[17]
ELLIPSE KEYCONCEPTS
1.
STANDARD EQUATION & DEFINITIONS: 2
2
Standard equation of an ellipse referred to its principal axes along the co-ordinate axes is — + a 2 b2 2 2 2 2 2 2 2 Where a > b & b = a (l - e ) => a - b = a e . Where e = eccentricity (0 < e < 1). FOCI: S = (ae 0)&S' = ( - a e 0). EQUATIONS OF DIRECTRICES: a p x =— & x e
=
j.
a e
VERTICES: A' = ( - a, 0) & A = (a, 0). MAJOR AXIS : The line segment A' A in which the foci S' & S lie is of length 2a & is called the major axis (a > b) of the ellipse. Point of intersection of major axis with directrix is called the foot of the directrix (z). MINOR AXIS: The y-axis intersects the ellipse in the points B ' s (0, - b ) & B = (0, b). The line segment B'B of length 2b (b < a) is called the Minor Axis of the ellipse. PRINCIPAL AXIS: The major & minor axis together are called Principal Axis of the ellipse. CENTRE: The point which bisects every chord of the conic drawn through it is called the centre of the conic. 2
2
C = (0,0) the origin is the centre of the ellipse _ + X— = i. a 2 b2 DIAMETER: A chord of the conic which passes through the centre is called a diameter of the conic. FOCAL CHORD: A chord which passes through a focus is called a focal chord. DOUBLE ORDINATE: A chord perpendicular to the major axis is called a double ordinate. LATUSRECTUM: The focal chord perpendicular to the major axis is called the latus rectum. Length of latus rectum 2 (minor axis) 2 x 2 b 2s „ ^ ,> , (LL) = = = 2a(l - e )=2e (distance from focus to the corresponding directrix) a
major axis
NOTE: (i) The sum of the focal distances of any point on the ellipse is equal to the maj or Axis. Hence distance of focus from the extremity of a minor axis is equal to semi major axis. i.e. BS = CA. (ii)
If the equation of the ellipse is given as 2L_ +
- 1 & nothing is mentioned' then the rule is to assume
that a> b.
Bansal Classes
Conic Section (Parabola, Ellipse, Hyperbola)
[10]
POSITION OFA POINT w.r.t. AN ELLIPSE
2.
X
V
l The point P(x, y,) lies outside, inside or on the ellipse according as; —— + —L- - 1 > < or=0. a b
3.
AUXILIARY CIRCLE / ECCENTRIC ANGLE:
A circle described on major axis as diameter is called the auxiliary circle.
Let Q be a point on the auxiliary circle x2 + y2 = a2 such that QP produced is perpendicular to the x-axis then P & Q are called as the CORRESPONDING POINTS on the ellipse & the auxiliary circle respectively '9' is called the ECCENTRIC ANGLE ofthe point P on the ellipse (0 < 9 < 2 Tt). i'(PN) _ b _ Semi minor axis Note that *(QN) a Semi major axis Hence " Iffromeach point of a circle perpendiculars are drawn upon a fixed diameter then the locus of the points dividing these perpendiculars in a given ratio is an ellipse of which the given circle is the auxiliary circle". PARAMETRIC REPRESENTATION: x2
y2 The equations x = a cos 9 & y = b sin 9 together represent the ellipse + —j - 1 . a b Where 9 is a parameter. Note that if P(9) = (a cos 9, b sin 9) is on the ellipse then ; Q(9) = (a cos 9, a sin 9) is on the auxiliary circle. LINE AND AN ELLIPSE : 2
2
x y . . . . The line y=mx+c meets the ellipse —r + —r = 1 in two points real, coincident or imaginary according a b 2 2 2 2 as c is < = or > a m + b .
6.
x2 y2 Hence y = mx + c is tangent to the ellipse — + —j = 1 if c2 = a2m2 + b2. a b The equation to the chord of the ellipse joining two points with eccentric angles a & P is given by x a+B y . a+B a-B —cosh —sm = cos • a 2 b 2 2 TANGENTS:
(i)
xx yy, — i + —J= 1 is tangent to the ellipse at (Xj y,).
Note :The figure formed by the tangents at the extremities of latus rectum is rhoubus of area (ii)
y = mx± J a
2
m
2
+b
2
2a e
is tangent to the ellipse for all values of m.
Note that there are two tangents to the ellipse having the same m, i.e. there are two tangents parallel to any given direction. (iii) (iv)
(v)
®+ = j is tangent to the ellipse at the point (a cos 9, b sin 9). a b The eccentric angles of point of contact of two parallel tangents differ by Tt. Conversely if the difference between the eccentric angles of two points is p then the tangents at these points are parallel. sin q + p cos a+J3 2 Point of intersection of the tangents at the point a & P is a - q-p q-3 COS cos
^B ansaIClasses
Conic Section (Parabola, Ellipse, Hyperbola)
[17]
7.
NORMALS :
(i)
Equation of the normal at (x, y.) is
(ii)
Equation of the normal at the point (acos 0' bsin 0) is; ax sec 0 - by cosec 0 = (a2 - b2).
(iii)
Equation of a normal in terms of its slope'm' is y=mx —( a )"L Va2.+ b 2 m 2 DIRECTOR CIRCLE: Locus of the point of intersection of the tangents which meet at right angles is called the Director Circle. The equation to this locus is x2 + y2 = a2 + b2 i.e. a circle whose centre is the centre of the ellipse & whose radius is the length of the line joining the ends of the major & minor axis.
8.
a2x x,
b2y = a2 - b2 = a2e2. y,
9.
Chord of contact, pair of tangents, chord with a given middle point, pole & polar are to be interpreted as they are in parabola. 10. DIAMETER: The locus of the middle points of a system of parallel chords with slope'm' of an ellipse is a straight line b2 passing through the centre of the ellipse, called its diameter and has the equation y = —-r— x. a~m X2 V 2 11. IMPORTANT HIGHLIGHTS : Refering to an ellipse —r+ Ar =1• 2 b2 i (SP) + i (S'P) = 2a. H - 1 If P be any point on the ellipse with S & S' as its afoci then H - 2 The product of the length's of the perpendicular segments from the foci on any tangent to the ellipse is b2 and the feet of these perpendiculars Y,Y' lie on its auxiliary circle.The tangents at these feet to the auxiliary circle meet on the ordinate of P and that the locus of their point of intersection is a similiar ellipse as that ofthe original one. Also the lines joining centre to the feet of the perpendicular Y and focus to the point of contact of tangent are parallel. H - 3 If the normal at any point P on the ellipse with centre C meet the maj or & minor axes in G & g respectively & if CF be perpendicular upon this normal' then (i) PF. PG = b2 (ii) PF. Pg = a2 (iii) PG.Pg = SP. S'P (iv) CG. CT = CS2 (v) locus ofthe mid point of Gg is another ellipse having the same eccentricity as that of the original ellipse, [where S and S' are the focii of the ellipse and T is the point where tangent at P meet the major axis] H - 4 The tangent & normal at a point P on the ellipse bisect the external & internal angles between the focal distances of P. This refers to the well known reflection property of the ellipse which states that rays from one focus are reflected through other focus & vice-versa. Hence we can deduce that the straight lines joining each focus to the foot of the perpendicularfromthe other focus upon the tangent at any point P meet on the normal PG and bisects it where G is the point where normal at P meets the major axis. H - 5 The portion of the tangent to an ellipse between the point of contact & the directrix subtends a right angle at the corresponding focus. H - 6 The circle on any focal distance as diameter touches the auxiliary circle. H - 7 Perpendicularsfromthe centre upon all chords which join the ends of any perpendicular diameters of the ellipse are of constant length. H - 8 If the tangent at the point P of a standard ellipse meets the axis in T and t and CY is the perpendicular on it from the centre then, (i) T t. P Y = a2 - b2 and (ii) least value of Tt is a + b. Suggested problems from Loney: Exercise-32 (Q.2 to 7,11,12,16,24), Exercise-33 (Important) (Q.3, 5, 6, 15, 16, 18, 19, 24, 25, 26, 34), Exercise-35 (Q.2, 4, 6, 7, 8, 11, 12, 15)
^B ansaIClasses
Conic Section (Parabola, Ellipse, Hyperbola)
[17]
EXERCISE-II Q. 1
Find the equation of the ellipse with its centre (1,2), focus at (6,2) and passing through the point (4,6).
Q.2
The tangent at any point P of a circle x2 + y2 = a2 meets the tangent at a fixed point A (a, 0) in T and T is joined to B, the other end of the diameter through A, prove that the locus of the intersection ofAP and BT is an ellipse whose ettentricity is
.
Q. 3
The tangent at the point a on a standard ellipse meets the auxiliary circle in two points which subtends a right angle at the centre. Show that the eccentricity ofthe ellipse is (1 + sin2a)"1/2.
Q.4
An ellipse passes through the points ( - 3,1) & (2, -2) & its principal axis are along the coordinate axes in order. Find its equation.
Q.5
If any two chords be drawn through two points on the major axis of an ellipse equidistant from the a B v 8 centre, show that tan — -tan — -tan — -tan— = 1, where a, B, y, S are the eccentric angles of the 2
extremities of the chords. Q.6
2
2
x2 y2 Ifthe normals at the points P,Q,R with eccentric angles a, p, yon the ellipse— + — =1 are concurrent, a b then show that sin a sinP siny
Q. 7
2
cos a cosp cosy
sin 2a sin2p sin2y
2 2 Prove that the equation to the circle, having double contact with the ellipse X hV— = 1 at the ends of a 2 b2 2 2 3 2 2 4 a latus rectum, is x + y - 2ae x = a (1 - e - e ).
Q. 8
X2 y 2 Find the equations of the lines with equal intercepts on the axes & which touch the ellipse— + — = 1.
Q.9
' 16 V ,CH
v ,2 ,l VI1 y ft J a c k p o t ©J x + y - 2x - 15 = 0. Find 6. Find also the equation to the common tangent. 2 2 W o © (£ ; • 11 4 x y "cKeQk i d . + f^aJ. /uj Q.IO A tangent having slope to the ellipse — + — = 1, intersects the axis of x & y in points A & B 3 18 32 respectively. If O is the origin,findthe area of triangle OAB.
Tr r, \\ .
Q.ll '
'O' is the origin & also the centre of two concentric circles having radii of the inner & the outer circle as 'a' & 'b' respectively. A line OPQ is drawn to cut the inner circle in P & the outer circle in Q, PR is drawn parallel to the y-axis & QR is drawn parallel to the x-axis. Prove that the locus of R is an ellipse touching the two circles. Ifthe focii of this ellipse lie on the inner circle,findthe ratio of inner: outer radii &findalso the eccentricity of the ellipse.
QI12 ABC is an isosceles triangle with its base BC twice its altitude. Apoint P moves within the triangle such o * • that the square of its distancefromBC is half the rectangle contained by its distancesfromthe two sides. [2 Show that the locus of P is an ellipse with eccentricity J— passing through B & C.
^Bansal Classes
Conic Section (Parabola, Ellipse, Hyperbola)
[13]
Xz
y
+ ~ = 1 to the tangent drawn at a Q.13 Let d be the perpendicular distance from the centre of the ellipse — a 2 ' b2
^
point P on the ellipse.If F, & F2 are the two foci of the ellipse, then show that (PF, - PF2)2=4a*'1\ \
Q.14
Common tangents are drawn to the parabola y2 = 4x & the ellipse 3x2 + 8y2 = 48 touching the parabola at A & B and the ellipse at C & D. Find the area of the quadrilateral.
Q-l 5 If the normal at a point P on the ellipse of semi axes a, b & centre C cuts the major & minor axes at G & g, show that a2. (CG)2 + b2. (Cg)2 = (a2 - b2)2. Also prove that CG = e2CN, where PN is the ordinate ofP.
, o)'
Q 1 6
x2 y2 Prove that the length of the focal chord of the ellipse — + ~ = 1 which is inclined to the maj or axis at a" b 2ab2
. >
angle 9 is ^ 5 ~ 5—. a sin 0 + b cos 0
x 2 v2 Q.17 The tangent at a point P on the ellipse — + - - j = 1 intersects the major axis in T & N is the foot of the a b perpendicular from P to the same axis. Show that the circle on NT as diameter intersects the auxiliary circle orthogonally. Q.18
x2 y 2 The tangents from (x, y,) to the ellipse —— + ~ = 1 intersect at right angles. Show that the normals at a b x y the points of contact meet on the line — = — . Yi x i x 2 y2 Find the locus of the point the chord of contact of the tangent drawn from which to the ellipse — + - = 1 a b touches the circle x2 + y2 = c2, where c < b < a.
v
Q.20 Prove that the three ellipse —j + —j = 1 af
bf
2
2
if a a2
2
2
a2
b2
x2 y2 1 and — + --— = 1 will have a common tangent
1
b 1 = 0. b2 1
EXERCISE-II Q.l
PG is the normal to a standard ellipse at P, G being on the maj or axis. GP is produced outwards to Q so 2 2 that PQ = GP Show that the locus of Q is an ellipse whose eccentricity is a - b & find the equation a 2 +b 2 of the locus of the intersection of the tangents at P & Q.
Q.2 Q.3
P & Q are the corresponding points on a standard ellipse & its auxiliary circle. The tangent at P to the ellipse meets the major axis in T. Prove that QT touches the auxiliary circle. x2 y 2 : 1 is joined to the ends A, A' ofthe major axis. Ifthe lines through The point P on the ellipse— + a b" P perpendicular to PA, PA' meet the major axis in Q and R then prove that /(QR) = length of latus rectum.
^B ansaIClasses
Conic Section (Parabola, Ellipse, Hyperbola)
[17]
Q.4
Q.5
Q.6
Q.7
Q.8
X2 V 2 Let S and S' are the foci, SL the semilatus rectum of the ellipse + and LS' produced cuts the a b (1-e 2 ) ellipse at P, show that the length of the ordinate of the ordinate of P is y a, where 2a is the length l + 3e of the major axis and e is the eccentricity of the ellipse. X2 V 2 A tangent to the ellipse — + Ar = 1 touches at the point P on it in the first quadrant & meets the a b coordinate axis in A & B respectively. If P divides AB in the ratio 3 :1 find the equation of the tangent. X2 V 2 PCP' is a diameter of an ellipse —— + = i (a > b) & QCQ' is the corresponding diameter of the a b~ auxiliary circle, show that the area of the parallelogram formed by the tangent at P, P', Q & Q' is 8a2b where (j) is the eccentric angle of the point P. (a-b)sin 2
Q.9
Q.IO
32aV +
x2 v 2 If (x^ y,) & (x2, y2) are two points on the ellipse —2 + 2 = 1, the tangents at which meet in a b (h, k) & the normals in (p, q), prove that a2p=e2hx, x2 and b 4 q=- e2ky,y2a2 where 'e' is the eccentricity. x2 y 2 A normal inclined at 45° to the axis ofthe ellipse — + = 1 is drawn. It meets the x-axis & the y-axis in P a b fa2
-b2)2
& Q respectively. If C is the centre ofthe ellipse, show that the area of triangle C.PQ is — sq. units. 2(a +b ) Q.ll
Q.12
Q.13
Q.14
Tangents are drawn to the ellipse
2
b2
= 1fromthe point
.
.2 a
2
\ 2
, V a +b . Prove that they •b2 intercept on the ordinate through the nearer focus a distance equal to the major axis. a2
+
.,2
x 2 v2 P and Q are the points on the ellipse — If the chord P and Q touches the ellipse a b 4 X 2 y 2 4x , prove that seca+sec(3=2 where a, (3 are the eccentric angles of the points P and Q. a2 b2 a x 2 y2 A straight line AB touches the ellipse — + 2 = 1 & the circle x2 + y2 = r2 ; where a > r > b. a b A focal chord ofthe ellipse, parallel to AB intersects the circle in P & Q, find the length ofthe perpendicular drawn from the centre of the ellipse to PQ. Hence show that PQ = 2b. Show that the area ofa sector of the standard ellipse in thefirstquadrant between the maj or axis and a line drawn through the focus is equal to 1/2 ab (9 - e sin 9) sq. units, where 9 is the eccentric angle of the point to which the line is drawn through the focus & e is the eccentricity of the ellipse.
^B ansaIClasses
Conic Section (Parabola, Ellipse, Hyperbola)
[17]
Q. 15 A ray emanating from the point ( - 4,0) is incident on the ellipse 9x2 + 25y2 = 225 at the point P with abscissa 3. Find the equation of the reflected ray after first reflection. x2 y2 Q. 16 If p is the length of the perpendicularfromthe focus' S' of the ellipse —- + = 1 on anv tangent at 'P', , i , b2 2a , then show that—T = 1. 2 p *(SP)
a
b
x2 y2 Q. 17 In an ellipse —j + — = 1, nt and n^ are the lengths of two perpendicular normals terminated at the maj or a b 1 + 1 a2+b2 axi s then prove that: ~T ~2 ~ z— n, n 2 b x2 y2 Q. 18 If the tangent at any point of an ellipse— + ~ r = 1 makes an angle a with the major axis and an angle a b P with the focal radius of the point of contact then show that the eccentricity 'e' of the ellipse is given by cosP the absolute value of . cos a x2 y2 Q. 19 Using the fact that the product of the perpendiculars from either foci of an ellipse — + ~ = 1 upon a a b 2 tangent is b , deduce the following loci. An ellipse with 'a' & 'b' as the lengths of its semi axes slides between two given straight lines at right angles to one another. Show that the locus of its centre is a circle & the locus of its foci is the curve, (x2 + y2) (x2 y2 + b4) = 4 a2 x2 y2. x2 y2 Q.20 If tangents are drawn to the ellipse —2 + = \ intercept on the x-axis a constant length c, prove that a b the locus of the point of intersection of tangents is the curve 4y2 (b2x2 + a2y2 - a2b2) = c2 (y2 - b2)2.
EXERCISE i n Q.l
2
If tangent drawn at a point (t , 2t) on the parabola y2 = 4x is same as the normal drawn at a point (V5 cos <{>, 2 sin <> | ) on the ellipse 4x2 + 5y2 = 20. Find the values of t &
Q.2
2
2
2
[ REE '96,6 ]
2
A tangent to the ellipse x + 4y = 4 meets the ellipse x + 2y = 6 at P & Q. Prove that the tangents at P & Q of the ellipse x2 + 2y2 = 6 are at right angles. [ JEE '97, 5 ]
Q.3(i) The number of values of c such that the straight line y = 4x + c touches the curve (x2/ 4) + y2 = 1 is (A) 0 (B) 1 (C) 2 (D) infinite (ii) If P = (x, y), F, = (3, 0), F2 = (-3, 0) and 16x2 + 25y2 = 400, then PFj + PF2 equals (A) 8 (B) 6 (G)1.0 (D) 12 [ JEE '98,2 + 2 ] Q.4(a) If Xj, x2, x3 as well as y,, y2, y3 are in G.P. with the same common ratio, then the points (x,, yj), (x 2 ,y 2 )&(x 3 ,y 3 ): (A) lie on a straight line (B) lie on on ellipse (C) lie on a circle (D) are vertices of a triangle, (b) On the ellipse, 4x2 + 9y2 = 1, the points at which the tangents are parallel to the line 8x = 9y are: / 2-> 11 \ ' 2 1N ' 2 O '2 (A) (B) ~ (C) - - , - (D) 5 5 5 5 5/ v5' \ 5 5-> J fa B ansa I Classes
Conic Section (Parabola, Ellipse, Hyperbola)
[21]
(c) Consider the family of circles, x2 + y2 = r 2 ,2 < r < 5. If in the first quadrant, the common tangent to a circle of the family and the ellipse 4 x2 + 25 y2 = 100 meets the co-ordinate axes at A & B, then find the equation of the locus ofthe mid-point ofAB. [ JEE '99,2 + 3 + 10 (out of 200) ] Q.5
Find the equation of the largest circle with centre (1,0) that can be inscribed in the ellipse x2 + 4y2 =16. [REE'99,6]
Q.6
Let ABC be an equilateral triangle inscribed in the circle x2 + y2 = a2. Suppose perpendiculars from A, x2 y 2 B, C to the major axis of the ellipse, - y + - y = 1, (a > b) meet the ellipse respectively at P, Q, R so that a b P, Q, R lie on the same side of the major axis as A, B, C respectively. Prove that the normals to the ellipse drawn at the points P, Q and R are concurrent. [ JEE '2000,7]
Q.7
Let C, and C2 be two circles with C2 lying inside C r A circle C lying inside C, touches C j internally and C2externally. Identify the locus of the centre of C. [ JEE '2001,5] 2 2 x y Find the condition so that the line px + qy = r intersects the ellipse —y + —y = 1 in points whose a b 7t eccentric angles differ by —. [ REE '2001,3 ]
Q.8
Q. 9
Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact must on the corresponding directrix. [ JEE' 2002, 5]
Q.l0(a) The area of the quadrilateral formed by the tangents at the ends of the latus rectum of the x2 y2 ellipse— + — = 1 is 9 5 (A)9^3 sq.units
(B)27-\/3 sq.units
(C)27sq.units
(D)none
(b) The value of 6 for which the sum of intercept on the axis by the tangent at the point X2
cos 9, sin 9),
2
0 < 9 < 7t/2 on the ellipse — + y = 1 is least, is : Tt
Tt
Tt
(Q-
Tt
(D)[JEE'2003 (Screening)]
Q.ll
The locus ofthe middle point of the intercept of the tangents drawnfroman external point to the ellipse x2 + 2y2 = 2, between the coordinates axes, is
i?V=l < c >l?v =1
(D)
2?+7=i
[JEE 2004 (Screening)] x2 y2 Q. 12(a) The minimum area of triangle formed by the tangent to the ellipse —y + • (A) ab sq. units
= 1 and coordinate axes is
a 2 +b 2 . ^ (a + b)2 . / T a 2 + a b + b2 (B) — - — sq. units (C) — sq. units (D) sq. units
[JEE 2005 (Screening)] (b) Find the equation of the common tangent in 1st quadrant to the circle x2 + y2 = 16 and the ellipse x2 y 2 — + — = 1. Also find the length of the intercept of the tangent between the coordinate axes. [JEE 2005 (Mains), 4] fa B ansa I Classes
Conic Section (Parabola, Ellipse, Hyperbola)
[21]
KEY CONCEPTS
(HYPERBOLA)
The HYPERBOLA is a conic whose eccentricity is greater than unity, (e > 1). STANDARD EQUATION & DEFINITION(S) Standard equation of the hyperbola is 2 2 2 L _ l L = i. Whereb2 = a 2 ( e 2 - 1) a b
Vx = (ae, b /a) 2
or a2 e2 = a2 + b2 i.e. e2 = 1 + CA T.A
1+
FOCI: S s (ae, 0) &
S' = ( - ae, 0).
EQUATIONS OF DIRECTRICES:
x= e
&
x=
a . e
V E R T I C E S : A = (a, 0 )
&
A ' = ( - a, 0).
2b2 _ (C.A.)2 T.A.
/ (Latus rectum)
:
2a (e2 - 1).
Note: / (L.R.) = 2e (distance from focus to the corresponding directrix) TRANSVERSE Axis : The line segment A'A of length 2a in which the foci S' & S both lie is called the T . A . O F T H E HYPERBOLA. CONJUGATE AXIS
: The line segment B'B between the two points B' = (0, - b) & B s= (0, b) is called as
t h e C . A . O F T H E HYPERBOLA.
2.
3.
The T.A. & the C.A. of the hyperbola are together called the Principal axes of the hyperbola. FOCAL PROPERTY: The difference of the focal distances of any point on the hyperbola is constant and equal to transverse axis i.e. ||PS | - |PS' 11 = 2a. The distance SS' = focal length. CONJUGATE HYPERBOLA: Two hyperbolas such that transverse & conjugate axes of one hyperbola are respectively the conjugate & the transverse axes of the other are called CONJUGATE HYPERBOLAS of each other.
X2 V 2 eg. V2 V 2 * —+ — = 1 are conjugate hyperbolas of each. a b a 2 b2 Note That: (a) If e,& e2 are the eccentrcities of the hyperbola & its conjugate then ej 2 + e2 2 = 1. 2
2
&
(b) (c) 4.
The foci of a hyperbola and its conjugate are concyclic and form the vertices of a square. Two hyperbolas are said to be similiar if they have the same eccentricity.
RECTANGULAR OR EQUILATERAL HYPERBOLA:
The particular kind of hyperbola in which the lengths of the transverse & conjugate axis are equal is called an EQUILATERAL HYPERBOLA. Note that the eccentricity of the rectangular hyperbola is -fl and the length of its latus rectum is equal to its transverse or conjugate axis.
fa B ansa I Classes
Conic Section (Parabola, Ellipse, Hyperbola)
[21]
5.
y
AUXILIARY CIRCLE:
A circle drawn with centre C & TA. as a diameter is called the AUXILIARY CIRCLE of the hyperbola. Equation ofthe auxiliary circle is x2 + y2 = a2. Note from thefigurethat P & Q are called
/ P(a sec0, b tanB)
(-aA
t h e "CORRESPONDING POINTS " o n t h e
1
Xe i c (0,0) J A
\(a,0)
hyperbola & the auxiliary circle. '9' is called the eccentric angle of the point 'P' on the hyperbola. (0<9<2tt). Note : The equations x = a sec 9 & y = b tan 9 together represents the hyperbola
N
. „
X V — =1 a 2 b2
where 9 is a parameter. The parametric equations: x = a cos h
POSITION OFAPOINT'P'w.r.t. AHYPERBOLA: ..2
The quantity M
b or without the curve.
2 =
i is positive, zero or negative according as the point (x, y,) lies within, upon
LINE AND A HYPERBOLA:
X
2
"V
2
(a)
The straight line y=mx + c is a secant, a tangent or passes outside the hyperbola — + = 1 according as: c2 > = < a2 m2 - b2. a" b"1 TANGENTS AND NORMALS : TANGENTS: xx, Equation of the tangent to the hyperbola x : 1 at the point (x^y,) is i. a- b z ' "* a' b Note: In general two tangents can be drawnfroman external point (x, y,) to the hyperbola and they are y - y, = m,(x - x,) & y - y, = m2(x - x2), where m, & m2 are roots of the equation (x,2 - a2)m2 - 2 x ^ m + y ^ + b2 = 0. If D < 0, then no tangent can be drawnfrom(x, y}) to the hyperbola.
(b)
Equation ofthe tangent to the hyperbola
8.
2
2
- ~ = 1 at the point (a sec 9, b tan 9) is
xsec9
-
ytan9
= 1.
9J + 9 ,
6 9 sin cos i " 2 Note: Point of intersection ofthe tangents at 9, & 9? is x = a,y=b 9,+9 2 9j+9 2 cos cos
(c)
(d)
y = mx ± Va2m2 - bz can be taken as the tangent to the hyperbola X" a z b2 Note that there are two parallel tangents having the same slope m. Equation of a chord joining a & p is x a - B y , a + B11 a+B — cos --—sm = cos a 2 b 2 2
fa B ansa I Classes
Conic Section (Parabola, Ellipse, Hyperbola)
= 1.
[21]
NORMALS:
(a)
(b)
(c) 9.
x2 y 2 The equation of the normal to the hyperbola — z - = 1 at the point P(x,, y,) on it is a b 2 2 ax by 2 , 2 2 2 + —- = a - b =a e • x i Yi x2 y2 The equation of the normal at the point P (a sec9, b tanG) on the hyperbola —2 - - 2 _2 = i is a b ax b X y „2 , L2 2 2 H =a +D =a e . sec B tan 9 Equation to the chord of contact, polar, chord with a given middle point, pair of tangentsfroman external point is to be interpreted as in ellipse.
DIRECTOR
CIRCLE:
The locus of the intersection of tangents which are at right angles is known as the DIRECTOR CIRCLE of the hyperbola. The equation to the director circle is: x2 + y2 = a2 - b2. Ifb 2 a2, the radius of the circle is imaginary, so that there is no such circle & so no tangents at right angle can be drawn to the curve. 10.
H-1
HIGHLIGHTS ON TANGENT AND
NORMAL:
x 2 y2 Locus of the feet of the perpendicular drawnfromfocus of the hyperbola —— = 1 upon any tangent a b is its auxiliary circle i.e. x 2 +y 2 = a2 & the product of the feet of these perpendiculars is b2 • (semi C A)2
H-2
The portion of the tangent between the point of contact & the directrix subtends a right angle at the corresponding focus.
H-3
The tangent & normal at any point of a hyperbola bisect the angle between the focal radii. This spells the reflection property of the hyperbola as "An incoming light ray " aimed towards one focus is reflected from the outer surface of the hyperbola towards the other focus. It follows that if an ellipse and a hyperbola have the same foci, they cut at right angles at any of their common point. x 2 y2 x2 Note that the ellipse — + - 1 and the hyperbola a 2 b2 a2-k2 and therefore orthogonal.
H-4 11.
y
Light ray f Tangent 0 . •3/
* S'
J
""pf\
\
S
v2 I = 1 (a > k > b > 0) Xare confocal k2-b2
The foci of the hyperbola and the points P and Q in which any tangent meets the tangents at the vertices are concyclic with PQ as diameter of the circle. ASYMPTOTES:
Definition: If the length of the perpendicular let fallfroma point on a hyperbola to a straight line tends to zero as the point on the hyperbola moves to infinity along the hyperbola, then the straight line is called the Asymptote of the Hyperbola. To find the asymptote of the hyperbola:
x2 y2 Let y = mx + c is the asymptote of the hyperbola —-= j = 1. £L 1) Solving these two we get the quadratic as (b 2 - a2m2) x 2 - 2a2 mcx - a2 (b2 + c2) = 0 ....(1) fa B ansa I Classes
Conic Section (Parabola, Ellipse, Hyperbola)
[21]
In order that y = mx + c be an asymptote, both roots of equation (1) must approach infinity, the conditions for which are: coeff of x2 = 0 & coeff of x = 0. b & b2- a2m2 0 o r m = + — a a2 mc 0 => c = 0. x y equations of asymptote are — + — = 0 a b and a b combined equation to the asymptotes —j -
Note
y
y=o
B
A'
yc
\
A
( a, 0)
-0.
PARTICULAR CASE: When b = a the asymptotes ofthe rectangular hyperbola. x2 - y2 = a2 are, y = ± x which are at right angles.
(i) (ii) (iii) (iv)
Equilateral hyperbola <=> rectangular hyperbola. If a hyperbola is equilateral then the conjugate hyperbola is also equilateral. A hyperbola and its conjugate have the same asymptote. The equation ofthe pair of asymptotes differ the hyperbola & the conjugate hyperbola by the same constant only. The asymptotes pass through the centre of the hyperbola & the bisectors of the angles between the (v) asymptotes are the axes of the hyperbola. (vi) The asymptotes of a hyperbola are the diagonals of the rectangle formed by the lines drawn through the extremities of each axis parallel to the other axis. (vii) Asymptotes are the tangent to the hyperbola from the centre. (viii) A simple method to find the coordinates of the centre of the hyperbola expressed as a general equation of degree 2 should be remembered as: Let f (x, y) = 0 represents a hyperbola. Find — & — . Then the point of intersection of — = 0 & — = 0 ox dy ox dy gives the centre of the hyperbola. 12.
H-l
H-2 H-3
H-4
HIGHLIGHTS ON ASYMPTOTES: If from any point on the asymptote a straight line be drawn perpendicular to the transverse axis, the product of the segments of this line, intercepted between the point & the curve is always equal to the square of the semi conjugate axis. Perpendicular from the foci on either asymptote meet it in the same points as the corresponding directrix & the common points of intersection lie on the auxiliary circle. x2 y2 The tangent at any point P on a hyperbola —y - ~ = 1 with centre C, meets the asymptotes in Q and R a b and cuts off a A CQR of constant area equal to ab from the asymptotes & the portion of the tangent intercepted between the asymptote is bisected at the point of contact. This implies that locus of the centre ofthe circle circumscribing the A CQR in case of a rectangular hyperbola is the hyperbola itself & for a standard hyperbola the locus would be the curve, 4(a2x2 - b2y2) = (a2+b2)2. If the angle between the asymptote of a hyperbola
fa B ansaI Classes
1 is 20 then e = secG.
Conic Section (Parabola, Ellipse, Hyperbola)
[21]
13. (a) (b) (c)
RECTANGULAR HYPERBOLA: Rectangular hyperbola referred to its asymptotes as axis of coordinates. Equation is xy = c2 with parametric representation x = ct, y = c/t, t e R - {0}. Equation of a chord joining the points P (tj) & Q(t2) is x + tjt2y = c(t, +12) with slope m = — — . t,t 2 X V X Equation ofthe tangent at PCXj'y,) is — + — = 2 &atP(t)is —+ ty = 2c. x,
y,
(d)
Equation of normal: y — = t2(x - ct)
(e)
Chord with a given middle point as (h, k) is kx + hy = 2hk.
t
Suggested problems from Loney: Exercise-36 (Q.l to 6, 16, 22), Exercise-37 (Q.l, 3, 5, 7, 12)
EXERCISE-I Q.l Q. 2
Find the equation to the hyperbola whose directrix is 2x+y= 1, focus (1,1)&eccentricity V3 • Find also the length of its latus rectum. 2 y2 The hyperbola = 1 passes through the point of intersection of the lines, 7x +13y - 87 = 0 and a 2 b2 5x - 8y + 7 = 0 & the latus rectum is 32 V2 /5. Find 'a' & V. X
2
V
2
Q. 3
•2— = 1 prove that 100 25 (i) eccentricity=75/2 (ii) SA. S'A=25, where S & S' are the foci & Ais the vertex.
Q. 4
Find the centre, the foci, the directrices, the length of the latus rectum, the length & the equations of the axes & the asymptotes of the hyperbola 16x2 - 9y2 + 32x + 36y -164 = 0.
Q.5
For the hyperbola
2 v 2 = 1 drawn at an extremity of its latus rectum is parallel to an The normal to the hyperbola x — a 2 b2 asymptote. Show that the eccentricity is equal to the square root of (1
Q.6
If a rectangular hyperbola have the equation, xy = c2, prove that the locus of the middle points of the chords of constant length 2d is (x2 + y2)(x y - c2) = d2xy.
Q.7
A triangle is inscribed in the rectangular hyperbola xy = c2. Prove that the perpendiculars to the sides at the points where they meet the asymptotes are concurrent. If the point of concurrence is (x,, y,) for one asymptote and (x2, y2) for the other, then prove that x2y , = c2.
Q. 8
v The tangents & normal at a point onx = j cut the y - axis at A & B. Prove that the circle on AB 2 a b2 as diameter passes through the foci of the hyperbola.
2
2
Find the equation ofthe tangent to the hyperbola x2 —4y2=36 which is perpendicular to the line x - y+4=0. x 2 2v Q. 10 Ascertain the co-ordinates of the two points Q & R. wrhere the tangent to the hyperbola = l at 45 20 the point P(9,4) intersects the two asymptotes. Finally prove that P is the middle point of QR. Also compute the area of the triangle CQR where C is the centre of the hyperbola.
Q. 9
Q.ll
If 6j & 9, are the parameters of the extremities of a chord through (ae, 0) of a hyperbola x 2 v2 °i e-1 - 2— —2 = 1, then show that tan-2L -tan^f+e— - =0. 2 +1 a b
fa B ansa I Classes
Conic Section (Parabola, Ellipse, Hyperbola)
[21]
.2
2
Q.12 If C is the centre of a hyperbola —— = 1, S, S' its foci and P a point on it. a 2 b2 Prove that SP. S'P = CP2 - a2 + b2. Q.13 Tangents are drawn to the hyperbola 3 x2 - 2y2 = 2 5 from the point (0,5/2). Find their equations. 2 2 Q.14 Ifthe tangent at the point (h, k) to the hyperbola X V = 1 cuts the auxiliary circle in points whose a 2 b2
ordinates are y, and y7 then prove that — + — = — .
y, y2 k
Q.15 Tangents are drawn from the point (a, (3) to the hyperbola 3x2 - 2y2 = 6 and are inclined at angles 9 and <)| to the x -axis. If tan 9. tan (j) = 2, prove that p2 = 2a 2 - 7. Q.16 If two points P & Q on the hyperbola
X2 V 2 — = 1 whose centre is C be such that CP is perpendicular a 2 b2
to CQ & a < b, then prove that —— + —-— = — —. CP2 CQ2 a 2 b2
2 Q.17 The perpendicular from the centre upon the normal on any point of the hyperbolax R. Find the locus of R. a2
2 v— = 1 meets at 2 b
x2 y2 Q.18 Ifthe normal to the hyperbola —- - ~ = 1 at the point P meets the transverse axis in G & the conjugate a b axis in g & CF be perpendicular to the normal from the centre C, then prove that IPF. PG | = b2 & PF. Pg=a2 where a & b are the semi transverse & semi-conjugate axes of the hyperbola. 2
2
Q.19 If the normal at a point P to the hyperbola - — = 1 meets the x - axis at G, show that SG = e. SP, a 2 b2 S being the focus of the hyperbola. Q.20 An ellipse has eccentricity 1/2 and one focus at the point P (1/2,1). Its one directrix is the common tangent, nearer to the point P, to the circle x2 + y2 = 1 and the hyperbola x2 - y2 = 1. Find the equation of the ellipse in the standard form. Q.21
Show that the locus of the middle points of normal chords of the rectangular hyperbola x2 - y2 = a2 is (y2 - x2)3 = 4 a2x2y2.
Q.22 Prove that infinite number of triangles can be inscribed in the rectangular hyperbola, x y=c 2 whose sides touch the parabola, y2 = 4ax. Q.23 A point P divides the focal length of the hyperbola 9x2 - 16y2 = 144 in the ratio S'P : PS = 2 :3 where S & S' are the foci of the hyperbola. Through P a straight line is drawn at an angle of 135° to the axis OX. Find the points of intersection of this line with the asymptotes of the hyperbola. x2 y2 Q.24 Find the length ofthe diameter ofthe ellipse — + — = 1 perpendicular to the asymptote ofthe hyperbola 25 9 • X2 V 2 — - ~ = 1 passing through the first & third quadrants. 2 2 Q.25 The tangent at P on the hyperbola —— — = 1 meets one of the asymptote in Q. Show that the locus of a 2 b2 the mid point of PQ is a similiar hyperbola.
fa B ansa I Classes
Conic Section (Parabola, Ellipse, Hyperbola)
[21]
EXERCISE-II 2
2
Q. 1
v = 1 whose equation is x cos a + y sin a = p subtends a right angle The chord of the hyperbola x 2 — a b2 at the centre. Prove that it always touches a circle.
Q.2
If a chord joining the points P (a secG, a tanG) & Q (a sec
Q. 3
Prove that the locus of the middle point of the chord of contact of tangents from any point of the circle 2 2 f-2 (x 2 + y 2 ) x y_ X V 2 2 2 x + y = r to the hyperbola —2— - 2 = 1 is given by the equation 2 i2 b r2 a b va j x2 v2 A transversal cuts the same branch of a hyperbola — = 1 in P, P' and the asymptotes in Q, Q'. a b Prove that (i)PQ = P'Q' & (ii)PQ' = P'Q
Q.4 Q.5
Find the asymptotes of the hyperbola 2x2 - 3xy - 2y2 + 3x - y + 8 = 0. Also find the equation to the conjugate hyperbola & the equation of the principal axes of the curve.
Q.6
An ellipse and a hyperbola have their principal axes along the coordinate axes and have a common foci separated by a distance 2-Vl3 , the difference of their focal semi axes is equal to 4. If the ratio of their eccentricities is 3/7. Find the equation of these curves.
Q. 7
The asymptotes of a hyperbola are parallel to 2x + 3y = 0 & 3x + 2y = 0. Its centre is (1,2) & it passes through (5,3). Find the equation of the hyperbola.
Q.8
Tangents are drawn from any point on the rectangular hyperbola x2 - y2 = a2 - b2 to the ellipse
Q.9
x2 y2 —j + —r = 1. Prove that these tangents are equally inclined to the asymptotes of the hyperbola, a b The graphs of x2 + y2 + 6 x - 24 y + 72 = 0 & x2 - y2 + 6 x + 16 y - 46 = 0 intersect at four points. Compute the sum of the distances of these four points from the point (-3,2).
Q.10 Find the equations of the tangents to the hyperbola x 2 - 9y2 = 9 that are drawn from (3,2). Find the area of the triangle that these tangents form with their chord of contact. Q. 11 A series of hyperbolas is drawn having a common transverse axis of length 2a. Prove that the locus of a point P on each hyperbola, such that its distance from the transverse axis is equal to its distance from an asymtote, is the curve (x2 - y2)2 = 4x2(x2 - a2). 2 2 Q. 12 A parallelogram is constructed with its sides parallel to the asymptotes ofthe hyperbolaX "V— = 1, and a 2 b2 one of its diagonals is a chord of the hyperbola; show that the other diagonal passes through the centre.
Q. 13 The sides of a triangle ABC, inscribed in a hyperbola xy=c 2 , makes angles a, (3, y with an asymptote. Prove that the nomals at A, B, C will meet in a point if cot2a + cot2p + cot2y = 0 Q. 14 A line through the origin meets the circle x 2 +y 2 = a2 at P & the hyperbola x2 - y 2 = a2 at Q. Prove that the locus of the point of intersection of the tangent at P to the circle and the tangent at Q to the hyperbola is curve a4(x2 - a2) + 4 x2 y4 = 0. 2 2 Q. 15 A straight line is drawn parallel to the conjugate axis of a hyperbola X v = \ to meet it and the a 2 b2 conjugate hyperbola in the points P & Q. Show that the tangents at P & Q meet on the curve ( 2 2^ 4xz y4 y = —— and that the normals meet on the axis of x. i 2 b a 2 J a" ^Bansal Classes
Conic Section (Parabola, Ellipse, Hyperbola)
[24]
Q.16 A tangent to the parabola x2 = 4 ay meets the hyperbola xy = k2 in two points P & Q. Prove that the middle point of PQ lies on a parabola. 2
2
Q.17 Prove that the part of the tangent at any point of the hvperbola —— — = 1 intercepted between the a 2 b2 point of contact and the transverse axis is a harmonic mean between the lengths of the perpendiculars drawnfromthe foci on the normal at the same point. x2
y2
Q.18 Let 'p' be the perpendicular distance from the centre C of the hyperbola — - —- = 1 to the tangent
^
drawn at a point R on the hyperbola. If S & S' are the two foci of the hyperbola, then show that ( (RS + RS')2 = 4 a2 V P j Q.19 P & Q are two variable points on a rectangular hyperbola xy = c2 such that the tangent at Q passes through the foot of the ordinate of P. Show that the locus ofthe point of intersection of tangent at P & Q is a hyperbola with the same asymptotes as the given hyperbola. Q.20 Chords of the hyperbola
X2 V 2
= \ are tangents to the circle drawn on the line joining the foci as a 2 b2 diameter. Find the locus of the point of intersection of tangents at the extremities of the chords.
2 2 Q.21 From any point of the hyperbola X V — = 1, tangents are drawn to another hvperbola which has the a 2 b2 same asymptotes. Show that the chord of contact cuts off a constant area from the asymptotes. 2
2
Q.22 The chord QQ' of a hyperbola - — ^ - = 1 is parallel to the tangent at P. PN, QM & Q' M' are a 2 b2 perpendiculars to an asymptote. Showthat QM • Q' M' = PN2. Q.23 If four points be taken on a rectangular hyperbola xy = c2 such that the chord joining any two is perpendicular to the chord joining the other two and a, p, y, 5 be the inclinations to either asymptotes of the straight lines joining these points to the centre. Then prove that; tana • tanp • tany • tanS = 1. Q.24 The normals at three points P, Q, R on a rectangular hyperbola xy = c2 intersect at a point on the curve. Prove that the centre of the hyperbola is the centroid of the triangle PQR. 2 2 Q.25 Through any point P ofthe hyperbola ^ = l a line QPR is drawn with afixedgradient m, meeting a
k a 2 b 2 (l + m 2 ) the asymptotes in Q & R. Show that the product, (QP) • (PR)=—, z , z, . b -a m
EXERCISE-III
Q. 1
Find the locus of the mid points of the chords of the circle x2 + y2 = 16, which are tangent to the hyperbola 9x2 - 16y2 = 144. [ REE '97,6 ]
Q.2
If the circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points P(x1; y,), Q(x2, y2), R(x3,y3), S(x4, y4), then ( A ) X , + X 2 + X3 + X4 = 0 (B)y,+y 2 + y3 + y4 = 0 (C) Xj x2 x3 x4 = c4 (D) y i y 2 y 3 y 4 = c4 [ JEE'98,2 J
Q.3 (a) The curve described parametrically by, x = t2 +1 + 1, y = t2 - 1 + 1 represents: (A) a parabola (B) an ellipse (C) a hyperbola (D) a pair of straight lines
fa B ansa I Classes
Conic Section (Parabola, Ellipse, Hyperbola)
[21]
(b)
71
Let P (a sec 9, b tan 9) and Q (a sec
(c)
If x = 9 is the chord of contact of the hyperbola x2 - y2 = 9, then the equation of the corresponding pair of tangents, is: (A) 9x2 - 8y2 + 18x - 9 = 0 (B) 9x2 - 8y2 - 18x + 9 = 0 2 2 (C) 9x - 8y - 18x - 9 = 0 (D) 9x2 - 8y2 + 18x + 9 = 0 [ JEE '99,2 + 2 + 2 (out of 200)]
Q.5
The equation of the common tangent to the curve y2 = 8x and xy=-1 is (A) 3y = 9x + 2 (B)y = 2x+1 (C)2y = x + 8 (D)y = x + 2 [JEE 2002 Screening] 2 2 x y Given the family of hyperbols r — = 1 for a e (0, rt/2) which of the following does not cos a sin a change with varying a? (A) abscissa of foci (B) eccentricity (C) equations of directrices (D) abscissa of vertices [ JEE 2003 (Scr.)]
Q.6
The line 2x + p y = 2 is a tangent to the curve x2 - 2y2 = 4. The point of contact is
Q. 4
(A)(4,-V6) Q.7
(B)(7,-2V6)
(C) (2,3) X
2
Tangents are drawn from any point on the hyperbola — locus of midpoint of the chord of contact.
(D)(^,l) V
[JEE 2004 (Scr.)]
2
— = 1 to the circle x2 + y2 = 9. Find the [JEE 2005 (Mains), 4]
X2 V 2 Q. 8(a) If a hyperbola passes through the focus of the ellipse -— + — = 1 and its transverse and conjugate axis 25 16 coincides with the major and minor axis of the ellipse, and product of their eccentricities is 1, then X2
y2
(A) equation of hyperbola — - — = 1
X2
y2
(B) equation of hyperbola — - — = 1
(C) focus of hyperbola (5,0) (D) focus of hyperbola is (sV3, o) [JEE 2006,5] Comprehension: (3 questions) Let ABCD be a square of side length 2 units. C2 is the circle through vertices A, B, C, D and C, is the circle touching all the sides of the square ABCD. L is a line through A PA 2 +PB 2 +PC 2 +PD 2 (a) If P is a Fpoint on C,1 and Q in anotherFpoint on C,, — — — " ~ r 2 then Q A ' + QB + QC + QD (A) 0.75 (B) 1.25 (C) 1 (D)0.5
is equal to
(b) A circle touches the line L and the circle C, externally such that both the circles are on the same side of the line, then the locus of centre of the circle is (A) ellipse (B) hyperbola (C) parabola (D) parts of straight line (c) A line M through A is drawn parallel to BD. Point S moves such that its distances from the line BD and the vertex A are equal. If locus of S cuts M at T 2 and T ? and AC at T P then area of A T j T 2 T 3 is (A) 1/2 sq. units (B) 2/3 sq. units (C)lsq.unit (D)2sq.units [JEE 2006,5 marks each]
^Bansal Classes
Conic Section (Parabola, Ellipse, Hyperbola)
[26]
ANSWER KEY PA RABC)LA EXERCISE-I Q.2 Q.5 Q.8
(a, 0); a Q.3 2 x - y + 2 = 0, (1,4) ;x + 2y+ 16 = 0,(16,-16) 3x - 2y + 4 = 0 ; x - y + 3 = 0 Q.6 (4 , 0); y2 = 2a(x - 4a) y = -4x + 72, y = 3x - 33 Q.9 7y±2(x + 6a) = 0
Q.15 x2 + y2 + 1 8 x - 28y + 27 = 0 '
Q.19
8<^2 y -9
Q.18 x - y = 1; 8 V2 sq. units
4f
(2 8^ 22^ x — , vertex Q.20 v9 9y V 9j
'
1 P
v ^ Q.16 Q(4, -8)
4
/
a 2 >8b 2
EXERCISE-II
Q.3 [a(t2o + 4), - 2at0] Q.IO (a)
15a2/4 Q.21. (2a, 0) Q.23
Q.5 (ax + by) (x2 + y2) + (bx - ay)2 = 0
;(b)y = - ( x 2 + x)
Q.21y2 = 8ax
Q.12 ( (x, - 2a), 2y, )
(x2 + y2 - 4ax)2 = 16a(x3 + xy2 + ay2)
Q.18
EXERCISE-III 2
Q.l x - 2 y + 12 = 0 Q.3 x = 3
y
2/3 +2
1 , Q.4 x - 2 y + 1 = 0; y = mx +-— where m 4m
-5±V30 10
Q.5 (a) C ; (b) B Q.6 (x + 3)y2 + 32 = 0 Q.7 (a) C ; (b) D Q.8 C Q.9 D Q.IO (a) C; (b) a = 2 Q.ll B Q.12 xy2 + y2 - 2xy + x - 2y + 5 = 0 Q.13 (a) D, (b) A, B, (c) (i) A, (ii) B, (iii) D, (iv) C
ELLIPSE 2
EXERCISE-I
2
Q.l
20x + 45y - 40x - 180y - 700 = 0
Q.4
3x2 + 5y2 = 32
Q.8
x + y - 5 = 0, x + y+ 5 = 0
Q.9
0
Q.IO 24 sq.units
Q.ll 4=v4=
or y
; 4x ± V33 y - 32 = 0
Q.14 55 V2 sq. units Q.19
+ z l = _L a4
V2 V2
b4
c2
EXERCISE-II Q. 1 (a2 - b2)2 x2y2 = a2 (a2 + b2)2 y2 + 4 b6x2 Q.13 ^/r2 - b2
Q.5 bx + a V3 y = 2ab Q.15 12x + 5y = 48; 1 2 x - 5 y = 48
EXERCISE-III 1
1
Tt
Q.l (j) = Tt - tan"1 2, t =—-= ; <> j = Tt + tan_12, t = - = ;<|>=±—,t = 0 -v 5 V5 2 Q.4(a)A; (b)B,D ; (c)25y 2 + 4x 2 = 4x 2 y 2
Q.3 (i)C; (ii)C
Q.5(x-l) 2 +y 2
11
Q.7 Locus is an ellipse with foci as the centres of the circles C}a nd C2. Q.8
Tt
a2p2 + b2q2 = r 2 sec 2 - = ( 4 - 2 V 2 ) r 2
fa B ansa I Classes
Q.10 (a) C ; (b)A
Q.11C Q.12(a)A,(b)AB=
Conic Section (Parabola, Ellipse, Hyperbola)
14 [21]
HYPERBOLA^ EXERCISE-I Q.l
7x 2 + 12xy-2y 2 -2x + 4 y - 7 = 0 ;
Q.2 a2 = 25/2 ; b2 = 16
Q.4
(-1,2); (4, 2) & (-6, 2); 5 x - 4 = 0 & 5x+ 14 = 0 ; ~ ; 6 ; 8 ; y - 2 = 0 ;
ft*
V 5
x+1 = 0 ; 4 x - 3 y + 10 = 0 ; 4x + 3 y - 2 = 0. Q.9
x + y±3V3=0
Q.10 (15,10) and (3,-2) and 30 sq. units
,2 Q.17 (x2 + y2)2 (a2y2 - b 2 x 2 ) = x2y2 (a2 + b2)
Q.13 3x + 2y - 5 = 0 ; 3x - 2y + 5 = 0 (x-i) 2
( y _l) 2
f 4
'x 2
3
y2^
=3
EXERCISE-II Q.5
x - 2y + 1 = 0 ; 2x + y + 1 = 0 ; 2x2 - 3xy - 2y2 + 3x - y - 6 = 0 ; 3x - y + 2 = 0 ; x + 3y = 0
Q.6
—+ ^=1 49 36
V
2
2
2
2
-^-=1 9 4
Q.7
Q.9 40
6x2 + 13xy + 6y 2 - 38x - 37y - 98 = 0
Q.10
; x - 3 = 0 ; 8 sq. unit 2
2
Q.19 xy = —c 9
Q.20
2
J— a 4 b4 a +b
Q.21 ab
EXERCISE-III Q.l (x2 + y2)2 = 16x2 - 9y2
Q.2 A, B, C, D 2
Q.5
A
fa B ansa I Classes
Q.6
A
Q.7
Q.3 (a) A; (b) D ; (c) B (' x 2 +y2 2 Q.8
Conic Section (Parabola, Ellipse, Hyperbola)
Q.4
D
(a) A, (b) C, (c) C
[21]
TARGET I1T JEE 2007
P ) Y \ ft RfeV" S o i ft t v \ K £
o S ] 06 / o j
MATHEMATICS XII (ALL)
QUESTION BANK ON
DEFINITE & INDEFINITE INTEGRATION
Question bank on Definite & Indefinite Integration There are 168 questions in this question bank. Select the correct alternative : (Only one is correct) 00
Q.l
The value ofthe definite integral, J(ex+1 + e3~x)"' dx is l n tz 1 f NR 7t _1 11\ (A)tj (B)(C) ^ - - t a n " 4e 4e e 2^ 2 J cosfe x j-2xevx dx is 0 (B) 1 +(sin 1) (C)l-(sinl)
Q.2
The value ofthe definite integral,
£
(A) 1
Q.3
TC (D)
2e 2
(D)(sinl)-1
•/2 Value ofthe definite integral j ( sin '(3x-4x 3 ) - cos '(4x J -3x) )dx -V2 n In n (A)0 (B)-(C)y (D)~ x
Q.4
r _ adtt _ Let /(x) = J I j and g be the inverse off. Then the value of g'(0) is 2 Vl + t^ (A) 1
Q.5 (Ve
(B) 17
1 , cot" 1 (e x ) (A) 7 In (e2x + 1) +x+c x
1 , cot _1 (e x ) (B) - In (e2x + 1) + +x +c x
1 , 2x cor 1 (e x ) (C) — /n(e + 1 ) - — - y — - - x + c
i , corV) (D) ~ / n ( e 2 x + l ) + ^ - x
e
2
g-
e
2
e
2
e
+c
1 1k Lim —J f(l + sin 2x) x dx k->0 k 0 ^
(A) 2
Q.7
(D) none of these
rcot _1 (e x ) , . I dx is equal to : e 2
Q.6
(C) V n
(B) 1
(C) e2
(D)nonexistent
(B) 6-71
(C) 5-71
(D)None
/n5ex / " ^ d x (A) 4-7t
feBansal Classes
Q. B. on Definite & Indefinite Integration
[2]
fx Q.8 S'^f \es \qOVi
Q.9
Q.IO
If x satisfies the equation
a 2 sin a
( B ) ±
2 sin a a — a
_ I a (C)± sina
sma a
(D)± 2.
x , 1, If/(x) = eg(x) and g(x) = J j then f' (2) has the value equal to : 2 1 +t (D) cannot be determined (C)l (A) 2/17 (B)0 J etan 9 (sec 9 - sin 9) d9 equals : (A) - etan 9 sin 9 + c
Q.ll
(Jl
value x is (A)±
dt ^ 3 rt -t 2 sin2t smz ' 2 x dt x - 2 = 0 (0 < a < TC), then the 2 2 +2tcosa + l \-3{ t ' + l J I—}fc>
(B) etan 9 sin 9 + c
J (x-sin 2 x-cosx)dx o (B) 2/9 (A)0 r=4n
(B)
35
(D) etan 9 cos 9 + c
(C) - 2/9
(D) - 4/9
is equal to
Q.12 The value of Lim ]>] n->oo r=i Vr(3Vr + (A)
(C) etan 9 sec 9 + c
1 14
(C)
1
(D)
10
b-c
Q.13
J / ( x + c)dx =
V '"
b-2c
(A) Jf(x)dx
(B) |f(x+c)dx
(C) Jf(x)dx
(D) Jf(x+2c)dx
a-2c
n/2 . 2n n/2 1 / ,, f Sinx-COSX , r f. t, . 3•> N , r I I 6 s , Q.14 Let lj = j " : dx; l 2 = j(cos x)dx ; I3 = J(sm x)dx & I 4 = f/ n - - 1 dx then
£ Q.15
t
o
1+ s i n x
-cosx
o
(A)I1 = I2 = I3 = I4 = 0 (C) I, =I 3 = I4 = 0 but I2 * 0
_„/2
0
Vx
(B) lj = I2 = I3 = 0 but I4 * 0 (D) I, = I2 = I4 = 0 but I3 * 0
f — —7 d x equals : x(l + x )
J
(A)/nx+ - /n (1 + x7) + c
( B ) / n x - - /n(l - x 7 ) + c
(C)/nx- - In (1 + x7) + c
(D)/nx+ - /n(l - x 7 ) + c
r
7t/2n
Q.16
£
0
dx l+tannnx
(A)0
fe Bans aIClasses
Tt (B) 4n
(C)
nTt
Tt (D) 2n
Q. B. on Definite & Indefinite Integration
[14]
Q.17
f(x) = Jt(t—l)(t—2) dt takes on its minimum value when: o (A) x = 0 , 1
(B) x = 1,2
(C) x = 0 ,2
(D)x =
3+
S
a
£
Q.18
Jf(x)dx = -a a
a
a
(A) J[f(x)+f(-x)]dx (B) J[f(x)-f(-x)]dx (C) 2 Jf(x)dx
Q.19
Let f (x) be a function satisfying f 1 (x)=f (x) with f (0) = 1 and g be the function satisfying f (x) + g (x) = x2 l The value of the integral jf (x)g(x) dx is (A)e-^e2-^
Q- 2 0
£ Q.21
(D)Zero
(B) e - e2 - 3
(C)~(e-3)
(D)e
—
.
J. w JI\xA V T T ] ^ d x e < i u a l s : 2 (A) - V l + / n | x | ( / n | x | - 2 ) + c
(B) —^l + ln | x | (/n | x | + 2) + c
(C) ^ l + ln\x\ ( / n | x | - 2 ) + c
(D) 2-yJl + /n | x | (3 /n | x | - 2) + c
3» 2 j | i ( | x - 3 | + | l - x | - 4 ) | dx equals:
(A)--
(B)
(D)
(C)
8
Where {*} denotes the fractional part function. Q.22
1 P 1—x.cos— dx has the value:
J 3x
X
0 ^ (A)
8V2
Xy
(B)
24 V2
/ \ _ \ / 7t 2 71 2 + Lim — sec + sec 2 n—>oo 6n UnJ V 6n, -
Q.23
(A)
V3
fe Bans a I Classes
(B)V3
(C)
32 V2
(D) None
2, 7t 4 + sec (n-1)—- + — has the value equal to 6n 3 (C)2
Q. B. on Definite & Indefinite
Integration
[14]
Q.24
Suppose that F (x) is an antiderivative of f (x) = (A) F (6) - F (2)
Q.26 £ Q.27
X
(A)
X
4
, +c
+X+1
J
(B) - —7 X
+X+1
+c
c ^ ^ ^
x+1
(D)
(C)— +c x +X+1
"tK^ X+ 1
x 4 +x+l
+c
tcppaootCfr)
(n-l)7t + cos, equal to 2n J
1
(B)
iog x 2-
can be expressed as
i
Cl^ec^
x
7t 2 71 Lim — 1 + cos— + cos— + 2n 2n 2n (A) 1
X
fsi"2x , x> 0 then J
(B) ~ (F (6) - F (2)) (C) | ( F ( 3 ) - F ( 1 ) ) (D) 2( F (6) - F (2))
3x4 - 1 Q.25 Primitive of — t •> w x t X is: v (x + x +1)
a
smx
(C)2
(D)none
(C)2
(D)4
('ogx2)2 dx : tn2
(B)l
(A) 0
Q.28 If m & n are integers such that (m - n) is an odd integer then the value of the definite integral cos mx -sin nx dx o 2n
(A) 0
B
(D) none
( Qn ^- T m
( ) „2
n —m 2
Q.29 Lety={x}M where {xjdenotes the fractional part ofx&[x] denotes greatest integer
£ipGi-^o, (A) 5/6 , Wm /f Q.30 If |
£U>,
x4+l x(x 2 +l) 2
(B) 2/3 ' dx=A/n x +
(A)A=4JB = - 1 D)'-ff O ^ 0 1 - sin x Q.31 dx= 1-cosx it/2
J
(A)l-/n2
(C) 1 B 1+x
^
(B)/n2
(D) 11/6
i-
+ c, where c is the constant of integration then :
(B) A = - 1 ; B = 1 ^
•
(C)A=1;B = 1 y (C)l + /n2
(D)A = - 1 ; B = - 1
foe (D)none f(x)
Q.32 Let f: R —» R be a differentiable function & f (1) = 4, then the value of; Lim f 2 —l is: X-»l J xY—1 1 (B)4f'(l) (D) 8 f ' ( l ) (C)2f'(l) (A)f'(l)
fe Bans aIClasses
Q. B. on Definite & Indefinite Integration
[14]
Q.33
f(x) t2dt
If j o
= x cos roc, then f'(9)
1 (A) is equal to - — (it/2)I/3 Q.34 Jx 5 sinx 3 dx = o (A) 1
1 (B) is equal to - -
1 (C) is equal to -
(D) is non existent
(B) 1/2
(C) 2
(D) 1/3
Q.35 Integral of ^/l+2cotx(cotx+cosecx) w.r.t. x is: X
x
(A) 2 In c o s - + c
(B) 2 In sin~ + c
1 x (C) — In cos — + c
(D) In sin x - /n(cosec x - cot x) + c 3
Q.36 P ^
VL cA
If/(x)= | x | + | x— 1 | + | x - 2 i ,x e Rthen j / ( x ) d x (A) 9/2
(B) 15/2
Q. 3 7 Number of values of x satisfying the equation (A)0
(C) 19/2
(D)none
V
N +, 2 28 8t + 3—-1 + 4 dt=l . -A J °g(x+i)^x + 1
(B) 1
(C) 2
7t/2 (B) JJ - ^ - d x 0 smx
n/2 (C) ± J — d x 2 * smx
, is
(D)3
1
Q.38
+ -i ftan x . j dx = o x
n/4 . (A) J ^ d x * x
1 7t/4
1
) dt Q. 3 9 Domain of definition of the function f (x) = j ~T=2 2 is 6 o Vx +t (A)R (B)R+ (C) R+ u {0} Q.40 If J e3x cos 4x dx = e3x (A sin 4x + B cos 4x) + c then: (A) 4A = 3B (B) 2A = 3B (C) 3A = 4B b Q.41 If / ( a + b - x) = f (x), then jx.f (a + b - x) dx =
(D) | J — d x 2 £ sinx
( D ) R - {0}
(D) 4B + 3A = 1
a
(A) 0
(B) |
(C)
J / ( x ) dx
(D)
a
fe Bans aIClasses
Q. B. on Definite & Indefinite Integration
J / ( x ) dx ~
a
[14]
Q.42
The set of values of'a' which satisfy the equation j(t - log2 a) dt = log.21 2 IS a y 0 (A)aeR
Q.43
(B)aeR+
(C)a<2
(D)a>2
The value ofthe definite integral J ^2x --J5(4x - 5) + ^2x + V5(4x - 5) dx 2 /
(A)
7V3 + 3V? ^
^ (B) 4 V2
4 (C) 4 V 3 + -
(D)
7V7-2V5
Q .44 Number of ordered pair(s) of (a, b) satisfying simultaneously the system of equation b b 3 2 jx dx = 0 and Jx2dx = - is a
(A) 0 _ .. Q.45
a
(B) 1
(C)2
(D)4
ftan -1 x-cot _1 x , . . —— — dx is equal to : J tan x+cot x 4 2 (A) - x tan"1 x + - /n (1 + x2) - x + c 71 7t
4 2 (B) - x tan"1 x - - In (1 + x2) + x + c 7t 7t
4 2 (C) - x tan"1 x + - In (1 + x2) + x + c 71 7T
4 2 (D) - x tan"1 x - - In (1 + x2) - x + c 7T 71 y
f dt d2y Q.46 Variable x and y are related by equation x = j / 7 . The value of —y is equal to oVl + t dx ( A ) ^
2y < p ) j —
(B)y
(D)4y
1 x+h dt Q.47 Let /(x) = Lim - f r = = 2, then Lim x • / ( x ) is h->o h ^ t + Vl + t (A) equal to 0
1 (B) equal to —
(C) equal to 1
(D) non existent
Q.48 If the primitive of f (x) = 7t sin 7ix + 2x - 4, has the value 3 for x = 1, then the set of x for which the primitive of f(x) vanishes is: (A) {1,2,3}
(B) (2, 3)
(C) {2}
(D) {1,2, 3, 4}
Q.49 Ifa f & g are continuous functions in [0, a] satisfying f (x) = f (a - x) & g (x) + g (a - x) = 4 then Jf(x).g(x)dx = 0 , a
(A) — J f (x)dx 9 fe Bans aIClasses
a
(B) 2 J f (x)dx 0
a
(C) j f(x)dx 0
a
(D)4_[ f(x)dx 0
Q. B. on Definite & Indefinite Integration
[14]
Q.50
In f x+^/l+x2 J x, — 7=—7—- dx equals : V1+? (A) ^ ^ / n f x + V l + x 2
(C) | . In2
+
+c
(7x-6)~ 1/3
31 (A) ~7
(D)
+x"
+c
/n (x+V^?) +x + c
z. 0
f Vl^
Q.51 If/(x) =
2 (B) J . In f
-x +c
(B)
32
(D)
21
55 42
i
Q.52 The value ofthe definite integral je6" (l + x-e x )dx isequalto (B) e e - e
(A)ee
Q-53
1 . J " sin 1/2 '
X
V
xj
(C) e e - 1
(D)e
(P>4
(D)2
dx has the value equal to 3
(A) 0
(B)T
Q.5 4 The value ofthe integral f e o (A) 1 (B) - 2 0
f
2x
(sin2x + cos2x)dx =
Ze
Tt
ft , ^
(B) ~/n2
(D) zero
(C) - n In 2
(D) Tt /n 2
-z
Q. 5 5 The value of definite integral j —— (A)--/n2
(C) 1/2 A
dz
.
Q. 5 6 A differentiable function satisfies 3/ 2 (x)/'(x) = 2x. Given/(2) = 1 then the value of/(3) is (A) 3/^4
(B)3/^
(C)6
(D)2
e
Q.57 For I n = J (/n x)ndx, n e N; which ofthe following holds good? (A)I„ + ( n + l ) I n + 1 = e (C)In + 1 + (n+l)I n = e
fe Bans aIClasses
(B)I n + 1 + nl n = e (D) I n+ j + (n - 1) ln = e
Q. B. on Definite & Indefinite Integration
[14]
r 1 Q. 5 8 Let f be a continuous functions satisfying f (In x)
x
for
0
for x > 1
and f (0) = 0 then/(x) can be
defined as - 1
if x < 0 (A) f(x)
:
(B)f(x) =
- l-ex
if x > 0
x
if x < 0
ex
if x > 0
(C)f(x) =
- e x - 1 if x > 0 r
(D)f(x) =
if x < 0
x
if x < 0
- e x - 1 if x > 0 f(x)
Q.59
Let f: R —» R be a differentiable function such that f (2) = 2. Then the value of
71/2 Q.60 (A)
Q. 61
(B) 12 f'(2)
(B)
2-Jl + a
+a
Lim Y n-*ȣrjn
1 (B)-/nx-ex + C
n
y-j +kx
x
dtis 2
(D) none
(C) 32 f' (2)
(C)
2 it
(D) none
Vi + a
1 0 (C)-/n2x-x + C
(D) — + C 2x
> 0 is equal to
1
(A) x tan (x)
(B)tan- (x)
(C)
tan ! (x)
(D)
tan '(x) 2^
2cos2x sin(2x) -sinx n/2 2 Let f (x) = sin2x 2sin x cosx then J [f(x) + f'(x)]dx 0 sinx -cosx 0 (A) 71
Q.64
_
1 Let f (x) = — /n — then its primitive w.r.t. x is ve x y
_1
Q.63
X
dx , 2 - 2 has the value: 1+a sin x
1 (A)-ex-/nx + C Q.62
3
j — 2
(A)6f'(2)
4t
(B) %/2
(C)2n
(D) zero
19
The absolute value of f s i n x is less than: J 1i 10 1 -f~ x
(B)IO- 11 (C) 10 ~7 (D) 10 -9 I! Q. 6 5 The value of the integral J (cos px - sin qx)2 dx where p, q are integers, is equal to: (A)10- 1 0
- JT
(A)-7i fe Bans a I Classes
(B)0
(C) 7t
Q. B. on Definite & Indefinite
(D) 27t Integration
[14]
Q.66
Primitive o f / ( x ) = x • 2 /n(x2+1) w.r.t. x is 2/n(x2+l) J-" + C v(A) ; 2(x +1)
( B )
( x 2 + 1 ) /n2 + l
(C)
Q.67
Lim f|
n->=o Jl
1+ -
n+1
dt is equal to (B)e 2
(A)0
+ c
(x 2 +l) / n 2 (D) +C 2(/n 2 + 1)
2(/n2 +1) \n
( x ^ 1 ) 2 ^ /n2 + l
(C) e2 - 1
(D) does not exist
x+h
Jfti 2 t dt - j>n 2 t dt
Q.68
S
Q.69
( C )
(B) ln2x
(A)0
mx
(D) does not exist
Let a, b, c be non-zero real numbers such that: l
J (1 + cos8x) (ax2 + bx + c) dx = J (1 + eos8x) (ax2 + bx + c) dx , then the quadratic equation ax2 + bx + c = 0 has : (A) no root in (0,2) (C) a double root in (0,2) Q .70
Letin=
jc/4 J tan"xdx,then
(A) A.P. Q.71
(B) atleast one root in (0,2) (D) none 1 I2 + I4
I3 + I5
(B) G.P.
,.... are in:
I4 + I6
(C) H.P.
(D) none
Let g (x) be an antiderivative for/(x). Then In(i + (g(x)) 2 ) is an antiderivative for 2/(x)g(x) (A)
l + (/(x)) 2
2/(x)
2/(x)g(x) (B)
l + (g(x)) 2
(C)
l + (/(x)) 2
(D)none
ji/4
Q.ll
J (cos 2x)3/2. cos x dx : 0 (A)
3tc 16
(B)
3tx 32
\6\ 2 1/V2
Q.73
The value of the definite integral
(A)
Tt
feBansa I Classes
(D)
!
16
x 2 dx
^ f IS o v 1 — X2 (1 + V1 — X2 ) 1
+
3 71 a/2
' 4 7J
(C)
Tt
1
4
V2
Q. B. on Definite & Indefinite
(D) none Integration
[14]
Q.74 The value ofthe definite integral j({x}2 + 3(sin 27tx))dx where { x} denotes thefractionalpart function. 19
(A)0
(C)9
(B)6
(D) can not be determined
IT/2
Q. 7 5 The value of the definite integral jV tan x dx, is (B)
(A) V2tt
7T
(C) 2V2 7T
VI
(D)
7t
2V2
Q.76 Evaluate the integral: Jf / n ( ^v6 x (B) |[/n 2 (6x 2 )] +C
(A)^[/n(6x 2 )] 3 +C o
1 (D) - ^ [ / n ( 6 x 2 ) ] 4 + C
(C) ^-[/n(6x2)] +C Q.77
to
j - ( 3 s i n 9 ) 2 - - ( l + sine) 71/6^ (A)TT-V3
d0 (C) 7C - 2-\/3
(B)n
(D) 71 +
V3
2
?dt Lim Q.78 Let /= L™ — j/ntdt then the correct statement is x->°o J— •> t and m= x->co- 7x/nx r (A) / m = /
Q.79
(B)/m = m
If f (x) = e"x + 2 e"2x + 3 e~3x +
(C)/ = m
+ 00 , then
(D) / > m
/n3 jf (x) dx = fa 2
(A)l
(B)-
(C)
1
(D) In 2
ju/4
n/2
Q.80 If I = J ^n(sinx) dx then J &i(sinx + cosx) dx : jc/4
(B)
(A)i I ( n
Q.81
r
(A)n
(C)
7J
^ / n
The value of Jj [ ] ( + ) 0 Vr=l
Q.82
x
I 1
Z—
(D)I
dx equals
j Vk=i x + k y
(B)n!
(C)(n+ 1)!
(D) n•n !
rcc cos3x+cos5x J- 2 dx • 4 sin x+sm x (A) sinx-6tan 1 (sinx) + c (C) s i n x - 2 (sinx)" 1 -6 tan"1 (sinx) + c
fe Bans a I Classes
(B) s i n x - 2 sin *x + c (D) s i n x - 2 (sinx)-1 + 5 tan"1 (sinx)+ c
Q. B. on Definite & Indefinite Integration
[14]
Q.83
J
^x
2
+ 4x + 4
+ yjx2 - 4x + 4 5 3 (B)ln- + —
CA)hf-f Q.84
Q-85
w
2
2
(C) In - + -
w
2
(D)none
2
The value ofthe function f(x)= 1+x+ J (ln2t + 2 lnt) dt where f' (x) vanishes is: i _1 (A) e (B) 0 (C) 2 e_1 (D)l+2e"1 n n n Limit I 11 +, +J +J + n \n + 1 yn + 2 \n + 3
(B) 2V2 — 1
+
n + 3 (n - 1)
has the value equal to
(D)4 00 Let a function h(x) be defined as h(x) = 0, for all x ^ 0. Also j/z(x) • / ( x ) dx = f (0), for every (A)2 V2
Q.86
dx =
(C) 2
OO
—CO
function f (x). Then the value of the definite integral J/?' (x) • sin x dx, is -00
(A) equal to zero
(B) equal to 1
(C) equal to - 1
(D) non existent
jc/4
Q.87
J (tan11 x + tan11 0 (A)
1 n-1
(B)
Q.89
n+ 2
(C) n - 1
(D) none of these
4 (C) In ~ e
(D)4
1/x
(\
Q.88
x)d(x — [x]) is: ([• ] denotes greatest integer function)
Lim J(l + x) x dx vo
is equal to
(A) 2 /n 2
(B)-
4
Which one ofthe following is TRUE. (A) x. Jf — = x / n | x | + C x
(B) x. Jf — = x/n | x | + Cx x
(C) cosx
(D) —-—• fnc v Jfcosx dx =x + C
jcosx dx = tanx+C
fe Bans a I Classes
Q. B. on Definite & Indefinite
Integration
[14]
Q.90
2
j x 2 n + , -e _ x dx is equal to (n e N). o (A)n!
(B)2 (n!)
i
n
(C) ^
(D)
(n + 1)!
o Q. 91 The true set of values of 'a' for which the inequality J (3 ~2x - 2.3"x) dx > 0 is true is: a
(A) [0,1]
(B) ( - o o , - l ]
(C)[0,oo)
(D)(-oo,-l]u[0,oo)
a
Q.92 If a e (2,3) then number of solution of the equation J cos (x + a 2 ) dx = sin a is : o (A) 1 (B)2 (C)3 (D)4. x2
Q.9 3 If x • sin nx = Jf (t) dt where/ i s continuous functions then the value of f (4) is o n (A)-
Q- 94
J
1 (C) -
(B) 1
(D) can not be determined
( x 2 + 4 x + l) 3/2 3
X +c (A) ( X 2 + 4X + 1)1/2
ru\ B
X 2 T7T 2 ( >(X + 4X + 1)1/2
x2 „ (C) 2 1/2+C ( X + 4x + 1 )
I ( )(X + 4X + 1)1/2 D
2
+c
+ C
2
Q.95 If the value of the integral { ex dx is a , then the value of | Jinx dx is: (A) e4 - e - a
Q-
96
V3 r i1d d tan J 7~ dx ji 2~2 dx{ 71
(A) 3
'
(B)2e4-e-a
e
(C)2(e4-e)-a
(D)2e4-l-a
2x H 1 -x 2 / equals 71
(B)--
71
n
(O-
(D)?
(C)-ae" a
(D)Aea
Q.97 Let A= f e d t then f-—— has the value J , t-a-1 oo 1 + t a~ l (A)Ae-a
(feBansal Classes
(B)-Ae _a
Q. B. on Definite & Indefinite Integration
[13]
ji/2
Q.98
j V s i n 2 e sin6 d0 is equal to : (B) 7t/4
(A)0
Q.99
(C) 7t/2
CD) 71
rx2 +2 —j dx is equal to x +4 2
2x
1
2x
(B) | tan -1 (x 2 +2)+ C (D) ± tan"1 2
2x
+C
Q.l 00 Ifp + 2 Jx2 e""2 dx = Je-"2 dx then the value of p is (A)e -1
(D) can not be determined
(C)l/2e
(B)e
Q. 101 A quadratic polynomial P(x) satisfies the conditions, P(0) = P(l) = 0 & J P(x) dx = 1. The leading o coefficient of the quadratic polynomial is: (C)2 (B)-6 (A) 6 (D) 3 Q. 102 Which one of the following functions is not continuous on (0,7t)? x (B)g(x)= | t sin - dt (A) f(x)= cotx 1
0
(C) h (x): L L
JV
Q.103 I f f ( x )
2
• —x 2O sin 9
371
371
x sin x ,
0
7t
(D) I (x): 7t . 7t — S i n ( X + 7t) , — < X < 7 t
< X < 7t
2
2
tsintdt
. r, • T for0
(A) f (0+) = - r t
H H
(B)f
(C) f is continuous and differentiable in
z
v
(D) f is continuous but not differentiable in
V
fe Bans aI Classes
7t
{4)
7t
8
y
L
J
Q. B. on Definite & Indefinite Integration
[14]
Q. 104 Consider f(x) -
; g(t) = J f(t) dt. Ifg(l) = 0 then g(x) equals
1 + x3
( A ) ^ n ( l + x3)
(B)^n
V ^ V
f' 1I + x A^ V 3
(C) 2
(D) - f t i
100 Q.l 05 The value ofthe definite integral, J -^ydx is equal to (A)y(l-e-10)
(B) 2(1 - e~10)
(C)|(e-10-1)
(w |
(.-.-)
Q.106 J [2 e x] dx where [x] denotes the greatest integer function is (B) In 2
(A)0 Q.l 07 The value of
C dx
1 2
(A)
(C) e2
(D) 2/e
(C)4
(D) undefined
3 1,1 (C) —+ - / n — 4 2 54
1 , 27 3 (D) - / n ^ 2 2 4
is (B)2
x Q.108 J x f a f l. + • dx 2) o r (A)
3s] 1-2/n-j 2J
v(B) ;
3 7,3 /n2 2 2
-qx"-' dx is + +2xP 1+l
pXP+2"-'
Q.l 09 The evaluation of J x2p+ 2q (A) -
Q.110
x p + q +l
+C
(B)
x p + q +l
+C
(C)
x p + q +l
+C
(D)
x +l p+q
+C
+1 J\ xx ++1x1 2|x| + 1 dx = a In 2 + b then: 1
3
2
(A) a = 2 ; b = 1 b
(B) a - 2 ; b = 0
(C) a = 3 ; b = - 2
(D)a = 4 ; b =
b
Q-111 j [x]dx+ j [—x] dx where [. ] denotes greatest integer function is equal to : a
a
(A) a + b
(B) b - a
(C) a - b
(D)
a +b
Q.l 12 If J 375 x5 (1 + x2) ~4 dx = 2n then the value of n is: (A) 4
fe Bans a I Classes
(B)5
(C)6
(D)7
Q. B. on Definite & Indefinite Integration
[14]
1/2
0.113
j
j
+ x
f ^ n - — dx is equal to : (j 1-x 2 1-x (A)
(B) ^ In2 3
M
(C)-^ln23
(D) cannot be evaluated.
J ( x 3 - 2 x 2 + 5 ) e 3 x d x = e3x (Ax3 + Bx2 + Cx + D) then the statement which is incorrect is
Q.l 14 If
(A) C + 3D = 5 (C) C + 2B = 0
(B)A+B +2/3 = 0 (D) A + B + C = 0 dx
"'f —: =/n 2. then the value ofthe def. integral. J • sin x + cos x
Q.l 15 Given J
7t
(B)--/n2
(A)^/n2 Q. 116
( C ) 7 - ^/n2
sinx
dx is equal to
(D)^+/n2
A function f satisfying f' (sinx) = cos2x for all x and f(l) = 1 is: x3
X3 2 (B) f(x)=— + —
1
(A)f(x) = x + — - —
x3 1 (D)f(x)=^-y + -
x3 1 (C)f(X) = X - y + V3/2
Q.117 ForO
7t
(A) 12
(C) ^-[(n/3-1 )f (sin-v/3-sinl)
(D) - V3-1 )-(sinv3-sinl J
f Q118
X cos
O
j
¥
X
o^ydxisequalto:
(A) Tt - 2
(B) - (2 + 7t)
(C) zero
(D) 2 -
Tt
Q.119 Jf —?=- (x + Vx) dx vx
Q.120
(A)2e Vx [ x - V ^ + l ] + C
(B) e V x [ x - 2 V ^ + l
(C)
(D) e ^ ( x + V^ + l ) + C
x/2
|
(x + Vx) + C dx is equal to: cos x + sin6x 6
(A) zero
fe Bans aIClasses
(B) Tt
(C) TT/2
(D) 2 Tt
Q. B. on Definite & Indefinite Integration
[14]
dz>x j sin 2 xdxis:
Q. 121 The true solution set of the inequality, 0
(A)R
(B) (1,6)
o
(C)(-6,l)
V /?nx * Q.l 22 If J r1 - x 2 dx = k f In (1 + cosx) dx then the value of k is : oV o (A) 2 (B) 1/2 (C) - 2
(D) (2,3)
(D)-1/2
Q.l23 Let a, b and c be positive constants. The value of 'a' in terms of 'c' if the value of integral I
J(acxb+1 + a 3 bx 3b+5 ) dx i s independent of b equals 0
m j f
(C)|
(D)JJ
Q.124 Jsec2G (secG+tanG)2 dG (A) ( s e c 9 + t a n e ) [ 2 + tanQ(sec6 + tane)] + C (B) (C)
(SeC 6
^ t a n 9 ) [2 + 4 tan 6 (sec 6 + tan 6)] + C
&!£Q±^>[2
(D)
Q.125
+
tan0(secG+tanG)] + C
[2 + tanG(secG + tanG)] + C
r X +1
J ~ T T dx is equal to:
1 X ii
(A) ± tan"1 V2
1
cor 1 2
(C) | tan-1 |
(D)
tan"12
(C) f(Xj)
(D) does not exist
X
X
Q.126 V™ 1 —
(B) ~
f
1
J f(t) dt is equal to : x i
f (x )
(A) -L-12. x i
(B)Xjf(Xj)
Q. 127 Which of the following statements could be true if, f" (x) = x1/3. I
II
f(x)=- 9 -x 7 / 3 + 9 28 (A) 1 only
f'(x)=—x7/3-2 28 (B) III only
fe Bans aIClasses
III
IV
f ' ( x ) = - x4/3 + 6 f(x) = — x4/3 - 4 4 4 (C) II & IV only (D) I & III only
Q. B. on Definite & Indefinite Integration
[14]
it/2
Q.l 28 The value ofthe definite integral J sinx sin2x sin 3xdx is equal to:
tan ' x Q.129
2
TO-3
(A) j
2
(1 + x )
sec
1
(D)-
(Q-
'l-x^' dx V 1 + x 2 ] + cos 1 : vl + x y
( X >0)
,tan- 1 ,X .(tan' 1 x) 2
(A) e tan_ x .tan -1 x + C
(B)
tan 1 Ax .1[ sec~*l -1 Vl + x z I I + C (C) el£Ul
(D)e
+ c
tan-lx f cosec A \ Vl + x 2 I I + c
2
Q.130
ofpositive solution ofthe equation, f (t ~ {*}) d t = 2 ( x - 1) where {} denotes the fractional o part function is: (D) more than three (C) three (B) two (A) one
Number
i
1
Q.131 If f(x) = cos(tan- x) then the value of the integral Jx f "(x) dx is
(A)
3-V2
(B)
3 + V2
Q.l32 If jJ 1 + s i n - dx=Asii V / v . (A) 2V2
.y
(C)l
then value ofA is:
(B) V2
(C) ^
1
(D) 4V2
1 1 n n Q.l 33 For U = j x (2 - x) dx; Vn = J xn (1 - x)n dx n e N, which ofthe following statement(s) 0 is/are ture? (B) Un = 2 - Vn (C) Un = 22" Vn (D) Un = 2 " 2« Vn (A)Un = 2"Vn
Q.134
J
(x ~ ! ) d x
(X 4 +3X 2 +1) tan
(A) In x + — V
X
fe Bans a I Classes
J
1
x2+l V
x
= /n | f (x) | + C then /(x) is
V
(B) tan
' V
1 ^ x +— X
J
(C) cot x + — v X; -1
f r 1 1 (D)/n tan x +— A V v JJ
Q. B. on Definite & Indefinite Integration
[14]
71/3 Q.l35 Let f(x) be integrable over (a, b), b > a> 0. If Ij = j f(tan0 + cot9). sec 2 9d0& 7t/6 7l/3 j 2 I2 = I f (tan 6 + cot 9). cosec 0 d 9, then the ratio j - : ti/6
2
(A) is a positive integer (C) is an irrational number
(B) is a negative integer (D) cannot be determined.
sin x
Q.136 f(x)= J (1 - t + 2t 3 )dt hasin[0,2rc] cosx 71 371 (A) a maximum at — & a minimum at — (C) a maximum at ~
371 771 (B) a maximum at — & a minimum at —
& a minimum at ~
(D) neither a maxima nor minima
Q.137 LetS (x) = J I n t d t ( x > 0 ) a n d H ( x ) = ^ - ^ . T h e n H ( x ) i s ; x2
X
(A) continuous but not derivable in its domain (B) derivable and continuous in its domain (C) neither deri vable nor continuous in its domain (D) derivable but not continuous in its domain. sinx
d f dt r . Q. 13 8 Number of solution of the equation —- J -—-y =2 v2 in [0, re] is "
(A) 4
X
(B)3
COSX
t
(C)2
(D)0
1./ x 2 s i n 2 x - 1 cosx(2sinx + l) Q.l39 Letf(x) = + -—; : then cosx 1 + sinx je x (f(x) + f'(x))dx (where c is the constant of integeration) (A) ex tanx + c
(B) excotx + c
(C) ex cosec2x + c x+3 Q. 140 The value of x that maximises the value of the integral jt(5 - 1 ) dt is X (A) 2 (B)0 (C)l
(D) exsec2x + c
(D) none
Q.141 For a sufficiently large value of n the sum of the square roots of the first n positive integers i.e.
+
+ sjr3+
(A)V/2
+vn (B)|n3/2
f dx Q.142 Thevalueof Jt, ZT is o (A) -2 (B) 0
fe Bans aIClasses
approximately equal to (C)~n 1 / 3
(D)|n 1 / 3
(C) 15
(D) indeterminate
Q. B. on Definite & Indefinite Integration
[14]
~ , »af dx "r8 2 tan 9 Q. 143 If 1 —7= = ——-d6 , then the value of 'a' is equal to (a > 0) „ Vx + a+Vx > sm20 h \ j 3 ( ) 4 A
B
( )
Tt
of
4
(D)
16
_ r sin(/n(2 + 2x)) , Q. 144 The value ofthe integral J T~--j- - d x [ s f (B) In sin-
(A) - cos In (2x + 2) + C
V
7IX
' X + ly
+c
/ O \
f 2 ^ (C) cos yX —+r \J +C Q.l45 If f(x) =Asin
\
(D) sin + B, f
vX + ly
+C
- ] = V2 and { f(x) dx = — , Then the constants A and B are
respectively. (A) —& — 2
(B) ! & ! 71 Tt
2
(C) 0& - — 7t
(D) - & 0 71
tc/2 2 k/1 2 V2 2 e_x X Q.l 46 Let I,= J sin(x)dx . j = J V dx ; I = Je~ x i (l + x)dx 0 and consider the statements I n III Ii
Q. 147 Let f (x) =
sinx x
2
(A)-Jf(x)dx
, then
( j f (x) f — - x dx = J n V2 (B) | f ( x ) d x 0
(QTtJf(x)dx 0
71/2 V/n(x +1) Q.148 Let u — J — j dx and v= J/n(sin2x)dx then x +1 0 0 (A) u = 4v (B) 4u + v = 0 (C) u + 4v = 0
Q. 149 If / ( x ) =
(ft sinx sin Ve f —" .de then the value of/!, _ 2 , l + cosW0 ^ tT/16
(A) ft
fe Bans aIClasses
I.-I3
(B) — Tt
(C) 2ti
1
(D)-Jf(x)dx
(D) 2u + v = 0
,1S
(D)0
Q. B. on Definite & Indefinite Integration
[14]
jt/2 . c r sin 5x Q.l 50 The value ofthe definite integral, J bin• x 0 (A)0
(B) |
dx
is
(C) 71
(D) 2it
Select the correct alternatives : (More than one are correct) b Q. 151 Jsgn x dx = (where a, b e R) a
(A) | b | -1 a |
(B) (b-a) sgn (b-a)
(C) b sgnb - a sgna
(D)|a|-|b
x^ Q.152 Jf— =A.tan-1 nrtan + C then: 5 + 4cosx V 2 (A) X = 2/3 (B) m = 3 (C)X=l/3
(D) m = 2/3
Q.l 53 Which ofthe following are true? Jt-a
Jt - a
(A) Jx . f(sinx) dx =—. J f ( s i n x ) d x 2
a
a
(B) Jf(x)2 dx = 2. Jf(x) 2 dx 0
n,t (C) Jf(cos 2 x)dx = n. *Jf(cos 2 x)dx
b-c b (D) Jf(x + c) dx= f f(x) dx
1 2 2x _ ,_. _ _f ZX +3x+3 +JX + J Q.154 Thevalueofl —T-r2 v dxis: 0 (x+l)(x +2x+2J
( A ) j +21n2-tan- 1 2
(B)j +21n2-tan-1|
(C) 2 ln2 - cor 1 3
(D) - 7 + ln4 + cor 1 2 4
Q.l55 Jf x
+
x
1+x
cosec2x dx is equal to :
(A) c o t x - c o t _ 1 x + c
(B) c - cot x + cot -1 x
COS CC X
(C) - t e n
- 1
_i
* - + c
(D) - e' n tan~
x
-cotx + c
where 'c' is constant of integration. Q. 156 Let f(x) = J — dt (x > 0) then f(x) has: 0 t (A) Maxima if x = n7t where n = 1,3,5,... (B) Minima if x = n7t where n = 2,4,6, (C) Maxima if x = n7t where n = 2,4,6,... (D) The function is monotonic
fe Bans aIClasses
Q. B. on Definite & Indefinite Integration
[14]
Q.l 57 If I = J /
dx
vn ; n e N, then which of the following statements hold good ?
0 (l + X |
TT
(A) 2n I n+ j = 2 _n + (2n - 1) In
-
7t _ J_ (C)I 2 = 8o - T 4 Q.158 J - ^ y
1
( B ) I 2 = ^8 + 4 +
—
equals:
1 . , x+1 ^ 1 , 2, x + 1 + c (D) +c (A) \ In2 j- +c v(B) 7 In2 j- +c v(C) 4- In2 v y - In v 4 x-1 ' 2 x+1 ' 4 x+1 ' 2 x— 1 n/2sin (2n - 1) x n/2 sm ( . nx \2 dx;Bn= J d x ; for n e N , then Q.159 IfA n = J sin x o v sin x y (B)B n + 1 = Bn (A)An + 1 =A n (D)Bn + 1 - B n = A n + 1 (QA n + 1 - A n = B n + 1 Q.160 }J 0
(1 + x) (1 +
X2)
dx: (B)
(A)?
(C) is same as Jj 0
Q.l61 J V1 +
cscx
dx (1 + x) (1 + x 2 )
71
(D) cannot be evaluated
dx equals
(A) 2 sin-1 Jsinx + c
(B) V2 c o s V c o s x + c (D) cos (1 - 2 sin x) + c
(C) c - 2 sin -1 (1 - 2 sin x) Q.l62 I f f ( x ) = n j ^ n ( 1 + x s i " 2 9 ) d 0 , x > 0 then: 0 sin 0 (A)f(t)=--7C ( V T T T - i )
(D) none of these.
(C) f (x) cannot be determined
Q.l 63 If a, b, c e R and satisfy 3 a + 5 b + 15 c = 0 , the equation ax4 + bx 2 + c = 0 has (A) atleast one root in (-1,0) (B) atleast one root in (0,1) (C) atleast two roots in (-1,1) (D) no root in (-1,1)
Q.164 Letu= JX4+7X2+1 (A) v > u
fe Bans aIClasses
x 2 dx &V =
(B) 6
JX4+?X2+1 V = 71
then: (C) 3u + 2v = 5n/6
(D)U
Q. B. on Definite & Indefinite Integration
+ V=
TI/3
[14]
Q.l 65 If J e u .sin2x dx can be found in terms of known functions of x then u can be: (A) x (B) sinx (C) cosx (D) cos2x Q. 166 If f(x) = | -—- dt where x > 0 then the value(s) of x satisfying the equation, 1
'
+
*
f(x) + f(l/x) = 2is: (A) 2 (B) e
(C) e "2
(D)e 2
, 2 &f J f(x) dx=— 19 can be: Q. 167 A polynomial function f(x) satisfying the conditions f(x) = [f' (x)] o 12 ,,. x 3 9 A + X + ( )T 2 4
( B )
x J T"2X
+
9 ?
, _v x (C>T-X
_ + 1
, x (D)T
+ X + 1
Q. 16 8 A continuous and differentiable function1 f1 satisfies the condition, X
j f (t) d t = f 2 (x) - 1 for all real' x'. Then: o (A)'f'ismonotonicincreasing V x e R (B)' f' is monotonic decreasing V x e R (C)1 f' is non monotonic (D) the graph of y=f (x) is a straight line.
fe Bans aIClasses
Q. B. on Definite & Indefinite Integration
[14]
ANSWER
a osrb a swb 3 OWb v serb a oerb a S3rb a 03ib 3 sirb a oirb a sorb v oorb a S6b 3 06'b 3 S8'b V 08'b a srb v orb a S9 b V 09 b v ssb v osb a st?b 3 Ofr'b a s£b D orb a S3b v orb d srb a orb d sb
fe Bans aIClasses
a'3 99I'b a'v 39lb a'a ssrb a'D'v frsrb V 6Hb Vw b V 6erb a Krb 3 63ib 3 pzrd V 6irb 3 wrb 3 6orb a wrb a 66 b a wb a 68'b a P8d a 6Lb a w/b a 69 b 3 wb 3 6Sb 3 wb a 6Pt) a ppb a 6eb a Kb a 63"b V pzd a 6lb 3 wb V 6b 3 pd
a'3' a'v S9I'b a'v I9ib a'v LS rb a'3'a'v £srb a 8H"b a ewb 3 8£lb 3 ££lb a 83I'b V £3l'b a sirb V £irb V sorb 3 £orb a 86 b V £6'b a 88 b 3 £8b V 8r b 3 £rb a 89 b V £9b a 8S'b V £sb 3 a 3 8£'b V ££'b a 83 b V £3'b V 8fb V £I'b a 8b a £b
KEY
a'D'a P9 rb 3'V 09lb a'v 9srb a'v 3srb V /.Hb a 3H"b a z,£ib a 3£I'b a Z,3l'b a zz rb V z,irb a nrb 3 /.orb A 30 rb a Z.6b a 36 b V Z,8'b 3 38 b a Z.Z/b 3 3r b 3 L9t> 3 39 b 3 z.sb V 3S b a z.rb a 3t?'b a z,£b a 3£'b V z.3"b 3 33'b 3 z,rb 3 3ib V rb 3 rb
Q. B. on Definite & Indefinite Integration
a'V 89 It) a'a L9i'0 3'a'v £9lt> a'v 6srb a'3'a ssrb 3'V isrb a 9Wb a iwb a 9£lb a i£rb a 93l"b a I3I'b 3 9irb 3 urb a 9orb a lorb V 96 b a i6b 3 98'b a i8b a 9r b a trb 3 99 b 3 i9b a 9Sb a isb a 9t?b 3 3 9£b V irb V 93 b 3 irb a 9rb a irb 3 9b V rb
[14]
8
BANSAL CLASSES TARGET ITT JEE 2007
MATHEMATICS X I I (ABCD)
DEFINITI 7 & INDEFINITE INTEGRATION
CONTENTS KEY-CONCEPTS EXERCISE I EXERCISE-II EXERCISE-III EXERCISE-IV ANSWER KEY
KEY 1.
CONCEPTS
DEFINITION:
If f & g are functions of x such that g'(x) = f(x) then the function g is called a PRIMITIVE ANTIDERIVATIVE O R INTEGRAL of f(x) w.r.t. x and is written symbolically as
f ci I f(x) dx = g(x) + c <=> — (g(x) + c) = f(x), where c is called the constant of integration, dx STANDARD RESULTS :
2. i)
OR
r J
\n+l
(ax+b)n (ax + b)°dx= , \ a(n+l)
iii) f e
ax+b
dx = - e a
ax+b
+c
(ii)
n*-l
j
ax+b
=-a
/n(ax + b) + c
r x+ 1 npx+q (iv) I aP i dx = — (a > 0) + c . p fna
+c
v) f sin(ax+b) dx = -— cos(ax+b) + c a
(vi) J cos(ax+b)dx= - sin(ax+b) + c a
vii) j tan(ax + b) dx = — In sec (ax + b) + c a
(viii) f cot(ax+b) dx = - /n sin(ax+b)+ c a
ix) J sec2 (ax + b) dx = — tan(ax + b) + c a
(x) j cosec2(ax + b) dx = _ 1 cot(ax + b)+ c
xi) J sec (ax + b) . tan (ax + b) dx = — sec (ax + b) + c a xii) J cosec (ax + b) . cot (ax + b) dx =
a
cosec (ax + b) + c In tan {
+yj
+ c
xiii) J secx dx=In (secx + tanx) + c
OR
xiv) J cosec x dx = In (cosecx - cotx) + c
OR In tan ^ + c OR - In (cosecx + cotx)
xv) J sinh x dx = cosh x + c (xvi) J cosh xdx = sinh x + c (xvii) J sech2x dx = tanh x + c xviii) J cosech2x dx = - coth x + c
(xix) J sech x . tanh x dx = - sech x + c
xx) J cosechx. cothxdx= - c o s e c h x + c
(xxi) j"
xxii) J
xxiv) JJ '
„ dx „ = - tan -1 — + c a +x2 72 =ln + L x + a2 dx
xxv) J
xxvi) |
(xxiii) J
2
dx a -x 2 2
x2 + a2
Tn x + l , a+x In +c 2a a-x -
dx dx 2
x-v/x -a
OR
sinh-1 - + c a
OR
cosh"1 - + c a (xxvii) j
. , x , = sin 1 — + c
dx x -a2 2
Definite & Indefinite Integration
2
1 x = — sec-l1 —^+1 a
1 , x-a In +c 2a x+a [2]
(xxviii) j .y]a2-x2 dx = 2 (xxix) {
A/a
2
- x 2 + — sin"1 - + c 2 a
dx= — ^ x 2 + a 2 + — sinlr1 - + c 2 2 a
*Jx2+a2
(xxx) j Jv x 2 - a 2 dx= - Jv x 2 - a 2 2
- — cosh"1 - + c 2 a
e3*
f
(xxxi) I eax. sin bx dx = —-—x- (a sin bx - b cos bx) + c a +b f
ax
(xxxii) J e ax . cos bx dx = — — - (a cos bx + b sin bx) + c a +b 3. TECHNIQUES OF INTEGRATION: (i) Substitution or change ofindependent variable. Integral 1= j f(x) dx is changed to } f(4> (t)) f'(t) dt, by a suitable substitution x = (J) (t) provided the later integral is easier to integrate. (ii)
Integration by part: { u.vdx=u j" vdx- {
d U du A . f vd? r1Y J dx
dx where u&varedifferentiable
function. Note: While using integration by parts, choose u & v such that (a) J v dx is simple &
(b) /
du f vdx r)v dx J
dx
is simple to integrate.
(iii)
This is generally obtained, by keeping the order of u & v as per the order of the letters in ILATE, where ; I-Inverse function, L-Logarithmic function, A-Algebraic function, T-Trigonometric function & E-Exponential function Partial fraction, spiliting a biggerfractioninto smallerfractionby known methods.
4.
INTEGRALS OF THE TYPE:
0)
j [f(x)]"f'(x)dx
(")
Jj
ax' + bx + c
OR {
dx
putf(x) = t & proceed
> 1 I , dX > f Vax2 + bx + c
Express ax2 + bx + c in the form of perfect square & then apply the standard results. ("0
Jf
q —?2 PX ^— dx, Jf i = = dx . ax + bx + c ' ^/ax2 + bx + c
Express px + q=A (differential co-efficient of denominator) + B. (iv)
j ex [f(x) + f'(x)] dx = e x . f(x) + c
(vi)
JJ
(vii)
J
f
r
dx ns N x(x n +l)
dx
(n l ) / n
x2'(xn+l) "^
(ilZ?a«sa/
(v)
J [f(x) + xf'(x)] dx - x f(x) + c
Take xn common & put 1 + x n = t .
n e N , take xn common & put 1+x n = tn
Classes
Definite & Indefinite Integration
[18]
(viii)
dx c»(l + x f
dx a+bsin z x
n
take xn common as x and put 1+x OR
r
Multiply N" & D dx a+bsin x
00
r
dx J" a+bcos
2
x
= t.
Jasin
2
dx x+bsin xcos x + ccos 2 x
by sec2x & put tanx = t . dx a+bcos x
OR
OR
n
OR
i
dx a + bsin x + ccos x
Hint: Convert sines & cosines into their respective tangents of half the angles, put tan — = t a.cos x + b.sin x + c ^ Lcos x + m.sin x + n
(xi)
x2+l x2-l dx OR | —; dx where K is anyJ constant . x 4 + K x 2 +1 x+Kx+1 Hint: Divide Nr & Dr by x2 & proceed .
(xii)
(xiii)
dx „ r dx 1 & J 7—t—; \—i 2 (ax + b) Vpx + q (ax + bx + cj ^px + q
(xiv) (ax
(xv)
Express Nr = A(Dr) + B — (Dr) + c & proceed . dx
; put px + q = t .
• j-u 1 ( dx • 1 dx , put ax + b = -1 ; . , , p u t x = -1 (ax + bx + cj yj px + qx + r + b) J-px + qx + r
x- a dx or p-x —^
°r
- a) (p - x) ;
put x = a cos2 0 + P sin2 0
I V(x ~ a ) ( x ~ P) >
put x = a sec2 0 - 0 tan2 0
J
dx ; put x - a = t2 or x - P = t 2 J(x - a) (x - p)
(ilZ?a«sa/
Classes
Definite & Indefinite Integration
[18]
DEFINITE 1.
INTEGRAL
f f(x) dx = F(b) - F(a) where J f(x) dx = F(x) + c a
b
VERY IMPORTANT N O T E
:
If
F
f(x) dx = 0 => then the equation f(x) = 0 has atleast one
a
root lying in (a, b) provided f is a continuous function in (a, b) . 2.
PROPERTIES O F DEFINITE INTEGRAL : b
P-L
b
b
a
j f(x) dx = J f(t) dt provided f is same
P - 2 f f(x) dx = - J f(x) dx
a
•
a
b
c
b
a
a
c
a
b
P-3 J f(x) dx= J f(x) dx+ J f(x) dx, where c may lie inside or outside the interval [a, b] . This property to be used when f is piecewise continuous in (a, b). a
P-4
J f(x) dx = 0 if f(x) is an odd function i.e. f(x) = - f ( - x ) . = 2 j f(x) dx if f(x) is an even function i.e. f(x) = f(-x) . 0 b
b
a
a
P-5
J f(x) dx = j f(a + b - x) dx, In particular J f(x) dx = J f(a - x)dx
P-6
j f(x) dx - J f(x) dx + J f(2a - x) dx = 2 J f(x) dx if f(2a - x) = f(x) 0 0 0 0 = 0 if f(2a - x) = - f(x) na
P-7
a
J f(x) dx = n Jf(x)dx ; where'a'is the period ofthe function i.e. f(a+x) = f(x) 0 0 b+nT
b
j f(x) dx = J f(x) dx where f(x) is periodic with period T& n e l .
P-8
a+nT
a
na
j f(x) dx = (n - m) aj f(x) dx if f(x) is periodic with period 'a'.
P-9
ma
0 b
b
P-10 If f(x) < ())(x) for a < X < b then J f(x) dx < J 4> (x) dx
P-LL
Jf (x)dx < ] I f(x) I dx.
P-12 If f(x) >0 on the interval [a, b], then J f(x) dx > 0.
(ilZ?a«sa/
Classes
Definite & Indefinite Integration
[18]
WALLI'S FORMULA: ji/2 J sm-x. c o s ^ dx - [ Q - l ) ( n - 3 ) ( n - 5 ) . . . . lot 2X(m-1)(m-3).... l o r 2 j (m+n)(m+n-2)(m+n-4).... lor2 Where K
=
7C
2
-1 4.
K
if both m and n are even (m, n e N) otherwise
DERIVATIVE OF AN TIDERIVATIV E FUNCTION : If h(x) & g(x) are differentiable functions of x then, m -dx p g(x) 1 m d t - f [ h (x)]. h'(x) - f [g (x)]. g'(x)
5.
DEFINITE INTEGRAL AS LIMIT OF A SUM : b
J f(x) dx = Lmut h [f (a) + f (a + h) + f (a + 2h) +
+ f ( a + n-1T»)]
a
=
h I f (a + rh) where b - a = nh f=0
If a = 0 & b = 1 then, Limit Limit 6.
f(rh) =
h
J f(x) d x .
where n h = l
0
R
2 f f - - ] = J f(x) dx . v n ; r=i v n y o
ESTIMATION OF DEFINITE INTEGRAL: b
(i)
For a monotonic decreasing function in (a, b); f(b). (b - a) < J f(x) dx < f(a). (b - a) &
(ii)
For a monotonic increasing function in (a, b); f(a).(b - a) < j f(x) dx < f(b).(b - a)
a
b
a
7.
SOME IMPORTANT EXPANSIONS :
...
,
0) (iii) , v
(v)
1
1
2 1
1
1
1
—2~I 2
2
1
3
—
1
1
2
4
1
-+ —
4
5
1
3
1 2
00
4
-+
6
(ilZ?a«sa/ Classes
„
7t2
00=7—
12
,
....
1
,
(iv)
1
1
1
- 52 - + —2 T + ~ T + — T + l 2 3 4
(u) .
,
1
1
1
1
-r-+—+—+.—+
1
3
5
7
00 :
7T
6 co=-
7t2
71 2
-1 1
22"
,
= In 2
8
2
24
Definite & Indefinite
Integration
[18]
EXERCISE-I tan 20
Qll
Vcos6 9 + sin6 0
d0
dx
Q.4 J
5x4 + 4x5
Q2f
5
(x + x + l)
eJ
dx
Q.5 Integrate j
+
by the substitution z=x + ^/x'2 +2x-1
X^X2 + 2 X - 1
K-'f Q.6 j
cos X dx 1 + tanx
dx
2
I s lx
cos 9+sin 9 d0 Q.7 | cos 20 . In cos 0-sin 9
^nxdx
a 2 sin 2 x+b 2 cos 2 x dx Q.9 j 4 - 2 a sin x+bi 4 cos2 x
Q.IO J
Q.12 f
sin(x-a) d x V sin(x+a)
Q.13 J (sinx)"11/3 (cosx)"1/3dx
Q.15 j
1-Vx dx !l+Vx
Q.16 j sin"-l
Q.17 |
^/x2+l[ln(x2+l)-21nx]
dx
Q.ll J V x ^ + 2 dx
(x + Vx(l + x) ) 2
X
cot xdx Q.14 j (1-sin x)(sec x+1)
dx
l+ X
dx Q.18 j ln(lnx)+
dx Q-8 f - y ^ J sin ^xx + sin 2x
(lnx)
dx
Q.19
j
x+1 4+xe*)2
dx
Q.20 Integrate ^ f'(x) w.r.t. x4 , where f(x) = tan -1 x + /n^/l+x - / n ^ l ^ (Vx + l)dx j Vx(Vx+1)
Q.21
Q.22 f
cosec x-cotx secx ]j cosec x+cotx^/l+2secx dx Q.27 j sinx+secx ^Q-30
\
\2
x cosx-sinx
dx
V
S m X
r e cosx v(x sin3 x + cosx) Q.36 j • ' dx sin2 x
,„ \ (7x-10-x )
2 3/2
2 - 3 x j1+x }Ei Q.42 j 2 + 3x 1-x
^Bansal Classes
x
Q- 23
cosx-sinx dx 7-9sin2x
„ _, r Q.26 j
2
dx dx secx+cos ecx
dx Q.29 j — f — sinx Jsin(2x + a)
r 3+4sinx+2cosx dx Q.31 J : 3+2sinx+cosx
. ^n(cosx+Vcos2x] Q.32 J[ —^ ~ dx sm x
'
Q.40 q.43 fJ
J 3
Q.28 J tan x.tan 2x.tan 3x dx
Q 37
dx
J:V(x + l) (x-l)
sin^ ,/cos3 ^
Q 35 \
Q.34 Jf — ^ sin x +1 -tan x
Q.33 f ^ dx V sinx + V cosx
Q.3'9 j
dx
dx
3x2 +1 (x2-l)3
dx
r (ax2 -b) dx J xVc2x2-(ax2+b)2
Q.38
J (^Jxlnx
Q.41 f J i z * dx v J Ali+x X
dx
f " " dx 1 + 3sm2x
r
e \z—a /= (l-x)Vl
dx
4x 5 - 7x 4 + 8x3 - 2x 2 + 4x - 7 dx Q.44 {J x 2 (x 2 +1)2
Definite & Indefinite Integration
m
X 2 Q 4 5 1 ((x 2+3x+3 I V>j^/x+1 r~T
^ r Q.48
Q.46 jJ ^ - xx-2 x
dx 3 Y x V(l + x)3
Q.49 v
f Jf
2
dx
Q.47 JJ (x-a)V(x-a)(x-(3) <• (l + x 2 )dx S — r a e (V0 , TJC) vQ.50 J I l-2x2cosa+x4
Jcos2x —s i n x dx
EXERCISE-II 71
Q2IVTH3I dx
q'!^™
!/n3xdx
0 TI/2 TI/4 x dx 4 2 Q.5 J cos 3x. sin 6xdx Q.6 j c o s x _ o • o (cosx + sinx)
ji/2 Q.4 J sin2x. arctan(sinx)dx o Q.7
QJ
Let h (x) = (fog) (x) + K where K is any constant. If
dx
(h(x)) =
^ ^ — then compute the cos (cos x)
f(x)
value of j (0) where j (x) = j" —— dt ; where f and g are trigonometric functions. g(x) S W n/2
Q.8
j 7(cos 2n_1 x - cos 2n+1 x) dx where n e N
-it/2
Q.9
Evaluate the integral: J (jx + 2 yflx - 4 + Jx-2 ^2x^4 j dx 3 00
Q.10 I f P =
00
2
i x r xdx -dx; Q = j l+x J 1+ X
0
(a) Q.ll
Q=
(b) P = R,
Provethat ? l
rx x(l-x) i i -4 x J 1+X ,
0
, dx
Q.15 f ^ - 1f+ xj d x 0 2ti / \ ' 7T X N x J e cosf — + — dx 2/ v4
r j
0
dx T 1+ X
then prove that
(C)P-V5Q + R = ^
. ( n - l K a , h)x + n a h ) ^ ^ b n . _ a n-! (x+ a) (x-t- b) 2(a + b) 2 ff vx2 l.In n vx
rf y x 2 " x dx Q. 14 Evaluate: J 7 2
, q.13 J 2LJE2L dx
0 Jl-x
_2"VX +4
2
^
Q.18
and R =
0
J a
Q.12
oo
Q.16
Q 19
n/2 J 0 sm(f+xj f V2
dx
2721-n , dx Q. 17 J j 2+sin2x ^ 0
2X7+3X6-1 0X 5 -7X 3 -1 2X 2 +X+1dx X2+2
Q.20 If for non-zero x, a f(x) + bf —j = f — I - 5; where a ^ b then evaluate j f(x) dx. vx ji/4 Q.21 J c o s x - s m x d x Q.22 fJ ( a X + b ) s £ C X t a n X dx (a,b>0) v V 7 0 10 +sin 2x 4 + tan x ' 0
(ilZ?a«sa/ Classes
Definite & Indefinite Integration
[18]
(2x+ 3 )
3/2 Q.24 J |x. sinTtxl dx
x
Q.23 Evaluate: f f dx. I (1 + cos x) p+qn
Q.25 Show that J | cos x| dx = 2q + sinp where q s N & -— < p <— o 2 2 "5 2/3 Q.26 Show that the sum ofthe two integrals J e (x+5 ^dx+3 J Q9(x-2/3f dx is zerQ 1/3 it/4 71/2 smx+cosx r sin x Q.27 dx v > 0, b > 0) Q.28 *J aq sin 2—72 2 4 < a m V4.h cos rr\c —yx dx (a ' ' x+b "o cos x+sin x 0 n/2 _ f . 1I -tX. + xX x3 Vl+sinx+ .^1-sinx Q.30 j tan" dx ^/l+sm x - ,/l-sin x
I
p
«
Q.31 (a) J Vto^dx; (b)
^
x.dx .1.UA
J
// 2 2 v, 2 Vlx -a Ab
H )
2
1 Q.3 2 j | x - t ! .cos7it dt where 'x' is any real number o Q.34 Provethat J =J - J * n 0 1 +x 0 (l-x ) X zi
x
dz = e ^ f e ' ^ 4 dz o
OO
4
1
—
Q.36
o x + 2xcos9 + 1
j
Q.42
J
dx
16
Q.44
sinB
(ilZ?a«sa/
dx
if 0 s (0,ic)
dx 2
n0 x + 2 x c o s 9 + 1
.
if
0
g
(k,2k)
271 (a+b) V^ab (a 2 +ab+b 2 ) ' X ,u
s, X r Prove that J f f (t) dt du= J f(u).(x-u)du. n J o VO J
Q.41
27t
(a) j t a n "
2x-7t
Q.43 Evaluate J /n(Vl-x + Vl + x)dx
o (5+ 4 cosx)2 1
dx
'^
? dx 2 2 Q.39 Prove t h a t j (x +ax+a ) (x2+bx+b V Q.40
a
r x 2 sin 2x,sin (y.cos x )
J
9 - 2 7i sin 9
x2(sin2x-cos2x) dx (1+sin 2x)cos2 x
2 V
dx 2
1
=2
2a-x
(n> 1)
Q . 3 7 ( a ) | l ^ , J » -3 , (b) f o l+x'Vx+x^x ' 0 1+x
Showthat f —
1
Q.33 j x sin-1
X
Q.35 Showthat J e " - e " o
Q.38
2a
d x , (b) Evaluate J
Classes
.
2
d0
a>b>0
Definite & Indefinite Integration
[18]
A
Q.45 Let f(x) = - j /n cos y dy then prove that f(x) = 2f
v4
2,
-2f
71
X _
4 2
-x/n2.
Jt/2
sec
Hence evaluate J
t dt
~ .^ , r x. lnx , , r a x. dx Q.46 Showthat J f ( - + - ) . dx = lna.Jf(-+-). x a x x a x 1
-(2x332 +x 998 +4x 1668 -sinx 691 ) dx Q. 47 Evaluate the definite integral, j 1 + x666 -l Q.48 Prove that
0» IM
(a) } V ( x - a ) ( P - x )
p (c)5{
p
dx xA/(x-a)()3-x)
x.dx
V^P
4 cos x Q.49 If f(x) = (cosx-1) 2 (cosx + 1)2
1 (cosx + 1)2 (cosx + 1)2
1 (cosx-1) 2 , find Jf(x) dx cos 2 x
1
Q.50 Evaluate:
/ n t a n xx -1, Je/ntan_ -sin _1 (cosx)dx. -1
o
EXERCISE-III Q.l
If the derivative of f(x) wrtx is — - then show that f(x)is a periodic function.
Q.2
Find the range of the function, f(x)= f
Q.3
A function f is defined in [ - 1 , 1 ] as f (x) = 2 x sin — - cos — ; x ^ 0 ; f(0) = 0 ;
f(x)
SmX
—7 .
1 — 2t c o s x + t
X
X
f (I/71) — 0 . Discuss the continuity and derivability of f at x=0. -1 if - 2 < x < 0 I and g(x) = J f(t) dt. Define g (x) as a function of x and test the x -* 1 if 0 < x < 2 _2 continuity and differentiability of g(x) in (-2,2).
Q.4
Let f(x) = [
Q.5
Prove the inequalities : (a) 0 < J 0
x 7 dx_
Ml
/3
1
(b) 2 e~1/4 < } ex2"x dx < 2e2.
8
2s (c) a< f — < b then find a & b. i 10+3cosx Q.6
dx (d) ^ < J 2 + x2 " 6 —2 0
Determine a positive integer n<5, such that J e x ( x - l) n dx= 16 - 6e.
(ilZ?a«sa/ Classes
Definite & Indefinite Integration
[18]
Q.7
Using calculus
(a)
If | x | < 1 then find the sum of the series _ L + 2 x + 4 x 3 +4 1 + X 1 + X2 1 + x
/(b) UN
1,1 / ! provethat .U * If |x|I <1
(c)
Prove the identity f (x) = tanx + - tan
Q.8
If (j)(x) = cos x - J (x -1) cj)(t) dt. Thenfindthe value of <|>" (x) + cj)(x).
Q.9
1x d2 If y = - f f (t) • sin a(x -1) dt then prove that —^ + a 2 y =/(x). A^t* a9 * 0 " dx
l - 2 x - + 2 x -r4 x 3 + 4x 3 -8x 7r + r 1-x+x 1 —x +x 1-x+x x
+
1
x tan^j +
8x?
+ 1+x 8
QO .
.00 = 1 + 2X - .2 l+x + x
l x + 77 tan
l
x = — 7 cot ^77 - 2cot 2x
X
jlntdt
Q.10 If y = x 1
, find 7^- at x = e. dx
1
Q. 11 If f(x) = x + j [xy2+x2y] f(y) dy where x and y are independent variable. Find f(x). Q. 12 If fw(9) = —
I
de
dx ~- COS0 cosx
. Show that f' (9). sin 9 + 2 f(9). cos 9 = ~
r f' fx") Q. 13(a) Let g(x) = x c . e2x & let f(x) = J e 2t . (312 + 1)1/2 dt. For a certain value of'c', the limit of — ^ 0
g' (x)
as x ->• 00 isfiniteand non zero. Determine the value of'c' and the limit. f t2 dt J JaTt (b) Find the constants 'a' (a > 0) and 'b' such that, Lim : — = 1. x-»o b x - sinx
Q. 14 If f: R -> R is a continuous and differentiable function such that, x
J
O
x
f (t) dt + f ' " (3) | dt -
-1
x
j
(t ) dt - f ' (1) 3
1
2
J
2
(t ) dt + f " (2)
X
x
J
(t) dt
3
thenfindthe value of f' (4). Q. 15 Given that Un = {x(l - x)}n & n > 2 prove that
- n (n - 1) Un_2 - 2 n(2n - l)Un_p
1 further if Vn - J e x . U n dx, prove that when n > 2, V n + 2n(2n- l).V n _ r n(n-1) Vn_2 = 0 0 f J?nt Ait Q. 16 If J —2—2 .2 0 X ~F~ T equation.
=
71 %£n22
"1-x
4
if
( x > 0) then show that there can be two integral values of 'x' satisfying this
0
Q.17 Let f(x)=
0 if l < x < 2 • Define the function F(x) = J f(t) dt and show that F is 2 (2-x) if 2 < x < 3 ° continuous in [0, 3] and differentiable in (0,3).
(ilZ?a«sa/
Classes
Definite & Indefinite Integration
[18]
Q.18 Let f be an injective function such that f(x) f(y) +2 = f(x) + f(y) + ffay) for all non negative real x& y w ±th £' (0) - 0 & f'' (1) = 2 # f(0). Find f(x) & show that, 3 J f(x) dx - x (f(x) + 2) is a constant. l/n 1/n L i m 1+ 1+ 1+ 1+ (b) Q.19 Evaluate : (a) ^ A
(c)
1 + 2 + n+1 n+2
Lim I
3n +— 4n
.. . . . bf dx 1 1 (d) Using ab initio prove that J — = — - —
Q.20 Prove that sinx + sin 3x + sin 5x + .... + sin (2k- 1) x : prove that,
71/2 1 1 1 j £HL^dx=: i+-+_+^+ sin x 3 5 7 .
k e N and hence
+ 1 2k-l
"J? • 2 Q.21 If Un= J dx, then show that U,, U 2 , U 3 , q sin x Hence or otherwise find the value of Un. Q. 22
sin 2kx sinx
, Un constitute an AP .
Solve the equation for y as a function ofx, satisfying X
X
x • J y (t) dt = (x +1) j t • y(t) dt, where x > 0. o o 1 m!n! m, n s N. Q.23 Prove that: (a) Iffl n = j x m . (1 - x)n dx = (m+n+1)! (b>
I m ; I 1 = jox m . ( l n x ) n d x = (-l) n
n! ^n+i m, n s N.
(m+1
Q. 24 Find a positive real valued continuously differentiable functions/on the real line such that for all x X
! 2 f (x)= / ( ( / ( t t f + C / ' W ^ j d t + e
Q.25 Let f(x) be a continuously differentiable function then prove that, J [t] f' (t) dt = [x], f(x) - V f (k) i k=i where [• ] denotes the greatest integer function and x > 1. Q.26 Let f be a function such that | f(u) - f(v) I < | u - v | for all real u & v in an interval [a, b]. Then: (i) Prove that f is continuous at each point of [a, b]. (ii)
Assume that f is integrable on [a, b]. Prove that, f f(x) dx - (b-a) f(c)
7t f dx Q.27 Establish the inequality : — < I 7 = =f 6 o V4- x - x Q. 28
(b-a) 2
, where a < c < b
<
Show that for a continuously thrice differentiable function f(x) f(x)-f(0) = x f ( 0 ) + ® ^ - + i}f"'(t)(x-t) 2 dt 2
m
o
n / v I 1 Q.29 Prove that k£= o("l) k kVK/ 1k +— mT + 1= Sk = (o" ^ ( i f\)k / , k + n + 1
(ilZ?a«sa/
Classes
Definite & Indefinite Integration
[18]
Q.30 Let / and g be function that are differentiable for all real numbers x and that have the following properties: (1) /'(x)=/(x)-g(x) (ii) g'(x)=g(x)-/(x) (iii) f ( 0 ) =5
g(
(iv)
(a) (b)
0)=1
Prove that/(x) + g (x) = 6 for all x. Find/(x) and g (x).
EXERCISE-IV Q
1
SJsn,if:
Ftad
Sn=^
^ +7 4 = + V4n -1 y4n -4
+—
+T
=
=
=
=
V 3 n '+2n-l
.
[REE'97,6]
X
Q.2
If g (x) = J c o s 4
(a)
tdt
, then g (x + %) equals :
(A)g(x) + g(7C) (b)
Limit i g _ ^ e q u r=l Vn 2 +r 2 (A)l + V5
(B) g(x)-g(7C) a l s
(C)g(x)g(7C>
(D) [g(x)/g(7l)]
( Q - 1 + V2
(D)l + V2
:
(B) - 1 + VJ
e" The value of J ————— dx is
(c)
I 4 H e sinx ? Let — F(x) = , x > 0 . If f — dx x ' x values ofkis .
(d)
Q.3
dx = F (k) - F (1) then one ofthe possible
2x
0 +sinx) d x 1 + COS X
(e)
Determine the value of J „
(a)
If ff(t)dt = x + f t f ( t ) d t , then the value o f / ( l ) is
[JEE '97,2 + 2 + 2 + 2 + 5]
(A)° 1/ 2
(B) 0 (C) 1 (D) -1/2 i f \ i Y 1 Prove that f tan"1 2 dx = 2 f tan x dx . Hence or otherwise, evaluate the integral i U-x+x ; i i Jtan _1 (l-x+x 2 )dx [JEE'98,2 + 8]
(b)
Q
4
E
r
a
l
u
a
t
e
(ilZ?a«sa/
[
Classes
Definite & Indefinite Integration
R
E
E
'
9
8
-
6
1
[18]
Q.5
(a)
If for al real number y, [y] is the greatest integer less than or equal to y, then the value of the 371/2 integral j [2 sinx] dx is: (A) - n
(B) 0
(C)
(D) *
2
2
3,1/4
rlY f — — — is equal to : L 1+ cosx
(b)
(A) 2
(C)
(B) - 2 x3 + 3x + 2 , — dx (x 2 +l) (x + 1)
/N (c)
Integrate: J
(d)
Integrate: J „cosx . -cosx -dx e +e
T
R
ii
[JEE '99, 2 + 2 + 7 + 3 (out of200)]
71/6
Q.6\\
Evaluate the integral J V 3 c o s 2 x cosx
Q.7
(a)
[REE'99, 6]
The value of the integral j (A) 3/2
logex
dx is:
(B) 5/2
(C) 3
(D) 5
x
(b)
r 1 ' 1 Let g(x)= J f(t) dt, where f is such that - < f(t) < 1 for t £ (0, 1] and0< f(t) < -
2
o
2
for t G (1,2]. Then g(2) satisfies the inequality: (A)-|
(d)
Q.8
(a) (b)
Q.9
(B) 0 < g (2) < 2
+
Sn - — l ~r= + Given
f sin t 0 1+t
iIT/: t/2
Classes
r
sm u t
— d t in terms of a . 471 - 2 4 7C + 2 t [ REE 2000, Mains, 3 + 3 out of 100]
dx. 51
9
cos x r r3~ d x . Evaluate } — 53 0 cos x + sin x
(ilZ?a«sa/
imit Find Ln— > coS_ n
n + Vn
d t = a , find the value of
2x + 2
1
+
471
W 4 X 2 + 8 X + 13
(a)
(D)2
j ecosx . sin x for |x| < 2 If f (x) = { 2 otherwise Then j f(x)dx : (A) 0 , (B) 1 (D)3 (C) 2 x t> t For x > 0, let f (x) = j dt. Find the function f(x) + f (1/x) and show that, 1 1+t [JEE 2000, 1 + 1 + 1 +5] f(e) + f(l/e)= 1/2.
Evaluate f sin
Q.10
(C)^
(b) /
Definite & Indefinite
r xdx Evaluate J : 0 1 + cos a sinx [REE2001, 3 + 5] Integration
[18]
X
Q. 11 (a)
Let f(x) = j ^ 2 - t 2 dt Then the real roots ofthe equation x2 - f ' (x) = 0are (A)±l
(h)
(B) ± -""
(C)±-
(D) 0 and 1
Let T > 0 be afixedreal number. Suppose/ is a continuous function such that for all x e R 3+3T
T
/ ( x + T) =/(x). If I = f /fx; dx then the value of J ' j W dx is 3 0 (B)21
(A) | I (c)
(C) 3 I
(D) 61
(C)l
(D) 2/n
The integral JI [x] + / n j j - ^ J dx equals (B)0
(A)--
[JEE 2002(Scr.), 3+3+3] (d)
For any natural number m, evaluate J (x3m + x2m + x m ) (2x2m +3xm + 6)m
dx, wherex>0
[JEE 2002 (Mains),4]
n/2 n/4 Q.12 If f is an even function then prove that J f (cos2x) cosx dx= V2 J f (sin2x) cosx dx 0 0 [JEE 2003,(Mains) 2 out of 60] l
Q-i3
(a)
f^dx
=
o
(B)f-.
(A)f + 1 (b)
(C)7t
(D)l
(O-f
(D)l
r4\ If j x f ( x ) d x = - ^ t 5 , t > 0 , t h e n f v25y 0 2 (A)7
(B)
5
[JEE 2004, (Scr.)]
2 cosxc
°2s ® DQ ^en i+sin Ve
(c)
If y(x) = Jf 2,
(d)
Evaluate J -it/3 2-cos
Q.14 (a)
X
TC +-
sinx
feBansal Classes
at x = TC.
(B) i/V3
[JEE 2004 (Mains), 2]
[JEE 2004 (Mains), 4]
-dx.
If Jt 2 (f(t))dt = ( 1 - s i n x ) , t h e n / (A) 1/3
find dx
vV3y
[JEE 2005 (Scr.)]
is (C)3
Definite & Indefinite Integration
(D)V3 [15]
(b)
J (x3 + 3x2 + 3x + 3 + (x +1) cos(x + l))dx is equal to
[JEE 2005 (Scr.)]
- 2
(A)-4 (c)
Q15
(B)0
Evaluate: fe 0
|cos
(D)6
(C)4
f
x'N — cosx sin x dx. ^ 2sin - c o s x +3 cos v2 j j v2 , V
[JEE 2005, Mains,2]
^W5?^TdXisequalt0 (A)
(C)
V2x4-2x2+l+c x -2xz+l
(B) V2X 4 -2X 2 +1 +C (D) 2x 2
+C
Comprehension Q.16
[JEE 2006,3]
^ ^
Suppose we define the definite integral using the following formula (f(x)dx = more accurate result for c e (a, b) F(c) = b
i_b - a
2
(f(a)+f(b)),for
^ (f (a) + f (c)) + — - (f (b) + f (c)). When c = p p , 2
^
Jf(x)dx = ^ ( f ( a ) + f ( b ) + 2f(c>) a it/2
(a)
| sin xdx is equal to o (A)|(l + V2)
+
jf(x)dx_i_i(f(t) (b)
(c)
+ f ( a))
0 for all a then the degree off (x) can
If/(x) is a polynomial and if Lim— t->a
atmost be (A)l
(B)2
71 (D) 4V2
71 ( C ) ^
(C)3
(D)4
If /"(x) < 0, V x s (a, b) and c is a point such that a < c < b, and (c,/(c)) is the point lying on the curve for which F(c) is maximum, then/'(c) is equal to (A)
f(b)-f(a) b-a
(B)
2(f(b)-f(a)) b-a
(C)
2f(b)~f(a) 2b-a
5050 } ( I - x 5 ° R dx
(D)0 [JEE 2006, 5 marks each]
Q. 17 Find the value of J ( I - x 5 ° R < dx
(ilZ?a«sa/
Classes
Definite & Indefinite Integration
[JEE 2006,6] [18]
ANSWER
KEY
EXERCISE-I Q.l In
l + Vl + 3cos 2 20 +C cos 20
^ _ Q.2 -
x+1 7 x + x +1
+ C
f x - iN 1 x j o x 3 1 Q.3 - /n(cosx + sinx) + - + - (sin 2x + cos 2x) + c Q.4 — -0 tan I all-1 x X - -——/ . 7 \- -——In +c 2 o 8 4(x - 1 ) 16 VX + V Q.5 2 tan -1 (x+Vx 2 +2x-l) + c Q.7 (c) - (sin 20) In f
Q.6 I * - .e/
cos9 + sinG ] 1 - - /n(sec20) + c cos9 - sin9j 2
/ a tanx -1 x + tan
V
JJ
Q.ll J ( x + V ^ " ) '
+c
(x+VX 2 + 2 f
cosx \
Q.12 cos a. arc cos v cosa y
1 Q.8 - In
Q.IO
tanx +c tanx+2
+
whent = x + V ^
1/2 + C
1 (sinx • + Jsin 2^x-sin 2""^ +, - sin a In C
^Q.13 —^ 3(l+4tan tttx)1 +c 8(tan x)
,1 , x x Q.14 - In tan- + - sec2 - +tan - + c
Q.15 V^Jl^x-2-Jl^
Q.16 (a + x) arc tan
Q.17
- ^ax + c
Q.18 xln (lnx) - —— + c lnx Q.20-/n(l-x4) + c
Q. 2 2
Q.29-
Q.23 -
Q.2«
VT+sin x-cos x + arc tan (sin x+cos x)+c V3 -sm x+cos x
1 In cotx+cota+^/cot 2 x+2cotacotx-l V sina
' x ^ Q.31 2x-3arctan tan—+1 +c v 2 ,
(ilZ?a«sa/
Classes
In
xe" x x + c v l+xe j l+xe
t4 t2 1 z -1 Q.21 6 — - — +1 + ~/n(l + t ) - tan 1 + C where t = x1/6 4 2 2
1 + -yjcOSj . +2tan" 1 Jcosf -In +c v ycosf 1 - ycosi
1 2V3
+ arc cos Vx+c
2 (x 2 + l \l/x +1 2-31nfl+-^ 9x V x
Q.19
~ 1 , (4+3sin x+3cos x) Q.25 —In—+c 24 (4-3sin x-3cos x) Q 27
+c
Q.32
2
vx-ly
+C
Q24. sin"1 —sec — v.2 2j
sin x - cos x —p=ln tan V2
F
X
71 ^
+ C
+ C
Q.28 -^n(secx) - ^ ^n(sec2x) +^n(sec3x) + c +c
Q.30
x sin x +cosx — +c xcosx-smx
Vcos2x - x - cotx . In (e (cosx + Vcos2x)j + smx
Definite & Indefinite
Integration
c
[18]
1 „ 1 t2 Q.33 /n(l +1) - - /n(l +1*) + ^ /n t2 +
-Jit
_ 1, X 1 2X Q.34 —lntan tan —+c 2 2 4 2
Q.35 c -
+ 1 1t a n . +- c - j *
where 1=
+ 1
,— Vcotx
(x2-l)2
ax +b +k Q.38 ex J ^ cx J
20>>
Q.36 c - ecos x (x + cosec x) Q.37
sin-1
_ lnx Q.40 arcsecx—, +c
I 2— /r l+2u + c where u = 3 Q.41 In , ' + V3tr~ V3tan 1 4 2 V 1+x V3 Vu +u + l
i
+ c Q.39
9^7x-10-x 2
+c
1+x 1 . f V51 - 1 - |sin 1 x - ^/l-x 2 j + c where t = h^~7= Q.42 - tan t + —r= x 2V5 vv5t + 1 Q.43 tan
7 Q.44 4 / n x + - + 6 t a n - 1 ( x ) +
.^2 sin 2x +c v sinx + cosxy
2
x
Q.45 —i=arctan—p —+c V3 V^+i)
-2
X
J2-X-X2
Q.46
x
x^
Q.47 — - J — - + c a-p y x-a Q.49
n
V2 l V 2 - t
Q.48
ITX
J 2 - x -x
j +C . sin
x
(2x+l +c ^ 3
n1 x4
- x - 22 15 , V T + x - i +c + —8 In 71 + X + 1 4x2 y l + x
'1-0 - —/n 2 vl + ty where t = cos9 and 9 = cosec'(cotx)
a cosec— tan -1 2 v,
Q.50
V2 „ f 4 - x + 2V2
+— 4
6x
2x
\ a cosec— 2/ y
EXERCISE-II 2 Q.l — ft
Q.2 /n2
Q.3 6 - 2e
Q.7 1 - sec(l) Q.8 — Q . 9 2n + l
Q.14 4V2-4/n(V2 + l)
_
Q.22
71
16V2
(a7t + 2b)7t 3V3
(ilZ?a«sa/ Classes
Q.4 - - 1 2
2 V2 + |
Q.15 ^
_
Q.23
Q.5
(3 V3 - 2V2)
7t(7t + 3)
64
Q.6 £ In2 8
f 22
Q.12 17
V2 Q.21 i arc tan 3 3 Q.24
^
7t J
7t Q.13-(l-ln4)
Q.17^=Q.18- ^ ( e 2 * + l )
Q.16
(aln2-5a+^)
^
371+1 7t2
l" arc tan3
Q 27
Definite & Indefinite Integration
[18]
Q.28
71
Q.29
2a(a+b)
57t 3
2 Q.32—5-cos7ixforO
Q.37(a)f
Q.30 ^ L 16 2 71
Q.31 (a) ^=[rc+21n(V2-l)] (b)
2 for x> 1 & — 2 f o r x ^ ° 71
Q.40 ^ - 7 /n2 16 4
( b ) ^
167t _ rr 271 i-J^b2 Q.44 (a) — 2V3 (b)
32 Q.49 -2ti - —
Q.50
Q.42
12
2 Q- 3 3 —
Q.36%
4
Q.43 - In 2 +
^ 27
2
Tt
Q.45 - / n 2
Q.47
1
71 +
4 666
—--(l+/n2) + 8 4 2 EXERCISE-III
Q.2 Q.4
,
71
TC ,
- - , - (•
Q.3 cont. & der. at x = 0
g(x) is cont. in (-2, 2); g(x) is der. at x = 1 & not der. at x = 0 . Note that; -(x + 2) 2 g(x) = - 2 + x - ^ ^--x-1
Q.7 (a) -—1—x
for - 2 < x < 0 for 0
Q.8 - cos x
Q.13 (a) c = 1 and Limit
willbe
Q.lOl+e R 2
Q.17 F(x)=
Q.ll
(b) a = 4andb =1 x2
if if
0
M . 1 k if 3 2
2
X
Q.16 x = 2 or 4
2TI
Q.5 a= —
-T \
27T
&b= —
f(x) = x +
119
Q.19
(a) - (b)2e( 1/2 )("- 4 ) (c) 3 - / n 4 e
Q.22 y = ^ e"1/x
Q.24
f (X) = e x + 1
Classes
119
Q.14 10
Q.18 f(x) = l + x 2
(ilZ?a«sa/
Q.6n=3
Q.21 U n =
^ 2
Q.30 f (x) = 3 + 2e2x; g (x) = 3 - 2e2x
Definite & Indefinite Integration
[18]
EXER CISE-IV Q.2 (a) A (b) B (c) 2 (d) 16 (e) %2
Q.l ti/6
Q.4 Q.5
I I
„
2vn
rln -C1I
Q.3 (a) A (b) hQ.
VN+I I
vn-i
3 1 1 x n (a) C, (b) A; (c) - t a n ^ x - - / n ( l +x) + - /n(l +x 2 ) + y f - r + c, (d) -
Q.6 y - 71 - 2 tan-1 V2
Q.7 (a) B, (b) B, (c) C, (d) - /n 2 x
Q.8 (a) 2/n2, ( b ) - a
Q.9 ( x + l ) t a n - 1 ^ j ^ " ^ n ( 4 x 2 + 8x + 13)+C
1 57t_J. Q.10 (a) 4 3
7ta
if a e (0,7t)
sin a
, (b)I
7t
sin a
(a - 27i) if a e (71,27:) m+l
Q.ll
+ 6 x
(a) A (b) C, (c) B, (d)
Q.13 (a) B, (b) A, (c) 2%, (d)
f
Q.14 (a) C, (b) C, (c) ^ ecos V
Q.15 D
47t
^ tan
f 1 \
Classes
m 1
•+
C
-1
e . fi\ A -1 + —sin 2 k2J J
Q.16 (a) A, (b) A, (c) A
(ilZ?a«sa/
1
Q.17
5051
Definite & Indefinite Integration
[18]
MATHEMATICS 1 .,,
X H (ALL)
::
QUESTION BANK ON
DETERMINANT & MATRICES
T i m e L i m i t : 4 Sitting Each of 75 M i n u t e s duration approx.
|
Question bank on Determinant & Matrices There are 102 questions in this question bank. Select the correct alternative : (Only one is correct)
Q.l
a2 a 1 The value ofthe determinant cos (nx) cos (n+1) x cos (n+2) x is independent of: sin(nx) sin(n+l)x sin(n+2)x (A) n
Q.2
0 1 A is an involutary matrix given by A= 4 - 3 3 - 3 (B)
Q.5
(B) a"1 b"1 c"1
IfA
(B) cos a cosp cosy (D) zero
cos9 -sin 6 sine cose}A-lissivenby (B)AT
(C)-A T
(D)A
Ifthe system of equations ax+y+z = 0, x+by+z = 0 & x+y+cz = Q (a, b, c ^ 1) has a non-trivial solution, then the value of (A) - 1
Q. 8
(D) - 1
1 cos(P~a) cos(y-a) If a, 3 & y are real numbers, then D = cos(a-p) 1 cos(y-P) cos(a-y) cos(p-y) 1
(A)-A
Q.7
(C) - a - b - c
If A and B are symmetric matrices, then ABA is (A) symmetric matrix (B) skew symmetric (C) diagonal matrix (D) scalar matrix
(A) - 1 (C) cos a + cos P + cosy
Q.6
(D)A2
(C)
1+a 1 1 If a, b, c are all different from zero & 1 1+b 1 = 0, then the value of a -1 + b -1 + c_1 is 1 1 1+c (A) abc
Q.4
-1 4 then the inverse of — will be 2 4
-l
(A) 2 A
Q.3
(D) a, n and x
(Ox
(B) a
1-a (B) 0
+ T~R + T— IS : 1-b 1—c (C) 1
3 6 -1" [2 4"! 0 2 ,B = 0 1 ,c= 1 2 1 -2 5 -1 2
4 Consider the matrices A= 3
(i) ( A B ) T C (ii)CTC(AB)T (A) exactly one is defined (C) exactly three are defined
(IBansal Classes
(D) none of these . Out of the given matrix products
(iii) CTAB and (iv)ATABBTC (B) exactly two are defined (D) all four are defined
Q. B. on Determinant & Matrices
[4]
Q. 9
Q.IO
The value of a for which the system of equations ; a 3 x+(a+1 )3 y + (a+2) 3 z = 0, a x + ( a + l ) y + (a+2)z = 0 & x + y + z = 0 has a non-zero solution is : (D) none of these (A) 1 (B) 0 (C) - 1 1 a^i IfA= q j j, then An (where n € N) equals
(A)
Q.ll
'1 v0
(\
na^l \j
l+sin 2 x cos2x Let f(x) = sin2x l+cos 2 x cos2x sin2x (A) 2
Q.12
IfA=
"3
4"
1
-6
andB =
Q.15
vo
na^
o,
CD)
n
V°
2
(D) 8
then X such that A + 2X = B equals 5 (C) - 1
2 0
(D)none of these
x + 3x x - 1 x+3 If px + qx + r x + s x + t = x+1 2 - x x - 3 then t = x - 3 x+4 3x 4
3
2
(B) 0
(C) 21
(D) none
IfA and B are invertible matrices, which one ofthe following statements is not correct (A) Adj. A = | A| A-1 (B) det (A-1) = |det (A)|_1 1 1 ! (C) (A + B)- = B- + A~ (D) (AB)-1 = B"1 A"1
If D
a" +1 ab ac 2 ba b +1 be then D = ca cb c2 +1
(A) 1 + a2 + b 2 + c2
Q.16
1
3 5 (B) - 1 0
(A) 33 Q.14
(\
(C) 6
"-2 5" 6
(C)
4sin2x : 4sin2x , then the maximum value of f (x) l+4sin2x
(B)4
2 3 (A) - 1 0
Q.13
nM 1 ,
(B) 0 v
IfA= , c
(B) a2 + b2 + c2
(C) (a+b +c) 2
(D) none
b^l ^ j satisfies the equation x2 - (a + d)x+k = 0, then
(A) k = be
(IBansal Classes
(B) k = ad
(C) k = a2 + b2 + c2 + d2
Q. B. on Determinant & Matrices
(D)ad-bc
[4]
(ax + a~x)~ (a x -a" x )" 1 Q, 17 If a, b, c> 0 & x, y, z e R, then the determinant (by + b~y)2 (b y -b- y ) 2 1 (c z +c" z ) 2 (A) axbycz
(B) a-xb-yc-
(c z -c" z ) 2
(C) a2xb2yc2z
1
(D) zero
Q. 18 Identify the incorrect statement in respect of two square matrices A and B conformable for sum and product. (A) tr(A + B) = tr(A) + tr(B) (B) tr(aA) = a t r (A\ a s R (C) tr(AT) - tr(A) (D) tr(AB) * tr(BA) (0 + (j)) —sin (9+4>) cos 2^) cos 6 sin4> is : sinG sin9 cos(f) -cos©
COS
Q. 19 The determinant
(B) independent of 9 (D) independent of 9 & (j) both
(A) 0 (C) independent of (f)
Q.20 IfAandB are non singular Matrices ofsame order then Adj. (AB)is (A) Adj. (A) (Adj. B) (B) (Adj. B) (Adj. A) (C) Adj. A+Adj. B (D) none of these a+1 a+2 a+p Q.21 If a + 2 a + 3 a + q= 0, then p, q, r are in : a+3 a+4 a+r (A) AP
Q.22
LetA-
(B) GP X-rX
X
X
X
x+X
X
x
X
x + X,
(C) HP
(D) none
, then A -1 exists if
(A) x 0 (C) 3x + X * 0, X * 0
(B) X * 0 (D)x*0,
1 logxy logxz l lOgy is Q. 23 For positive numbers x, y & z the numerical value ofthe determinant lOgy X logzx logzy (A) 0 Q.24
(B) 1
If K g Rq then det. {adj (KIn)} is equal to (A) K" ~1 (B)K n ( n _ 1 )
(IBansal Classes
(C) 3
(D) none
(C) Kn
(D)K
Q. B. on Determinant & Matrices
1
[4]
Q.25
bj + Cj Cj + aj aj + b, The determinant b, + c2 c2 + a2 a2 + b2 b3 + c3 c3 + a3 a3 + b3 a
i b i ci (B)2 a2 b2 c2 a3 b3 c3
a
i b, Cj a (A) 2 b2 c2 a 3 b3 c3
a
i b i ci (C)3 a2 b2 c2 a3 b3 c3
a
i b, c, (D)4 a2 b2 c2 a3 b3 c3
Q.26 Which ofthe following is an orthogonal matrix 6/7 (A) 2/7 3/7 -6/7 (C) -2/7 3/7
2/7 3/7 -6/7
6/7 (B) 2/7 3/7
-3/7 6/7 2/7
-2/7 3/7 6/7
"6/7 (D) -2/7 6/7
-3/7 6/7 2/7
2/7 -3/7 6/7
3/7 6/7 -2/7
-2/7 2/7 2/7
3/7' -3/7 3/7
1+a+x a+y a+z Q. 27 The determinant b+x 1+b+y b + z c+x c+y 1+c+z (A) (1 +a + b + c) (1 +x+y + z) - 3 (ax+by+cz) (B) a(x+y) + b(y+z) + c (z+x) - (xy+yz+zx) (C) x(a + b) + y(b + c) + z(c + a) - (ab + bc + ca) (D) none of these Q. 2 8 Which of the following statements is incorrect for a square matrix A. (| A | ^ 0 ) (A) IfA is a diagonal matrix, A -1 will also be a diagonal matrix (B) IfA is a symmetric matrix, A-1 will also be a symmetric matrix (C) IfA -1 = A => Ais an idempotent matrix (D) IfA-1 = A => Ais an involutary matrix x
Q. 29 The determinant
c1 y c, z
Cj
X
X
y
C,
c
y
z
c
C: C; Z C:
(A) \ xyz (x+y) (y+z) (z+x)
(B) ~ xyz(x+y-z) (y+z--x)
(C) ™ x y z ( x - y ) ( y - z ) ( z - x )
(D) none
Which ofthe following is a nilpotent matrix "1 0" (A) 0 1
(IBansal Classes
cos9 (B) sin 9
-sin9 cos 9
"0 0" (C) 1
Q. B. on Determinant & Matrices
"1 1 (D) 1 1
[4]
a3 a - 1 3 4 If a, b, c are all different and b b b - 1 = 0, then : a
Q.31
(A) abc(ab + bc + ca) = a + b + c (C) abc(a + b + c) = ab + bc + ca Q.32
Q.33
Give the correct order of initials T or F for following statements. Use T if statement is true and F if it is false. Statement-1: IfAis aninvertible 3 x 3 matrix and B is a 3 x 4 matrix, then A"'B is defined Statement-2 : It is never true that A+ B, A - B, and AB are all defined. Statement-3: Every matrix none ofwhose entries are zero is invertible. Statement-4: Every invertible matrix is square and has no two rows the same. (A) TFFF (B)TTFF (C) TFFT (D)TTTF 1 CO3 CO2 If co is one ofthe imaginary cube roots ofunity, then the value ofthe determinant (O3 1 a o 2 CO 1 (A) 1
Q.34
(B) (a + b + c) (ab+bc + ca) = abc (D) none of these
(B) 2
(C) 3
(D) none
Identify the correct statement: (A) If system of n simultaneous linear equations has a unique solution, then coefficient matrix is singular (B) If system of n simultaneous linear equations has a unique solution, then coefficient matrix is non angular 1 (C) IfA"1 exists, (adjA)"1 may or may not exist b C c <)f;tXL ^ J A' I -"> Sivnpl« ( I aJjJJLscosx - s i n x 0 (D) F(x) = sinx 0
cosx 0
0 , then F(x) . F(y) = F(x - y) 0
a+p 1+x u+f Q. 3 5 If the determinant b+q m+y v+g splits into exactly K determinants of order 3, each element of c+r n+z w+h r l n n , .. which contains only one term, then the value of K, is J r °£ f (A) 6 (B)8 (C)9 (D)12 J
Q. 3 6
A and B are two given matrices such that the order ofA is 3 x 4, if A' B and B A' are both defined then (A) order of B' is 3 x 4 (B) order of B'A is 4 x 4 (C) order of B'A is 3 x 3 (D) B'A is undefined
Q.37
Ifthe system of equations x+2y+3z = 4, x+py+2z = 3, x+4y+pz = 3 has an infinite number of solutions, then : (A) p = 2 , p = 3 (B) p = 2 , p = 4 (C) 3p = 2 p (D) none of these
faBansal Classes
Q. B. on Determinant & Matrices
[10]
Q.38
cos a If A = sina cosa
sina cosa sin a
; B
=
cos 2 p sin P cos P sinP cosp sin 2 p
are such that, AB is a null matrix, then which of the following should necessarily be an odd integral 7t
multiple of —. (A) a
(C)a-p
(B)P
(D) a + p
a b a+b a c a+c b + d then the value of Ei where b * 0 and Q.39 Let Dj = c d c + d and D-JL = b d D„ a b a-b a c a +b+c ad * be, is (A)-2
(B)0
Q. 40 For a given matrix A=
(C) - 2b
cos0 -sin 9 sin 9 cosO
(D)2b
which of the following statement holds good? 71
(A) A = A -1 V9eR (C) A is an orthogonal matrix for 9 6 R
(B) A is symmetric, for 9 = (2n + 1) — , n e l (D) A is a skew symmetric, for 9 = nTt; n € I
1 + a x (1 + b )x (1 + c )x 2 2 2 Q.41 If a + b + c =-2and f(x) = (l + a )x l + b x (l + c )x then f (x) is a polynomial of degree (l + a 2 )x (l + b 2 )x l + c 2 x 2
2
2
(A)0
(B) 1
(C)2
(D)3
x 3 2 Q.42 Matrix A= 1 y 4 , if x y z = 60 and 8x + 4y + 3z = 20, then A (adj A) is equal to 2 2 z 64 0 0 (A) 0 64 0 0 0 64
88 0 0 0 (B) 00 88 0 88
68 0 0 (C) 0 68 0 0 0 68
34 0 0 (D) 0 34 0 0 0 34
Q. 43 The values of 9, X, for which the following equations sin9x - cos9y + (A+1 )z = 0; cos9x + sin9y - Xz = 0; Xx +(A, +1 )y + cos9 z = 0 have non trivial solution, is (A) 9 = mr, X e R - {0} (B) 9 = 2mr, X is any rational number (C) 9 = (2n + l)7t, X e R+, n e I
Tt (D)9 = (2n+ 1)—, X e R, n e I Zi
Q.44 IfA is matrix such that A 2 +A+2I = 0 , then which of the following is INCORRECT? (A)Aisnon-singular
(B)A ^ O
(C) Ais symmetric
(D) A -1 = -— (A+1)
(Where I is unit matrix of order 2 and O is null matrix of order 2)
faBansal Classes
Q. B. on Determinant & Matrices [10]
[4]
Q. 45 The system of equations: 2x cos20 + y sin20 - 2sin0 = 0 x sin20 + 2y sin20 = - 2 cosO x sin0 - y cos0 = 0, for all values of 0, can (A) have a unique non - trivial solution (C) have infinite solutions
(B) not have a solution (D) have a trivial solution 1 1 v 2z J3 / (C) 1
Q. 46 The number of solution ofthe matrix equation X2 (A) more than 2
(B)2
IS
(D)0
Q. 47 If x, y, z are not all simultaneously equal to zero, satisfying the system of equations (sin30)x-y + z = O (cos20)x + 4y + 3z= 0 2x+7y+7z=0 then the number of principal values of 0 is (A) 2 (B)4 (C)5 (D)6
Q.48 Let A+2B
1 2 6 - 3 -5 3
0 2 - 1 5 3 and 2 A - B = 2 - 1 6 1 0 1 2
then Tr (A) - Tr (B) has the value equal to (A) 0 (B) 1 a 2z +, bu2 c Q.49 For a non - zero, real a, b and c
a
c u2 , Jl b +c a b
(A)-4
(B)0
(D)none
(C)2
c a
= a abc, then the values of a is
c_2 +, _a 2
(C)2
"l 3" "l o" Q.50 Given A= 2 2_ ;I = 0 1 . IfA - XI is a singular matrix then (A) X e
(D)4
(D)X 2 -3X~6 = 0
Q. 51 Ifthe system of equations, a2 x - ay = 1 - a & bx+(3 - 2b) y=3 + a possess a unique solution x = 1, y = 1 then: (A) a= 1; b = - 1 (B) a = - l , b = l (C) a = 0, b = 0 (D) none
faBansal Classes
Q. B. on Determinant & Matrices
[10]
1 Q.52 Let A -
~S1\11U
(A) Det (A) = 0
sinB 1* - s~n i n Ae6
1 " 1
S1 D sin6
, where 0 < 9 < 2n, then
(B) Det A e (0, QO)
(C) Det (A) e [2, 4]
(D)DetAe [2, oo)
Q. 53 Number of value of 'a' for which the system of equations, a2 x + (2 - a ) y = 4 + a2 ax + (2a - i ) y = a5 - 2 possess no solution is (B)l (A) 0 (0)2 "0 1 2 [1/2 Q.54 IfA= 1 2 3 , A -1 = - 4 3 a 1 5/2 (A) a - 1, c = - 1
-1/2 3 -3/2
(B) a = 2, c = -
(D) infinite
1/2] c , then 1/2 1
1 1 (D)a=-,c=-
(C)a = - 1, c= 1
Q. 5 5 Number of triplets of a, b & c for which the system of equations, ax - by = 2a - b and (c + l)x + cy = 10 - a + 3 b has infinitely many solutions and x = 1, y = 3 is one of the solutions, is: (A) exactly one (B) exactly two (C) exactly three (D) infinitely many Q.5 6 D is a 3 x 3 diagonal matrix. Which of the following statements is not true? (A) D' = D (B) AD = DAfor every matrix Aof order 3x3 1 (C) D if exists is a scalar matrix (D) none ofthese Q. 57 The following system of equations 3x - 7y + 5z = 3; 3x + y + 5z - 7 and 2x + 3y + 5z = 5 are (A) consistent with trivial solution (B) consistent with unique non trivial solution (C) consistent with infinite solution (D) inconsistent with no solution
Q.58 IfAj, A3,
f'
at
A2n_| are n skew symmetric matrices ofsame order then B = ]T(2r~ l ) ( A 2 r l ) 2 r i will r=l
be (A) symmetric (C) neither symmetric nor skew symmetric
(B) skew symmetric (D) data is adequate
x 2 x -l Q. 59 The number of real values of x satisfying 7x-2 (A) 3
(B)0
3x + 2 4x 17x + 6
2x-l 3x + l = 0 is 12x-l
(C) more than 3 X-l
X
Q.60 Number of real values of X for which the matrix A= 2 - 1 X+3 X-2 (A) 0
tl Bansal Classes
(B) 1
(D)l X+ l
3
(C) 2 Q. B. on Determinant & Matrices
X+l
has no inverse
(D) infinite
[9]
Q.61 IfD =
1 z (y + z) x2 y(y + z) x2z
1 z 1
(x + y) z2 1
X
x + 2y + z xz
X
y(x + y) xz2
then, the incorrect statement is
(B) D is independent of y (D) D is dependent on x, y, z
(A) D is independent ofx (C) D is independent of z
Q.62 If every element of a square non singular matrix Ais multiplied by k and the new matrix is denoted by B then | A-11 and | B_11 are related as (A)|A- , | = k|B- 1 |
(B)| A_1| =
B_1|
(C)| A -1 | = k n |B _ 1 |
(D) | A_1| =k _ n |B _ 1 |
where n is order of matrices, mx mx-p mx+p n n+p n-p theny = f(x) represents Q.63 Iff'(x) = mx + 2n mx + 2n + p mx + 2 n - p (A) a straight line parallel to x« axis (C) parabola 1 -1 2 1 Q.64 Let A= 1 1 (A)-2
"4 1 - 3 and 10B = - 5 1 1 (B)-l
(B) a straight line parallel to y- axis (D) a straight line with negative slope 2 2" 0 a . If B is the inverse of matrix A, then a is -2 3 (C)2
(D) 5
„3 x — 1 (x-1)2 X x-1 x 2' (x + 1) Q.65 If D(x) = then the coefficient ofx in D(x) is 2 (x + 1) (x + 1)
(A) 5
(B)-2
(C)6
(D)0
Q. 66 The set of equations Xx-y + (cos9)z = 0 3x + y + 2z =0 (cos9)x + y + 2z = 0 0 < 9 < 271, has non- trivial solutions) (A) for no value of X and 9 (B) for all values of X and 9 (C) for all values of X and only two values of 9 (D) for only one value of X and all values of 9 Q.67 Matrix A satisfies A2 = 2A-1 where I is the identity matrix then for n > 2, An is equal to (n s N) (A)nA-I (B) 2 n _ 1 A - ( n - 1 ) 1 ( C ) n A - ( n - l ) I (D^^A-I
faBansal Classes
Q. B. on Determinant & Matrices
[10]
a 2 +1 ab ac 2 ab b +1 be Q. 68 If a, b, c are real then the value of determinant = 1 if ac be c2+l (A) a + b + c = 0
(B) a + b + c = 1
(C) a + b + c = - 1
(D) a = b = c = 0
Q. 69 Read the following mathematical statements carefully: I. There can exist two triangles such that the sides of one triangle are all less than 1 cm while the sides of the other triangle are all bigger than 10 metres, but the area of thefirsttriangle is larger than the area of second triangle. n. If x, y, z are all different real numbers, then 1
+-
1
1
l x-y —
+
l y-z —
+
l \ z-x
2
—
( x ~ y) (y-z) in. log3x • log4x • log5x=(log3x • log4x) + (log4x • log5x) + (log5x • log3x) is true for exactly for one real value of x. IV. A matrix has 12 elements. Number of possible orders it can have is six. Now indicate the correct alternatively. (A) exactly one statement is INCORRECT. (B) exactly two statements are INCORRECT. (C) exactly three statements are INCORRECT. (D) All the four statements are INCORRECT. Q. 70 The system of equations (sinG)x + 2z = 0, (cos0)x+(sin0)y = 0, (cos9)y + 2z = a has (A) no unique solution (B) a unique solution which is a function of a and 0 (C) a unique solution which is independent of a and 0 (D) a unique solution which is independent of 0 only "1 2 3" 0" 0 2 5 3 Q.71 LetA= andb = . Which of the following is true? 0 2 1 1 (A) Ax=b has a unique solution. (C) Ax=b has infinitely many solutions.
(B) Ax=b has exactly three solutions. (D) Ax = b is inconsistent.
Q. 72 The number of positive integral solutions of the equation x3 + l xy xz2
x2y
x2z
y3 +1 y2z = 11 is yz z3+l
(A)0
(B)3
(C)6
(D)12
Q.73 If A B and C are n x n matrices and det(A) = 2, det(B) = 3 and det(C) = 5, then the value ofthe det(A 2BC~') is equal to (A) 7
faBansal Classes
(B)
12
(C)
18
Q. B. on Determinant & Matrices
CD)
24
[10]
(1 + x)2 Q.74 The equation 2x + l x+1
Q. 75
(1 + x)2 2x + l x + 1 -(2 + x ) 2 3x 2x l~5x + (1-x) l-2x 3x-2 2x-3 2-3x
(1-x) 3x 2x
(A) has no real solution (C) has two real and two non-real solutions
(B) has 4 real solutions (D) has infinite number of solutions, real or non-real
a a+2b The value of the determinant a+b
a + 2b a+b is a
(A) 9a2 (a + b)
a+b a a+2b
(C) 3b2 (a + b)
(B) 9b2 (a + b)
"3 "3 4" "2 l" Q. 76 Let three matrices A= 4 1 ; B = 2 3 andC = -2 tr(A) + t r
'ABC
N
+ t.
(A) 6 Q. 77
' A ( B C )
2
^
+ t.
'A(BC) 3 ^
(B) 9
8
+
(D) 7a2 (a + b) -4 3 _then
+ 00 =
( Q 12
(D)none
(C)3
(D)l
The number of positive integral solutions 1-X 2 -3 X 2 -2
1 - 2 = 0 is 1+X
(A)0
(B)2
Q. 78 P is an orthogonal matrix and Ais a periodic matrix with period 4 and Q==PAP PAPT then X = p T Q 2 0 0 5 p will be equal to (D)A4 (A)A (B)A2 (C) A3 a-x Q.79 If x = a + 2b satisfies the cubic (a, b eR) f (x)= b b
b b =0, then its other two roots are a-x
(B) real and coincident (D) such that one is real and other imaginary
(A) real and different (C) imaginary Q.80 A is a 2 x 2 matrix such that A J j = (A)-l
b a-x b
(B)0
-1 2 1 2^ and A J j = ^ , The sum ofthe elements ofA, is (C)2 (D)5
Q. 81 Three digit numbers xl7,3y6 and 12z where x, y, z are integers from 0 to 9, are divisible by a fixed x 3 1 constant k. Then the determinant 7 6 z must be divisible by 1 y 2 (A)k
faBansal Classes
(B)k2
(C)k 3
Q. B. on Determinant & Matrices
(D) None
[10]
Q.82 In a square matrix A of order 3, a; ;'s are the sum of the roots of the equation x2 - (a + b)x + ab= 0; a; j + 5's are the product of the roots, {_ j's are all unity and the rest of the elements are all zero. The value of the det. (A) is equal to (A) 0 ' (B) (a + b)3 (C) a3 - b3 (D) (a2 + b2)(a + b) 28 25 38 Q,83 Let N = 42 38 65 , then the number of ways is which N can be resolved as a product of two 56 47 83 divisors which are relatively prime is (A) 4 (B) 8 Q. 84 If A B, C are the angles of a triangle and the triangle is (A) a equilateral (C) a right angled triangle Q.85
x L L e t a =x—/nx ^ . d=
1 x/nx
(C)9
(D)16
1 1 1 1 + sin A 1 + sinB 1 + sin C = 0, then sin A + sin A sin B +sin B sin C + sin C (B) an isosceles (D) any triangle
T.
x — 16x /n (1 + sin x) T. Lim ; c = Lim —^ and ->o 4x + x x-^o x x
3
a b (X+1) — — r then the matrix c d is x->-i 3(sin( xK+1) - (x +1)) + l)-[X + l)) T• Lim — j —
(A) Idempotent
(B) Involutary
Q.86 Ifthe system of linear equations x + 2ay + az = 0 x + 3by + bz — 0 x + 4cy + cz = 0 has anon-zero solution, then a, b, c (A) are in G .P. (C) satisfy a + 2b + 3c = 0
(C) Non singular
(D) Nilpotent
(B) are in H.P. (D) are in A. P.
Q. 87 Give the correct order of initials T or F for following statements. Use T if statement is true and F if it is false. Statement-1: If the graphs of two linear equations in two variables are neither parallel nor the same, then there is a unique solution to the system. Statemeat-2: If the system of equations ax + by = 0, cx + dy = 0 has a non-zero solution, then it has infinitely many solutions. Statement-3: The system x + y + z= l,x = y, y = l + z i s inconsistent. Statement-4: Iftwo of the equations in a system of three linear equations are inconsistent, then the whole system is inconsistent. (A) FFTT (B)TTFT (D)TTTF (C) TTFF Q.88 Let A
1 + x 2 - y2 ~ z2 2(xy + z) 2(zx-y) 2 2 2 2(xy-z) 1+y -z -x 2(yz + x) then det. Ais equal to 2 2 2 2(zx + y) 2(yz-x) 1+z -x -y
(A)(l +xy + yz + zx)3 (C)(xy + yz + zx)3
faBansal Classes
(B) (1 +x 2 + y2 + z2)3 (D) (1 +x 3 + y3 + z3)2 Q. B. on Determinant & Matrices
[10]
Select the correct alternatives : (More than one are correct) Q.89 The set of equations x - y + 3z = 2 , 2 x - y + z = 4 , x - 2 y + az = 3 has (A) unique soluton only for a = 0 (B) unique solution for a * 8 (C) infinite number of solutions for a = 8 (D) no solution for a = 8 Q.90
Suppose al5 a
(A)A=
i a4 a5
real numbers, with at * 0. If a1? a
2 a5 a6
aj,
are in A.P. then
a
3 a6 is singular a7
(B) the system of equations a,x + a ^ + agZ = 0, a4x + a5y + a6z = 0, ayX + a8y + a9z = 0 has infinite number of solutions (C)B
a, ia2
ia2 a, j
is non singular; where i - V-l
(D) none of these a2 a2 - (b-c) 2 be Q. 91 The determinant b2 b2 - (c-a) 2 ca is divisible by : c2 c2 - (a-b) 2 ab (A) a + b + c (C) a2 + b2 + c2
(B) (a+b) (b + c) (c + a) (D) ( a - b ) ( b - c ) ( c - a )
Q. 92 IfA and B are 3 x 3 matrices and | A | * 0, then which of the following are true? (A) | AB | = 0 => | B | = 0 (B) j AB | = 0 => B = 0 -1 _1 (C) | A | = |A| (D) | A + A| = 2 | A| TC 7T It Q.93 The value of 9 lying between -— & — and 0
2 sin 40 2 sin 40 = 0 are l+2sin40 7t
(B)A=^=e
"i 71
I
Q.94 IfAB =AandBA=B, then (A) A2B =A 2 (B)B 2 A=B 2
(C)ABA=A
(D) BAB = B
x a b Q. 95 The solution(s) of the equation a x a = 0 is/are : b b x (A) x = - ( a + b )
faBansal Classes
(B) x = a
(C) x = b
Q. B. on Determinant & Matrices
(D) - b
[10]
Q. 96 If Dj and D2 are two 3x3 diagonal matrices, then (A) DjD2 is a diagonal matrix (B) D,D2 = D2Dj 2 2 (C) D j + D2 is a diagonal matrix (D) none of these 1 a a2 2 Q.97 If 1 x x = 0, then b2 ab a2 (A)x = a
(B) x = b
(C) x = -
a
(D)x=-
Q.98 Which ofthe following determinant(s) vanish(es)? 1 be bc(b + c) (A) 1 ca ca (c + a) 1 ab ab(a + b)
1 ab a + b (B) 1 be b + c 1 ca 1c + 1 a
0 (C) b - a c-a
logxxyz logxy logxz 1 logyz (D) logyxyz logzxyz logzy 1
Q.99 IfA=
a- b a- c 0 b-c c-b 0
"a b" (where be 7= 0) satisfies the equations x2 + k = 0, then _c d_
(A) a + d = 0
(B) k = - | A|
(C) k = |A|
(D) none of these
Q.100 The value of 6 lying between 9 = 0 & 9 = 7t/2 & satisfying the equation : 1+sin 2 9 COS29 4sin49 2 sin 9 1+COS 9 4sin49 = 0 are: sin 2 9 l+4sin49 COS29 lbt (C) 24 Q.101 If p, q,r, s areinAP. and f(x)
71 (D) 24
p + sinx q + sinx p - r + sinx q + sinx r + sinx - 1 + sinx such that { f(x)dx = - 4 then r + sinx s + sinx s - q + sinx
the common difference of the A.P. can be: (A) - 1
(B)
1
(C) 1
(D) none
1 2 2 Q.l02 LetA= 2 1 2 , then 2 2 1 (A) A2 - 4 A - 5I3 = 0
(B) A -1 = - (A - 4I3)
(C) A3 is not invertible
(D) A2 is invertible
faBansal Classes
Q. B. on Determinant & Matrices
[10]
ANSWER KEY
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a 88 b
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a £8 b
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V I8b
a £8'b
a
a 08'b
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v 8 Lb
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3
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3
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a
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3
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3
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Lib
a 9i b
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faBansal Classes
Q. B. on Determinant & Matrices
[10]
BANSAL CLASSES TAR GET TIT JEE 2007
MATHEMATICS XII (ALL)
mimmemm & MATRICES
C O N T E N T S
DETERMINANT KEY CONCEPT. EXERCISE-I EXERCISE-II EXERCISE-III
Page -2 Page -5 Page -9 Page -10
MATRICES KEY CONCEPT. EXERCISE-I EXERCISE-II EXERCISE-III
Page -13 Page -18 Page -20 Page -22
ANSWER KEY
Page -23
KEY
CONCEPTS
D E T E R M I N A N T
1.
bl
The symbol
is called the determinant of order two.
Its value is given by : D = a 1 b 2 -a 2 b 1 a, bj q 2.
The symbol a 2 b2 02 is called the determinant of order three. a3
bj
c3 b2
bx
c2
c,
bi Ci
Its value can be found as : D = a b c - «2 b c + ag b 3 3 3 3 2 b2 =
a
i
c2
c3
a
2 - b, a 3
a2
c
2 c3
+
Cl a
3
b2 b
3
OR
C
2
and so on.
In this manner we can expand a determinant in 6 ways using elements of; R j , R^, R3 or Cj, C 2 , C3 . 3.
Following examples of short hand writing large expressions are: The lines : ajX + bjy + Cj = 0. (1) (0 a2X + b2y + c2 = 0. (2) a3x + b3y + c3 = 0. (3) are concurrent i f ,
(ii)
'
= 0 .
Condition for the consistency of three simultaneous linear equations in 2 variables. ax2 + 2hxy + by2 + 2gx + 2fy+ c = 0 represents a pair of straight lines if : a h g 2
2
2
abc + 2 fgh - af - bg - ch = 0 = h b f g f
(iii)
c
Area of a triangle whose vertices are (x r , y r ); r = 1 , 2 , 3 is : 7i 1
•H
y2 1 y3 1
If D = 0 then the three points are collinear. x
(iv)
y1
Equation of a straight line passsing through (Xj, y,) & (x^ y2) is Xl yi 1 = 0
y 1
2 MINORS : The minor of a given element of a determinant is the determinant of the elements which remain after deleting the row & the column in which the given element stands. For example, the minor of a} in (Key a
i c, »3 c3 3 3 Hence a determinant of order two will have "4 minors" & a determinant of order three will have "9 minors". COFACTOR : If Mjj represents the minor of some typical element then the cofactor is defined as : Cy = (-1 )1+J. Mjj ; Where i & j denotes the row & column in which the particular element lies. Note that the value of a determinant of order three in terms of 'Minor' & 'Cofactor' can be written as : D = anM n - a12M,2 + a13M13 ORD = ajjCjj + a12C12 + a13C13 & so on Concept 2) is
fc Ban sal Classes
b
c
& the minor of b2 is
Determinant & Matrices
[2]
6.
P-l
PROPERTIES OF DETERMINANTS : The value of a determinant remains unaltered, if the rows & columns are inter changed. A
e.g. if D =
A
b, CL b2 c2 b3 c3
I 2
a3
A,
A
3
:
b2 b3 c3 C
BI
=
A
2
CL
D'
2
D & D' are transpose of each other . If D' = - D then it is S K E W SYMMETRIC determinant but D' = D => 2D = 0 => D = 0 => Skew symmetric determinant of third order has the value zero. P - 2 : If any two rows (or columns) of a determinant be interchanged, the value of determinant is changed in sign only. e.g. a
Let D =
A A
P - 3 :
bi Cl b2 c2 b3 C
i 2 3
B2
C2
& D' = i
BI
CJ
A3
B3
C3
A
2
a
3
Then D' = - D .
If a determinant has any two rows (or columns) identical, then its value is zero, B,
e.g. Let D =
then it can be verified that D = 0.
B, I3
B3
P - 4 : If all the elements of any row (or column) be multiplied by the same number, then the determinant is multiplied by that number. If D
egP-5:
and
2 b2 3 b3
KA,
KB,
2
t)2
D' =
KC,
Then D'=KD
If each element of any row (or column) can be expressed as a sum of two terms then the determinant can be expressed as the sum of two determinants. e.g. AJ+X
FY+Y
a2 a 3
b2 b3
a
l b, Cl 2 b2 C a 3 b3 C
C, + Z
c2 c3
a
=
2
3
X
+ a2
a3
y Z b2 c2 b3 C 3
P - 6: The value of a determinant is not altered by adding to the elements of any row (or column) the same multiples of the corresponding elements of any other row (or column), e.g. Let D =
AJ+MA,
D' =
a
B1+MB b
2
A3+NA2
and
2
CJ+MC^
2
C3+NC2
C
2
B3+NB
2
Then D ' = D
Note : that while applying this property ATLEAST ONE ROW (OR COLUMN) must remain unchanged . P - 7: If by putting x = a the value of a determinant vanishes then (x-a) is a factor of the determinant. 7.
MULTIPLICATION OF TWO DETERMINANTS (i)
bi
li m. in.
A1l1+B1L2
A1M1+B1M
A2lj+B2L2
A2M1+B2M2
2
Similarly two determinants of order three are multiplied.
fc Ban sal Classes
Determinant & Matrices
[3]
(ii)
If D =
b, b2 b3
A2 A
3
CL 2
then, D
=
C3
Aj B, c, A 2 B2 c 2 A3
b, CL Consider 2 b2 c2 PROOF : b3 c33 Note : ajA2 + bjB 2 + CjC2 = 0 etc. A
I
A
X
A
A,
therefore, D x
A2
A,
A3
BI
B2
B3
CI
C2
C3
3
= D =>
where A- ,B
?
B3
C3
AI
A2
A3
1) 0 0
B,
B2
B3
0
D
0
CI
c2
c3
0 0
D
A2
A3
B,
B2
B3
c,
c2
C3
2
=D
OR
AI
B>
c,
A2
B2
C2
CA3
B3
c3
= D2
SYSTEM OF LINEAR EQUATION (IN TWO VARIABLES) : (i) Consistent Equations : Definite & unique solution. [ intersecting lines ] (ii) Inconsistent Equation : No solution . [ Parallel line ] (iii) Dependent equation : Infinite solutions . [ Identical lines ] Let ajX + bjy + 0 , ^ 0
a2X + b2y + c2 = 0
&
then :
£L = i ^ i i B.'j bo Cn
Given equations are inconsistent
bi
Given equations are dependent
&
CRAMER'S RULE : [ Simultaneous Equations Involving Three Unknowns ] Let ,ajX + bjy + c,z = dj ...(I); a ^ + b2y + c2z = d2 ...(II); ajX + b3y + c3z = d3 ...(III) Then,
X x
-Bl D
'
y-^1 Y
D
'
z-D^ D
a a d, bi Cl i di ci i bi d, i bi Cj Where D = a2 b2 c2 ; D i = d2 b2 c. ; D2 = a2 d2 c2 & D3 = a2 b2 d2 d3 b3 c3 a3 d3 c3 a3 b3 d3 a3 b3 c3 NOTE: If D * 0 and alteast one of D t , D 2 , D3 ? 0 , then the given system of equations are consistent and (a) have unique non trivial solution. I f D ^ O & Dj = D2 = D3 = 0, then the given system of equations are consistent and have trivial (b) solution only. If D = D, = D 2 =D 3 = 0, then the given system of equations are consistent and have infinite solutions. (c) a
(d) 10.
ajX + bjy+ C!Z=d! a 2 x + b 2 y + c 2 z=d 2 In case represents these parallel planes then also a 3 x + b3y + c3z = d 3 d=d, = d2= D3 = 0 but the system is inconsistent. If D = 0 but atleast one of Dj, D 2 , D3 is not zero then the equations are inconsistent and have no solution. If x , y , z are not all zero , the condition for aLx + b,y + CjZ = 0 ; a2x + b2y + c2z = 0 & a
i W Cj a3x + b3y + c3z = 0 to be consistent in x , y , z is that a2 b2 c2 =0. a 3 b3 c3 Remember that if a given system of linear equations have Only Zero Solution for all its variables then the given equations are said to have TRIVIAL SOLUTION.
(!IBansal Classes
Determinant & Matrices
[4]
EXERCISE-II Q. 1
Without expanding the determinant prove that: 0
b
(a) -b
0 -a
c
-c a -o 0
ax by cz 2 2 2 (d) x y z
1
Q.2
I
0 (b) q - p r-p
a X
=
1
yz
p-q 0 r-q
p-r q-r = 0 0
-7
(c
5 + 3i —-4i 3 5 — 3i 8 4 + 5i is real —+ 4i 4 - 5 i 3
9
1 a a2-be 2 (e) 1 b b - ca = 0 1 c c2 - ab
b y
c z zx xy
Without expanding as far as possible, prove that : (a)
a 2 +2a 2a+l 3
2a+l 1 a+2 1 = ( a - l ) 3 3 1
(b)
1
1 1
x
y
x
y
3
3
z = [(x-y) (y-z) (z-x) (x+y+z)]
z3
x 3 +l x 2 x y 3 +l y 2 y = 0 and x , y , z are all different then, prove that xyz = - l . z3 + l z 2 z
Q.3
If
Q.4
18 40 89 Using properties of determinants or otherwise evaluate 40 89 198 89 198 440
Q.5
a-b-c Prove that 2b 2c
Q.6
If D = c a b
a b c b c a
Prove that
-2a b+a c+a
2a b-c-a 2c
2a 2b = (a + b + c)3 c-a-b
b+c c+a a+b b+c and D' = c+a a+b
a+b c+a b+c
then prove that D' = 2D.
a+b a+c - 2 b b+c = 4 [(a+b) (b+c) (c+a)] c+b - 2 c
Q,
1 + a2 - b 2 2ab 2ab 1-a 2 +b2 Prove that 2b -2a
Q.9
Prove that
a a+c a-b
Q.10
tan(A+P) tan(B+P) tan(C+P) Show that the value ofthe determinant tan(A+Q) tan(B+Q) tan(C + Q) vanishes for all values of A tan(A+R) tan(B+R) tan(C+R)
b-c b b+a
c+b c-a c
-2b 2a = (1 +a 2 + b2)3 2 2 1-a - b
(a + b + c) (a2 + b2 + c2).
B, C, P, Q & R where A + B + C + P + Q + R = 0 (!^Bansal Classes
Determinant & Matrices
[5]
Q.ll
be be' + b'c b'c' Factorise the determinant ca ca' + c'a c'a' ab ab'+a'b a'b'
Q.12 Prove that
((3 + y - a - 5)4
(p + y - a - 5)2 1
(Y+A-p-8)4
(Y+OC-|3-8)2
(a + p -
Q. 13
Y
- 5)
4
(a + p -
Y
- 8)
2
1
= - 64(a- (3)(a - y)(a - 5)(P - y) (P - 5) (y- 6)
1
n! (n+1)! (n+2)! - 4 (n+1)! (n+2)! (n+3)! then show that For a fixed positive integer n, if D = 3 (n!) (n+2)! (n+3)! (n+4)!
is divisible by n. Q.14
x + 2 2x + 3 3x + 4 Solve for x 2x + 3 3x + 4 4x + 5 = 0. 3x + 5 5x + 8 10x + 17
Q 15 If a + b + c = 0 , solve for x: --—r).16
a-x c b
c b b- x a = 0. a c- x
If a2 + b2 + c2 = 1 then show that the value of the determinant a 2 +(b 2 +c 2 )cos9 ba(l-cosG) .22 , /_2 ab(l-cosG) b +(c 2 +a 2 )cos0 ac(l-cosG) bc(l-cosO)
ca(l-cosQ) cb(l-cosG) c2 +(a 2 +b 2 )cosG
simplifies to cos2G.
a b c pa qb rc Q. 17 If p + q+r = 0 , prove that qc ra pb = pqr c a b b c a rb pc qa Q.'l8
a a3 a 4 - 1 If a, b, c are all different & b b3 b 4 - 1 = 0, then prove that: c c3 c 4 - 1
abc (ab + be + ca) = a + b + c. Q.19
Showthat a 2 +X ab ac
Q.20 (a)
(b)
ab b +X be 2
ac be is divisible by A,2 andfindthe other factor. c 2 +X
be a a 2 2 Without expanding prove that ca b b ab c c2 a2 (a + I)2 (a - I)2
fc Ban sal Classes
b2 (b + l)2 (b-1) 2
a 2 b2 c2 (c + l)2 = 4 a b 1 1 (c - l)2
=
1 a 2 a3 1 b2 b3 1 c2 c 3
c2 c 1
Determinant & Matrices
[6]
Q.21
x +x Without expanding a determinant at any stage, show that 2x + 3 x - l x 2 +2x+3 2
x+l 3x 2x-l
x-2 3x-3 = Ax + B, 2x-l
where A & B are determinants of order 3 not involving x.
Q.22
Q.23
-be b2 + be c2 + be 2 -ac c2 + ac = (ab + be + ca)3 Prove that a + ac a 2 + ab b2 + ab -ab x2-b2
X -C
Solve (x-a) 3
(x-b) 3
(x-c)
3
(x+b)
3
(x+c)
x-2
2x - 3 3x - 4 2x - 9 3x - 16 = 0. 2x - 27 3x - 64
(x+a)
Q.24
2 2
x 2 - a2
Solve for x : x - 4 x-8
1
a+x
1 Q.25 If a+y 1
a+z
1
= 0 where a, b, c are non zero and distinct
1
b+x
c+x
b+y
c+y
1
where Q is the product of the denominator, prove that
b+z
c+z
1
y
1 1
P = (a - b) (b - c) (c - a) (x - y) (y - z) (z - x) 2r_1
Q.26 If Df =
2^3r_1j 4(5 r -')
x
y
n
n
2 -l
3 -l
z
then prove that X Dr = 0. r= l
n
5 -l
Q.27 If 2 s = a+b + c then prove that
(s-b) 2 (s-c) 2
(s-a) 2 b2 (s-c) 2
(s-a) : (s-b)'
= 2 s3 (s - a) (s - b) (s-c).
cotf
cotf
cot£
Q.28 In a A ABC, determine condition under which tan| + tan-| tan-- + tan|- tan^ + tanf
-b 2 c 2
Q.29
ab(c 2 +a 2 )
ca(b 2 +c 2 j
ac(a 2 +b 2 ) 2 2
Show that ba(b 2 +c 2 ) ' -c 2 a 2
= 0
2 2
2 2 3
bc(a 2 +b 2 ) = (a b + b c + c a ) .
cb(c 2 +a 2 )
-a 2 b 2
Q.30 Prove that be-a2 ca-b2 ab - c2 ~bc + ca + ab b c - c a + ab bc + c a - a b = 3 .(b-c) (c-a) (a-b) (a+b + c) (ab + bc + ca) (a+b) (a+c) (b+c) (b+a) (c+a) (c+b)
fc Ban sal Classes
Determinant & Matrices
[267]
2bc - a Q.31
„2
Show that
2ac-b 2
b2 = (a3 + b3 + c3 - 3 abc)2. a2 2ab - c'
a! 1, + b) m] a! 12 + bj m2 a, 13 + bj m3 a2 li + b2 m i a2l2 + b2m2 a 2 l 3 +b 2 m 3 = 0. a 3 l 1 +b 3 m 1 a 3 l 2 +b 3 m 2 a 3 l 3 +b 3 m 3 (ai-bi)2 (ai-b 2 ) 2 (aj-b,) Q.33 ^rove that (a 2 -bi) 2 (a 2 -b 2 ) 2 (a 2 -b 3 ) (a 3 -bi) 2 (a 3 -b 2 ) 2 (a 3 -b 3 ) 2 Q.34 Prove that oc + p + y+5
2 ( a r a ^ - 83X83- a j X S - b2)(b2- b 3 )(b 3 - b t )
a+p+y+8
ap + y5
2 (a + P)(y +5) aP(y+5) + y 8 ( a + P)
a p + y8
ap (y+5) + yS (a+P) =
0.
2aPy8
Q.35 If ax2 + 2 hxy + by2 + 2 gx + 2 fy + c = (lLx + rrijy + n1) (l2x + ir^y + n2), then prove that a h g h b f =0. g f c Q.36 Prove that 1 2 cos ( B - A ) cos2 ( C - A )
cos2 (A - C) cos2 (B-C) 1
: 2sin2(A- B)sin2(B - C)sin2(C - A)
axj2 + byt2 + czj2 = ax^ + by22 + cz22 = ax32 + by32 + cz32 = d and ax^xg + by2y3 + cz2z3 = ax3xt + by3yj + cz3zj = axjx2 + byjy2 + CZ[Z2 = f, then prove that
Q.37 If x
cos2 ( A - B ) 1 2 cos (C - B )
i yi
x2 y 2
z
i
z2 = (d-f>
d + 2f abc
1/2
(a,b,c*0)
x3 y3 z3 Q.38 If ( x r x 2 ) 2 + ( y r y2)2 = a 2 , (x,-^) 2 + (y 2 -y 3 ) 2 = b2 and (X3-Xl)2 + (y 3 - Yl ) 2 = c2 xi yi 1 prove that 4 x2 y2 1
(a + b + c) (b + c - a) (c + a - b) (a + b - c ) .
x3 y 3 1
50 Sj S2 Q.39 If Sr = a r + pr + y r then show that 51 s2 S3 = (a - 3) (P - y)2 (y - a) 2 . 52 S3 S4 Q.40 ). 40 If u = ax2 + 2 bxy+cy2 , u' = a'x2 + 2 b'xy + c'y2. Prove that y2 -xy x 2 a b c a' b' c'
fc Ban sal Classes
ax+by bx+cy a'x+b'y b'x+c'y
u u ax+by a'x+b'y
Determinant & Matrices
[8]
EXERCISE-II 4
Q. 1
Solve using Cramer's rule :
Q.2
Solve the following using Cramer's rule and state whether consistent or not. x + y + z - 6 = 0
(a) 2x + y - z - 1 = 0
x + 2y + z = 1
(b) 3x + y + z = 6
x + y - 2z + 3 = 0
Q.3
+ —— = - 1
& —
x+5 y + 7
x + 2y = 0
x+5
— =- 5 .
y +7
x - 3 y + z = 2
(c) 3x + y + z = 6 5x + y + 3z = 3
7 x - 7 y + 5z = 3
(d)
3x + y + 5z = 7 2x+3y+5z
=5
z + ay + a2x + a3 = 0 Solve the system of equations ; z + by + b2x + b3 = 0 z + cy + c2x + c3 = 0
Q.4
For what value of K do the following system of equations possess a non trivial (i.e. not all zero) solution over the set of rationals Q? x+Ky + 3 z = 0 , 3 x + K y - 2 z = 0 , 2 x + 3 y - 4 z = 0, For that value of K, find all the solutions of the system.
Q.5
Given x = cy+bz ; y = az + cx ; z = bx+ay where x, y, z are not all zero, prove that a2 + b2 + c2 + 2 abc = 1.
Q.6
Given a = —— ; b = —— ; c =
X
y- z
Y
Z
z - x x - y
where x, y, z are not all zero , prove that :
1 + ab + be + ca = 0. Q.7
If sin q ^ cos q and x, y, z satisfy the equations x cos p - y sin p + z = cos q + 1 x sin p + y cos p + z = 1 - sin q x cos(p + q) - y sin (p + q) + z = 2 thenfindthe value of x2 + y2 + z2.
Q.8
IfA, B and C are the angles of a triangle then show that sin 2A-x + sin C*y + sin B-z = 0 sin C x + sin 2B-y + sin A-z = 0 sin B x + sin A-y + sin 2Cz = 0 possess non-trivial solution.
Q.9
Investigate for what values of A , u the simultaneous equations x + y + z = 6 ; x + 2 y + 3 z = 1 0 & x + 2 y + A,z = p, have ; (a) A unique solution . (b) An infinite number of solutions . (c) No solution .
Q.IO For what values of p, the equations : x + y + z = l ; x + 2y+4z = p & 2 x+4y+10z = p have a solution ? Solve them completely in each case . Q.ll
Solve the equations : K x + 2 y - 2 z = l , 4x + 2 K y - z = 2 , 6x + 6y+Kz = 3 considering specially the case when K = 2 .
Q.12
Solve the system of equations: ax + y + z = m s x + ay + z = n and x + y + az = p
Q.13 Find all the values of t for which the system of equations ; ( t - l ) x + ( 3 t + l ) y + 2tz = 0 (t - 1) x + (41 - 2) y + (t + 3) z = 0 2 x + ( 3 t + l ) y + 3 ( t - l ) z = 0 has non trivial solutions and in this contextfindthe ratios of x: y: z, when t has the smallest of these values. (!^Bansal Classes
Determinant & Matrices
[9]
Q.14
Solve: (b + c)(y+z)-ax = b - c , (c + a)(z+x)-by = c - a and (a+b)(x+y)-cz = a - b where a + b + c^O.
ap a p Q.15 If bc + qr = ca+rp = ab + pq = - 1 showthat bq b q cr c r
= 0.
Q. 16 If x, y, z are not all zero & if ax+by + cz = 0, bx+ cy + az = 0 & cx + ay + bz = 0, then prove that x: y: z= 1 : 1 : 1 OR 1: co: © 2 O R 1 : CD2: co, where© is one ofthe complex cube root of unity. Q.17 If the following system of equations (a - t)x + by + cz = 0 , bx + (c - t)y + az = 0 and cx+ay+(b - t)z = 0 has non-trivial solutions for different values of t, then show that we can express product of these values of t in the form of determinant. Q. 18 Show that the system of equations 3x - y + 4z = 3 , x + 2y - 3z = - 2 and 6x+5y + A-z = - 3 has atleast one solution for any real number X. Find the set of solutions of X = -5.
EXERCISE-III Q. 1
Q.2
Q.3
Q.4 (i)
Using determinants solve the equations, x + 2y + 3z = 6 2x + 4y + z = 17 3x + 2y + 9 z = 2 [REE'94,6] cos(A-P) cos(A-Q) cos(A-R) For all values of A, B, C & P, Q, R show that cos(B-P) cos(B-Q) cos(B-R) =0. cos(C-P) cos(C-Q) cos(C-R) [ IIT '94, 4] For what values of p & q, the system of equations 2 x + p y + 6 z = 8 ; x + 2y + qz = 5 & x+y + 3z = 4 has ; (i) no solution (ii) a unique solution (iii) infinitely many solutions [REE'95, 6] Let a, b, c positive numbers. The following system of equations in x, y & z. x!
. iL _Z_
2 '2 a- 2 + b2 c2 (A) no solution (C) infinitely many solutions
(ii)
+
±_ 2
x2
y2
z2
(B) unique solution (D)finitelymany solutions
1 l+i+o 2
fc Bansal Classes
Determinant & Matrices
[IIT '95,1+1]
[10]
Q. 5
Let a > 0, d > 0 . Find the value of determinant 1
1
(a + d) (a + 2d)
(a + d)
(a + d) (a + 2d)
(a + 2d) (a + 3d)
(a + 2d)
(a + 2d) (a + 3d)
(a + 3d)(a + 4d)
1
Q.6
Q.7
Q.8
1
1
1
[ IIT '96 , 5 ]
1
Find those values of c for which the equations : 2x+ 3 y = 3 (c + 2)x+ (c + 4)y = c + 6 (c + 2)2x +(c+4) 2 y = (c + 6)2 are consistent. Also solve above equations for these values of c.
[ REE'96, 6 ]
For what real values of k, the system of equations x + 2y + z = 1; x + 3y + 4z = k; x + 5y + 1 Oz = k2 has solution ? Find the solution in each case. [ REE' 97, 6 ] 1 a a The parameter, onwhich the value ofthe determinant cos(p-d)x cospx cos(p+d)x doesnot depend sin(p-d)x sin px sin(p+d)x upon is: (A) a
Q.9
1
a (a + d)
(C) d
(B)p
6i -3i If 4 3i 20 3
1 - 1 = x + iy, i
(A) x = 3 , y = 1
(D)x [ JEE '97, 2 ]
then :
(B) x = 1, y = 3
(C)x = 0 , y = 3
(D)x = 0 , y = 0 [ JEE '98 , 2 ]
Q.IO (i)
1 x x+1 If f(x) = 2x then f(100) is equal to : x (x-l) (x+l)x 3x (x-l) x(x-l)(x-2) (x+1) x (x-l) (A) 0
(ii)
(B) 1
(C) 100
(D) -100
Let a, b, c, d be real numbers in G.P. If u, v, w satisfy the system of equations, u + 2v + 3w = 6 4u + 5v + 6w = 12 6u + 9v = 4 then show that the roots of the equation, i + i + 1 ] x2 + [(b - c)2 + ( c - a)2 + ( d - b ) 2 ] x + u + v + w = 0 and
•.U
V
wJ
20 x2 + 10 (a - d)2 x - 9 = 0 are reciprocals of each other. [JEE'99, 2+10 out of200]
fc Ban sal Classes
Determinant & Matrices
[271]
Q.ll
Ifthe system of equations x - K y - z = 0 , K x - y - z = 0 and x + y - z = 0 has a non zero solution, then the possible values of K are (A)-l, 2 (B) 1,2 (C)0, 1 (D)-l,l [ JEE 2000 (Screening)] sin 9
Q.12 Prove that for all values of 6, sin
sin 20
cos 0
+ ^-J
sin [o -
cos 1^0 +
sin j^29 + ^
cos f9 -
sin (^29 -
=
0
[ JEE 2000 (Mains), 3 out of 100 ] Q.13 Find the real values of r for which the following system oflinear equations has a non-trivial solution. Also find the non-trivial solutions: 2 rx - 2y + 3 z = 0 x + ry + 2z - 0 [ REE 2000 (Mains), 3 out of 100 ] 2x + rz = 0 Q.14
Solve for x the equation .2 a" sin(n + l)x cos(n + l)x
a 1 sin nx sin(n - l)x cos nx cos(n - l)x
=0 [ REE 2001 (Mains), 3 out of 100 ]
Q.15 Test the consistency and solve them when consistent, the following system of equations for all values of A: x+y+z =1 x + 3y-2z = A [ REE 2001 (Mains), 5 out of 100 ] 3x + (A + 2)y-3z = 2A + 1 Q.16 Let a, b, c be real numbers with a2 + b2 + c2 = 1. Show that the equation ax - by - c
bx + ay
cx + a
bx + ay
-ax + by - c
cy + b
cx + a
cy + b
-ax - by + c
= 0
represents a straight line.
[ JEE 2001 (Mains), 6 out of 100 ]
Q.17 The number of values of k for which the system of equations (k+ l)x+ 8y = 4k kx + (k + 3)y = 3k - 1 has infinitely many solutions is (A) 0 (B) 1 (C) 2
(D)inifinite [JEE 2002 (Screening), 3]
Q.18 The value of A for which the system of equations 2x - y - z = 12, x - 2y + z = -4, x+y + Az = 4 has no solutionis (A) 3 (B)-3 (C) 2 (D)-2 [JEE 2004 (Screening)]
fc Ban sal Classes
Determinant & Matrices
[12]
KEY CONCEPTS MATRICES USEFUL IN STUDY OF SCIENCE, ECONOMIC SAND ENGINEERING Definition :
Rectangular array of mn numbers. Unlike determinants it has no value. Ml
12
21
l
a
l
"22
2n
or
m2
Special Type Of Matrices :
(a)
Row Matrix : A = [ a n , a12, (or row vectors)
(b)
Column Matrix : (or column vectors)
a ln ]
»2I
A=
l
in^ 2n
'm/
2.
<
n, i denotes the row and
having one row. (1 x n) matrix.
having one column, (m x 1) matrix
ml
Zero or Null Matrix: (A = Om x n)' An m x n matrix all whose entries are zero v
0 0 0 0 0 0 (d)
l
a
ml
Abbreviated as :A = [ a ^ j 1 < i < m ; 1 < j j denotes the column is called a matrix of order m x n.
(c)
a
i2 a 2*! a22
Mn
a
is a
Horizontal Matrix:
3x2
null matrix
& B=
0
0
0'
0
0 0
0
0 0
is 3 x 3 null matrix
A matrix of order m x n is a horizontal matrix if n > m. "1 2 3 4" 2
5
(e)
Verical Matrix:
(1)
Square Matrix : (Order n)
11
A matrix of order m x n is a vertical matrix if m > n.
(i)
3 6
a square matrix.
In a square matrix the pair of elements a. & a. are called Conjugate Elements. eg-
(ii)
1 1 2 4
If number of row = number of column => Note
2 5'
faa
n
V a21
a
12
^
a 22 y
The elements a n , a22, a33, am are called Diagonal Elements . The line along which the diagonal elements lie is called "Principal or Leading" diagonal. The qty E a H = trace of the matrice written as, i.e. tr A
fc Ban sal Classes
Determinant & Matrices
[13]
Square Matrix Triangular Matrix I
Diagonal Matrix denote as
all elements except the leading diagonal are zero 0 '1 3 ! , A = 0 2 4 ; B = 2 -3 0 diagonal Matrix Unit or Identity Matrix ,4 3 3, ,0 0 5 , I Lower Triangular Upper Triangular d, 0 0 1 if i = j" a.. = 0 V i < j a.. = 0 V i > j 0 d2 0 0 if ij 0 0 d, Note that: Minimum number of zeros in Scalar Matrix Note: (1) If d l = d2 = d3 = a a triangular matrix of Unit Matrix (2) If d1 = d2 = d3 = 1 order n = n(n-l)/2 Note: Min. number of zeros in a diagonal matrix of order n = n(n- 1) d
dia ( d i '
d
2'
> dn)
"It is to be noted that with square matrix there is a corresponding determinant formed by the elements ofAin the same order." 3.
4.
5.
6.
Equality Of Matrices: A = [a ; j ] & B = [b i ; ] are equal if, Let both have the same order. (ii) (0
ai j = b
for each pair of i & j.
Algebra Of Matrices: Addition : A + B = [ a ;j + b;j j where A& B are of the same type, (same order) (a) Addition of matrices is commutative. A=mxn B=mxn i.e. A+B = B + A (b) Matrix addition is associative. Note : A, B & C are of the same type. (A+ B) + C = A + (B + C) (c) Additive inverse. A = mxn If A + B = O = B + A Multiplication Of A Matrix By A Scalar a b ka kb kc kA= kb kc ka If A= b c c a kc ka kb Multiplication Of Matrices: (Row by Column) AB exists if, A=mxn & B = nxp 2x3 3x3 AB exists, but BA does not => AB * BA \ A = pre factor Note: In the product AB. [ B = post factor A =
a
a
( i> 2>
a
n)
1xn AB = [a1b1 + a 2 b 2 +
&
nx1 + anbj
If A = aAj j m x n & B = [ b^ ]
^Bansal Classes
"b
B
n x p matrix, then
Determinant & Matrices
(ABX- = I
a ir . b
[14]
1.
Properties Of Matrix Multiplication : Matrix multiplication is not commutative. A=
;
; AB =
B =
BA =
AB * BA (in general) 2.
"l 1" AB = _2 2_
"-1 1
1" -1
"0 0" 0 0
=> AB = O
=t> A = 0
or B = 0
Note: IfA and B are two non- zero matrices such that AB = O then A and B are called the divisors of zero. Also if [AB] = O => | AB | =>|A| | B | = 0 = > | A | = 0 or | B | = 0 but not the converse. IfA and B are two matrices such that (i) AB = B A => A and B commute each other (ii) AB = - B A => A and B anti commute each other Matrix Multiplication Is Associative: If A, B & C are conformable for the product AB & BC, then (A.B).C = A.(B.C) Distributivity : A (B + C) = AB + AC" Provided A, B & C are conformable for respective products (A + B) C = AC + BC_ POSITIVE INTEGRAL P O W E R S O F A 2
SQUARE MATRIX : 3
For a square matrix A, A A = (A A) A = A (A A) - A . Note that for a unit matrix I of any order, Im = I for all m e N. 6.
(a) (b)
MATRIX POLYNOMIAL: If f (x) = a0xn + ajX"-1 + a2xn_2 + + anx° then we define a matrix polynomial n n n 2 f (A) = a0A + atA -' + a2A " + + aV where A is the given square matrix. If f (A) is the null matrix then A is called the zero or root of the polynomial f(x). DEFINITIONS: Idempotent Matrix: A square matrix is idempotent provided A 2 =A. Note that A" = A V n > 2 , n e N. Nilpotent Matrix: A square matrix is said to be nilpotent matrix of order m, m e N, if Am = 0 , A m_1 ^0.
(c)
Periodic Matrix: A square matrix is which satisfies the relation AK+1 =A, for some positive integer K, is a periodic matrix. The period of the matrix is the least value of K for which this holds true. Note that period of an idempotent matrix is 1.
(d)
Involutary Matrix: IfA 2 = I , the matrix is said to be an involutary matrix. Note that A=A - 1 for an involutary matrix.
7.
The Transpose Of A Matrix : (Changing rows & columns) Let Abe any matrix. Then, A = a;j of order m x n T => A or A' = [ a.. ] for 1 < i < n & 1 < j < m of order T
Properties of Transpose: If A & (a) (A± B)T = AT ± BT IMP. (b) (AB)T = B T A T (c) (AT = A (d) (k A)T = kAT General :
^Bansal Classes
(A^A^,,
n xm
T
B denote the transpose of A and B , ; note that A & B have the same order. A & B are conformable for matrix product AB. k is a scalar .
An)T = A j ,
, A j , A^
(reversal law for transpose)
Determinant & Matrices
[15]
8.
Symmetric & Skew Symmetric Matrix : A square matrix A = [ a ;j j is said to be, symmetric if, a.. = a.. V i & j (conjugate elements are equal) (Note A=A 1 ) . n(n +1) Note: Max. number of distinct entries in a symmetric matrix of order n is — - — . and skew symmetric if, &..= -&.. V i & j (the pair of conjugate elements are additive inverse of each other) (NoteA = - A T ) Hence If A is skew symmetric, then a11 . = - a.1..1 => a.li = 0 V i Thus the digaonal elements of a skew symmetric matrix are all zero, but not the converse.
Properties Of Symmetric & Skew Matrix : P - 1 A is symmetric if AT = A A is skew symmetric if AT = - A P - 2 A + AT is a symmetric matrix A - AT is a skew symmetric matrix. Consider (A+A T ) T = AT + (AT)T = AT + A = A + AT A + AT is symmetric. Similarly we can prove that A - AT is skew symmetric. P - 3 The sum oftwo symmetric matrix is a symmetric matrix and the sum of two skew symmetric matrix is a skew symmetric matrix. Let AT = A ; BT = B where A & B have the same order. (A+B) T = A + B Similarly we can prove the other P - 4 If A& B are symmetric matrices then, (a) A B + B A is a symmetric matrix (b) AB - BA is a skew symmetric matrix. P - 5 Every square matrix can be uniquely expressed as a sum of a symmetric and a skew symmetric matrix. A = ^ (A + AT) + ~ (A - AT) J
P Symmetric
Skew Symmetric
Adjoint Of A Square Matrix : /
Let
a
ll A - [ ij ] = 21 ^31 a
a
a
i2 22 a 32 a
a
23 33>
be a square matrix and let the matrix formed by the
a
c Ml C21 IC31 r
cofactors of [a ] in determinant !A| is = f Ml c
Then (adj A) = C 12 c V 13
C 21 C 31^ r 22 r 32 C 23 c 33 J
^12 22 ^32
r
c 13^ C23 C
33;
V. Imp. Theorem: A (adj. A) = (adj. A).A= |A| I n , If Abe a square matrix of order n. (!%Bansal Classes
Determinant & Matrices
[16]
Note : If A and B are non singular square matrices of same order, then ® | adj A| = | A| n ~' (ii) adj (AB) = (adj B) (adj A) (iii) adj(KA) = K""1 (adj A), K is a scalar Inverse Of A Matrix (Reciprocal Matrix) : A square matrix A said to be invertible (non singular) if there exists a matrix B such that, AB = I = B A B is called the inverse (reciprocal) ofA and is denoted by A - ' . Thus A-' = B O A B = I = B A . We have,
A. (adj A) = | A | IN A- 1 A (adj A) = A - 1 In |A| In ( a dj A) = A - 1 A- 1 =
| A | In
(adj A) |A|
Note : The necessary and sufficient condition for a square matrix Ato be invertible is that IA | * 0. Imp. Theorem : If A & B are invertible matrices ofthe same order, then (AB)"1 = B 1 A -1 . This is reversal law for inverse. Note (0
If A be an invertible matrix, then AT is also invertible & (A 1 ) -1 = (A_1)T.
(ii)
If A is invertible, (a)
(iii)
IfA is an Orthogonal Matrix. AAT = I = ATA
(iv)
A square matrix is said to be orthogonal if, A - 1 = A T .
(v)
1 I A"11 = 7T7 S Y S T E M
O F
(A"1)"1 = A ; (b)
E Q U A T I O N
&
(A*)"1 = (A"])k = A-k, k e N
C R I T E R I A N
F O R
C O N S I S T E N C Y
GAUSS - JORDAN METHOD x +y+ z = 6 x-y +z=2 2x + y — z = 1 or
' x+y+z N (6} x-y+z _ 2 ,2x+y- Z > 10
(\
1 o 1 -1 1 1 -ij u
f \ X y w
AX = B X = A1B =
fa BansaIClasses
(
2 10 A' 1 A X = A 1 B (adj. A).B
Determinant & Matrices
[17]
Note (1)
system is consistent having unique solution
If IAI * 0,
(2)
If | AI * 0 & (adj A). B * O (Null matrix), system is consistent having unique non-trivial solution.
(3)
If | A | * 0 & (adj A) . B = O (Nullmatrix), system is consistent having trivial solution.
(4)
If
IAI = 0 , matrix method fails 1
r
1
If (adj A) . B = null matrix = O
If (adj A) , B * O
Consistent (Infinite solutions)
Inconsistent (no solution)
EXERCISE-I 2 3 3
10 2 1 1 2 2 1 D= 13 and that Cb=D. Solve the matrix equation Ax=b. c= 9 1 1 1
Q.l
"l 2 Given that A= 2 2 1 -1
Q.2
Find the value of x and y that satisfy the equations 3 3 2
Q.3
-2"
0 4
r
?
~i
y
y
X
X
?
^
"3 3" 3y 3y 10 10
0 0 0 0 1 0 0 0 1 andF = 1 0 0 calculate the matrix product EF & FE and If, E = 0 1 0 0 0 0 showthat E 2 F + FE2 = E .
Q.4
Q.5
If A is an orthogonal matrix and B = AP where Pisa non singular matrix then show that the matrix PB_1 is also orthogonal. 0 1 Define A= 3 0
0 Find a vertical vector V such that (A8 + A6 + A4 + A2 +1)V = 11
(where I is the 2 x 2 identity matrix). Q. 6
cos2 0 sin 0 cos 0 cos2 <|) sin
Q.7
TC
2"
1 0 2 0 2 1 , then show that the maxtrix Ais a root of the polynomial f (x)=x 3 - 6x2 + 7x+2. If, A= 2 0 3
fa B ansaIClasses
Determinant & Matrices
[18]
Q. 8
(a)
For a non zero X, use induction to prove that: (Only for XIICB SE)
X 0 0
n
1 ol 1 _ 0 X
X
XD
nAT1
0 0
A" 0
n(n-l) 2 nXn~
, for every n € N
Xn
(b)
n n 1 If, A= 00 01 , then (aI+bA) = a I + na"~ b A, where I is a unit matrix of order 2, V n € N.
Q. 9
Find the number of 2 x 2 matrix satisfying (i) a . i s l o r - 1
;
(ii) ^
+ a 2 2 = a221 + a222 =2 ; (iii) a n a21 + a12 a22 = 0
Q.10 Prove that (AB)T=BT. AT, where A& B are conformable for the product AB . Also verify the result for 1 2 the matrices, A= -1
2 -3 and B = 2 - 3 1 2 2
5 3
1 2 5 2 3 - 6 asa sum ofalower triangular matrix& an upper triangular matrix with zero Q. 11 Express the matrix -10 4 in its leading diagonal. Also Express the matrix as a sum ofa symmetric & a skew symmetric matrix. Q. 12 Find the inverse of the matrix: cosa -sina (i)A= sina cosa 0 0
0" "1 1 0 1 w ;(ii) 1 1 w2
"a 0 0" 1" 2 w where w is the cube root of unity, (iii) A= 0 b 0 w 0 0 c
"2 l" A "3 2" Q.13 Find the matrix A satisfying the matrix equation, _3 2_ .A. _5 —3_
'2 4" 3 -1
Q. 14 A is a square matrix of order n. I = maximum number of distinct entries ifAis a triangular matrix m=maximum number of distinct entries if Ais a diagonal matrix p = minimum number of zeroes ifAis a triangular matrix If / + 5 = p + 2m, find the order of the matrix. Q. 15 IfA is an idempotent matrix and I is an identity matrix of the same order, find the value of n, n e N, such that ( A+ I )n = I + 127 A. a b Q. 16 IfA= c d then prove that value of f and g satisfying the maxtrix equation A2 + f A + g I = Oare equal to - t r (A) and determinant of A respectively. Given a, b, c, d are non zero reals and 1=
"l 0"
"0 o" ; o = _0 0 2
-1
Q.17 Matrices A and B satisfy AB = B"1 where B = 2 0 , Find (I) withoutfindingB"1, the value of K for which KA - 2B"1 +1 = 0 (ii) WithoutfindingA-1, the matrix X satisfying A~'XA= B (iii) the matrixA using A -1
fa B ansaIClasses
Determinant & Matrices
[19]
4 - 4 3 Q.18 For the matrix A= 2 3 - 3
Q.19
1 1 1 2 4 1 Given A = 2 3 1
5 - 3 find A_: 4
1 0 1 2 3 B = 3 4 . Find P such that BPA = 0 1 0
Q.20 Use matrix to solve the following system of equations. x+y+z=3 (i) x+2y+3z=4 x+4y+9z=6
x+y+z=6 (ii) x-y+z=2 2x+y-z=l
x+y+z=3 (iii) x+2y+3z=4 2x+3y+4z=7
x+y+z=3 (iv) x+2y+3z=4 2x+3y+4z=9
EXERCISE-II Ql
Q.2
"9 3" "2 l" GivenA 2 1_ ;B= 3 1 . I is a unit matrix oforder 2. Find all possible matrix X in the following cases. (ii) XA = I (!) XB = 0 butBX^O. (l) AX = A
Q3
IfA& B are square matrices of the same order & Ais symmetrical, show that B' AB is also symmetrical. -l 1 -tanf tanl 1 cos 8 -sinG Show that, tan§ sinG cosG 1 -tanf 1
Q.4
1 2 If the matrices A = 3 4
a b and B = c d
(a, b, c, d not all simultaneously zero) commute, find the value of matrix which commutes with Ais of the form Q.5
a-p p
d-b . Also show that the a + c-b
2p/3" a
1 1 If the matrix A is involutary, show that — (I + A) and — (I - A) are idempotent and i(I+A)^(I-A)=0.
Q.6
Prove that
CO
| adj (adj A) | = | A
(ii)
adj (adj A) = |A|n 2 . A, where | A| denotes the determinant of co-efficient matrix.
Q.7
"-5 1 Find the product oftwo matrices A& B, where A= 7 1 1 -1
/
„ I
, where Ais a non-singular matrix of order 'ri.
3" [1 1 2 -5 & B = 3 2 1 and use it to 2 1 3 1
solve the following system of linear equations, x + y + 2z = 1 ; 3x + 2y + z = 7 ; 2x + y + 3z = 2 . Q.8
1 2 If A = 2 4 then, find a non-zero square matrix X of order 2 such that AX = O. Is XA = O. 1
2]
If A= 2 3 J, is it possible tofinda square matrix X such that AX = O. Give reasons for it.
fa BansaIClasses
Determinant & Matrices
[20]
Q.9
cosa - s i n a cos2p sin2p 71 If A= sina cosa ;B= sin 2p -cos2p Where 0 < P < - then prove that BAB = A-1. Also find the least positive value of a for which B A4 B = A-1.
a b Q.10 If c 1 - a is an idempotent matrix. Find the value of f(a), where f(x)=x- x2, when be = 1/4. Hence otherwise evaluate a. Q.ll
If A is a skew symmetric matrix and I + A is non singular, then prove that the matrix B = (I - A)(I + A)-1 is an orthogonal matrix. Use this tofinda matrix B given A= - 5° 0 cosx sinx 0
Q.12 If F(x)
-sinx cosx 0
0 0 1
then show that F(x). F(y) = F(x + y)
Hence prove that [ F(x) ] 4 = F(- x). Q.13
1 2 3 1 1 2 IfA : 3 4 ;B = 1 0 ;C= 2 4 andX= (a) AX = B - 1
(b) (B - I)X = IC
X
1
X3
X
2
X4
then solve the following matrix equation.
(c) CX = A
"3 - 2 Q 14 Determine the values of a and b for which the system 5 - 8 2 1
1 9 a
X
y z
-
b 3 -1
(i) has a unique solution ; (ii) has no solution and (iii) has infinitely many solutions 0 1 Q.15 Let X be the solution set ofthe equation AX = I, whereA= 4 - 3 3 - 3
-1 4 and I is the corresponding 4
unit matrix a n d x c N thenfindthe minimum value of ^ ( c o s x 6 f sm x 0), 0 s R . Q .16 Determine the matrices B and Cwith integral element such that A=
Q.17 IfA=
0 2P a B a -p
Q.18 If A=
k m /
n
-1
1
0
- 2
= B3 + C3
y - y is an orthogonal matrix,findthe values of a , P, y. y and kn * Im ; then show that A2 - (k + n)A + (kn - /m) I = O.
Hence find A 1 . X
1
Q. 19 Evaluate
Lim
n
X
1
n
~1
f " 3 -3 2 y 5;B = - 3 2 Q.20 Given matrices A= z -3 1 y 3_ Obtain x, y and z if the matrix AB is symmetric. X
X
z -3 1
Determinant & Matrices
[21]
EXERCISE-III Q. 1
a b c b c a where a, b, c are real positive numbers, abc = 1 and A T A = T , then find the c a b
If matrix A
value of a3 + b3 + c 3 . Q.2
IfA = a2
2 a
[JEE 2003, Mains-2 out of 60]
and | A31 =125, then a = (B) ±2
(A) ±3 Q.3
"
(D) 0
(C) ± 5
[JEE 2004(Scr)]
If M is a 3 x 3 matrix, where MTM = I and det (M) = 1, then prove that det (M -1) = 0. [JEE 2004, 2 out of 60] "a2" "f" a 1 l" a 0 ll A = 1 c b , B = 0 d c > u = g ,V= 0 0 f g h h 1 d b If there is vector matrix X, such that AX=U has infinitely many solution, then prove that BX=V cannot have a unique solution. If afd * 0, then prove that BX=V has no solution. [JEE 2004,4 out of 60]
Q.5
1 0 0 1 A= 0 -2
1 0 0 0 0 1 0 and A 1 = •g (A2 + cA + dl) , then the value of c and d are 1 4 ,1 = 0 0 1 (B) 6, 11
(A)-6,-11 "V3 Q.6
IfP = 1 2
2
A 2 .
1 2005 1
1 2 +a/3 -1
(D) 6, - 1 1 [JEE2005(Scr)]
1 1 T T 2005 P, then x is equal to > A= 0 1 and Q = PAP and x = P Q
(B)
(A) 0 (C)
(C)-6, 11
1 2-V3
4 + 2005V3 6015 2005 4-2005a/3
(D)
2005 2-V3 2 + V3 2005
[JEE 2005 (Screening)]
Comprehension (3 questions) Q.7
1 0 0
A = 2 1 0 , U,, U2 and U3 are columns matrices satisfying. AU5 3 2 1
1~ 2 0 ;AU2= 3 0 0
2
,AU3=
3 1
and U is 3 x 3 matrix whose columns are Uj, U2, U3 then answer the following questions (a) (b)
(c)
The value of | U | is (A) 3
(B)-3
(C) 3/2
(D)2
(C)l
(D)3
(C)4
(D) 3/2 [JEE 2006,5 marks each(-2)]
1
The sum of elements o f U is (A)-l (B)0 The value of [3 2 o]U (A) 5
^Bansal
Classes
(B) 5/2
is
Determinant & Matrices
[22]
A N S W E R
K E Y
DETERMINANT
EXERCISE-I Q4. - 1
Q l l . (ab'-a'b)(bc'-b'c)(ca'-c'a)
Q 15. x = 0 or
x = ±
Q 14. x = - l
j | ( a 2 + b 2 + c2)
orx=-2
Q19. X,2 ( a2 + b2 + c2 + A)
Q 23. If ab+bc+ca<0, then x =0 is the only real root; If ab+bc+ca >0, then x =0 or x = ± Q 24. x = 4 Q28. Triangle ABC is isosceles.
ab + be + ca
EXERCISE-II Q 1. x = - 7 , y = - 4 Q 2.. ( a ) x = l , y = 2 , z = 3 ; consistent 13
7
(b) x = 2 , y = - l , z = l ; consistent
35
(c) x = y , y = - - , z = - — ; consistent (d) inconsistent Q 3. x = -(a + b + c), y = ab + be + ca, z = -abc
Q 4. K= y , x: y: z = - y : 1 : - 3
Q7. 2
Q9. (a) X±3 (b) 1 = 3, p=10 (c)X = 3, \x± 10 Q 10. x= 1+2K, y = - 3 K , z = K, whenp= 1 ; x = 2K, y= 1 - 3 K , z=Kwhenp = 2 ; whereK G R x _ y z 1 2 Q l l . If K*2, 2(K+6) ~ 2K+3 ~ 6(K-2) ~ 2(K +2K+15)' 1-2X
If K=2,then x = X,y =—-— andz = 0 where X e R Q 12. If a * 1 or-2, unique solution; If a = - 2 & m + n + p = 0, infinite solution; If a = - 2 & m + n + p ^ 0, no solution ; If a = 1, infinite solution if m = n = p ; If a = 1, no solution if m * n or n ^ p or p t- m Q13. t = 0 o r 3 ; x :y : z = 1 : 1 : 1 Q18.
Q14. x =
4 9 If X*-5 thenx= ~;y = -~ andz = 0 ;
a+b+c
y=
, , 4-5K 13K-9 , If X = 5 then x = — - — ; y = — - — and z = K
a+b+c
z=
a+b+c
a b c
Q17. b c a c a b
where K e R
EXERCISE-III Ql.
x= 1, y = 4 , z = - 1
Q 3.
Q 4.
(i) d (ii) a
Q 5.
Q6.
for c = 0 , x = - 3 , y = 3 ; for
Q 7.
k = 1 : (5t+l, -3t, t); k = 2 : (5t-1,1 -31, t) for t e R ; no solution
Q 8.
B
Q9. D
c
a(a+d) 2 (a+2d)3 (a+3d)2 (a+4d)
=-10,x =- y
y= j
Q10. (i) A
k Q 13. r = 2 ; x = k ; y = - ; z = - k
fa B ansa I Classes
( i ) p * 2 , q = 3 ( i i ) p * 2 & q * 3 (iii) p = 2 4d4
where k e R - { 0 }
Determinant & Matrices
Qll.D Q 14.
x = n7t, nel
[23]
Q 15. If A = 5, system is consistent with infinite solution given by z = K, y= — (3K + 4) and x = - - - (5K + 2) where K e R 2
1 1 5, system is consistent with unique solution given by x= -(1-X,);x= ~(A + 2)andy = 0.
Q17. B
Q.18 D ****************************************************
MATRICES EXERCISE-I Q . l x 1 = l , X2 = - 1 , X3 = l
Q.2x=|, y=2
0 Q.5
.11.
cosa Q.12(i) -sina 0
sina 0 cosa 0 0 1
l/a 1 0 w > (iii) w2 0
1 1 1 w2 1 w Q.15
Q.14 4
?
~0 2 5 " 1 0 0 0 2 3 + 0 0 -6 Q.ll -1 0 4 0 0 0
Q.9 8
V = J_
"0 0 o" 1 0 o" 0 , FE = 0 1 0 0 1 Q.3EF = 0 0 1 0 0 0
?
0 0" 1/b 0 0 1/c
J_ 48 Q.13 19 -70
0 3" 0 -3 3 0
-25 42
Q.16 f = - (a + d); g = ad - be
n=7
17 4 -10 0 Q.18 -21 - 3
1 " - 2 2" Q.17 (i) K=2, (ii) X = B, (iii) A = ~ - 4 2_ Q.20
"0 2 -3 + 0 -3 4
1 2 2 3 2 -3
-19 13 25
-4 7 3 - 5
Q.19
-7 5
(i) x = 2, y = 1, z = 0 ; (ii) x = 1, y = 2, z = 3 ; (iii) x = 2 + k, y = 1 - 2k, z = k where k s R ; (iv) inconsistent, hence no solution
EXERCISE-II a Q.l(i)X= 2 - 2 a 3b + d * 0
a b l - 2 b for a, b G R; (ii) X does not exist; (iii) X = c
Q.4 1 Q.7 x = 2, y= 1, z = - l
Q.8 X -
1 -12 Q.IO f(a)= 1/4,a= 1/2 Q.ll 13 5
-2c c
-3a - 3 c a, c e R and3a + c*0;
-2d d , where c, d G R-{0}, NO
-3 -5 ® -12 Q. 13(a) X= 2
-3 2 ,(b)X = 1 - 1
2 - 2
Q.9
2K
, (c) no solution
Q.14 (i)a * - 3 ,b e R; (ii) a = - 3 and b * 1/3 ; (iii) a = -3 ,b = 1/3 Q.15 2
^ ^
0 Q.16 B = 0
n 1 kn-/m - /
-m k
1 -1 - 1 and C = 0
Q.19
cosx -sinx
0 -1
sinx Q.20 cosx
Q.17
3
a = ±
^
J_ s
*
y=±
1
,(3,3,-1)
3
EXERCISE-III Q.l
Q.2
fa B ansa I Classes
A
Q.5
C
Q.6
A
Q.7
(a)A, (b)B, (c)A
Determinant & Matrices
[24]
5 BAN SAL ^
CLASSES
TARGET IIT JEE 2007
XII & XIII
(With Hints and Solutions at the End)
• • • •
Q. 1 to Q.29 are of 6 Marks Problems Q.30 to Q.66 are of 8 Marks Problems Q.67 to Q.82 are of 10 Marks Problems Q.83 to Q. 100 are Objective type problems.
Advise: Do not spend more than 10 minutes for each problem and then read the solution and then do it.
ALL TH€ BEST FOR JEE - 2 0 0 7
SUBJECTIVE: Q.l
If the sum ofthe roots ofthe equation 2 3 3 3 x _ 2 +2 l l l x + 1 = 2 222x+2 +1 is expressed in the form q find 2
S j + S2, where g is in its lowest form.
[6]
Q.2
Let K is a positive integer such that 36 + K, 300 + K, 596 + K are the squares of three consecutive terms of an arithmetic progression. Find K. [6]
Q. 3
Find the number of 4 digit numbers starting with 1 and having exactly two identical digits.
Q.4
A chord ofthe parabola y2 = 4ax touches the parabola y2 = 4bx. Show that the tangents at the extremities of the chord meet on the parabola by2 = 4a2x. [6]
Q. 5
Consider a circle S with centre at the origin and radius 4. Four circles A, B, C and D each with radius unity and centres (-3,0), (-1,0), (1,0) and (3,0) respectively are drawn. A chord PQ of the circle S touches the circle B and passes through the centre of the circle C. If the length of this chord can be expressed as
, find x.
[6]
[6]
Q.6
x72 s Integrate J (j _ x )
Q.7
n/2 . r 1 — sm 2x a If J " ^ s j n 2 x ) 2 d x = — where a, b are relatively prime find a + b + ab.
dx
t6l
[6]
Q.8
A bus contractor agrees to run special buses for the employees of ABC Co. Ltd. He agrees to run the buses if atleast 200 persons travel by his buses. The fare per person is to be Rs. 10 per day if200 travel and will be decreased for everybody by 2 paise per person over 200 that travels. How many passengers will give the contractor maximum daily revenue? [6]
Q.9
If the point P(a, b) lies on the curve 9y2=x3 such that the normal to the curve at P makes equal intercepts with the axes. Find the value of (a + 3b). [6J
Q.10
Let x(t) be the concentration of glucose per unit volume of blood at time t, p being the amount of glucose being injected per unit volume per unit time. If the glucose is disappearing from the blood at a rate proportional to the concentration of glucose (K being the constant of proportionality),findx(t). Also find the ultimate concentration of glucose as t —> oo. [6]
Q. 11 Find the value (s) of the parameter 'a' (a > 0) for each of which the area ofthe figure bounded by the straight line, y=
fa B ansaIClasses
g2 ^ X -r- & the parabola y= 1+ a
+ 2 & X -f- 3 ci ^ . is the greatest. 1 +a
Determinant & Matrices
[6]
[2]
Q.12
Mr. A is a compulsive liar. He lies 2/5 ofthe time. However a clue to his validity is that his ears droop 2/3 ofthe time when he is telling a lie. They only droop 1/10 ofthe time when he is telling the truth. Mr. A tells his friend Mr. B that "certain event has occured" and his ears were dropping as noticed by Mr. B. Find the probability that Mr. A was telling the truth. [6]
Q.13 Five persons entered the lift cabin on the ground floor of an eightfloorhouse. Suppose that each of them, independently & with equal probability can leave the cabin at any floor beginning with the first,findout the probability of all 5 persons leaving at different floors. [6] Q.14 Let u and v be non zero vectors on a plane or in 3-space. Show that the vector w =] u | v+1 v | u bisects the angle between u and v .
[6]
Q.15 Find the distance from the line x = 2 + t , y = l + t , z= -
-~t to the plane x + 2y+6z= 10. [6]
Q.16
If 9 is the angle between the lines in which the planes 3x - 7y - 5z = 1 and5x-13y + 3z + 2 = 0cuts the plane 8x - 11 y + 2z = 0,findsinB. [6]
Q.17
Suppose u, v and w are twice differentiable functions of x that satisfy the relations au + bv + cw = 0 u u' where a, b and c are constants, not all zero. Show that
Q.18
Q.19
v v'
w w' = 0.
[6]
(A) (B) fc} In any triangle ABC, prove that, cos A • sin2 + cosB sin2 + cos C • sin2 12 J k2) U J —
—
—
f
TN
Ifthe normals to the curve y = x2 at the points P, Q and R pass through the point 0, - ,findthe radius v 1) of the circle circumscribing the triangle PQR. [6 j
Q.20 Let A = {a e R | the equation (1 + 2/)x3 - 2(3 + i)x2 + (5 - 4i)x + 2a2 = 0} has at least one real root. Find the value of Y a 2 .
[6]
aeA
Q.21
Find the equation of a line passing through (- 4, -2) having equal intercepts on the coordinate axes. [6]
Q.22
Let S be the set of all x such that x4 - 1 Ox2 + 9 < 0. Find the maximum value of f (x) = xJ — 3x on S. [6]
Q.23
Solve the differential equation, (x4y2-y)dx + (x2y4-x)dy = 0
Bansal Classes
Problems for JEE-2007
(y(l) = l)
[6]
Q.24
All the face cards from a pack of 52 playing cards are removed. From the remaining pack half of the cards are randomly removed without looking at them and then randomly drawn two cards simultaneously P(38C2Q) from the remaining. If the probability that two cards drawn are both aces is 40 r . 20r ,findp. [6]
[N e> •
^ x 2 y2 Q.25 A circle intersects an ellipse - y + — = 1 precisely at three points A, / a b v / B, C as shown in the figure. AB is a diameter of the circle and is perpendicular to the major axis of the ellipse. If the eccentricity ofthe ellipse is 4/5, find the length of the diameter AB in terms of a. [6] Q.26
Suppose R is set of reals and C is the set of complex numbers and a function is defined as f: R -» C, -ft / f (t) = -—— where t e R , prove that / is injective.
[6]
Q. 27 Circles A and B are externally tangent to each other and to line t. The sum of the radii of the two circles is 12 and the radius of circle A is 3 times that of circle B. The area in between the two circles and its \l brc external tangent is a v 3 - ~ thenfindthe value of a + b. [6] 0 0 1 8 6 4 2 3 0 . Find a vertical vector y such that (A + A + A + A +1) y = 11 where I is a unit matrix of order 2. [6]
Q. 2 8 Define a matrix A r
Q. 29 A circle is inscribed in a triangle with sides of lengths 3,4 and 5. A second circle, interior to the triangle, is tangent to the first circle and to both sides of the larger acute angle of the triangle. If the radius of teh second circle can be expressed in the form (0,90°), find the value of k + w.
sink where k and w are in degrees and lie in the interval cos w [6]
ax2 - 24x + b = x Q.30 If the equation 5 > has exactly two distinct real solutions and their sum is 12 then find x2-l the value of (a-b). [8] Q.31 If a, b, c and d are positive integers and a < b < c < d such that a, b, c are in A.P. and b, c, d are in G.P. and d - a = 30. Find the four numbers. [8] Q .32 Let the set A = {a, b, c, d, e} and P and Q are two non empty subsets ofA. Find the number of ways in which P and Q can be selected so that P n Q has at least one common element. [8] Q. 3 3 If the normals drawn to the curve y = x2 - x + 1 at the points A, B & C on the curve are concurrent at the point P (7/2,9/2) then compute the sum ofthe slopes ofthe three normals. Also find their equations and the co-ordinates of the feet of the normals onto the curve. [8]
fa B ansaIClasses
Determinant & Matrices
[4]
Q. 3 4 A conic passing through the point A (1,4) is such that the segment j oining a point P (x, y) on the conic and the point of intersection of the normal at P with the abscissa axis is bisected by the y - axis. Find the equation of the conic and also the equation of a circle touching the conic,at A (1,4) and passing through i its focus. [8] Q.3 5 y k hyperbola has onefocusat the origin and its eccentricity = Find the equation to its asymptotes.
and one ofits directrix is x + y + 1 =0. [8]
Q. 3 6 Let A, B, C be real numbers such that ^(i) (sin A, cos B) lies on a unit circle centred at origin, (ii) tan C and cot C are defined. ' I ^ If the minimum value of (tan C - sinA)2 - (cot C - cos B)2 is a + bV2 where a. b e N, find the value ^ o f a 3 + b3" ' 1*1 Q.37 For a > 2, if the value of the definite integral f — equals " . Find the value of a. J / ,\2 5050 n0 27 1 I a
f (7t 4 6 ) t a n 6 d0 = rc /n k - — , find the value of (kw), where k, w e N. ,„ 1-tanO w 71/4 J
-
[8]
1 Q. 3 9 Given a function g, continuous everywhere such that g( 1) = 5 and j g (t) dt = 2. 0 1 x If f(x) = - J (x -1) 2 g (t) dt, then compute the value of f"' (1) - f " (1). 20 Q.40
Let f: [0,1 ] —> R is a continuous function such that Jf(x)dx = 0. Prove that there is c e (0,1) such
that Jf(x)dx =f(c). 0 Q. 41
Q.42
[8]
[Bj
Consider the equation in x, x j - ax + b = 0 in which a and b are constants. Show that the equation has only one solution for x if a < 0, for a=3,findthe values of b for which the equation has three solutions. f8] Atank consists of 50 litres offreshwater. Two litres ofbrine each litre containing 5 gms of dissolved salt are run into tank per minute; the mixture is kept uniform by stilting, and runs out at the rate of one litre per minute. If'm' grams of salt are present in the tank after t minute, express'm' in terms of t andfindthe amount of salt present after 10 minutes. [8]
Q.43 Urn-1 contains 3 red balls and 9 black balls. Urn-II contains 8 red balls and 4 black balls. Urn-Ill contains 10 red balls and 2 black balls. A card is drawnfroma well shuffled back of 52 playing cards. If a face card is drawn, a ball is selected from Urn-I. If an ace is drawn, a ball is selected from Urn-II. If any other card is drawn, a ball is selected from Urn-Ill. Find (a) the probability that a red ball is selected. (b) the conditional probability that Urn-I was onefromwhich a ball was selected, given that the ball selected was red. |8]
MBansal Classes
Problems for JEE-2007
fSJ
Q .44 The digits of a number are 1 , 2 . 3 , 4 , 5 , 6 , 7 , 8 & 9 written at random in any order. Find the probability that the order is divisible by 11. [8] Q.45 A number is chosen randomly from one of the two sets, A = {1801, 1802, ,1899, 1900} & B = {1901,1902, ,1999,2000}. If the number chosen represents a calender year. Find the probability that it has 53 Sundays. [8] Q.46 A box contains 2 fifty paise coins, 5 twenty five paise coins & a certain fixed number N (> 2) of ten &fivepaise coins. Five coins are taken out of the box at random. Find the probability that the total value of these five coins is less than Re. 1 & 50 paise. [8] Q.47
A hunter knows that a deer is hidden in one of the two near by bushes, the probability of its being hidden inbush-I being 4/5. The hunter having a rifle containing 10 bullets decides to fire them all at bush-I or II. It is known that each shot may hit one of the two bushes, independently of the other with probability 1/2. How many bullets must he fire on each of the two bushes to hit the animal with maximum probability. (Assume that the bullet hitting the bush also hits the animal). [8]
Q.48 ABCD is a tetrahedron with A(- 5, 22, 5); B(l, 2, 3); C(4, 3, 2); D(- 1 , 2 , - 3). Find AB x FEE x BE)) • What can you say about the values of (AB X BE) X BD and (AB X BD) X BC. Calculate the volume of the tetrahedron ABCD and the vector area ofthe triangle AEF where the quadrilateral ABDE and quadrilateral ABCF are parallelograms. • [8] Q.49
Find the equation ofthe line passing through the point (1,4,3) which is perpendicular to both of the lines x— 1 y+3 z 2 x+2 y-4 z+l -—— - —— = — a n d —-— = —-— - —— 2 1 4 3 2 -2 Also find all points on this line the square of whose distance from (1,4,3) is 357.
[8]
Q.50
Find the parametric equation for the line which passes through the point (0,1,2) and is perpendicular to the line x = 1 +1, y = 1 - 1 and z = 2t and also intersects this line. [8]
Q.51
Suppose that r ; * r2 and r ^ = 2 (r,, r, need not be real). If r, and r, are the roots of the biquadratic x4 - x3 + ax2 - 8x - 8 = 0 find r,, r2 and a. [8]
Q.52
Express
x2+y2+a2 2ax + xy 2ay + x
2ax + xy a2+2x2 2ax + xy
2ay + x 2ax + xy x2+y2+a2
as a product of two polynomial.
[8]
"1 2 2 2 1 1 10 2 2 3 2 2 1 Q.53 Given the matrices A = and D = 13 and that Cb = D. 1 -1 3 ; C = 1 1 1 9 Solve the matrix equation Ax = b. Q.54 Prove that Q.55
a b+c
[8]
b c 3 + 7~T > T for a, b, c > 0. c+a a+b 2
[8]
x 2 +y 2 31-2 Given x, y e R, x + y > 0. If the maximum and minimum value ofthe expression 3 x +xy + 4y 2 M and m, and A denotes the average value of M and m, compute (2007)A, [8] 2
2
Q.56 Prove that the triangle ABC will be a right angled triangle if A B C A B C cos— cos~cos— - sin— sin— sin — 2
fa B ansaIClasses
2
2
2
2
2
J. 2
Determinant & Matrices
[8]
[6]
Q. 5 7 A point P is situated inside an angle of measure 60° at a distance x and y from its sides. Find the distance of the point Pfromthe vertex of the given angle in terms of x and y. [8] Q.58
In AABC, a = 4 ; b = 3 ; medians AD and BE are mutually perpendicular. Find 'c' and 'A'.
[8]
Q.59 The lengths ofthe sides of a triangle are log,012, log1075 and logj0«, where n e N. Find the number of possible values of n. [8] Q.60 A flight of stairs has 10 steps. A person can go up the steps one at a time, two at a time, or any combination of 1 's and 2's. Find the total number ofways in which the person can go up the stairs. [8] b
Q.61
Let a and b be two positive real numbers. Prove that
fe
J a
x a
/ _eb/x dx = 0 x
[81
Q.62 Let/(x) = 2 kx + 9 where k is a real number. If 3/(3) =/(6), then the value of / ( 9 ) - / ( 3 ) isequalto N, where N is a natural number. Find all the composite divisors of N. [8] Q. 6 3 Line / is a tangent to a unit circle S at a point P. Point A and the circle S are on the same side of I, and the distancefromA to / is 3. Two tangents intersect line I at the point B and C respectively. Find the value of (PB)(PC). [8] Q. 64 A triangle has one side equal to 8 cm the other two sides are in the ratio 5:3. What is the largest possible area ofthe triangle. [8] R
Q.65 Q.66
In triangle ABC, max {ZA, ZB} = ZC + 30° and — = V3 + 1, where R is the radius of the circumcircle and r is the radius of the incircle. Find ZC in degrees. [8] 2 The parabola P: y = ax where 'a' is a positive real constant, is touched by the line L: y = mx - b (where m is a positive constant and b is real) at the point T. Let Q be the point of intersection of the line L and the y-axis is such that TQ = 1. If A denotes the 1 maximum value of the region surrounded by P, L and the y-axis, find the value of — . [8]
Q.67 A point moving around circle (x+4)2 + (y + 2)2=25 with centre C broke awayfromit either at the point A or point B on the circle and moved along a tangent to the circle passing through the point D (3, - 3). Find the following. (i) Equation of the tangents at A and B. (ii) Coordinates of the points A and B. (iii) Angle ADB and the maximum and minimum distances of the point Dfromthe circle. (iv) Area of quadrilateral ADBC and the AD AB. (v) Equation of the circle circumscribing the ADAB and also the intercepts made by this c ircle on the coordinate axes. [10] 7
Q.68
7
7
If £ » 2 x . = l and ]T(/ + l) 2 x. =12 and £ ( z + 2) 2 x, =123, ;=i ;=i ;=i 7
then find the value of ^T(/ + 3) 2 x. . 1=1
fa B ansaIClasses
[10]
Determinant & Matrices
[7]
Q.69 The normals to the parabola y2 = 4x at the points P, Q & R are concurrent at the point (15,12). Find (a) the equation of the circle circumscribing the triangle PQR (b) the co-ordinates of the centroid of the triangle PQR. [10] Q.70 The triangle ABC, right angled at C, has median AD, BE and CF. AD lies along the line y = x + 3, BE lies along the line y = 2x + 4. If the length of the hypotenuse is 60, find the area of the triangle ABC. [10]
Q.71
2
2
2
2
Let W, and W2 denote the circles x + y + lOx - 24y - 87 = 0 and x + y - lOx - 24y + 153 = 0 respectively. Let m be the smallest positive value of 'a' for which the line y=ax contains the centre of a P circle that is externally tangent to W2 and internally tangent to W,. Given that m1 = ~ wherep and q are
q
relatively prime integers, find [p + q). n
Q.72
If
3 rdx = a 5j J 6 (l + sinx)
a + b + c + abc. l dX Q.73 If f . r~= V
[10]
b-j3 where a, b, c e N and b, c are relatively prime, find the value of c [10]
J Vl + x + Vl —x + 2
= Va - Vb --JL where a,b,c e N, find the value a2 + b2 + c2. VC
[10]
Q. 74 Suppose/(x) and g (x) are differentiable functions such that xg(/(x))/'fe(x)k'(x) = /(g(x)M/(x))Ax) for all real x. Moreover,/(x) is nonnegative andg(x) is positive. Furthermore, J / ( g ( x ) ) d x = 1 o ^ for all reals a. Given that g(f(0)) = 1. If the value of g{ f(4)) = e"k where k e N. find k. Q.75
[10]
Let f (x) be a differentiable function such that f (x) +/(x) = 4xe~x • sin 2x and/(0) = 0. Find the value ofLim ^f(kTi). k=i
[10]
f x) f(x) Q. 76 Let f be a differentiable function satisfying the condition f ^ ^ J =
(y ^ 0, f(y) ^ 0) V-x, y e R and
f ' (1) = 2, then find the area enclosed by y = f(x), x2 + y2 = 2 and x - axis.
[10]
Q.77 The equation Z10 + (13 Z - l)10 = 0 has 5 pairs of complex roots a p b,, a2, b2, a3, b3, a4, b4, a5, b5. Each pair a;, bi are complex conjugate. Find
i i
•
[10]
Q.78(i) Let Cr's denotes the combinatorial coefficients in the expansion of (1 + x)n, n e N. If the integers an = C0 + C3 + C6 + C9 + bn = c, + c4 + c7 + c10 + and cn = C2 + C5 + Cg + C,, + , then prove that (a) a 3 + b3 + c3 - 3anbncn = 2", (ii) Prove the identity: Bans a/ Classes
(C0 - C2 + C4 - C6 +
(b) (an - bn)2 + (bn - cn)2 + (cn - an)2 = 2. )2 + (Cj - C3 + C5 - C7 +
Problems for JEE-2007
[10]
)2 = 2n [81
Q.79
Given the matrix A =
- 1 3 1 -3 - 1 3
where x e N - {1}. Evaluate J v
Q.80
Q.81
5 - 5 and X be the solution set of the equation Ax = A, 5
x3-ly
where the continued product extends V x e X.
If a, b, c are the sides of triangle ABC satisfying log
(,1
[10]
c)
+ log a - log b = log 2. Also V a/ a(l - x 2 ) + 2bx + c(l + x ) = 0 has two equal roots. Find the value of sin A + sin B + sin C. [10] 1 ^ sin(nx) For x g (0, n/2) and sin x = —, if Zu 3 3 n=0
=
a
+ bVb ~~ then find the value of (a + b + c), c
where a, b, c are positive integers. e„ix - e„ - i x (You may Use the fact that sin x = ) 2i
[10]
Q.82 /Two distinct numbers a and b are chosen randomly from the set {2, 2 2 ,2 3 , 24, probability that logab is an integer.
, 225}. Find the [10]
OBJECTIVE Select the correct alternative. (Only one is correct): Q.83 A child has a set of96 distinct blocks. Each block is one oftwo material (plastic, wood), 3 sizes (small, medium, large), 4 colours (blue, green, red, yellow), and 4 shapes (circle, hexagon, square, triangle). How many blocks in the set are different from "Plastic medium red circle" in exactly two ways? ("The wood medium red square" is such a block) (A) 29 (B)39 (C) 48 (D)56
Q.84
The sum ^ ( - ^ ( i k j (A) - 298
Q.85
where
( rj =
e(
(B) 298
*mls
(C) - 249
(D) 249
If A > 0, c, d, u, v are non-zero constants, and the graphs of / (x) = j Ax + c | + d and u+c g (x) = - j Ax + u | + v intersect exactly at 2 points (1,4) and (3,1) then the value of ~~~~~~ equals (A) 4
(B)-4
(C) 2 4
3
(D)-2
2
Q. 8 6
Consider the poiynomial equation x - 2x + 3x - 4x + 1 = 0. Which one of the following statements describes correctly the solution set of this equation? (A) four non real complex zeroes. (B) four positive zeroes (C) two positive and two negative zeroes. (D) two real and two non real complex zeroes.
Q.87
The units digit of 31001 • 71002 • 131003 is (A) 1 (B) 3
fa B ansa I Classes
(C)7
Determinant & Matrices
(D)9
[9]
Q. 8 8 The polynomial f (x) = x4 + ax3 + bx2 + cx + d has real coefficients and /(2/) = / (z + /) = 0. The value of (a + b + c + d) equals (A) 1 (B) 4 (C) 9 (D) 10 ^ Q.89 If the sum 2L ^ equals (A) 6
1 +
Va+ Vb
o ) ^ +fc-^/k+ 2 (B) 8
=
7 c — where a, b, c e N and lie in [1, 15] then a + b + c (C) 10
(D) 11
Q.90 Triangle ABC is isosceles with AB =AC and BC = 65 cm. P is apoint on BC such that the perpendicular distances from P and AB and AC are 24 cm and 36 cm respectively. The area of triangle ABC in sq. cm is (A) 1254 (B) 1950 (C)2535 (D)5070 Q.91
The polynomial function f(x) satisfies the equation f ( x ) - f ( x - 2 ) = (2x- l) 2 forallx. Ifpandqarethe coefficient of x2 and x respectively in f (x), then p + q is equal to (AO 0 (B) 5/6 (C) 4/3 (D) 1
Q.92 Three bxes are labelled A, B and C and each box contains four balls numbered 1,2,3 and 4. The balls in each box are well mixed. A child chooses one ball at randomfromeach of the three boxes. If a, b, and c are the numbers on the balls chosen from the boxes A, B and C respectively, the child wins a toy helicopter when a = b + c. The odds in favour of the child to receive the toy helicopter are (A) 3:32 (B)3:29 (C)1 : 15 (D)5:59 \ f ( 41 - arc cosf arc sin is equal to Q. 9 3 The value of tan 5J 1 13 J/ \ —
25 (A)-
—
3 (B)--
33 ( Q - -
16 CD)-
Select the correct alternatives. (More than one are correct): Q. 94 Three positive integers form thefirstthree terms of an A.P. If the smallest number is increased by one the A.P. becomes a G.P. In original A.P. if the largest number is increased by two, the A.P. also becomes a G.P The statements which does not hold good? (A) first term ofA.P. is equal to 3 times its common difference. (B) Sn = n(n+ 11) (C) Smallest term of the A.P. is 8 (D) The sum of thefirstthree terms of an A.P. is 36. Q.9 5 If the line 2x + 9y + k = 0 is normal to the hyperbola 3x2 - y2 = 23 then the value of k is (A) 31 (B) 24 (C) - 31 (D) - 24 Q. 96 The line 2x - y = 1 intersect the parabola y2 = 4x at the points A and B and the normals at A and B intersect each other at the point G. If a third normal to the parabola through G meets the parabola at C then which of the following statements) is/are correct. (A) sum of the abscissa and ordinate of the point C is - 1. (B) the normal at C passes through the lower end of the latus rectum ofthe parabola. (C) centroid of the triangle ABC lies at the focus of the parabola. (D) normal at C has the gradient - 1.
fa B ansaIClasses
Determinant & Matrices
[10]
Q.97 If (j) (x) =/(x 2 ) +/(1 - x2) and/" (x) > 0 for x 6 R then which ofthe following are correct? (A) (j) (x) attains its extrema at 0, ± ^
(B) (j) (x) increases in (- l/V2 , o)u (l/V2,00)
(C) cf> (x) attains its local maxima at 0.
(D) (j) (x) decreases in
\/4l, o)u ( l / , 00)
. lit (in x \ sm 3 sin x Q.98 If tan — ~ = - — where 0 < x < n, then the value of x is V 3 , cos 271 cosx 3
it (A)^
5n (B)-
77t ( O -
1 Itc (D) —
MATCH THE COLUMN: Q.99 Column-I (A) The smallest positive integeral value of n for which the complex
Column-II (P) 4
number (l + j s i j 1 ' 2 is real, is (B) (C)
(D)
Q.l 00
Let z be a complex number of constant non zero modulus such that z2 is purely imaginary, then the number of possible values of z is 3 whole numbers are randomly selected. Two events A and B are defined as A: units place in their product is 5. B : their product is divisible by 5. If p, and p2 are the probabilities ofthe events Aand B such that p9 = kpj then 'k' equals
(Q)
6
(R)
8
(S)
9
For positive integers x and k, let the gradient of the line connecting (1,1) and (x, x3) be k. Number of values of k less than 31, is Column-I For real a and b ifthe solutions to the equation Z9 - 1 = 0 are written in the form of a+ib then the number of distinct ordered pairs (a, b) such that a and b are positive, is
Column-II 0 (P) (Q)
i
(B)
Lim ( e x + l ! *
(R)
2
(C) (D)
Let A, B be two events with P(B) > 0. If B c A then P(A/B) equals A real number x is chosen at random such that 0 < x < 100. 1 a The probability that x - [xl > —i Sis —, where a and b are relatively 3 bb primes and [x] denotes the greatest integer then (b - a) equals
(S)
e
(A)
x-»oo
Bansal Classes
Problems for JEE-2007
[U]
HINTS AND SOLUTIONS Let 2 1Ilx = y lo g2y so that ]og2y= 111 X 111 equation becomes ,3
+ 2y = 4y2 + 1 4 y3 - 16y2 + 8y - 4 = 0 sum of the roots of the given equation is iog2 yj + log2 y2 + iog2 y3 _ l o g ^ y ^ y g ) X,+X2 + X3 = ill ill Let the 3 consecutive terms are a - d , a, a + d d>0 2 2 ....(1) hence a - 2ad + d = 36 + K a2 = 300 + K ....(3) a2 + 2ad + d2 = 596 + K now (2)-(1) gives d(2a - d) = 264 ....(4) (3)-(2) gives d(2a + d) - 296 ....(5) (5)-(4) gives 2d2 = 32 => d2 = 16 Hence from (4) 4(2a - 4) = 264 => => 2a - 4 = 66 => K = 352 - 300 1225-300 = 925 Ans.]
iog 2 4_ 2 ill i l l
Sj + S2 = 113 Ans.]
d = 4 (d = - 4 rejected) 2a = 70
=>
a = 35
Case-I: When the two identical digits are both unity as shown. x[y j 1 | any one place out of 3 block for unity can be taken in 3 ways and the remaining two blocks can be filled in 9 • 8 ways. Total ways in this case = 3 • 9 • 8 = 216 Case-II: When the two identical digit are other than unity. 1 Mx|y
UMy|x|;li|y|x|x|
two x's can be taken in 9 ways and filled in three ways and y can be taken in 8 ways. Total ways in this case = 9- 3- 8 = 216 Total of both case = 432 Ans. ] h = a(t,t2) k = a(t 1 +t 2 ) Equation to the variable chord 2x - (tj + t2)y + 2atj t2 = 0 t,+t 2
+
A(t,)
2at,t2 t,+t 2
2a 2h V ....(1) y -— x + —a k k Since (1) touches y2 = 4bx, using the condition of tangency 2ah _ bk k ~ 2a Locus is by2 = 4a2x ]
fa B ansaIClasses
Determinant & Matrices
[12]
5.
Note that triangles BCM and OCN are similar now let ON = p. N will be mid point of chord PQ P 1 now
Alternatively:
I 2
=
P=
1
R = 2-\jr2 - p2 for large circle = 2Vl 6 - (1/4) - V63 Equation of large circle as x2 + y2 = 16 1 now
C = (1, 0) with slope PQ = - ^
equation of PQ : ^ y + x ^ l 1 P (from origin) = —
J( 1 - x ' )
dx =
2 \ 5
J
f
J
- ^ U d x (x-'-l) 1
r dt "2 t5
J
i
,10
,v
result ]
-dx
Taking x2 out of the bracket
- 2
Put x "2 - 1 = t = —r dx = dt X
2 -
- J -
_1
-1
1 xz
7.
Using sin2x
(think !)
-l
+ C] J
2 tanx I + tan"x
2 tan x 7t /2 ,, .-j , " ( l + tan22 x)dx l + tan~ x dx = r (i l - t a n x )V i- J 2 tanx ' (I + tanx) 4 o l+ l + tan2 x n/2 l-
put y = tan x
=>
dy = sec2x dx
-Ci - y) dy io+y)" now put l + y = z
f
(2-z)
l Alternatively:
*?(!-tanx) 2 sec xdx / (l + tanx) 4
-dz
dy = dz =
-
CO -6z + 4 I1 i_3 JZ 1 13
a = l , b = 3 => l +3 + 3 = 7 Ans. ]
n/2, nu _ . a I = f - — — — m . dx Q (cosx +sinx) 4
fa B ansaIClasses
Determinant & Matrices
[13]
tc/2 1 dx I = - — f (cos x - sin x) • i J» ' v d x v (cosx + sinx)J J) integrating by parts rc/2 ti/2 (cos x - sin x) (sin x + cos x) dx (cos x +sin x)~ 0 + 01 (cosx + sinx)3 using sin 2x : -2> 3
*/2
{(-!)-(!)}+ J -
dx + sin 2x
2tanx 1 + tan x
lt V2 r sec2 x . dx 3 ^ (1 + tanx)2
2 1 3 ~3
2 i3 3
1 °fdt 2 1 i 3f ? - — 3 +33 h
j + jtWMD
a = 1, b = 3 => 1 + 3 + 3 = 7 Ans. ]
Let the number of passengers be x (x > 200) Fair changed per person = 10 (x -100) Total revenue = x f(x)= 14xf'(x) = 14 f"(x) <0
10-(x-200)
100
2 100
2x , 2x' = lOx - — ( x - 2 0 0 ) = lOx + 4x 100 100
2x 100 4x 100
=>
=0
x = 350 x = 350 gives maxima]
Given 9y2 = x3 Let the point on the curve be x = t2 and y : — = /t • dt ' d x
— = +2t' dt d y
dy _ dy dt _ t 2 t dx dt dx 2t 2 normal makes equal intercept hence
- 1
Hence P = ( 4 , - )
^Bansal
Classes
slope of the normal = - ~
t=2 =>
a + 3b = 4 + 3 • - = 4 + 8 = 12 Ans. ]
Problems for JEE-2007
[14]
10.
Amount of glucose in blood at time t is x (t) dx p-Kx
hence
dx dt
p-Kx
dt
1 /n(p-Kx) = t + C K /n(p-Kx) = -Kt + C p - K x = r K t+ c p - e -Kt+C x: K Lim x(t) = ^ ] t-»00 K
11.
A =
"| 2r (a2 - a x ) - (x2 + 2ax + 3a2)
dx
where Xj & x7 are the roots of, x2 + 2ax + 3a 2 = a 2 - a x x = - a or x = - 2 a A=
12.
a3
dA 4
6(1+a )
da
A: ears of Mr A formed to be drooping B,: Mr A was telling a truth P(B,) = 3/5 B,: Mr B was telling a false P(B,) = 2/5 P(A/B,)= 1/10 P(A/B2) = 2/3 3 J_ 5 10 P(B,/A) = 3 J_ 2 2 5 10 + 5 3
13.
3+
40
49
Ans. ]
E: all the 5 persons leave at different floors n(S) = 85 n(A) = 8C5 • 5! j
m 14.
0 gives a = 3I/4Ans. ]
U-W cos a = wjjuj cos a cos j3
C 5 -5!
u • (j u 1 v+1 v|u) I u II w!
(u • v)+ j V 11 u v•w Iw V
fa B ansaIClasses
105 ans. 512
G-fioor
(u • v) 1 u j + I V j I u j UI w! -
....(1)
v-(|u| v+| v|u) [ Vi2|uI +(v-u) I VI |w||v| ~ ! W11 VI Determinant & Matrices
[15]
v||u|+(v-u) ....(2) w from (1) and (2) cos a = cos p a = p] COS
15.
P
The line is — 1
=^
z' = —2. - t
....(1)
\
1
2 A
A
|
A
A
A
J
A'
line passes through 2i + j——k and is parallel to the vector V = i + j - — k vector normal to the plane x + 2y + 6z = 10, is n = i + 2 j + 6k V.n = 1 + 2 - 3
line (1) is 11 to the plane
2+2—3—10 d= Vl + 4 + 36 16.
(2,1 ,-1/2)
V41
/ ]
Ans
Vector v, along the line of intersection of 3x - 7y - 5z = 1 and 8x - 1 ly + 2z = 0 is given by i v, = nj xn, = 3
j -7
8
-11
k - 5 = -23(3i+ 2 j - k ) 2
|||ly vector v2 along the line of intersection of the planes 5x-13y + 3z = 0 and 8x - 11 y + 2z = 0 is i J k v2 = n 3 x n 4 = 5 -13 3 8
-11
7 (i + 2j + 7k)
2
now v, -v2 = 0 => angle is 90° => sin90° = l 17.
Given and
]
au + bv + cw= 0 -...(I) au' + bv' + cw' = 0 ....(2) au" + bv" + cw" = 0 ....(3)
u v w u' v' w' For non trivial solution (non zero) solution of a, b and c . We must have u" v" w" 18.
Let
y = cos A • sin2f
A
l
+ cosB sin2
= 0]
C
+ cos C sin2 f ' <2, ,2,
= — [cos A (1 - cos A) + cosB (1 - cosB) + cos C (1 - cos C)] = — [(cosA - cos2A) + (cosB - cos2B) + (cosC - cos2C)]
—
fa B ansaIClasses
<
cosA-— ] - — I 2J 4
I
2 —0 _ I 2J 4
» — <
( cosC I
Determinant & Matrices
—0 2y
4
[16]
cosA--
y=
cosB- — 2
cosC
, \2
now y will be maximum if cosA = cos B = cos C = — hence
19.
y max =3/8 ]
y = x2; x = t; y = t2 dy = 2x = 2t dx slope of normal m
2t
equation of normal 1 y -1 2 = - — (x -1) 2X if
2t(y -1 2 ) = - x +1
or
x = 0; y = -
' 3 22 t=0 2t — t v2 or 3 - 2t2 = 1 => t= 1 or - 1 hence one of the point is origin and the other two are (-1, l)and(l, 1) => PQR is arighttriangle radius of the circle is 1 its equation is x2 + (y - 1 )2 = 1 x2 + y2 - 2y = 0 ] 20.
Let x be a real root. Equating real and imaginary part x3 - 6x2 + 5x + 2a2 = 0 ,(1) 3 2 and 2x - 2x - 4x = 0 .(2) 2x(x2 - x - 2) = 0 2x(x - 2)(x + 1) = 0 the given x = 0,2 or - 1 if x=0 a =0 x = - 1 =>
a2 = 6 = >
a=
x=2
a2 = 3 = >
a = ± V^
=>
±
V6
a e {O,V6,-a/6,V3,-V3} S = 0 + 6 + 6 + 3 + 3 = 18 Ans.] 21.
For non zero intercepts slope = - 1 y=- x+c point ( - 4 , - 2 ) - 2=4+c =>
^Bansal
Classes
c= -6 Problems for JEE-2007
117]
linesisy = - x - 6 x + y+ 6= 0 • for zero intercept
lineisy=mx - 2 = m(- 4) => m= 1/2 2y = x lines are 2y = xandx + y +6 = 0 ] 22.
23.
x4 - 1 Ox2 + 9 < 0 (x 2 - 9)(x2 - 1 ) < 0 1
"V \ -3 / -1
/ ~2" V
/
'
/
x4y2dx + x2y4dy = xdy + ydx x2y2(x2dx + y2dy) = xdy + ydx x2dx + y2dy =
d(xy)
(xy)2
r 2 r , fd(xy) Jx dx + Jy dy = j,
Integrating,
3
xy
3
(x3 + y3) +
52
Let
+C
= C; now if x = 1; y = 1
hence x3 + y3 + 3(xy) 24
18
face card removed
C = 5,
5 Ans. ] 20 drawn randomly
40
E0:
20 cards randomly removed has no aces. 20 cards randomly removed has exactly one ace. 20 cards randomly removed has exactly 2 aces. E: event that 2 drawn from the remaining 20 cards has both the aces P(E) = P(E n E0) + P(E n E,) + P(E n E,) = P(E0) • P(E / E0) + P(Ej) • P(E / Ej+ P(E2) • P(E / E2) = 40 ( f
4 aces 36 other
a
36,
40,
'20
'C 20
fa B ansaIClasses
20 p
+
2
1
4
36p '
19 20 p
40 ,
2
"20
36
4p
40
19'
c 20
20,
2'
+ 2
"20
2'
C,- 3 6 C 18 20
40, "20
c„
18'
Determinant & Matrices
[18]
6- 36C20 + 12- j6 C 19 +6- 36C,8 40
C20
20,
40,
6( 38 C 20 ) 40,
20/ '20
25.
=>
20/
p = 6 Ans. ]
e= b _ JL _ 2 ....(1) a" 25 ~ 25 ' a ~ 5 now radius of the circle r = a - X (where X, 0 is the centre of the circle) also r=AC = b sin 9 a - X = b sin 9 where X = a cos 9 a(l -cos9) = b sin 9 a2(l - cos 9)2 = b2(l - cos 9)(1 + cos 9) a2(l - cos 9) = b2(l + cos 9) 1 -cos9 _ 9 l + cos9 ~ 25 25 - 25 cos 9 = 9 + 9 cos 9 16 = 34 cos 9
8 cos 9 = — ;
sin 9
AB = 2b sin 9 = 2 • Let
=>
27.
C-
"20
1
26.
20 '20
6( 37C 20 + 37C 19' 40,
6[ 36 C 20 + 36C 19 + 36C,"19n + 36C18]
^a cos 8,
^°
C(X,0l)(a,0)
— — i
3
15 17 3a 15 5 ' 17
18
17
a Ans. ]
a, b e R, such that f(a) = f(b) 1 + ai 1 + bz 1-a/ 1 - bz 1 - bi + ai + ba =1 + bi - ai + ab 2ai = 2bi = a=b f is injective.
Let r be the radius of circle A and R be the radius of circ le B r + R = 12 and r = 3R 4R= 12; R = 3 and r = 9 1 Area of trapezium ABCD = - (3 + 9) ^(\2) 2 - 6 2 =
6 Vl08 = 36^3
A \
Q
fi
\ 3
-i
j IE ^THs. 3J
1 Tt 27rc Area of arc ADC = ~ x 81 x — = — 2 3 2
fa B ansaIClasses
Determinant & Matrices
[19]
1 _ 271 Area of arc BCE = - x 9 x — = 3TI 2 3 27tt
required area = 36-73
+ 371
33tt
= 36V3
a = 36, b = 33 a + b = 69 Ans. ] 28.
"3 0" "o r "0 1" A2 = _3 0_ 0 0 0 3_ = 31 A4 = 91; A6 = 27; A8 = 811 (A8 + A6 + A4 + A2 + I) = 121 I "1 0" hence 121 0 1 V "121 & 121 b_
29.
=
"o" 11
.
"0" _11_ => a = 0, b =
0
"0" 11_
0" a _ A
121
11'
V=
0]_ 11.
A 6 Radius of the first circle = — = — = 1 S 6 sur
1-r ....(1) (r < 1) 1+r
also
sin C :
now
2sin2 • sin2 c
1 - cos C = 1 =
1 - r \2 l+tj
i I 5
V5 - 1 = ( V 5 + l ) r 30.
"121
5(1 - r)2 = (1 + r)2 ^
Vs-i V5+1
j< — ———
V5(l-r) = 1 + r s sin 18—„. => cos 36°
, ,...
k + w = 54° Ans. ]
Cross multiplication and rearranging gives the cubic, x3 - ax2 + 23x - b = 0 <—ct 2a + (3 = a a 2 + 2ap = 23 and a2P = p Also given a + P = 12 from (2) and (4) a 2 + 2a(12 - a) = 23 a 2 + 24a - 2a 2 = 23
fa B ansa I Classes
..(1) ••(2) ..(3) ..(4)
J i , ,
Determinant & Matrices
^ ^ o i o i - W .
[20]
and => 31.
a 2 - 24a + 23 = 0 a = l (rejected) since a = 23; .'. p =—11 a = 35 from (4) b = a 2 p = 529 x - l l b =-5819 => a - b = 35-(-5819) = 5854 Ans.]
Let the numbers be G.P.
A - D , A, A + D, ^ ± D ) 1 A . (a) (b) (c) , (d) Given d - a = 30 => =>
- ( A - D ) = 30
=>
(A + D)2 - A(A-D) = 30A
D2 + 3AD = 30 A D2 = 3A(10-D)
D2 A ~ 3(10-D) •'" (1) since A' is a + ve integer 0 < D < 10 ....(2) Also since '3' is prime and A is an integer D2 must be divisible 3 ;=> D must beoftheformof3K possible values of D are 3,6,9 D = 3 =>
3 A = - (rejected)
D = 6 => A = 3 (rejected) D = 9 => A = 27 Numbers are 18,27,36.48 Ans. 32.
]
Total number of ways in which P and Q can be chosen simultaneously = (25-l)(25-l) = 45 - 26 + 1 number of ways when P and Q have no common element = 5C,(24 - 1) + 5C2 (23 - 1) + 5C3(22 - 1) + 5C4(2' - 1) + 5C5(2° - 1) = 5Cj • 24 + 5C2 • 23 + 5C3 • 22 + 5C4 • 2 + 5C5 - (5C, + 5C2 + 5C3 + 5C4 + 5C5) = (5C0 • 25 + 5C, • 24 + 5C2 • 23 + 5C3 • 22 + 5C4 • 2 + 5C5 - 25) - (25 - 1) = (35 - 25) - (25 - 1) = 35-2fi+ 1 Hence P and Q have atleast one common element = (45 - 26 + 1) - (35 - 26 + 1) = 45 - 35 Ans. ]
fa B ansaIClasses
Determinant & Matrices
[21]
33.
34.
3m2 + 1 Slope of the normal m = Yl 4m 2 ' 2x, 5m2 -2m 3 + 1 equation of the normal in terms of slope of the normal is y = mx + 4 m2 3 2 It passes through (7/2, 9/2) => 12 m - 13 m + 1 = 0 sum = 13/12. Also (m - 1) (3m - 1) (4m + 1) = 0 => mx = 1 ; m2 = 1/3 ; m3 = -1/4 => the normals are x - y + 1 = 0 ; x - 3y + 10 = 0 & 2x + 8y - 43 = 0 Point A(0,1); B ( - 1 ; 3 ) ; C (5/2,19/4)] x = i1
m-1 2m
Equation of normal, 1
Y-y =
X = 0 gives Y= x2 +
=
Y=0 gives X = x + my and
(X-x)
m
x + m 1y m
Hence
x + x + my 2
C ; passes through (1,4):
-=0
„
dy
=>2x + y-^-=0 dx
C-9
• is • — *2 + — y2 =11 with e 1 focii are (0, 3) & (0, - 3) conic 9 18 V2 Equation of the circles are; (x - 1 )2 + (y - 4)2 + A (x + 2 y - 9) = 0 where x + 2 y - 9 = 0 is the tangent to the ellipse at (1,4)] Equation to the hyperbola where S = (0,0); directrix is x + y + 1 = 0 and e = J 2 is /
x + y + lN V2 2 2 x + y = (x + y + l) 2 2xy + 2x + 2y + 1 = 0 Let the combined equation of the asymptotes is 2xy + 2x + 2y + c = 0 put D = 0 to get c = 2 hence combined equation of the asymptotes are xy + x + y+ l = 0 ^
7
= ^2'
(x+l)(y+l) = 0 36.
=>
x + 1 = 0 and y + 1 = 0
]
Note (tan A)22 + (cot C - cos B)2 denotes the square of the distance PQ now that d2pQ =C ( Q- -sin OP) Q(tanC, d2PQ
cotC)
yj (tan2 C + cot2 C) - 1 I ^-^(sinA, cosB)
d2pQ= [ V ( t o C - c o t C ) 2 + 2 - l d2
=>
min= (V2-1)2 ^ 3 - 2V2 a = 3; b = 2 => a3 + b3 = 27 - 8 = 19 Ans. ]
fa B ansaIClasses
Determinant & Matrices
[22]
37.
dx
i-Jo x 2 + ^ - + (a2 - 2 ) x x z dx - J - x4+kx2+l
4
x 2 dx I-x 4 + (a2 -2)x 2 +1
(a2 - 2 = k > 0)
1 co f (x 2 +l) + ( x 2 - l ) dx 211 x 4 + k x 2 + l
20 q^x 2 + (l/x 2 ) + k
2 q x + (l/x ) + k
71
now proceed, I,1 = — and L1 = 0 2a 1=
38.
Let
71 2a
71
2a
a = 2525 Ans. ]
5050
G=- +x 4
d0 = dx
or
46 = n + 4x
x(l + tan x) , dx = - 4 °f ^ " . d x M -J/2 1 - 1 + t a n x -V2 1-tanl —+ x 1 - tan x U o (-4x) tan
^ 71
—+.X
7i - 49 = - 4x
^
4
0
/
°f x(l + tanx) ( 1 - t a n x ) ^ tan:x -Ti/2 1 ~ t a n x (-2) tan
x\
X
2 f^i±^ldx=9 f + x dx V tan x j L tan x J -t/2 -ji/2 o X -dx I - x l° + jJ Tij 2 ' -*/2 t a n X 71/2 2
I=
now
n/2
n
„ f +2 -dt 4 J 0 tant
x =—t
k/2 1 = f t cott.dt = t/nsint n/2 0 oi n
n/2
J/n sin t dt
o
71
I,1 = 0 + - In 2 2
Hence 2 - - J / n 2 - — = 7 i / n 2 - — 2 4 4
fa B ansa I Classes
=>
k = 2, w = 4
Determinant & Matrices
kw = 8 Ans. ]
[23]
39.
g(l) = 5and j g(t)dt = 2 o x
X X X 2f(x)= J (x - 2xt +1 ) g(t) dt = x Jg(t)dt-2xJ"tg(t)dt+ Jt 2 g(t)dt 2
2
2
0
0
0
0
Differentiating X
2 f'(x) = x2 g(x) + Jg(t)dt-2x - 2 x / g(x) + Jtg(t)dt
+ x2g(x)
A
2 f'(x) = 2x Jg(t)dt- 2 Jt g(t) dt 0 0 x
x
f" (x) = xg(x)+ Jg(t)dt-xg(x)= Jg(t)dt
hence f " ( l ) = jg(t)dt = 2 o also f"'(x) = g(x) f"' (1) = g (1) = 5 f ' " ( l ) - f " ( l ) = 5 - 2 = 3 Ans.] x
40.
x
Consider a function
g (x) = e~ Jf (t)dt in [0,1 ] o obvious continuous and derivable g(0) = 0 and g(l) = 0 (given) hence 3 some c e (0, 1) such that g' (c) = 0 x
now
x
g' (x) = e~ f (x) - e
_x
Jf(t)dt o c
c
g'(c) = e - f ( c ) - e ^ Jf(t)dt =0 41.
Jf(t)dt =f(c)]
Consider f (x) = x 3 - ax + b f'(x) = 3 x 2 - a if a < 0 then f' (a) > 0 for all x hence f is strictly increasing hence f (x) = 0 has exactly one root for a = 3 f'(x) = 3 x 2 - 3 = 0 x = 1 or - 1 in order that f (x) may have 3 roots f(Xj) • f(x 2 )<0 where Xj and and the roots of f' (x) = 0 hence (1 - a + b)(- 1 + a + b) < 0 put a = 3 (b - 2)(b + 2) < 0 or - 2
^Bansal Classes
Problems for JEE-2007
[24]
42.
Let m gms of salt is present at time t differential equation of the process is dm T
m(l) 50+ t
= 10 /
dm "dT
1 m = 10; v 50 + t
I.F = e
50+1
dt
= 50 +1; m(50 + t)= f(50 + t)dt = 10
(50+ty +
J
m(50 +1) = 5(50 +1)2 + C; t = 0; m = 0, C = -5.(50) 2 m(50 +1) = 5(50 +1)2 - 5 (50)2 2 5(50y m = 5(50 +1) 50 + t m(t= 10) = 5 - 6 0 -
6 -
250 11 = 50 60
A : red ball is selected Bj : Face card is drawn B 2 : ace card is drawn B 3 : neither face nor ace is drawn 12 3
P(B,/A) = 44.
2
5(50^ 60
25x11 2 m= — = 91-=50 43.
+C
12
52 12
4
8
B2
36 10
107
iff 83
Bi
156 9 Ans, ] 107 ~ 107
1,2, 3, 4, 5, 6, 7, 8,9 x + y = 45 ; x - y = l l = > x = 28; y = 17 Now to realise a sum 17 using 4 digits we can have different cases, 9 4 3 1 9 5 2 1
8
6
2
1
8 5 3 1 8 4 3 2
7 6 3 1 ; 7 5 4 1 7 5 3 2
; 6 5 4 2]
If we usefivedigits then
7,1,2,3,4 6,5,3,2,1 4!x 5 !x 9 + 5!x4!x2 Hence p = 9! [ odd in favour 11 :115]
45.
/
A^JJJ
A ={1801, 1802, ,1899, 1900} B = {1901, 1902, ,1999, 2000} • 24 leap year B 76 ordinary
fa B ansaIClasses
(9 cases)
(2 cases) 11 x 5 ! x 4 ! 9!
U_ 126
25 leap year 75 ordinary
Determinant & Matrices
[25]
E: randomly chosen year has 53 sundays P (E) = P (E n L) + P (E n O) = P (L). P(E/L) + P (O). P(E/0) 24 2 76 1 100 7 100 7
+
1 25 2-+• 75 1 2 100 7 100 7
249 Ans.l 1400 46.
P(E) = 1 - P (value of 5 coins is more than or equal to Rs. 1.50) = 1 - P(A A B B B or A A B B C or A B B B B ) ]
47
6 on bush-I & 4 on bush-II
48.
AB X(BC
x
BE>) = 0 ;(AB
X
Be) XBD = 0 ;(AB
X
• A A ( 5 0 P) B B B B (25 P) •CCC DD
Box
(10P+5P)
N coins
BE)) xBC = 0 ;
Note that AB ;BC ;BD are mutually perpendicular E> BC x BD is collinear with AB and so on 1 r -> -» -» 1 220 Volume = - AB, BC, BD = — cu. units 6
J
Vector area of triangle AEF = — AF x AE = ~ BC x BD = - 3 i + lOj + k ] 49.
Equation ofthe line passing through (1,4,3) x—1 y-4
z-3
x—1 y+3 z-2 since (1) is perpendicular to —r~ = —:— - —7— and 1 hence 2a + b + 4c = 0 and 3a + 2b - 2c = 0 •2-8
12 + 4
4-3
-10
16
x+2
y-4
- 2
1
x— 1 y-4 z-3 hence the equation of the lines is —— = —— = —— ....(2) Ans. —10 16 1 now any point P on (2) can be taken as 1-10X ; 16X + 4 ; >. + 3 distance of P from Q (1,4,3) (10X)2 + (16^.)2 + X2 = 357 (100 + 256+l)X 2 = 357 X=l or - 1 Hence Q is (-9, 20, 4) or (11,-12, 2) 50.
z+1
Ans.]
Equation of the line through (0,1,2) x-0
now given line
feBansal Classes
y-1
z-2
x — 1 y-1
z-0 =t
....(2)
Problems for JEE-2007
[26]
(2) is along the vector V = i - j + 2k a - b + 2c = 0 ....(3) since (1) and (2) intersect; hence must be coplanar hence
1
0
- 2
1
-1
2
a b c
=0
2a + 4b + c = 0 solving (3) and (4),
y-1
required equation is 51.
....(4) a : b : c = - 3 : 1:2 z-2
-3
:
t Ans. ]
Since r,r2 = 2, x2 + px + 2. = 0
r r
t 2r3r4
=
-
8
=>
r
3r4
x4 - x3 + ax2 - 8x - 8 = (x2 + px + 2)(x2 + qx - 4) compare coefficient of x3 and x => p+q=-1 (1) and 2 q - 4 p = - 8 => q - 2 p = - 4 ....(2) => p = 1 and q = - 2 on comparing coefficient of x2; a = - 4 p = l => x2 + x + 2 = 0 =>
-l±iV7
ru 2 =
52.
X
y a
X
a
X
a
X
X
X
a
X
a
y
y a
X
y
_ Ans. ]
=
X
a
X
y a
X
a
X
y
= [x (xy - ax) - a(y2 - a2) + x (xy - ax) ]2 = [2x2 (y - a) - a (y - a) (y + a) ]2 = (y - a)2 [2x2 - a(y + a)]2 Hence D = (y2 + a2 - 2ay) (2x2 - ay - a2)2
53.
]
Let
1 f 2 1 1 1 1
[2 2
2a, 2a, _ai i.e.
a
i
=
a
2 2a, a
a
i a2
_
_a3 _ a,J a
3
=
10 13 9 "10" 13 9
2 a3_ 1 ; a2 = 3 ; a3 = 5
fa B ansa I Classes
Determinant & Matrices
[27]
x
2] i 3 X2 3 _X3 .
1 2 2 2 1 -1 X
I
2x, X
i.e.
54.
I
2X2
2X3
2X2
3X3
~
X
2
[l] 3 5
-
=
[l] 3 5
3X3_
x, = 1 ; x, = - 1 ; x. = 1
Ans.
]
a+b+c a+b+c a+b+c 9 + + > b+c c+a a+b 2 M 3 Consider AM between the numbers x,, x2, x3
....(1)
TPT
a +b +c
1 1 1 +. b+c c+a a+b now HM between the numbers Xj, x2, x3 . +
b+c a+b+c AM > HM a+b+c
(a + b + c) 55.
_
3 c+a a+b+c 1 b+c
1 c+a
• +
1 b+c
—
- +
—
3(a + b + c) _ 3 2(a + b + c) 2
a+b a+b+c
+
1 c+a
-
-
1 3 > — a+b ~ 2
+
-
1 a+b
Hence proved ]
Letx = rcos0 and y = rsin9
y
r2 = x2 + y2; tan 0 = x N:
0 e (0, n/2)
r 2 [cos2 0 + sin 0 cos 0 + 4 sin2 0]
(1 + cos 20) + sin 20 + 4(1 - cos 20)
5 + sin 20 + 3 cos 20
Nmax = — " 7 = = — 15 (s " + VTo)' = M N„ A= 56.
= — (5 - VlO) = 5 + VTO 15 M + m _ 2vK) _ 2 » "15-2-3
m
2007 x - = 1338 Ans.]
Transposing 2 on RHS using 2 cos A • cos B relation, B+C B-C A — + COS -sinCOS" cos
fa B ansaIClasses
cos
B-C
cos
B+C
Determinant & Matrices
[28]
or cos— sin— + cos-y cos
B-C
. A B-C sin—cos+ sinz 2
1=0
2
B+C . A = sin—); ( cos2
2
B-C A . A A . A A coscos sin — + cos—• sin— -cos z — = 0 2 2 2 2 2 A . A B-C coscos sin — - COS" 2 2 cos
57.
A
. A sin—
2
cos
2
B-C '
A A cos— - sin— =0 2 2
if
COS" — - c o s y
B-C
. A sin —
2
2
A
COS—
2
r
A
cos
2
A tan— = 1
A = 90°
2
A
B- C =A => B=C+A B - C = - A => B + A = C = 90° hence triangle must berightangled. OAMB is a cyclic quadrilateral using sine law in A OBM and A OAM
B = 90°
x
and
sin 90°
sin(6O-0)
d sin 90°
y sin©
(l)and (2) x y
r^ w
sin(60 - 6) sin© + 1 = V3cot6
y from (2) d = y cosec 6 2
B/>
....(2) 'P
sin(6O-0)
2x
2
•d)
y
/
sin© sine
V3 2
/Pf
1
„
d
^
X
)60°-e
2 :
2
d = y (l + cot 6)
2x + y V3y
2
2
d =y
cote
1+
(2x+_y) 2\ 2
3y
d2 = y:, ,
(2*+yr
,, 4x,2 +4y +4xy dz = 3
,, 3y + 4x + y +4xy dz = 3 d= ^ V x 2 + y 2 + xy Ans. ]
fa B ansaIClasses
Determinant & Matrices
[29]
58.
Let G be the centroid : AD = x ; BE = y 2x x 2y y AG = — ; GD = — ; BG = ; GE = ir 3 3 3 3 4x z yz 9 , , , „ , „, • + — = - or 16x2 + 4y2 = 81 In AAGE : 9 9 4
..(1)
„.. 2 x.2 4v In ABGD : — + — = 4 or 9 9
.(ii)
•
2
x2 + 4y2 = 36
'2
D
(i)-(ii) , 15x2 = 45 In AADC, cosC =
9+4-3 2(2)(3)
20 = 25 - c2
or
5 6
9+16-c 2(4)(3)
c = V5
1 1 5 A=-absinC=-(3)(4) J1=VlT sq. units ] v6y 59.
From triangle inequality log1012 + log1075>logI0n log]0900>log10n => also log1012 + log10n>log1075 log1012n>log1075 12n> 75 n>
75 12
0
or
6.25
n>
n < 900
....(1)
25
900
Hence no. of values = 900 - 7 = 893 Ans. ] 60.
x + 2y= 10 where x is the number of times he takes single steps and y is the number of times he takes two steps Cases Total number of ways I: x = 0 and y = 5
5! = 1 (22222) 5!
II: x = 2 and y = 4
6! 2!-4!
III: x = 4 and y = 3 IV: x = 6 and y = 2
15 (1 1 2 2 2 2 )
7! - ^ J j =35 (1 1 1 1 222) ^ 8!= 2 8 (1 1 1 1 1 1 22)
9 V: x = 8 and y = 1 Ct = 9 ( 1 1 1 1 1 1 1 1 2 ) VI: x= 10andy = 0 1(1111111111) hence total number of ways = 1 + 15 + 35 + 28 + 9 +1 = 89 Ans. ]
fe Ban sal Classes
Problems for JEE-2007
[30]
-e re 1= J b
61.
x/a
V
a
let
_eb/x
, dx
x = at =>
dx = a dt
at V-ea/t 1= J T put
t=
1
dJt
a
y
....(1) , dt = - —r dy a
=>
y
f(ey -e a/ - v )dy
a
or
y from(1)and(2)
I=
=
f (e
l
~
21 = 0 =>
f(3) 23k +9 J ^ j = 2^+9
62.
(where b/a = a)
1 3;
-e a / t )dt
....(2)
i 1 = 0 Ans.]
/(9)
~/(3)
=
^
+ 9)
~(23k
+ 9) =
^~23k
"" ( 1 )
3(23k + 9) = 26k + 9 26k-3(23k)-18 = 0 23k = y y2 - 3y - 18 = 0 (y-6)(y + 3) = 0 y = 6; y = - 3 (rejected)
=>
2
now
3k
= 6
f (9) - f (3) = 29k - 23k 3k 3
= (2 ) - 2
{ from (1)}
3k
= 63 - 6 = 2 1 0
hence N = 210 = 2 • 3 • 5 • 7 Total number of divisor = 2 • 2 • 2 • 2 = 16 number of divisors which are composite = 16 - (1,2,3,5,7) = 11 Ans. ] 63.
Radius of the circle is 1 t£U1
B
A _ _L _ T ~ PB ~ s(s-b)
|||ly
PC = (s - c) (PB)(PC) = (s - b)(s - c) =
fe Ban sal Classes
A
s(s-a)(s-b)(s-c) s(s-a) Problems for JEE-2007
[31]
A'A s(s-a)
r
(s-a)
A _ A (s-a) A-a 3a J 3a 2 a I 2 64.
r = —= 1 => s = A
3 = 3 Ans. ] 3-2
5x + 3x> 8 => x>l 5x + 8 > 3x => x>-4 and 3x + 8 > 5x => x<4 Hence, x e ( l , 4 ) . Now perimeter of the triangle = 8(x + 1) s = 4x + 4 2 A ( X ) = ( 4(x + 4)(4 - x)(4x - 4)(x + 4)) = - 16(x 2 - l)(x 2 - 16) A2(t) = -16(t - l)(t -16), where x2 = t, t e (1,16) A2 (t) = - 16[t2 - 17t + 16] = f (t) f'(t) = 0 A 2 (t)
17 =
17
t:
=>
2
17
1
- 1 6
15
15
= 1 6 x y x Y = (2 X 15)2
•16
V
(Area)max = 30 sq. units ] 65.
From the identity .A B . C r=4R sin— - sin—- - sin — 2
or let
then
Let
2
2
A C . B sin—sin — sin— 2 2 2
A B C 4(V3+l) r-sin— -sm— - sin— or 2 2 2 ZA>ZB ZA - ZC = 30° A-C
V3-1 4
cos
V3-1
V6+V2 4
B siny =x yields
whose solutions are x =
cos
A+C
B sin-
. B B sm — sm— 2 2 x
V6-V2
V6+V2
•X +
V3-1
= 0,
V2 B B and x = — . It follows that — = 15° or — = 45°. The second
solution is not acceptable, because A > B. Hence B = 30°, A = 90° and C = 60° ]
Bansal Classes
Problems for JEE-2007
[32]
66.
y = ax2 dy dt , = 2ax0 = m hence line is y = (2ax0)x - b
•d)
(x0' 0, a xo0 ) lies on parabola and the line (1) 9 0 axQ = 2ax 0 - b
T(x0, ax 0 2 )
b = ax 2 . Hence Q = (0, - b) = (0, - ax 2 ) now using (TQ)2 = 1 x2 + 4a2 Xq = 1 •2 =
now
•(2)
4x4
u ax 3 A = J(ax2 - m x + b)dx = axQ3
3
mx2
3
+ bx
axn
2
mx„ 5. + bxo
ax„3
3
= —- - ax^o + ax„o = —^ 3
a 2 ,x, 6
2
A
3
f1 V
let
_ A2 = f (x0) =
2^
4x 4
)
36
36
This is maximum when x^ = — A l••max
?
i7 ' 2 36
1 144'
Amax = — |2
= 12 Ans. ]
67.
(i)
Equation of tangentfrompoint (3,-3) to the given circle is y + 3 = m(x - 3) mx - 3 m - y - 3 = 0
fe Ban sal Classes
Problems for JEE-2007
[33]
and also =>
-4m-3m+ 2-3
vr+m 2
(1 + 7m)2 = 25(1 +m 2 ) => 1 + 49m2 + 14m = 25 + 25m2 => 12m2 + 7 m - 1 2 = 0 (4m - 3)(3m + 4) = 0 m = 3/4 or m = -4/3 equation of tangent at point A and B are 4 y+3=--(x-3)
(ii)
=5
and
3y + 9 = - 4 x + 1 2 4x + 3y = 3 Equation of normals to these 2 tangents are 3 y + 2 = - ( x + 4)
and
4y + 8 = 3x + 12 3(3x-4y+ 4 = 0) 9 x - 12y = - 12 16x + 12y = 12 x = 0;
4y+12 = 3 x - 9 3x-4y = 21 4 y + 2 = - - ( x + 4) 3y + 6 = - 4 x - 1 6 4(4x + 3}= - 22) 16x+ 12y = — 88 9 x - 12y = 63
25x = - 2 5 x = - 1; points A and B are (0, 1) and (-1,-6) Ans. (iii)
/.
3 y+3= -(x-3)
y= 1
y:
angle between the 2 tangents = 90° ZADB = 90° I AD |max = CD + radius CD = V50
!AD|max=5V2 +5 (iv)
IAD | min = 5V2 " 5 Area of quadrilateral ADBC = AC x AD A D = V 7 2 + l 2 - 2 5 =V25 =5 area of quadrilateral ABCD = 5 x 5 = 25 sq. units. 1 area of triangle DAB = — x 25 =12.5 sq. units.
(iv)
Circle circumscribing A DAB will have points A and B as its diametrical extremities x2 + y2 — x(—1) — y(—5) — 6 = 0 x2 + y2 + x + 5y - 6 = 0 Ans. x-intercept = 2-yJg2 - c = 2 ^(1/4) + 6 = 5 y-intercept = 2-s/f 2 -c
fe Ban sal Classes
=
Ans.
2 -7(25/4)+ 6 = 7 Ans. ]
Problems for JEE-2007
[34]
68.
Let, f (x) = x2 XJ + (x +1 )2X2 + + (x + 6)2x7 [if x = 1, we get 1st relation, and so on] note that degree of f (x) is 2 hence f (x) = ax2 + bx + c where f (1) = 1, f (2) = 12 and f (3) = 123 to find f (4) = ? hence a + b + c = 1 4a + 2b + c = 12 9a + 3b + c= 123 solving a = 50, b = - 139, c = 90 f(4) = 16a + 4b + c = 800-556 + 90 = 334 Ans. ]
69.
Suppose, circle x2 + y2 + 2gx + 2fy + c = 0 Solving with x = at 2 , y = 2at a ¥ + 4a2t2 + 2gat2 + 4aft + c = 0 0 ....(1) t l + t 2 + t3 + t 4 3 N : y + tx = 2at + at passing through (h, k) at3 + t(2a - h) - k = 0 ....(2) t, +12 +13 = 0 ....(3) from (1) and (3) t4 = 0 hence circle passes through the origin ==> c = 0 equation of the circle after cancelling -at at3 + 4at + 2gt + 4f =0 at3 + 2(2a + g)t + 4f = 0 ....(3) Now (2) and (3) must be represent the same equation 2(2a + g) = 2a - h => 2g = - (2a + h) 4 f = - k => 2f = - k / 2 and equation of circle is x2 + y2 - (2a + h)x - (k/2)y = 0 x2 + y2 - 17x - 6y = 0 Ans. Centroid of APQR X
a(tf + t 2 + t | )
2a (t[ +t 2 +t 3 )
a = T [(tl+t2 + t 3 ) 2 - 2 X t l t 2 ]
2a 26
Z4!12 =
( 2 a - h ) 2a
a
'3
(2a-h)
(a = 1 ; h = 15) 26 n C: I y.0
70.
1 Area = - ab ; also a2 + b2 = 3600 AD:y=x+3 solve to get G = (-1,2) BE : y = 2x + 4 acute angle a between the medians is given by m
tan a •
^ Bansal Classes
i~m2 l + m,m-,
2 - 1 _ ]_ 1+2 ~ 3
tan a = ~
Problems for JEE-2007
[35]
In quadrilateral GDCE, we have (180 - a) + 90° + 0 + p = 360° => a = 0 + p - 90° cota = -tan(0 + P) tan 0 + tan p 1 - tan 0 tan p
71.
or
-3
1 - a b =400
9ab = 2 x 3600 Area = 400 sq. units ]
2b 2a _a b_ j_2b.2a a ' b —
+
—
9=
2(a +b ) ab
W2: C2 = (5, 12) W,: Cj = (-5,12) r2 = 4 r, = 16 now, CC2 = r + 4 CCj = 16 - r let C(h, k) = c(h, ah) CC, 2 = (16 - r)2 (h + 5)2 + (12- ah)2 = (16 - r ) 2 CC22 = (4 + r)2 => (h - 5)2 + (12 - ah)2 = (4 + r)2 By subtraction 20h = 240 - 40r => h'= 12-2r - => 12r = 72-6h ...(1) By addition 2[h2 + 25 + a2h2 - 24ah + 144] = 272 - 24r + 2r2 2
2
2
h (l + a ) - 24ah + 169 = 136 - 12r + r = 136 + (6h- 72) +
[using (1)]
4[h2(l + a2) - 24ah + 169] = 4[64 + 6h] + (12 - h)2 = 256 + 144 + h2 h2(3 + 4a2) - 96ah + 105 • 4 - 36 • 4 = 0 h2(3 + 4a2) - 96ah + 69 • 4 = 0; for 'h' to be real D > 0 2 2 (96a) - 4 • 4 • 69 (3 + 4a ) > 0 576a 2 -69.3 -276a 2 >0 o 69 13 300a2 >207 => a- > — ; hence m (smallest) = 100' 10
=> =>
So,
72.
12 — h\2
m2 =
69
p + q = 169 Ans. ]
100'
n 1 = 3 [(1-sinx) sec xdx = 3 J(l-2sinx + sin2 x)sec4 xdx 5n/6 5JC/6 2
4
Jsec2 x(l + tanz x)dx - 2 jsec x tan x - sec2 x dx + jsec2 x(tanz x)dx 5n/6 5n/6 5n/6 =3
n '(
\
j((l + 2tan 2 x)sec2 x)dx-2 j(secxtanxsec 2 x)dx
5U/6
fe Ban sal Classes
571/6
Problems for JEE-2007
[36]
=3
=3
o -1 j"(l + 2t 2 )dt-2 j ? d t - 1/V3 -2/V3 1 a/3
(0)-
V
2 1 3 3V3
IV3
9V3 }
6V3-5 _ 3V3 1 73.
+
3
5V3
11 + 6V3-16 2 3V3-8 =3 9V3 3 3V3
11 9a/3
3^3 bV3
a = 2, b = 5, c = 9 =>a + b + c + abc= 106 Ans.]
dx
0 Vl + X + Vl-X +2 put x = cos20 1= 2
J-1/V3
--(t3)"1 3 " >-1IS
'.JL^ V 3V3j
1 +
3
dx = - 2 sin 20 d0
Jt/4 j sin 26 d6 j V2 cos 6 + V2 sin 0 + 2
TT/ 4 = a 2
tc/4 JJ
sin 20 d0
J 0 cos0 + sin0 +V2
0
sin 20 d0 ^CQS V4
71/4 1=
0 cos TT/ 4
r
V4
sin 20 d0 v4
+
71/4
1
=
-
2
+ V2
7c/4
l-2sin 2 0 d0 JQ 1 + COS0
j ^COS0 j Q 11 Q +1
1
y
71/4
J(1-C°s8>de
n_ m 1 9 = J ' sin" ' 0
n/4
~ 2 l(1-cos9>de
jt/4 71/ 4 it/4 = {(cosec2 0 - c o t 0 cosec 0)d0 - 2 J(l-cos0)d0 = - cot 0 + cosec 0 — 2[0 — sin 0 ] | n ^ 1-COS0 = til - l ) - Lim Q->o eos0 2^2-1-
74.
7t
n 4
1_ = ( V 2 - l ) - - + V2 V2 ' 1 a = 8, b = l , c = 4 => a2 + b2 + c2 = 81 Ans. ]
x • g (f(x))f'(g(x))g'(x) = f(g(x))g'(f(x))f'(x) x g ( / ( * ) ) £ f(g(x)) = f ( g ( x ) ) £ g ( / ( x ) )
x
^Bansal
£a*o0)
£g(/(x))
/fe(*)>
*(/"(*))
Classes
Problems for JEE-2007
[37]
x-A/ dx
n
(/(
g
(x))) = —/n(g(/(x))) dx
a n o w ,,
...(1)
-2a
j/(g(x))dx =
l -
0
d i f f e r e n t i a t e w.r.t. 'a'
/fe(x))=e-2x
a))=e-2a
f{g(
/n/(g(x))=-2x
....(2)
f r o m (1) and (2) w e get -2x=f-(/ng(/(x))) dx put
x = 0, C = 0
g(/(x))= 75.
Let
In ( g ( / ( x ) ) ) = - x 2 + C
e
g ( / ( 4 ) ) = e~ 1 6 k = 16 A n s . ]
Hence
5
f(x) = y dy dx
+ y = 4xe
(linear d i f f e r e n i a l e q u a t i o n )
• sin 2 x
I.F. yex = 4 | x s i n 2 x dx I
IT cos2x
yex = 4 x
+ — Jcos 2x dx
xcos2x• + sin2x
yex = 4
2
4
x
ye = (sin 2x - 2 x cos 2x) + C f(0) = 0
=>
C = 0
x
y = e~ (sin 2 x - 2 x cos 2 x ) f ( k u ) = e~k7C ( s i n 2k7t - 2k7t • c o s 2 k r t ) = e""k7t ( 0 - 2k7i)
now
f ( k u ) = - 271 ( k • e _ k 7 t ) J]f(k7t)
=-'271 jTke" kn k=l
= 1 •
1+ . 3 7 t3 + 2e~ 2 7 t + 3e~-3
+ oo
+e
+ oo
,-2n
S e-*
+ 2e~
371
+
S ( 1 - e _7C ) = e~ n + e ~ 2 n + e ~ 3 n +
oo
1
S ( 1 - e" 71 ) 1-e"*
e71-!
1 S
(e 71 - l ) ( l - e _ 7 t )
-2nan
(e71-!)2
Ans. ]
(e 71 - l ) 2
fe Ban sal Classes
Problems for JEE-2007
[38]
76.
f ' ( x ) = Limit f(x + h ) - f ( x ) h^o h f(x) Limit h->0
f(x + h) f(x).
' = f(x)- Limit h-»0
x + h^ _ i VX
Jf 1+
-1 f(X)T . . f(l + t ) - l xj = Limit -i '-— h x t->0 t x x Now putting x = 1, y = 1 in functional rule :
f(x) • Limit I h->0
—
f(l) f(x) f'(x)= — -f'(l) x
2f(x) = —— x
f(x) _ 2 f(x) ~ X /n (f(x)) = 2/nx + C x = 1; f(l) = 0 => C = 0 Now solving y = x2 and x2 + y2 = 2 y2 + y - 2 = 0 (y + 2) (y - 1 ) - o y=l
;
f(x) = x2
A= 2j(V2r7-Vy)dy - |Vy dy
= 2
i 2 d 2 = now JVy y = -y^ n 3 3 i y 2 ^ 7 dy y=V2sin0 0 x/4 JV2 cos Q-Jl cos 0d9
71 1 4 2
9 + — sin 28 2 Hence feBansal Classes
A=2 4
n/4 jr/4 2 J2 cos 9 de = J(1 + cos29) d0
—
+
I_2 2~3
+
—
fn 1 A =
U " J ,
sq. units ]
Problems for JEE-2007
[39]
77.
10
slO 1 13 =0 Zy
10
Z +Z
NIO
= - 1 = cos Tt + i sin Tt
'.3-2 V
J_ Z
13
(cos(2m +1)71 + i sin 2mTt + 7t)1,/10 ,(2M+L)7T
=e
10 .(2M+L)FL
J_ Z
13- e
10
substituting m = 0,1,2, 1
9 we get
. Tt
= 13- e
10
,3rc = 1 3 - e ' 10 1 F 3=
1 3
J
io
= 13- e 1 a
Let
r- note — and —— are complex conjugate ^10
/,3Jt— , 10
i— ! 10
J_ and vJ io
and so on
. TC
.
TI
1 169-13 [e 'T10o + e 'T10o ] + 1 ab. 1 1 3+TT
:
169-13 [e
10
= 170-26 Ree a.b. 1 1
_ 3+7T
+e
10
] +1
10
,3JI
and
170-26 Ree' 10 etc
a2b2
Tt 3ti 5n 3u 9Tt 1- COS + COS + COS— 850 - 26 cos — + cos 10
aibj :
^Bansal
Classes
10
10
10
10
850 - 26[cosl8° + cos54° + cos90° + cos 126° + cos 162°] 850 Ans. ]
Problems for JEE-2007
[40]
78.(i) now put
+ bn + cn = C0 + C1 + C2 + C3 + C4 +
a
n
a
n + bn
+ C
n=
2
"
--(I)
(1 + x) = c0 + Cj x + c2 x2 + c3 x3 + n
X = CO
(1
+ co)n = c0 + Cj CO
+ c2 CO + c3 OD + c4 CO + 2
= (C 0 + c 3 + c 6 +
llliy
n + b n + Cn -
3a b C n n n
4
) + co(C, + c 4 + c 7 +
(1 + co)n = an + cobn + co2cn (1 + co2)n = an + co2bn + cocn a
3
) + CO2(C2 + c 5 + c 8 +
....(2) ....(3)
= (an
+ b
n +
n
C
n)
K + ®bn + ® 2 c n ) (an + ®2bn + ®Cn)
= 2 (l + co) (1 + co2)n = 2 n (- co2)n (- co)n = 2n
now
n
X (an - bn )2 = 2(an + cobn + co2cn) (an + co2bn + cocn) also 78.(ii) Let and
Z ( a n - b n ) 2 = 2 Ans. x
Cq C2 + C4- C6 + - c77 + y i c,j + c, 5 3 (1 + if = c0 + C, / + c2 i1 + c3 / equating the real and imaginary part = c
+ c4 /4 +
x
n + y n = 0 + 0" n n/2 | xn + iyn | = | 1 + i | = 2
^
•••
79.
I
=
hence x 2 + y 2 = 2 n
hence proved ]
-1 A = 1 -1
-1 1 -1
3 5 -3 -5 3 5
2
3 5 -3 -5 3 5
Hence A 2 =A 3 =A 4 = x = 2, 3,4, 5,
-1 1 -1
3 5 -3 -5 3 5
;
A =5> matrix A is idempotent
=A oo
.3
now
Lim f l x J 3 + 1 X=2 x - l n—>oo
=2
r3 4 5 Lim n->co 1 2 3'
n(n + l) ' (n-1)
Lim
n(n +1)
n->oo
80.
Given log =>
X
x=2
log
1-2
n + n +1
fe Ban sal Classes
+1 3 1_ \3 7 13 21
3
a + c^ + log a j ''a + c^
+X
2
Ans.
9 \ n -n +1 n +n + l ]
log 2
= log2 Problems for JEE-2007
[41]
=> also
a + c = 2b (1) a - ax2 + 2bx + c + cx2 = 0 (c - a)x2 + 2bx + (c + a) = 0 has equal roots D=0 4b2 - 4(c2 - a2) = 0 b2 = c2 - a2 ....(2) b2 = (c - a)(c + a) from (1) and (2) b2 = (c - a) 2b 2(c - a) = b ....(3) 2 triangle is a right at C. from (2) c a + b2 => Z C = 90° A + B = 90° from (3) using sine law 2(sin C - sin A) = sin B C = 90° • sin C = 1 A + B = 90° => B = 90° - A 2(1 - sin A) = sin(90 - A) = cos A squaring both sides => 4( 1 - sin A)2 = cos2 A=(1 - sin2A) 4(1 - sinA) = (1 + sinA) 3 = 5 sinA sin A=3/5 B = 90 - A sinB = cos A = 4/5 and sin C = 1 12 3 4 sinA+sinB + sinC= — + — + 1 : Ans. ]
81. n=0
put
sin(nx) -jn sin (nx) =
I n=0
emx
sin(nx)
_1_ 2i
_ glllX 2i 00 „n;x
1
e
2l n=0
„n;x
-e 3n
J_ 2i
_3 3 - eix
1—
\
I
n=0V
n
- I n=0\
3 3 - e~ix
2/ sin x 3 3 "(3--e~ i x )-(3-e i x )~ ix ix 2i 9--3(e +e~ ) + l ~ 2i 10-6cosx 3 sin x 2(5-3 cosx)
fe Ban sal Classes
=
1 2(5-3^/1-(1/9))
=
1 2(5-2V2)
Problems for JEE-2007
[42]
5+2V2 34
=
=> 82.
a = 5, b = 2 , c = 37
a + b + c = 5 + 2 + 37 = 41 Ans.]
logb log a (let b = 2m and a = 2" where m and n denotes the exponents on the base 2 in the given set) log b =
m n hence logab is an integer only if n divides m now total number of ways m and n can be chosen = 25 x 24 = 600 For favourable cases let n=1 hence m can take values 2,3,4,5,6, ,24 if n=2 m = 4, 6, 8, 10, 12,14, 16, 18, 20, 22, 24 n=3 m = 6, 9, 12, 15, 18,21,24 n=4 m = 8, 12, 16,20,24 n=5 m = 10, 15,20, 25 n=6 m = 12, 18, 24 n=7 m =14,21 n=8 m = 16, 24 m = 1 for each n = 9,10, 11, 12
= 24 = 11 = 7 = 5 = 4 = 3 = 2 = 2 = 4 62
62
Hence P :
600
31 Ans. ] 300
Select the correct alternative. (Only one is correct): Q.83 A [Sol. There are 4C2 = 6 ways a block can differ from the given block in exactly two ways (1) material and size, (2) material and colour, (3) material and shape, (4) size and colour, (5) size and shape, and (6) colour and shape. Since there is only 1 choicefordifferent material, 2 choices for different size, 3 choices for a different colour, and 3 choices for a different shape, it follows that the number of blocks in each of the above categories is (1 x 2), (1 x 3), (l x 3), (2 x 3), (2 x 3) and (3 x 3), respectively. The answer is the sum of these six numbers = 29 Ans ] Q.84 [Sol.
C Consider the expansion of (1 + x)99 and put x = i and equate the real part to get C]
Q.85 [Sol.
B f (x) = | Ax + c | + d g (x) = -1 Ax + u | + v (-u/A, v) \ 0 , D
( M K j / d - / - - ^ ( - c / A d) -c/A
-u/A
figure is parallelogram and diagonals bisect each other
fe Ban sal Classes
Problems for JEE-2007
[43]
u\ + A Q.86 [Sol.
(
V
c A
u+c = 3 +
1;
= - 4 Ans. ]
D f (x) = x4 - 2x3 + 3x2 - 4x + 1 = 0 f 1 (x) = 4x3 - 6x2 + 6x - 4x + 1 = 2(x - l)(2x2 - x + l ) f' (x) = 0 only at x = 1 also f"(x) = 12x 2 - 12x + 6 = 6(2x 2 -2x + 1)>0 => / i s concave up f" (1) > 0 => x = l is minima also f (0) > 1; f(l) = - l =>
two positive and two non real complex roots
VxeR
=>
(D) ]
Q.87 D [Sol. Notice the patterns Power of 3 go like this : 3,9,7,1 Power of 7 go like this : 7,9,3,1 Power of 13 go like this : 3,9,7,1 so knowing that we can see that 31001 ^
3
-71002
9
1 3
Q.88 [Sol.
1003
^
7
so therefore 3 • 9 • 7 = 9 ]
C If a polynomial has real coefficients then roots occur in complex conjugate and roots are 2i, - 2i, 2 + i, 2 - i hence f (x) = (x + 2i)(x - 2i)(x - 2 - i)(x - 2 + i) f ( l ) = ( l + 2 i ) ( l - 2 i ) ( l - 2 - i ) ( l - 2 + i) f (1) = 5 x 2 = 10 .Also f ( l ) = 1 + a + b + c+ d l + a + b + c + d=10 => a + b + c + d = 9 Ans. ] Q.89 D [Sol.
(k + 2 ) V k - W k + 2 Fk
T
T
k(k + 2)2 - k 2 ( k + 2)
_1 i = I .vr
1_ V3_
j 2 ,V2 J
i_ V4.
2
(k + 2)Vk-kVk + 2 2k(k + 2)
=
T1 3 .= — 2
V3
as k -»
V5. 00,
fe Ban sal Classes
1 Vk
Vk + 2
and so on
sum
1+
V2
1 + V2 _ Vl+V2 => a + b + c = 11 Ans. ] 2V2 ~ V8 Problems for JEE-2007
[44]
Q.90 C [Sol.
A = — b2sin29 = b2 sin 9 cos 9 ....(1) now
x 65-x 24 36 60 x = 24 • 65 sin 9
x=26 65-x
12 5 sin 9 = — and cos 9 = — again,
65 sin 29
sin 9
from(l)
A=
13
b=
65 2 sin 9
65-13 2-5
132 2
12 = 169- 15 = 2535 Ans.] 13 13
Q.91 B [Hint: Let f (x) = ax3 + px2 + qx + r nowuse f ( x ) - f ( x - 2 ) = x 2 - 4 x +1 compare the coefficients to get a = 2/3; p = 1; q = - 1/6; hence p + q = 5/6 Ans. ] Q.92 B [Sol. n(S) = 4 x 4 x 4 = 64 n(A) = 211 or 312 or 413 or 431 or 422 6 3 3 P(E) = 77 = 64 32 3 + 29 odds in favour 3 : 29 Ans. ] Q.93 D [Sol.
tan •sin = - tan
r
A \
J
• n + cos
r« Vl3 j)
, . 4 .,5 + sm — cos — 5 13 v TC
4 5 = - tan(a - B) where sin a = — and cosy B = — 5 13 4 _ 12 3 "5" '' tan a - t a n p = , 4 12 1 + tan a tan P 1 + - — 3 5 N
fe Ban sal Classes
20-36 63
^6 Ans. ] 63
Problems for JEE-2007
[45]
Select the correct alternatives. (More than one are correct): Q.94 A 3 [Sol. Let the numbers are a - d, a , a + d ( a - d + 1), a, (a + d) inGP. and (a - d), a, (a + d + 2) also in G.P. (a - d + l)(a + d) = a2 hence (a2 - d2) + (a + d) = a2 ....(1) and (a - d)(a + d + 2) = a2 (a2 - d2) + 2(a - d) = a2 ....(2) (2)-(l) 2a-2d-a-d =0 a = 3d => 2nd term ofA.P. = 3 times its common difference 2 2 from(l) d - d - 3d = 0 => d=4 d=4 A.P. is 8,12,16
a = 12
]
Q.95 B,D [Sol. Let the line is normal to the hyperbola at P(x,, y,) hence 2x 1 +4y 1 +k = 0 ....(1) dy differentiate the curve dx slope of normal = _ZL _ ~3x, line
2 -9
3Xj yi ZL
;
2 slope of line=- —
2x, = 3y(
(x j, y j) lies on the hyperbola 3x2 - y 2 =23
3x^
=> Xj = 3 or - 3 hence Pis (3,2) or (-3,-2) k = 24 or - 24 = Q.96 A,B,C [Sol. Equation ofAB y = 2x - 1 solving it v2 = 4x y2 = 2(y+l) = but y, + y2 + y3 = 0 but y} + y2 = 2 ••• y3 = - 2 putting in y2 = 4x x3 = 1
fe Ban sal Classes
4x
23
23 xf =23 • 9
B, D Ans. ]
y2 - 2y - 2 = 0
(1,2)
Problems for JEE-2007
\
y
2
=4ax
[46]
hence coordinates of c are (1, - 2) sum of the x and y coordinates of C are - 1 => (A) is correct, obviously normal at C passes through the lower end of the latus rectum => (B) is correct again centroid of AABC =
X|
X3
y = 2x - 1 with y2 = 4x (2x- l) 2 = 4x 4x2 - 8x + 1 = 0
now solving
centroid of the AABC =
2+1
Xj + x2 = 2;
also x3 = 1
(1,0)
again equation of the normal at C y+2 =
v ^y
(x-1)
y + 2 = x - 1 :> x-y-3 =0 hence gradient of chord at C is 1 =>
(D) is incorrect ]
Q.97 A,B,C [Sol.
=>
1
=> =>
so (j)' > 0
x > ^
or
1
x < - ^
(- l/V2, o ) u (l/ V2,00) increasing
-1/V2
t min.
0 t max.
4—h 1/V2 t. min.
]
Q.98 B,D
[Sol.
tan
f fn 7t X 2 cos — + x sin 71 J ~ 2 13 , S =-cot - + X / ^ 71 ^ 7t X 2 sin —+ X sin J J ~ 2 •y
2 71
- X
5tt 2n — + x = n7t+ — - x x=
n7t
Tt ~ 12
71 71 tan - + —+ X 2 3
571
^
tan — + X 6 y
2x = n7t + 571
n =
:
1 ITC r
12 ° 1 Y
Ans ]
-
MATCH THE COLUMN: Q.99 Ans. (A) P, (B) Q, (C) R, (D) P [Sol.
(A)
T T ' 71 . . 7T7^ 2 cos— + isin — 3 3 2n/2
is real
n7t . . nTt cos— + zsin— is real 6 6
fe Ban sal Classes
Problems for JEE-2007
[47]
(B)
mi mi hence s i n — = 0 ; .'. — =kn ; :. n = 6k 6 o smallest positive n is 6 Ans. => (Q) Let z = x + iy, x, y <= R and x2 + y2 = 2 (say) 2 z is purely imaginary (x + iy)2 is purely imaginary x2 - y2 + 2xyi = 0 + ki keR-{0} x2 = y2 and 2xy = k [If k = 0 thenx = 0andy = 0] let k > 0 say 2 x y = l => y=l/x x4 = 1 => x2 = 1 => x = 1 or - 1 y = 1 or - 1 z is 1 + i or - 1 - i if k < 0 say - 2 then xy = - 1; y= -l/x 4 2 x = 1 • => x =1 x = 1 or -1 y = - 1 or 1 z is 1 - i or - 1 +i there are four values of z which are ± 1 ± i
(D)
=>
(P) •
3 k = xx --1l =x 2 + x + l < 3 1
x2 + x - 3 0 < 0
(x + 6)(x-5)<0 => - 6 < x < 5 => numberof+vehitegeris4.]
Q.l00 Ans. (A) R, (B)S,(C)Q, (D)Q [Sol. (A) z = 1 is one solution remaining 8 lie on a circle 2 in each quadrant symmetrically situated on a circle => number of solutions are 2 Ans. -X x (B) Let / = Lim (e +1) ^ X—>00
In I = L i m - ^ — - / n ( e x + l ) = L i m ^ ^ - - / n e x ( l + e~x) = Lim 1 + C x-»oo X x->oo x x-»oo x
[x+/n(l + e~x)]
= Lim[l+/n(l + e" x ) 1/x ] X->00
(C)
In I = 1 B c A =>
/ = e Ans. P(A n B) = P(B) P(AnB) P(B) P(A/B)= p ( B ) => ^ j ^ = lAns.] 2
(D)
2
Break the interval into 100 identical cases favourable length — => Probability = — a - b = 1 Ans. ]
fe Ban sal Classes
Problems for JEE-2007
[48]
MATHEMATICS ,
XII (ALL)
QUESTION
BANK
ON
FUNCTION, LIMIT, CONTINUITY & DERimBILITY
Time Limit: 4 Sitting Each of 75 Minutes duration approx.
Question bank on function limit continuity & derivability There are 105 questions in this question hank. Select the correct alternative : (Only one is correct) Q. 1 If both f(x) & g(x) are differentiable functions at x = x0, then the function defined as, h(x)=Maximum (f(x), g(x)} (A) is always differentiable at x=Xq (B) is never differentiable at x = x 0 . (C) is differentiable at x = x0 provided f(x0) * g(x0) (D) cannot be differentiable at x = x0 if f(x0) = g(x 0 ). Q2
Q.3
If Lim (x -3 sin 3x + ax -2 + b) exists and is equal to zero then: X—>0 (A) a = - 3 & b = 9/2 (B) a = 3 & b = 9/2 (C) a = - 3 & b = - 9 / 2 (D) a = 3 & b = - 9 / 2 xmsin-i x # 0 , m e N The least value ofm for which f'(x)is A function f(x) is defined as f(x) = 0 if x = 0 continuous at x = 0 is (A)l (B)2
(C) 3
(D) none
1
Q.4
ifx=£ where p&q>0 q q 0 if x is irrational
For x > 0, let h(x) =
are relatively prime integers
then which one does not hold good? (A) h(x) is discontinuous for all x in (0, oo) (B) h(x) is continuous for each irrational in (0, QO) (C) h(x) is discontinuous for each rational in (0, QO) (D) h(x) is not derivable for all xin (0, QO) . n
- 3 ,x Q.5
The value of Limit X->°0
(where n e N )is
(A) In
-J
Q.7
V
(B)0
(C) n In
V-V
(D) not defined
5 c For a certain value of c, X—>-00 L i m |Yx + 7X4 + 2) - xl is finite & non zero. The value of c and the value ^ '
ofthe limit is (A) 1/5,7/5
(B) 0, 1
(C) 1,7/5
(D)none
Consider the piecewise defined function r /(x)
if 0
L
x-4
x
if 0 < x < 4 if
x>4
choose the answer which best describes the continuity of this function (A) The function is unbounded and therefore cannot be continuous. (B) The function is right continuous atx = 0 (C) The function has a removable discontinuity at 0 and 4, but is continuous on the rest ofthe real line. (D) The function is continuous on the entire real line
feBansal Classes
Q. B. onVector,3D&ComplexNo.&Misc.
[2]
Q.8
If a, P are the roots of the quadratic equation ax + bx+c = 0 then Lim 2
(B)^(a-p)2
(A) 0 Q. 9
(C) y (a-P) 2
1 - cosjax2 + bx + c) (x-a)2
equals
2
(D)-^-(a-P)2
2 Which one of the following best represents the graph of the function f(x) = Lim — tan (nx) n—7t y (
,(0,1)
(B)
0
(A)'
Lim
0 (0,-1 >
0 1 - 3x + X 1-x3
2
fO.l) ,
•(D) 0
3 x4
( -0 1
x^x"
(B)3
(A)i Q.ll
(
(C)
>(0-1)
Q.IO
(0,1)
(0,1) ,
(Q-
(D) none
ABC is an isosceles triangle inscribed in a circle of radius r. IfAB = AC & h is the altitudefromAto BC and P be the perimeter ofABC then Linl —equals (where A is the area of the triangle) (A)
(C)
32r
128r
(D)none
Q.12 Let the functionj,\ g and h be defined as follows x sin
r)
0
for - 1 < x < 1 and x * 0 for
x 2 sin
m vxj
0
x—0
for - 1 < x < 1 and x * 0 for
x =0
h(x) = |x| 3 for - 1 < x < 1 Which of these functions are differentiable at x = 0? (A)/and g only (B)/and/i only (C) g and h only Q.13 If [x] denotes the greatest integer < x, then Limit -L jjy x j +123 x] + (A) x/2
feBansal Classes
(B) x/3
(C) x/6
(D)none + [n3 x]j equals (D) x/4
Q. B. onVector,3D&ComplexNo.&Misc.
[2]
Q. 14 Let /(x) =
g(x)
where g and h are cotinuous functions on the open interval (a, b). Which of the
following statements is true for a < x < b? (A)/ is continuous at all x for which x is not zero. (B)/ is continuous at all x for which g (x) = 0 (C) / is continuous at all x for whichg (x) is not equal to zero. (D) / is continuous at all x for which h (x) is not equal to zero. I sin x I "I-1 cosx1 I Q. 15 The period of the function f (x) = 1 — - is | sinx-cosx | (A) 7C/2
Q. 16 If f(x) =
(B) tc/4 X —
(C)TC
(D) 27t
H~ GOS 2x , x ^ 0 is continuous at x = 0, then x2
(A) f (0) = |
(B) m ] = - 2
(C) {f(0)} = -0.5
(D) [f(0)]. {f(0)} =-1.5
where [x] and {x} denotes greatest integer andfractionalpart function
Q.17 The value ofthe limit ^ J
X
\
— — is
n=2
(A)l
(B)7
Q. 18 The function g (x) =
r
(C)^
(DW
x + b, x < 0
. cosx, x > 0
can be made differentiable atx = 0.
(A) if b is equal to zero (C) if b takes any real value
(B) if b is not equal to zero (D) for no value of b
Q. 19 Let / be differentiable at x = 0 and/' (0) = 1. Then / ( h)-/(-2h)_ h-»0 h (B)2
L i m
(A) 3
(C)l
(D)-l
Q.20 If f(x) = sin-1 (sinx) ; x e R thenfis (A) continuous and differentiable for all x
Tt (B) continuous for all x but not differentiable for all x=(2k + 1) — , k e I K (C) neither continuous nor differentiable for x = (2k - 1)— ; k€ I (D) neither continuous nor differentiable for x e R -[-1,1]
feBansal Classes
Q. B. onVector,3D&ComplexNo.&Misc.
[2]
Q.21
Limit x
~*2 c o s
sin x — ^-(3sinx-sin3x)
2 (A)-
where [] denotes greatest integer function, is
4 (Q-
(B) 1
Q.22 If Lim fa(3 + x ) - f a ( 3 - x ) x->0
=
^
(D) does not exist
v a l u e i g
x
(A) |
(B)-j
(C)-|
(D)0
x2n-l Q.23 The function / (x) = Lim — — i s identical with the function n->co +1 (A) g (x) = sgn(x - 1) (B) h (x) = sgn (tan^x) (C) u (x) = sgn( | x | - 1) (D) v (x) = sgn (coHx) Q. 24 The functions defined by f(x) = max {x2 (x - 1 ) 2 ,2x (1 - x)}, 0 < x < 1 (A) is differentiable for all x (B) is differentiable for all x excetp at one point (C) is differentiable for all x except at two points (D) is not differentiable at more than two points. x /nx Q.25 f ( x ) = — and g (x) = — . Then identify the CORRECT statement 1 (A) ^ ^
I
1
and f(x) are identical functions
(C)f(x).g(x)=l
V x>0
Q.26 If f(3) = 6 & f' (3) = 2, then (A) 6
(B) 4
(B) y ^ y and g (x) are identical functions = 1
(D) xf 3
( )~3f(*)
isgiyen by
(C) 0
Vx>0
. (D) none of these.
Q. 27 Which one of the following functions is continuous everywhere in its domain but has atleast one point where it is not differentiable? (A) f (x) = x1/3
(B) f (x) =
(C) f (x) = e~x
(D)f(x) = tanx
„ . _ 2V2- ("cosx + sinx) 3 . % . Q. 28 The limiting value of the function f(x) = when x —> — is 1 - sm 2x 4 (A) VI
feBansal Classes
(B)-jL
(C) 3 J 2
(D)-jt
Q. B. onVector,3D&ComplexNo.&Misc.
[2]
2
x
i: Q.29 Let
+ 2
3-X
_
6
LL-X
_
if x>2 then
f(x): x2 - 4 x-V3x-2
if
x<2
(A) f (2) = 8 => f is continuous atx = 2 (C)f(2-)*f(2 + ) f is discontinuous Q.30
(B) f (2) = 16 => f is continuous at x = 2 (D) f has a removable discontinuity at x = 2
e On the interval I = [-2,2], the function f(x) = J . lo then which one of the following does not hold good? (A) is continuous for all values of x e I (B) is continuous for x e I - (0) (C) assumes all intermediate values from f(- 2) & f(2) (D) has a maximum value equal to 3/e.
hH]
Q.31 Which ofthe following function is suijectivebutnotinjective ( A ) f : R - » R f(x) =X 4 + 2 X 3 - X 2 + 1 (B) f: R ^ R (C) f: R
x w 1. L
f(x) = x 3 + x + 1
(D) f : R - » R f(x) = x3 + 2 x 2 - x + l
R+ f(x) = V n V
Q. 3 2 Consider the function f (x):
(x*0) (x = 0)-
if l < x <2 if x = 2 if 2 < x < 3
where [x] denotes step up function then at x = 2 function (A) has missing point removable discontinuity (B) has isolated point removable discontinuity (C) has nonremovable discontinuityfinitetype (D) is continuous Q. 3 3 Suppose that / is continuous on [a, b] and that/(x) is an integer for each x in [a, b]. Then in [a, b] (A) / is injective (B) Range of / may have many elements (C) {x} is zero for all x e [a, b] where { } denotes fractional part function (D) f (x) is constant Q.34 The graph of function / contains the point P (1,2) and Q(s, r). The equation of the secant line through f si r+.2 s - 3 P and Q is y = s — 1 x - 1 - s. The value of / ' (1), is (A) 2
feBansal Classes
(B)3
(C)4
(D) non existent
Q. B. onVector,3D&ComplexNo.&Misc.
[2]
e x /nx5 ( x 2 + 2 ) (x 2 -7x + 10) . Q.3 5 The range of the function f(x) = 5 is 2x - 1 l x + 12
(A) (-00,00)
(B)[0,oo)
Q.36 Consider f(x) =
3 2'"
(Q
2
sinx - sin3xj + sinx - sin 3 x
2
smx - sin3xj
-
sinx - sin 3 x
, x* -
(D)
for x e (0, tt)
f(n/2) = 3 where [ ] denotes the greatest integer function then, (A) f is continuous & differentiable at x = n/2 (B) f is continuous but not differentiable at x=%/2 (C) f is neither continuous nor differentiable at x-%/2 (D) none of these Q.37 The number of points at which the function, f(x) = | x - 0.5 | + I x - 1 | + tanx does not have a derivative in the interval (0,2) is : (A) 1 (B) 2 (C) 3 (D) 4 Q. 3 8 Let [x] denote the integral part of x e R. g(x) = x - [x]. Let f(x) be any continuous function with f(0)=f(l) then the function h(x) = f(g(x)): (A) hasfinitelymany discontinuities (B) is discontinuous at some x = c (C) is continuous on R (D) is a constant function . Q.39 Given the function f(x) = 2x^x3 - I + 5\/x jl-x4 + lx2 sjx - 1 +3x + 2 then: (A) the function is continuous but not differentiable at x = 1 (B) the function is discontinuous at x= 1 (C) the function is both cont. & differentiable at x = 1 (D) the range of f(x) is R+. Q.40 I f / ( x + y ) = / ( x ) + / ( y ) + | x | y + xy2, V x, y 6 R and/' (0) = 0, then (A)/ need not be differentiable at every non zero x (B)/ is differentiable for all x e R (C)/ is twice differentiable at x = 0 (D) none Q.41 For
sin(x-10} (10-x) (where {} denotesfractionalpart function)
(A) LHL exist but RHL does not exist (C) neither LHL nor RHL does not exist
Q.42
T
• I2 n+22 (n-l)+32 (n-2)+.... ,+n2.l
tZ
I 3
(A)j feBansal Classes
+ 2
3
+ 3
3
+
+ N
CB)f
3
(B) RHL exist but LHL does not exist. (D) both RHL and LHL exist and equals to 1
is equal to :
(Qj
Q. B. onVector,3D&ComplexNo.&Misc.
[2]
- x - 61 + 16_xC2x-i + 20_3xP 2x-5 IS
Q. 43 The domain of definition ofthe function f (x) = log 3
(B) 4'°° -{2,3}
(A) {2}
(C) {2, 3}
(D)
'
1
^
V
4
j
Where [x] denotes greatest integer function. x 2 - b x + 25 Q 44 If/(x) = —— — forx^ 5 and / is continuous at x = 5, then/(5) has the value equal to x - 7 x + 10 (D)25 (C)10 (A) 0 (B) 5 Q.45 Let / be a differentiable function on the open interval (a, b). Which of the following statements must be true? I. / is continuous on the closed interval [a, b] n. / is bounded on the open interval (a, b) III. If a
vQ.46
The value of
LimU
(A) 1
cot
1
(x
a
log a X)
\ ( a > l )7 is equal to v H s e c " 1 ^ log x aj v
(B) 0
(D) does not exist
(C) Till
Q.47 Let f: (1,2) R satisfies the inequality cos(2x-4)-33 x I 4x-811 < f(x) <—1 , V x e(l,2). Then Lim f(x) is equal to x-2 x-»2 (A) 16 (B)-16 (C) cannot be determined from the given information (D) does not exists sinxcosx n T . Q.48 Let a = min L[x2 + 2x + 3,, x s R]J and b = Lim —X _ — Then the value of Y arbn~r is X r=0 2n+1 + 1
2n+1 - 1
4n+1 - 1 (D) 3-2n
2n - 1
Q.49 Period of f(x) = nx + n - [nx + n], (n e N where [ ] denotes the greatest integer function is : (A) 1 (B) 1/n (C) n (D) none of these (\ lh 1- X Q.50 Let f be a real valued function defined by fix) = sin-1 v 3 , is given by: (A) [-4,4] (B) [0,4] (C)[-3,3]
feBansal Classes
+ COS
1
"xl -
\
J
j
. Then domain of J(x)
(D) [-5,5]
Q. B. onVector,3D&ComplexNo.&Misc.
[2]
Q.51 For the function f (x) = Lhn ^ ^ ^ ^
^ which of the following holds?
2
(A) The range of f is a singleton set (C) fis discontinuous for allx s i .
(B) f is continuous on R (D) fis discontinuous for somexeR
Q.52 Domain of the function f(x) = , is -y/Zncot-1 x (A) (cotl, oo)
Q.53 The function
(B)R- {cotl} r 2x
f(x)=
+l
(C) (-oo,0)u(O,cotl) (D) (-oo, cotl)
,xeQ
• x2 - 2 x + 5 , x g Q
is
(A) continuous no where (B) differentiable no where (C) continuous but not differentiable exactly at one point (D) differentiable and continuous only at one point and discontinuous elsewhere Q.54 For the function f (x) =
—, x * 2 which of the following holds?
x+ 2 ^ (A) f (2) = 112 and f is continuous at x =2 (C) f can not be continuous at x = 2
O 55
^
T.
LIM
x->i/V2
(B) f (2) ^ 0,1/2 and fis continuous at x = 2 (D) f (2) = 0 and f is continuous at x = 2.
x-cos(sin _1 x)
— -
l-tan(sin
(A)^
x) (B)-"jj
(C)V2
(D)-V2
Q.56 Which one of the following is not bounded on the intervals as indicated 1 -X_1 1 (A) f(x) = 2 x on (0, 1) (B) g(x) = x c o sX- on (-oo, oo) (C) h(x) = xe~ on (0, oo) (D) I (x) = arc tan2x on (-oo, oo) 3T0 COtX
Q. 57 The domain of the functionfix)= , '
'
x rx2
VM J
x,is: (A) R (C) R - j±Vn : n eI + u {0}j
feBansal Classes
, where [x] denotes the greatest integer not greater than
(B) R - {0} (D)R-{n:neI}
Q. B. onVector,3D&ComplexNo.&Misc.
[2]
Q.58 If f(x) = cosx, x = n7t, n = 0, 1,2, 3, = 3, otherwise and x 2 + 1 when x^3, x^O
m=
3
when x = 0
5
when x = 3
then Limit f ( ( K x )) =
(B)3
(A) 1
Q.59 Let Lim sec"1 x-»0
'
X
^
Vsmxy
= / and Lim sec 1 x—>0
(A) I exists but m does not (C) I and m both exist 1
Q.60 Range of the function f (x) = function and e= Limit(l + a) a— (A)
Q.61
' o , £ ± r w{2}
v
/n(x + e)
= m, then vtanx; (B) m exists but / does not (D) neither I nor m exists +
1
Vi+ x
is, where [*] denotes the greatest integer
l/a
(B) (0,1)
e j
(D) none
(C) 5
(C)(0,l]u{2}
(D)(0,l)u{2)
(C) I - i
Lim sin '[tanx] = I then { / } is equal to x-»0~
, 71
(A) 0
ft
where [ ] and { } denotes greatest integer and fractional part function. Q. 62 Number of points where the function f (x) = (x2 - 1 ) | x2 - x - 21 + sin( | x |) is not differentiable, is (A) 0 (B) 1 (C)2 (D)3 Q.63
T.
. cor^Vx+T - V x 7) Limit \ ' r X->00 ^2x + lV.X sec ,x-l
(B)0 if x < x 0
(A) 1
-x Q.64 I f f ( x ) =
. is equal to
2
- ax + b if x > x 0
(A) 2x 0 , - xc
derivable V x e R then the values of a and b are respectively (C) - 2x0 , -
(B)-x0,2x0 1 + cos
2llX
1 - sin 7tx
Q.65 Let f(x) =
(D) non existent
(C) TC/2
x <
XQ
(P)2xjj , - x 0
2
x
J4 + [l2x - 1 - 2
(A) p e R - {4}
feBansal Classes
= 2 • If f(x) is discontinuous at x= — , then x> 1
( B ) p s R - y
(C) p 6 R,
(D) p e R
Q. B. onVector,3D&ComplexNo.&Misc.
[2]
Q. 66 Let f(x) be a differentiable function which satisfies the equation f(xy) = f(x) + f(y) for all x > 0, y > 0 then f'(x)is equal to (A)
f(l)
•
(B)
1 x
(C)f'(l)
(D)f'(l)(lnx)
f(x) = b ([x]2 + [x]) +1 for x > - 1 = Sin (k (x+a)) for x < - 1 where [x] denotes the integral part of x, then for what values of a, b the function is continuous at x = -l? (A) a = 2n + (3/2); b e R ; n e l (B) a = 4n + 2; b e R; n e l + (C) a = 4n + (3/2); b e R ; n e I (D) a = 4 n + l ; b e R+ ; n e l
Q.67 Given
/n(x2 + e x ) Q 68 Let f(x) = . If Limit f( x ) = / and Limit f( x ) = m then : /n(x + e ) (A) l=m (B)/=2m (C) 21=m (D) / + m = 0 Q.69
Lim cos( W n 2 +n I when n is an integer: n->00 ^ J • (A) is equal to 1
Q.70 x
. ™ x^o
Ll
lt;
(B) is equal to - 1
(C) is equal to zero
(D) does not exist
(sin x - t a n x) 2 - ( l - c o s 2 x ) 4 +x 5 — _! 7 _i . 5— is equal to H 7.(tan x) +(sin x) +3sin x
(B) 7
(A) 0
v(C) y
(D)l
3
Q.71 Range ofthe function, f(x) = cot"1 (log4/5 (5x2 - 8x + 4)) is : 7C
(B) ~4
(A) ( O . t c )
• • rxi2 Q. 72 Let y ^ f ~ ~ = / & X
'
(C)
0,
K
(D) [ 0 , ^
• • Tx12] = m, where [ ] denotes greatest integer, then: X
(A) / exists but m does not (C) I & m both exist
(B) m exists but / does not (D) neither / nor m exists .
tan x _1 sin x Q.73 The value of limit t 1 " 1 ' 1 ( (;{ , }/ f ); /) . { ) where { x} denotes thefractionalpart function: {x}({x}-l)
(A) is 1
(B) is tan 1
(C) is sin 1
(D) is non existent
In (V 2 + 2Vxj Q.74 If f(x) = — p is continuous at x = 0, then f (0) must be equal to tan (A) 0
feBansal Classes
(B) 1
(C) e2
(D) 2
Q. B. onVector,3D&ComplexNo.&Misc.
[2]
Q.75
2 + 2x + sin 2x is: (2x + sin 2x)esu! (A) equal to zero (B) equal to 1 Lim
(C) equal to - 1
(D) non existent
(cos ax ) cosec2bx is
Q.76 The value of
A 2l 2b J
8a"
8b
(B)
(A)
(C)
(D)
Select the correct alternative : (More than one are correct) Q,77
Lim f(x) does not exist when: x->c
(A) f(x) = [ [ x ] ] - [ 2 x - l ] , c = 3
(B) f(x) = [x] — x, c = l (D)f(x)=^^,c=0. sgnx
(C) f(x)={x} 2 -{-x} 2 , c = 0 where [x] denotes step up function & {x}fractionalpart function.
tan2 {x} for x > 0 x2 - [x]2 for x = 0 where [ x ] is the step up function and { x} is the fractional Q.78 Let f(x) = 1 V{x} cot {x} for x < 0 part function of x, then: (A)
x
L
(B)
™l f (x) = 1
s
L
™t f(x) = 1
\2
(C) cot"
Q.79
If f(x)=
Limit f(x) V.X H> 0 "
x . t a (cosx) 2
'"(i + x )
[0
=1
(D) f is continuous at x = 1
x*0 " ' " then: x=0
(A) fis continuous at x = 0 (C) fis differentiable at x = 0
(B) fis continuous at x=0 but not differentiable at x=0 (D) fis not continuous at x = 0.
Q. 80 Which ofthe following function (s) is/are Transcidental? (A) f (x) = 5 sin
^
( Q f ( x ) = Vx2 + 2x+ 1
2sin3x ® - ? + 2 x - l (D) f (x) = (x2 + 3).2X , W
Q. 81 Which ofthe following functions) is/are periodic? (B) g(x) = sin(l/x) , x ^ 0 & g(0) = 0 (A) f(x) = x - [ x ] (D) w(x) = sin-1 (sinx) (C) h(x)=xcosx
^Bansal Classes
ft B. on FLCD
[12]
Q. 82 Which of following pairs of functions are identical: (A) f(x) = e' nseo_lx &g(x) = sec-1x (B) f(x) = tan (tan-1 x) & g(x) = cot (cot1 x) (C) f(x) = sgn (x) & g(x) = sgn (sgn (x)) (D) f(x) = cot2 x.cos2x & g(x)= cot 2 x- cos2x Q. 83 Which ofthe following functions are homogeneous ? (A) x siny + y sinx (B) x ey/x + y e ^ (C) x2 - xy
(D) arc sinxy
Q. 84 If 0 is small & positive number then which of the following is/are correct ? , . . sinG
(A) —
e
Q.85
(B) 9 < sin0
=1
(C) sin0 < 0 < tan6
Let f(x)
x . 2X - x . (to. 2 & g(x) = 2xsin —- then: 1 - cosx . V 2 J
(A) Limit
f(x) = /n 2
(B) Limit
(C) Limit
f(x) = /n 4
(D)
Limit
(C)
Limit f ( x ) =
Q.86 Let f(x) = (A) Limit
g(x)
=/n
(D) — 0
>
Sm6
6
4
g(x) = /n 2
x-1 . Then: 2x - 7x + 5 2
f(x)
=_I
(B)
Limit f ( x ) = _ I
0
(D) Limit does not exist
Q.87 Which ofthe following limits vanish? (A) Lhmt
x
i
sin
1
( B ) x L ^2 ( l - s i n x ) . t a n x
Vx
2x + 3 • sgn(x) x +x- 5 where [ ] denotes greatest integer function (C) Limit
2
Limit [x]2 - 9 (D) x x -9
Q. 88 If x is a real number in [0,1 ] then the value of Limit Limit [l +cos 2m (n! % x)] is given by (A) 1 or 2 according as x is rational or irrational (B) 2 or 1 according as x is rational or irrational (C) 1 for all x (D) 2 for all x. Q.89
If f(x) is a polynomial function satisfying the condition f(x). f(l/x) = f(x) + f(l/x) and f(2) = 9 then: (A) 2 f(4) = 3 f(6) (B) 14f(l) = f(3) (C) 9f(3) = 2f(5) (D) f(10) = f(ll)
Q. 90 Which of the following function(s) not defined at x = 0 has/have removable discontinuity at x=0 ? rx 1 sinx (C)f(x) = x s i nn(D) f(x) = (B) f(x)=cos (A) f(x) = C in |xi 1+ 2 x
feBansal Classes
Q.B.onFLCD
[13]
Q.91 The function f(x):
"|x-3|
(A) continuous at x = 1 (C) continuous at x = 3
Q.92 If f(x) — cos continuous at : (A)x = 0
, x>l
(B) diff. at x = 1 (D) differentiable at x = 3
cos ( J ( x ~ OJ ; where [x] is the greatest integerr function of x, then f(x) is (B) x = 1
(C) x = 2
(D) none of these
Q. 93 Identify the pair(s) of functions which are identical. jl _ x2 (A) y = tan (cos x); y = ^ — x
1 (B) y = tan ( c o t x ) ; y = x-
(C) y = sin (arc tan x); y = , X V1 + x2
(D) y = cos (arc tan x); y = sin (arc cot x)
Q. 94 The function, f (x) = [ | x | ] - | [x] | where [ x ] denotes greatest integer function (A) is continuous for all positive integers (B) is discontinuous for all non positive integers (C) hasfinitenumber of elements in its range (D) is such that its graph does not lie above the x - axis. Q.95 Let f (x + y) = f (x) + f(y) for all x, y e R. Then: (A) f(x) must be continuous V x e R (B)f(x) may be continuous V x e R (C) f(x) must be discontinuous V x e R (D) f(x) may be discontinuous V x e R Q. 96 The functionf(x) = A/] _ V l - x 2 (A) has its domain -1 < x < 1. (B) hasfiniteone sided derivates at the point x= 0. (C) is continuous and differentiable at x = 0. (D) is continuous but not differentiable atx = 0. Q. 97 Let f(x) be defined in [-2,2] by f(x) = max(4-x 2 , 1+x 2 ), - 2 < x < 0 = min(4-x 2 , 1 + x2), 0 < x < 2 The fix) (A) is continuous at all points (B) has a point of discontinuity (C) is not differentiable only at one point. (D) is not differentiable at more than one point
feBansal Classes
Q. B. onVector,3D&ComplexNo.&Misc.
[2]
Q.98
The functionf(x) = sgnx.sinx is (A) discontinuous no where. (C) aperiodic
(B) an even function (D) differentiable for all x
Q .99 The functionfix)= x / n x (A) is a constant function (C) is such that lim it f(x) exist x-»l
(B) has a domain (0, 1) U (e, oo) (D) is aperiodic
Q. 100 Which pair(s) of function(s) is/are equal? (A) f(x) = cos(2tan 1 x); g(x) /
1
(B) f(x) =
X
1 ~F~ X
; g(x) = sin(2cot-'x)
(C)f(x)= e ^ - " 1 * ) ; g(x)= e ^ (D)f(x) = %fa , a>0; g(x)= a », a > 0 where {x} and [x] denotes the fractional part & integral part functions. Fill in the blanks:
i
smx
x
where c is a known quantity. If fis derivable at x = c, then the values of'a' & 'b' are ax+b& if x > crespectively. Q. 102 A weight hangs by a spring & is caused to vibrate by a sinusoidal force. Its displacement s(t) at time t is given by an equation of the form, s(t) = — 2
c -k2
(sin kt - sin ct) where A c & k are positive constants
with c^k, then the limiting value of the displacement as c —» k is ^
Q 103
T IMIT
x x (cosa) - v(sina) - cos2a W H ,E R E / /
Limit v
n is.
O< a < -
x- 4
2
Q.l 04 Limit ( cos 2x) has the value equal to Q.105 If f(x) = sinx, x * m t , n = 0, ±1, ±2, ±3,.... = 2, otherwise and g(x) = x 2 + 1, x * 0 , 2 = 4, x=0 = 5, x=2
.
then Ljmh g [f( x )] is
feBansal Classes
Q. B. onVector,3D&ComplexNo.&Misc.
[2]
[91]
Q.l Q.6 Q. 11 Q.16
(LYIJ uo a f )
C A C D
Q.21 A Q.26 C Q.31 D Q.36 A Q.41 B Q.46 A Q.51 C Q.56 B Q.61 D Q.66 A
Q.2 Q.7 Q.12 Q.17 Q.22 Q.27 Q.32 Q.37 Q.42 Q.47 Q.52 Q.57 Q.62 Q.67 Q.72
A D C D A A B C A B
C
Q.4
C D D
Q9 Q.14
Q.5 Q.10 Q.15
Q.23 Q.28 Q.33 Q.38
C D D C
Q.43
A D
Q.44
Q.20 Q.25 A Q.30 A Q.35 A Q.40 B Q.45 D
C C A B
Q.73
A. D Q.19 A Q.24 C Q.29 C Q.34 c Q.39 B Q.49 Q.54
D B
Q.69 Q.74
D
B
Q.76 Q.77 Q.81 Q.85
C B, C A, D C, D
Q.78 Q.82
A, C B, C, D
Q.86
A, B, C, D
Q.79 A, C Q.83 B, C Q.87 A, B, D
Q.89 B, C Q.93 A, B, C, D
Q.90 Q.94
Q.97 B, D
Q.98
B, C, D A, B, C, D A, B, C
Q.91 A, B, C Q.95 B, D Q.99 A, C
Q.80 Q.84 Q.88 Q.92
: -At
Q.l02
Q.103 cos4 a / n cos a - sin4 a / n sin a
Q.l04 e •6
Q.l05 1
V3ALSNV
coskt 2k
B C B
C A
Q.50 A Q.55 B Q.60 D Q.65 A
C
Q.70
D
Q.75
A,B,D C, D B,D B, C
Q.96 A, B, D Q.100 A, B, C
Q.lOl cosc& sinc- C COSC
A 3)1
A B
Q.59 Q.64 A
A A
Q.71
A
B
Q.3 Q.8 Q.13 Q.18
Q.48 Q.53 Q.58 Q.63 Q.68
D
S3SSVIJ jvsuvqlj)
C D
MATHEMATICS XII (ABCD)
HOME ASSIGNMENT Objective: Vector, 3D & Complex Subjective: Misc. Topics NOTE: This assignment will be discussed on th<£very first day after Deepawali Vacation, hence come prepared.
Question bank on Vector, 3D and Complex Number & Misc. Subjective Problem Select the correct alternative : (Only one is correct) Q.l
If jaj = 11, |b| =23, |a-bj =30,then |a + b| is : (A) 10
Q.2
(B) 20
(D) 40
The position vector ofa point P moving in space is given by OP = R = (3cost)i+ (4cost) j + (5sint)kThe time't' when the point P crosses the plane 4x - 3y + 2z = 5 is TC
(A) — sec Q. 3
(C) 30
71
71
(B) — sec
(C) — sec
71
(D) — sec
Indicate the correct order sequence in respect of the following: x - 4 y+6 y+6 x-1 y-2 z-3 The lines —— = —— = —— and —— = ^ are orthogonal. ?
I.
II. The planes 3x - 2y - 4z = 3 and the plane x - y - z = 3 are orthogonal. III. The function f (x) = /n(e~2 + ex) is monotonic increasing V x e R . IV. If g is the inverse of the function, /(x) = /n(e-2 + ex) then g(x) = /n(ex - e~2). (A)FTFF (B)TFTT (C)FFTT (D)FTTT Q.4
Zj +z 2 rh. If is purely imaginary then ZJ Z 2
2
(A) 1 Q.5
(B) 2
is equal to : (C) 3
(D) 0
In a regular tetrahedron, the centres of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is —, where m and n are relatively prime n positive integers. The value of (m + n) is (A) 3 (B)4 (C) 27 (D)28
Q.6
If a, b, c are non-coplanar unit vectors such that a x (b x c) = -4= (b+c) then the angle between a & b is V2 (A) 3 TT/4 (B) TI/4 (C) 7i/2 (D) n
Q. 7
The sine of angle formed by the lateral face ADC and plane of the base ABC of the tetrahedron ABCD where A s (3, -2,1); B = (3, 1, 5); C s (4, 0, 3) and D = (1, 0, 0) is 2
5
^
(C)
3^3
v^
(D)
-2
7i?
Q.8
Let z be a complex number, then the region represented by the inequality | z + 2 | < | z + 4 | is given by : (A) R e ( z ) > - 3 (B) Im(z)<-3 (C) Re(z) < - 3 & I m ( z ) > - 3 (D) R e ( z ) < - 4 & I m ( z ) > - 4
Q.9
The volume of the parallelpiped whose edges are represented by the vectors a = 2 i - 3j + 4k, b = 3i - j + 2k , c = i + 2j - k is : (A) 7 (B) 5
feBansal Classes
(C) 4
(D) none
Q. B. on Vector, 3D & Complex No. & Misc.
[2]
Q.10 Let u,v,w be the vectors such that u + v + w = 0 , if |u j= 3 ,|v j= 4 & |w u.v + v.w + w .u is : (A) 47 (B) -25 Q.ll
Let a= i - j , b= j - k , c= (A)
(C)0
(B)±^(i+j-k)
+1
(D)25
. If d is aunit vector such that a.d=0= [b,c,d] then d
Q. 12 If z be a complex number for which (A)
5 then the value of
( C ) ±-^(i+j+k) 1
z + Z
(D)± k
= 2, then the greatest value of | z | is :
(B) V3 + l
(C) 2V2 - 1
(D) none
Q.13 If the non - zero vectors a & b are perpendicular to each other, then the solution ofthe equation, ? x a = b is :
Q.14
(A) v 7 ? •= xa + —— fa x b) a. a V >
K(B) J
r = xb -
(C) r = xa x b where x is any scalar.
(D) r = xb x a
b.b \
(a x b) '
x-2 y-3 z-4 x—1 y-4 z-5 Thelines—:— = —:— = —— a n d - — = -—— = ——are coplanar if 1
(A) k = 1 or - 1
1
K
K
2*
(B) k = 0 or - 3
1
(C)k = 3 o r - 3
(D)k = 0 o r - 1
Q. 15 Which one ofthe following statement is INCORRECT ? (A) l f n . a = 0 , n . b = 0 & n . c = 0 f o r some non zero vector n, then[ a b c J = 0 (B) there exist a vector having direction angles a = 30° & P = 45° (C) locus of point for which x = 3& y = 4isa line parallel to the z - axis whose distance from the z-axis is 5 (D) In a regular tetrahedron OABC where'O'is the origin, the vector OA + OB + OC is perpendicular to the plane ABC. Q. 16 Given that the equation, z 2 +(p+iq)z+r+is = 0 has a real root where p, q, r, s e R. Then which one is correct (A) pqr = r2 + p2s (B) prs = q2 + r2p (C) qrs = p 2 +s 2 q (D) pqs = s 2 + q2r Q.17 The distance of the point (3,4,5) from x-axis is (A) 3
(B)5
(C)V34
(0)^41
Q. 18 Given non zero vectors A, B and C, then which one of the following is false? (A) A vector orthogonal to Ax B and C is± (AxB)xC (B) A vector orthogonal to A+B and A - B i s ± A x B (C) Volume of the parallelopiped determined by A, B and C is | A x B • C | (D) Vector projection of A onto B is
feBansal Classes
AB ^
Q. B. on Vector, 3D & Complex No. & Misc.
[2]
Q.19 Given three vectors a , b , c such that they are non - zero, non - coplanar vectors, then which of the following are non coplanar.
0.20
(A) a + b , b + c , c + a
(B) a - b , b + c , c + a
(C) a + b , b - c , c + a
(D) a + b, b + c, c - a
The sum / + 2/2 + 313 + + 2002/2002, where i = A/_i is equal to (A)-999+ 1002/ (B)- 1002+ 999/ (C) - 1001 + 1000/ (D) - 1002 + 1001/
Q.21 Locus of the point P, for which OP represents a vector with direction cosine cos a = ( ' O' is the origin) is : (A) A circle parallel to y z plane with centre on the x - axis (B) a cone concentric with positive x - axis having vertex at the origin and the slant height equal to the magnitude of the vector (C) a ray emanating from the origin and making an angle of 60° with x - axis (D) a disc parallel to y z plane with centre on x - axis& radius equal to OP sin 60° Q.22
A line with direction ratios (2,1,2) intersects the lines f = - j + X(i + j + k) and f = - i + p(2i + j + k) at A and B, then / (AB) is equal to (A) 3
( Q 2V2
(B)V3
(D)V2
Q.23
The vertices ofa triangle are A (1,1,2),B(4,3,1) & C (2,3,5). The vector representing the internal bisector ofthe angle A is : (A) 1 + j + 2k (B) 2i - 2j + k (C)2i + 2 j - k (D)2i+2j + k
Q.24
Lowest degree of a polynomial with rational coefficients if one of its root is, V2 + 1 is (A) 2 (B)4 (C)6 (D) 8
Q.25 A plane vector has components 3 & 4 w.r.t. the rectangular cartesian system. This system is rotated 7T
through an angle — in anticlockwise sense. Then w.r.t. the new system the vector has components :
(B)^.jj
(Q-j-.j.
(D) none
Q.26 Let a =xi+ 12j - k; b = 2i + 2xj + k & c = i + k. If the ordered set b c a ] is left handed, then: (A) x e (2,00) Q.27
(B) x e (-00, - 3)
(C)xe(-3,2)
(D)xe{-3,2}
If cosai + j + k , i + cospj + k & i + j + cosyk (a ^ p * y * 2 n Tt) are coplanar then the value of 2 ot
?p
2
2
y
cosec —t-cosec — + cosec — equal to (A)l
2
2
(B)2
(C) 3
(D) none of these
Q.28 The straight line (1+2/)z+(2/-l)z = 10/ on the complex plane, has intercept on the imaginary axis equal to (A) 5 Bansal Classes
(B)|
(C)-|
(D)-5
Q. B. on Vector, 3D & Complex No. & Misc.
[4]
Q.29 The perpendicular distance of an angular point of a cube of edge 'a'fromthe diagonal which does not pass that angular point, is (A)V3a
Q.30
Q.31
(B)V2a
(C)
a
. (D)
a
Which one of the following does not hold for the vector V = a x (b x aj? (A) perpendicular to a
(B) perpendicular to b
(C) coplanar with a & b
(D) perpendicular to a x b.
Let z,,z 2 &z 3 be the complex numbers representing the vertices ofa triangleABC respectively. If P is a point representing the complex number z0 satisfying; a (Zj - z0) + b (z2 - z0) + c (z3 - z0) = 0, then w.r.t. the triangle ABC, the point P is its : (A) centroid (B) orthocentre (C) circumcentre (D) incentre
Q.32 Given the position vectors ofthe vertices ofa triangle ABC, A=(a) ;B=(b) ; C = (c). A vector r is parallel to the altitude drawn from the vertex A, making an obtuse angle with the positive Y-axis. If I r | -2V34 ;a = 2i—j—3k ;b =i + 2 j - 4 k ; c = 3 i - j - 2 k , t h e n f is (A) — 61 — 8j — 6k
(B) 61 - 8 j + 6k
( C ) - 6 i - 8 j + 6k
(D) 6i+8j + 6k
Q.33 The complex number, z = 5 + i has an argument which is nearly equal to: (A) 7t/32 (B) 71/16 (C) 7t/12 (D) 71/8 a.a a.b a.c Q.34 If a = i + j + k ,b = i - j + k ,c = i + 2 j - k , then the value of b.a b.b b.c equal to c.a c.b c.c (A) 2
(B)4
(C) 16
(D) 64
Q.35 If the equation x2 + ax + b = 0 where a, b e R has a non real root whose cube is 343 then (7a + b) has the value (A) 98 (B) - 49 (C) - 98 •'(D) 49 Direction for Q.36 to Q.40. Let
A = i + 2j + 3k and B = 3i + 4 j + 5k
Q.36 The value of the scalar (A) 8
AxB| 2 +(A-B)2 is equal to (B)7V!0
(C)10V7
(D)64
Q.37 Equation ofa line passing through the point with position vector 2i + 3j and orthogonal to the plane containing the vectors A and B, is (A) r = (A, + 2)i-(2?i-3)j + A,k
(B) r = (X -
(C) f = XI+ (2X - 3) j - Xk
(D) none
^Bansal Classes
2)1 - (2X-3)j
0. B. on Vector, 3D & Complex No. & Misc.
+ Xk
[5]
Q.3 8 Equation of a plane containing the point with position vector (i - j + k) and parallel to the vectors A and B , is (A) x + 2y + z = 0 (C) x -- 2y + z - 4 = 0
(B) x - 2 y - z - 2 = 0 (D) 2x + y + z - 1 = 0
Q.3 9 Volume of the tetrahedron whose 3 coterminous edges are the vectors A, B and C = 2i + j - 4k is (A) 1 (B) 4/3 (C) 8/3 (D) 8 Q.40
Vector component of A perpendicular to the vector B is given by (A)
Bx (Ax B)
(B)
Ax(AxB) \
Bx(AxB)
2
CP)
Ax(AxB) pi
Select the correct alternatives : (More than one are correct) Q.41 If a, b, c are different real numbers and ai + bj + ck ; bi + cj + ak & c i + a j + bk are position vectors of three non-collinear points A, B & C then: A. . A a +J}b++Cc !/r
T
(A) centroid of triangle ABC is — - —
J*.
+ j + kj
(B) i + j + k is equally inclined to the three vectors (C) perpendicular from the origin to the plane of triangle ABC meet at centroid (D) triangle ABC is an equilateral triangle. Q.42
The vectors a , b , c are ofthe same length & pairwise form equal angles. If a = i + j & b = j + k, the pv's of c can be :
(A,0,0,,)
.(B)(-i,i,-i)
(C)Q,-fij
Q.43
Which of the following locii of z on the complex plane represents a pair of straight lines? (A) Rez 2 = 0 (B) Imz 2 = 0 (C)|z|+z =0 (D) | z - 1 | = I z - i |
Q.44
If a , b , c & d are linearly independent set of vectors & K; a + K2 b + K3 c + K4 d =0 then: (A) K, + K2 + K3 + K4 = 0 (B) K, + K3 = K2 + K4 = 0 (C) K, + K4 = K2 + K3 = 0 (D) none of these
Q.45
Ifa , b , c &d are the pv's of the points A, B, C & D respectively in three dimensional space & satisfy the relation3a - 2 b + c - 2d=0, then: (A) A, B, C & D are coplanar (B) the line joining the points B & D divides the line joining the point A & C in the ratio 2:1. (C) the line joining the points A & C divides the line joining the points B & D in the ratio 1:1 (D) the four vectors a , b , c & d are linearly dependent
Q.46
If z3 - i z2 - 2 i z - 2 = 0 then z can be equal to : (A) 1 - i (B) i (C) 1 + i
feBansal Classes
(D) - 1 - i
Q. B. on Vector, 3D & Complex No. & Misc.
[2]
Q.47 If a & b are two non collinear unit vectors & a , b, x a - y b form a triangle, then: (A) x = - 1 ; y = 1 &
a_b
2 cos
v
f
2
j
A
x = - 1 ; y = 1 & cos a b + 1I a + b I cos i,-(a + b) k
I =-2 cot
/
f A > a b
2
COS
(aAb^ v
2
&x = - l , y = 1
(D)none
y
Q.48 The lines with vector equations are; r, = -3i + 6j + A, (-4 i + 3 j + 2 kj and r2 =-2i + 7j + p. [~4i + j + kj are such that: (A) they are coplanar (C) they are skew Q.49
(B) they do not intersect (D) the angle between them is tan"1 (3/7)
Given a, b, x, y e R then which of the following statement(s) hold good? (A) (a + ib) (x + iy)-1 = a - ib =>x2 + y 2 = l (B) (1—ix) (1 + ix)"1 = a - ib ^ a2 + b2 = 1 (C) (a + ib) (a - ib)-1 = x - iy => |x + iy| = 1 (D) (y - ix) (a + ib)"1 = y + ix |a - ib| = 1
Q.50 The acute angle that the vector 2i - 2j + k makes with the plane contained by the two vectors 2i + 3 j - k and i - j + 2k is given by: (A) cos"1 Q.51
Q.52
(B) siff
(C)tan-'(V2)
(D)cot"1(V2)
The volume of a right triangular prism ABCA, B ,Cj is equal to 3. If the position vectors of the vertices of the base ABC are A( 1,0,1); B(2,0,0) and C(0,1,0) the position vectors of the vertex Aj can be: (A) (2,2,2) (B) (0,2,0) (C) (0,-2, 2) (D) (0,-2,0) I f x i= C i S (» — 1I for for 1 < r < n r, n e N then: IV?/ / / '
n
Re F K = - 1 1 y /
Im n * r | V .r = l y
feBansal Classes
(B) Limit
Re
f" > r u Vr- 1 ( » ^
= 0
(D) L™* Im r i M = 0 \r = 1 /
Q. B. on Vector, 3D & Complex No. & Misc.
[2]
A
/A.
^
Q.53 Ifa line has a vector equation, r = 2i + 6j + X i - 3j) then which of the following statements holds good? (A) the line is parallel to 2 i + 6 j
(B) the line passes through the point 3 i + 3 j
(C) the line passes through the point i + 9 j
(D) the line is parallel to xy plane
Q.54 If a , b, c are non-zero, non-collinear vectors such that a vector p = ab cos^rc - (a A bjj c and a vector q = ac cos [ft - (a
A
cjj b then p + q is
(A) parallel to a
(B) perpendicular to a
(C) coplanar with b & c
(D) coplanar with a and c
Q. 5 5 The greatest value ofthe modulus of of the complex number ' z' satisfying the equality z' (A)
(B)
(C) I ^ H
+
z
= 1 is
(D)
SUBJECTIVE: Q.l
Let a = V 3 i - j and b = - i + — j and x = a + (q 2 -3)b, y = - p a + qb-If x l y , then express;? as a function of q, say p =f(g), (p^O&q^O) and find the intervals of monotonicity of /(q).
Q.2 v
/nx ex - 1 Using only the limit theorems Lim = 1 and Lim = 1. EvaluateT 6 17 x->o x x->i x - 1
Q.3
The three vectors a - 4 i - 2 j + k ; b - 2i - j - k and c = 2 i + k are all drawn from the point with
x -x 7 77. /nx-x + 1
p.v. i - j . Find the equation of the plane containing their end point in scalar dot product form.
Q.4 Q.5
jV(cos 2n-1 x - cos2n+1 x) dx where n e N Let points P, Q & R have position vectors, ij = 3i — 2j — k; r2 = i + 3j + 4k & r3 = 2i+j-2k respectively, relative to an origin O. Find the distance of Pfromthe plane OQR. 3
Q.6
Evaluate: f| (x - l)(x - 2)(x - 3) |dx 1
Q.7
Given that vectors A, B,C form a triangle such that A = B + C ,finda,b,c,d such that the area of the triangle is 5-^6 where A = ai + bj + ck; B = di + 3j + 4k & C =3i + j - 2k.
feBansal Classes
Q. B. on Vector, 3D & Complex No. & Misc.
[2]
k+1 n-1
Q.8
n k+1 Lim n V f f x - i l f -x dx n— J k=o k W n ;V n n
Q.9
Find the distance ofthe point P(i + j + k) from the plane L which passes through the three points A(2i + j + k), B(i + 2j + k), C(i + j + 2k). Also find the pv of the foot of the perpendicular from P on the plane L.
rVsm vr~sir4x + cosr x dx , x Q.IO Evaluate: (a) J : sm xcosx Q.ll
H \
1
j
Vsin4 x + cos4 x dx smxcos x
Find the equation ofthe straight line which passes through the point with position vector a, meets the line ?=b + tc and is parallel to the plane r. ii = 1.
dx Q.12 Integrate: f- 3 cos x - s i •n 3 x Q.13 Find the equation of the line passing through the point (1,4,3) which is perpendicular to both of the lines x-1 y+ 3 z-2 x+2 y-4 z+1 _ = —:— = —:— and —-— = —— = —— 2 1 4 3 2 -2 Alsofindall points on this line the square of whose distance from (1,4,3) is 357.
Q.14
Lim n-»co
+ n -1 n
\2
+n-l
Q.15 If z-axis be vertical, find the equation of the line of greatest slope through the point (2, -1,0) on the plane 2x + 3y - 4z = 1. tc/2 k/2 • f cosx f sinx , Q.16 Let I = Jf dx and J = j — d x , where a> 0 and b> 0. 0 acosx + bsinx 0 acosx + bsinx Compute the values of I and J.
feBansal Classes
Q. B. on Vector, 3D & Complex No. & Misc.
[2]
[oil
•jsijtf y 'Otf xajdtuoj y (j£ Uops/i uo 7/ Q
sdssvjj
psuvfj^
Select the correct alternative : (Only one is correct) Q.l
B
Q.2
B
Q.3
C
Q.4
A
Q.5
D
Q.6
A
Q.7
B
Q.8
A
Q.9
A
Q.10
B
Q.ll
A
Q.12
A
Q.13
A
Q.14
B
Q.15
B
Q.16
D
Q.17
D
Q.l 8
D
Q.19
A
Q.20
D
Q.21
B
Q.22
A
Q.23
D
Q.24
B
Q.25
B
Q.26
C
Q.27
B
Q.28
A
Q.29
D
Q.30
B
Q.31
D
Q.32
A
Q.33
B
Q.34
C
Q.35
A
Q.36
C
Q.37
A
Q.38
C
Q.39
B
Q.40
A
Select the correct alternatives : (More than one are correct) Q.41 A,B,C,D
Q.42
A,D
Q.43 A,B
Q.44
A,B,C
Q.45
A,C,D
Q.46
B,C,D
Q.47
A,B
Q.48
B,C,D
Q.49
A,B,C,D
Q.50
B,D
Q.51
A,D
Q.52
A,D
Q.53
B,C,D
Q.54
B,C
Q.55
B,D
SUBJECTIVE: Q.l Q.2 Q.5 Q.8
_ q(q3-3)
,decreasing in q e (-1, 1), q •*• 0
- 2
3 units
(2i + 2 j - k ) . r = 3
Q.4
Q.6
1/2
Q.7
2n +1 (-8,4, 2,-11) or (8, 4, 2, 5)
Q.14
e"
n
8
(b)
'
1m V t C i - i 4
s 2
Q.15
(
' z where t = corx; Vt + l + l
1. i r + 1 - 1 + -/n + C, where t = tan2x 2 4 Vt +1-4-1
r = a + A. ( i - b ) - ^ c ' c.n
Q.12 2 [tan_1(sinx + cosx) +
Q.13
'4 4 4X T P v3'3'3y
Q 9
Q.10 (a)
Q.ll
Q.3
x—1 y-4 -10
16
x-2
y+i 12
2V2
In
V2+ si sin x +cosx +C A/2 -• sin x - cos x
z-3 , ; (-9, 20, 4); (11, -12,2) 1 z 13
Q.16
1 an + b/n a 2 a +b v /y 2
A 3)1
V3MSNV
1
bn
-a/n
v a yy
KEY CONCEPTS THINGS T o REMEMBER :
L
LOGARITHM OF A NUMBER : The logarithm of the number N to the base 'a' is the exponent indicating the power to which the base 'a' must be raised to obtain the number N. This number is designated as logaN. x
Hence:
log N = x <=> a = N
,a>0,a*l&N>0
REMEMBER
log, 0 2 = 0.3010
logI03 = 0.47 71 In 2 = 0.693
In 10 = 2.303 If a= 10, then we write logb rather than log10b . If a = e, we write In b rather than loge b . The existence and uniqueness of the number logaN follows from the properties of an experimental functions . From the definition of the logarithm of the number N to the base 'a', we have an identity:
a
log a N
=N
a>0 , a*1 & N>0
This is known as the FUNDAMENTAL LOGARITHMIC N O T E :log a l=0 (a>0 , a * l ) log a a=l ( a > 0 , a*l)and log1/aa = - l (a > 0 , a * 1)
IDENTITY .
THE PRINCIPAL PROPERTIES OF LOGARITHMS : Let M & N are arbitrary posiitive numbers, a > 0 , a ^ l , b > 0 , b ^ l and a is any real number then ; loga(M/N) = logaM - logaN (ii) CO loga (M. N) = loga M + logaN
2.
(iii)
log M a = a . log M
NOTE: Y
(iv)
(i) (ii) (iii) (iv)
log b
y log.a. log b. log c = 1
logba. logab == I o logba=T/logab.
y
y logz>y v x. log 7 y.' log oz^ " oaa z = logx. ©a
3.
log M
log.M
eIn a
x
PROPERTIES OF MONOTONOCITY OF LOGARITHM : For a> 1 the inequality 0 < x < y & log a x< logay are equivalent. For 0 < a < l the inequality 0 < x < y & log a x>log a y are equivalent. => 0 < x < ap If a > 1 then loga x < p If a > 1 then logax > p
=>
x > ap
M
If 0 < a < 1 then log a x
=>
x > ap
(vi)
If 0 < a < 1 then log. x > p
0 < x < ap
NOTETHAT:
y
If the number & the base are on one side ofthe unity, then the logarithm is positive; If the number & the base are on different sides of unity, then the logarithm is negative. The base of the logarithm 'a' must not equal unity otherwise numbers not equal to unity will not have a logarithm & any number will be the logarithm of unity.
y
For a non negative number 'a' & n > 2, n e N
Bansal Classes
Va = a1/n.
Logarithm
[2]
EXERCISE-I \
Q.l J Show that :
/
/
1,
2. log(8/45) + 3 . log(25/8)-4.log(5/6) = log2.
0.2 v
log N Prove that -——— = 1 + log b & indicate the Hpermissible values ofthe letters. loti , N •
Q.3
(a\f
Given: log1034.56= 1.5386, find log103.456 ; log100.3456 & log100.003456.
(b)
Find the number of positive integers which have the characteristic 3, when the base ofthe logarithm is 7.
Q.4y
If log102 = 0.3010, logI03 = 0.47 71. Find the number of integers in : (a) 5200
Q.5 J
(b) 615
(c) the number ofzeros after the decimal in 3-100.
&
If log102 = 0.3010 & log]03 = 0.4771, find the value of log10(2.25).
Q.67 Find the antilogarithm of 0.75, i f the base of the logarithm is 2401. Solve for x : Q.7J
(a)jlf log 10 (x 2 - 12x + 36) = 2
(b) 9 1+1 °s x -3 1+, °s x -210 = 0 ; where base of log is 3. log
b(logbN)
Q.8
Simplify:
1.(a) l o g ^ f o . ^ T ^
7 7
(b) a
'°g»a
Solve for x : Q.9
(a) If log4 log3 log2 x = 0
Q.10
Which is smaller?
Q.ll
Prove that logjS is irrational. )og 4
Jj
(b) If log, log4 log5 x = 0
2 or (logn2 + log27i).
lo8 9
^ j
VA V
log S3
Q.12 V If. 4 ^ +9 3 =10 " , then find x. Q.13 Find a rational number which is 50 times its own logarithm to the base 10. Q.14
Calculate:
Q.15
Simplify the following : (a)
(a^ j ^ ^
7
- 5 ^ .
_
(b)
-J
81
(b) 5 , 0 8 - y + l o g
[ v
y
Q.l 6
Show that
logic =20,
Q.17
If loge log, [v'2x~- 2 + 3] = 0, then find the value of x.
where
4
:+log,, 1/2 v 7+V3 10+2V2T '
2 /
the base of log is 10.
Q.l 8 Express log4a + log s (a) ,/3 + [l/loga8] as a logarithm to the base 2. Q. 19 Find the value of Q.20
+ 5~ logs4 .
Given that log23 : a , log35 = b, log72 = c, express the logarithm ofthe number 63 to the base 140 in terms of 3, b & c.
QBansal Classes
Logarithm
[3]
Q.21 Prove that
log, 24
log., 192
=3.
Q.22 Prove that ax - b>' = 0 where x = ^og^b & y = / l o § b a , a > 0 , b > 0 & a , b * 1. log N.log N.log^N Q.23 Prove the identity : logaN . loghN + logbN . logcN + logcN . Iog,IN= — ~T7— 8
Q.24
abc
log ]0 (x-3) _ 1 (a) S o l v e f o r x , l o g J x 2 _ 2 1 ) - 2 (b) log(logx) + log (log x 3 -2) = 0 ; where base of log is 10 everywhere. (c) logx2 . log2x2 = log4x2 (d) 5logx + 5 x'°e5 = 3 (a > 0) ; where base of log is a.
Q.25
Solve the system of equations: log a x loga(xyz) = 48 log. y log. (xyz) = 12 , a > 0, a * 1. a log'z « logd (xyz) = 84
EXERCISE-II Note : From Q.l to Q.9, solve the equation for x : Q.l
log ]0 [3 + 2.1og10(l +x)] = 0.
Q.3
XIOGX+4 =
Q.5
x+log 10 (l+2 x ) = x.log 10 5 + log106.
Q.6
5iogx_3iogx-i ^3iogx+i _5iogx-i;
22, where base of logarithm is 2.
Q.2
(1 /12) (log,0x)2 = (1 /3)-(1 /4) (log,0x)
Q.4
logx+1 (x2 + x - 6)2 = 4
where the base of
iogarithm
is
10
__ l + log 2 (x-4) Q 7
'
log
(VTTs-^i)
= 1
Q.8
log5120 + (x-3) - 2. log, (1 -5X"3) = -log5(0.2-5X"4)
Q.9
1N . + — log 4 + 2x log 3 = log ( ^ + 27). '
Q. 10 Prove that log710 is greater than log, ,13. Q.ll
If ^ U ^ g b . b i i , b-c c-a a-b
s h Q w t h a t aa bb
Cc=1
Q.12
If a = log1218 & b = log2454 then find the value of ab + 5(a-b).
Q.13 If x = 1 + logabc, y = 1 + logbca, z = 1 + logcab, then prove that xyz = xy + yz + zx. Q.14
If p = logabc, q = logbca, r = logeab, then prove that pqr = p + q + r + 2.
Q.15
If logba. logca + logab. logcb + logac. logb c = 3 (Where a, b, c are different positive real numbers * 1), then find the value of abc.
43Bansal Classes
Logarithm
[4]
Q. 16 Given a2 + b2 = c2 & a > 0 ; b > 0 ; c > 0 , c - b * l , c + b * l . Prove that: logc+ba + logc_ba = 2. log c+b a. logc.ba. Q.17
If
loga
N_
log c N
lo g l l
N -• logfcN
logbN-logcN
where
n >o &
N
a,b,c>
1,
0
& not equal to
1,
then
prove that b2 = ac. Q. 18 Find all the s o l u t i o n s of the e q u a t i o n
1x |
~
(logxf-logx m'°8
2
, where base of logarithm is 10.
Q.19
Solve the system ofthe equations (ax)loga = (by)logb; b,ogx = a'°s-v where a > 0, b > 0 and a*b,ab*l.
Q.20
Solve the system ofthe equations log9(x2 + l) r log 3 (y-2) = 0and log 2 (x 2 -2y 2 + 1 0 y - 7 ) = 2.
Q.21
Find x satisfying the equation log'
Q.22
Solve : log-(Vx + V x - 1 ) = log9
Q.23
Prove that : 2
Q.24
Solve for x: log 2 (4-x) + log (4-x) . log
M 4
+log 2
2 log"
X+4
—1 x —1
Vx-1 ) 2
if b>a>l
log b
2 a r ~J
V
Q.25
V
if l
•2 log2! x
0.
If P is the number of integers whose logarithms to the base 10 have the characteristic p , and Q the number of integers the logarithms of whose reciprocals to the base 10 have the characteristic - q, then compute the value of log!0 P - log10Q in terms of p and q.
EXERCISE-TII Q.l
Solve the following equations for x & y :
logi001 x+y |
2
log 1 0 y-log 1 0 lxblog 1 0 0 4. Q.2
[REE'96,6]
Find all real numbers x which satisfy the equation, 21og 2 log 2 x + log ]/2 log 2 (2V2x)= 1.
[ REE '99, 6 ]
Q.3
log3/4log8 (x2 + 7) + log1/2 log1/4 (v + 7)"1 = - 2.
Q.4
Number of solutions of log 4 (x- 1) = lug 2 (x-3) is (A) 3 (B)l (C) 2 " (D) 0
' feBansal Classes
[REE 2000, 5 out of 100]
Logarithm •I 1 • » • — w
[JEE 2001 (screening)]
[5]
iEM JPll
ANSWER
KEY
EXERCISE-I Q 2. a > 0 , a * l , N > 0 , N * l , b > 0 , b * l / a Q 3. (a) 0.5386 ; 1.5386 ; 3.5386 (b) 2058
Q 4. (a) 140
Q 5. 0.3522
Q
Q7. (a) x = 16 or x = - 4 (b) x = 5
Q 8 . ( a ) - 1 (b) logbN
Q 9. (a) 8 (b) 625
Q 10. 2
Q 12. x = 10
Q 13. N = 100
Q 14. (a) 0 (b)
(c)
47
(b) 6
Q 18. — .log, a 18
2
1 + 2ac
25
Q 20. 2c + a b c + 1
Q 24. (a) x = 5 (b) x = 10 (c) x = 2 r i or Q 25.
12
343
Q 15.(a) 1
9
Q 17. x = 3 Q19.
6.
(b)
(a4, a, a 7 )
(d) x = 2"losa where base of log is 5
J_ l_ J_ 4 ' a 'a 7 J Va EXERCISE-II 10 or 10~4
Q l . x = -0.9
Q 2.
x =
Q 4. x = 1
Q 5.
x —1
Q 7. x = 5
Q 8. x = 1
Q12. 1
Q 15.
abc = 1
Q 20.
x -
Q 19. x =
a
&
y =
b
Q 21. x = V2 or V 6 Q.25
Q3. x = 2 or ^
Q
.
6. x =
-
32
100
Q 9. x s (j)
V3
1, ; y = 4
or
x = S
Q 18. x = 2 or — or 1000 ; y = 4
Q22. [0, 1] u {4}
p - q
EXERCISE-III Q.l
{-10,20}, {10/3,20/3}
Q.3 x — 3 or - 3
Q.2 x = 8
Q.4 B
43Bansal Classes
Logarithm
[6]
BANSAL CLASSES TARGET IIT JEE 2007
MATHEMATICS XII (ALL)
LIMIT,
CONTINUITY &
DIFFERENTIABILITY W- OF FUNCTION
LIMIT KEY CONCEPT EXERCISE-I.. EXERCISE-II EXERCISE-III
.
«
Page -2 Page -3 Page -4 Page -6
CONTINUITY KEY CONCEPT EXERCISE-I EXERCISE-II EXERCISE-III
Page-7 Page -9 Page -13 Page -16
DIFFERENTIABILITY KEY CONCEPT EXERCISE-I..., EXERCISE-II EXERCISE-III
Page Page Page Page
-17 -18 -20 -23
ANSWER KEY
Page—25 -27
KEY CONCEPTS (LIMIT) THINGS T O REMEMBER :
1.
Limit of a function f(x) is said to exist as, x—>a when Limit f j - ) = Limit ^ = finite qua ntity.
2.
FUNDAMENTAL THEOREMS O N L I M A ' S :
Let Lx™11 f (x) = I & ^ 0) (iii)
L
^ f ( x ) ± g ( x ) = /±m (ii) Limit f(x) g(x) = /.m f Cx) g Limit ^ = - , provided m * 0
(iv) (v)
k f(x) = k
f(x) ; where k is a constant.
L-it f [g(x)j = ffLimit g(x)] = f(m); provided fis continuous at g (x) = m. f (x) /n/(/>0). /n(f(x) = /n Limit x-»a
For example 3.
STANDARD LIMITS :
,o)
Limit
V"/
g (x) = m. If / & m exists then:
t
x->0
s m x
=
i1 = Limit
x->0 „X
X
=
Limit t a n - ' X
x-»0
= L i m i t
x—>0
SUT'X
[ Where x is measured in radians ] (b)
Limit (1 + x)1/x = e = Limit , Limit
and
(c)
If
(
If
Limit however there h-» o (1 - h ) n = 0
,.
(1 + h ) n - » oo
Limit f(x) = 1 and Limit [ f ( X ) f 0 0 = e x ™ a
(d)
note
Limit
= A >
<>| (x) = oo, then; +(x)[f(x)-i]
0 & Limit <> | ( X ) = B ( a f in i te quantity)then
;
L
(e) m
2 L !t 5? j (x). ln[f(x)] = e BlnA =A B x™ <> xx— >a I.[f(Vx)] VJ = e where z = x->a x 1 T * ..." j o^ . Limit = In a (a >0). In particular Limit £ _ J = i
~ Limit
v
4.
x-a
SQUEEZE PLAY THEOREM :
If f(x) < g(x) < h(x) V x & L«mt f(x) = / = Limit h ( x ) then Ljmit g(x) = /. 5.
INDETERMINANT FORMS :
- , — , Oxoo , 0°, 00° , oo-oo and l00 0 00 Note : (i) We cannot plot oo on the paper. Infinity (oo) is a symbol & not a number. It does not obey the laws of elementry algebra. (ii) oo + oo = oo (iii) o o x o o = oo (iv) (a/00) = 0 if a is finite (v)
^ is not defined, if a 0.
(vi)
ab = 0,if&onlyifa = 0orb = 0 and a & b are finite.
^<§BansalClasses
Limits, continuity & Differentiability of Functions
[5]
The following strategies should be born in mind for evaluating the limits: Factorisation Rationalisation or double rationalisation Use of trigonometric transformation ; appropriate substitution and using standard limits Expansion offunction like Binomial expansion, exponential & logarithmic expansion, expansion of sinx, cosx, tanx should be remembered by heart & are given below:
6.
(a)
xlna
x 2 ln2a
x3 ln3a
1!
2!
3!
(i) ax =1 + —— + w (iii) In (1+x) = / \
^
(V) cosx=l v
'
X
+ 2
2!
+
/ * -1 (vii) v 7 tan'x=x
, ....
X
X
4
3
for-l
X
4! 3
6
6!
+
5
X
5
7
+X
(ii) ex =1+—+—+—+., v ' 1! 2! 3!
.a >0
3!
+
5
7
5!
7!
x3 3
2x5 15
2
4
+
(vi) tanx= x+—+—+.. v '
7
+ •i2 o2 c2
V 3 12.32 51 lz.3'.5" 7 -x +x +.. 3! 7!
. ,
(vm) sin *x= x+—x3+
3
(iv) sinx=x v '
, x 5x 61x® 1 •>'v (ix) sec" x = l+—+-—+ =
^
2!
4!
6!
EXERCISE-I Q/
2 Lim x -x.lnx +lnx - 1 X->1 x—1
Lim Vx-1
r „ Lim x—>1v i - x p Q/7. ^ X )
_
Tjm
2-v/x + 3x1/3 + 5x1/5
i-x ;
Lim tan
-i a
1 +A/tanx
x X 6 ' Lim 1 - 2 cos x
q
where a eR
Plot the graph ofthe function f(x) = Lim — tan - r vn t y "100 s * k -100 _K=1 . Lim x-»l X—1
06)
^
Find the sum of an infinite geometric series whosefirstterm is the limit of the function f(x)=
g(x)
>bey the
1 -Vx "(cos-1 x)2
as x
L (U se °f series expansion or L' Hospital's rule is not allowed.)
L^m ( x _ / n cogh x ) w here cosh t :
^10. .^HL
tanx-sinx — as x—>0 and whose common ratio is the limit ofthe function sin x
Lim
x
e' + e"1
Lim 1 ~ t a n x cos-1 [cot x] where [] denotes greatest integer function ^ Q 12. x_>| ~ — ^ s i n x
Lim [/n (1 + sin2x). cot In2 (1 + x)] x—>0
Lim V2 - cos 0 - sin 0 (49 — 7i)2
i 0 - > f
[2]
^Bansal Classes
16.
Lim f
sec 4x-sec 2x sec3x-secx 01-COSX 1 x(x-f)
Limits, continuity & Differentiability of Functions
[31
OT17. If Lim a sinx-sin 2x . -sr-^n * 3x__ tan ..2
n 115. S
X2
Lim A 1-cos X x->0 x8 2
f,
, & ^ limit LfoocL' y f Lim (ln(l + x) - ln2)(3.4x~1 - 3x) x-^1 [ ( 7 + x ) l_ ( 1 + 3x )i]. sm ( x _l)
finite t h e n f m d t h y a l u e
2
X2
X
cos— + cos—cos— 4 2 4
1 1 ^TT + J 2 + j +
Using Sandwich theorem to evaluate
-
2+
+
Vn 2 +2n
jt 1 021. Givenf(x)= Lim tan"1 (nx); g(x) = Lim s in 2n xandsin(h(x))=-[cos7t(g(x))+cos(2f(x))] Find the domain and range ofh (x). T im (x3 +27)In (x- 2) x2 - 9
Lim ->/l-V sin2x 7C-4x ^24.
Lim (cosa) x + (sin a ) x - 1 X—>2 x-2
0^25.
1 / 0 2 6 . Let'f(x) = - r — , x > 0 sinx =2-x, x<0
and
T im 27*-9*-3*^1 x->0
7 2 - V l + cosx
x <1
g(x) = x+3, 2
=x -2x-2, l
1
1
Q 27. Let P = a "" - 1 , V n = 2, 3, n
Lim ^ 2 8 . X-^-W Lj Q.29
4
2
x
and Let P, = a - 1 where a e R+ then evaluate Lim — x->0
X
3
(3x + 2x )sinL+| x | +5 |x|3+|x|2+|x|+l
If f(x) = /ncosec (xn) 0
2f(x)+l g( ) 3 f ( x ) + 1 x =
^
,
then
At the end-points and the midpoint of a circular arc AB tangent lines are drawn, and the points^ are joined with a chord. Prove that the ratio ofthe areas of the two triangles thus formed tends to • arc AB decreases indefinitely.
EXERCISE-II ^
L
JjS
2x2 + 3 2x2 +5 sm'
-V
Lim
x-^00
2
8x +3
,21
Tt
f
n
2-bx V^
2-axJ
cos 271
'
x ^ .l + x;
f\ q Lim ^ x - l + cosx^ t-/V x-»0 ^<§BansalClasses
\l/x
(1+x1
X +c =4 thenfindc , 02. Lim Vx- c
aeR
5
Lim x sin £n jcos
71 X
tan-*
7tX
tan •
/
x ,t Lim a, + a,* +a, + +ar x->co
1 \nx
where aj,83,83,
Limits, continuity & Differentiability of Functions [5]
V
2 W. Letf(x)= sm " 1(1 . {x}) c ° S 1(1 {x}) thenfindx xL j f(x) and ^ / V2{x} . (l-{x» part function. Find the values of a, b & c so that Lim x->0 f
2n )
a 2 +x 2 - 2 sin ax U —
))]
f(x), where {x} denotes the fractional «
aeX
J
~~ b cos x + ce" • = 2 x. sin x N ^ TlX ^ sin U J y where a is an odd integer
Lim tan 2 x-x 2 x2tan2x
T im X n f ( x ) + g ( x ) n—>0
x n +l
°
Q p . Lim [1.x ]+ [2.x ]+ [3.x ] +
+[n.x] ^
where
^
denotes the greatest
Q 16. Without using series expansion or LHospital's rule evaluate, Lim x—>1
exp(xln(l + Limit-
xsR integer function.
1 — x "f" /n x 1 + COS71X
j -expfxln(l +
X-><»
Q 18. If sn be the sum of n terms of the series, sin x + sin 2x + sin 3x + Limit s i + s 2 + Q 19.
^
n 1+x \l+ in (1 + x) x X
2
+ sn = 1
cpt
|
+ sin nx then show that
(x;£2k7t)kGl)
1 X
23 -1 33 - 1 43 -1 5 ^ 0 i. Let P„n = 2 + 1 3+1 4 + 1
n3-l .Evaluate Lim n—>oo p n 3 +l 7T
Q 21. A circular arc of radius 1 subtends an angle ofx radians, 0 < x < — as shown in itsAandB to 4 as the
thefigure.The point C is the intersection of the two tangent lines at A& B. Let T(x) be the area of triangle ABC & let S(x) be the area of the shaded region. Compute: (b) S(x)
&
-|l/x
j I = yx + yx + Vx
fljk
&
^ '
(c) the limit of
S(x)
as x -> 0.
0
Lim ^x + Vx+Vx -Vx L
e
If f(n, 9 ) = n ^ " t a n 2 ^ r j , then compute Lim / ( n , 6) Q 24.
Let/=Lim
x- a
f
.2
Q 2 y Lim >0
X 7t
V
^
cosh — cos—
&
m
=
Lx-»a im
where cosh t =
ax - xa where a > 0. If / = m thenfindthe value of ' a'. x- a e'+e" 1 2
Lim 2 (tanx - sinx) - x3 x 1x1 -°
XJ
<§Bansal Classes
Limits, continuity & Differentiability of Functions
[5]
f[
Q 27^ Through a point A on a circle, a chord AP is drawn & on the tangent at A a point T is taken such t ^ AT = AP. If TP produced meet the diameter through Aat Q, prove that the limiting value ofAQ whe moves upto Ais double the diameter of the circle. J^f28. Using Sandwich theorem, evaluate 1 2 2 + ^ -*=° 1 + n^ 2 + n2 +.*.
n
v*
n + ^n + n22
' Lim n n w 00 v +b )n ,0
x 2 +l Lim ^ j Q 29. Find a & b if :Ji) X-»00 X + l - a x - b = 0 \x + h
Q30. Showthat Lim (sm(x + h))x + h
(sinxf
Lim (ii) X-»-00
x+ l -ax-b
=(sinx)X[xcotx + /nsinx]
EXERCISE-III l + 5x2 _l + 3x A ^
l [ IIT'96,1 ]
j]-cos2(x-l) X-^l x-l (B) exits and it equals - ^2 (A) exits and it equals ^2 (C) does not exist because x - 1 -» 0 (D) does not exist because left hand limit is not equal to right hand limit, Lim
j j m xtan2x - 2xtanx (1 - cos2x)2 (A) 2 ^£4
x-»0
v/
0.6 JQ. 7
(C)I
(B)-2
[JEE2000, Screening] (C)e-
sin(7tcos2x) —2 equals
(A)-71
^
[JEE'99, 2 (out of200)]
1S :
rx - 3 F o r x e R , Lim vx + 2j (B)e-' (A)e Lim
[ IIT '98, 2 ]
(D)e5 [JEE2001, Screening]
(Of
(B) %
(D)l
^tanx _ gSinx
Evaluate Lim , a > 0. x->o tanx-sinx ^cos x lVcos X The integer n for which Lim & n
[REE 2001,3 out of 100] ^
is afinitenon-zero number is
x
(A)l , 0f8
If Lim x-»0
(B)2 sin nx
(
(A)i n
(C)3
)[(a-n)nx~tanx]
(D)4 [JEE 2002 (screening), 3 ]
( n >0) then the value of'a'is equal to
= 0
(C)
2
(B) n +1
n 2 +1 n
(D) None [JEE 2003 (screening)]
1
Find the value of Lhn — (n + l)cos f - 1 - n 7t ^<§BansalClasses
[JEE'2004, 2 out of 60]
Limits, continuity & Differentiability of Functions
[5]
ikensuchtl ofAQ wher
KEY CONCEPTS
7
il
1.
A function f(x) is said to be continuous at x = c, if Limit f(x) = f(c). Symbolically
2.
fis continuous at x - c if Limit f(c-h)=Limit f(c+h) = f(c). h—>0 * h-»0 i.e. LHL at x = c = RHL at x = c equals Value of ' f at x = c. It should be noted that continuity of a function at x=a is meaningful only if the function is defined in the immediate neighbourhood of x=a, not necessarily atx=a. Reasons of discontinuity:
(i)
Limit f(x) does not exist
x—>c
;
x->c
x-»c
5,2]
(CONTINUITY)
THINGS T O REMEMBER :
2\~
i.e. Limit f(x) ^ Limit f(x)
,1]
v
_ _ ,
(ii) (iii)
x->c
+
1
f(x) is not defined at x= c Limit f(x)*f(c)
0
x-»c
rf
3
4
Geometrically, the graph ofthe function will exhibit a break at x= c. The graph as shown is discontinuous atx = 1, 2 and3. 3. Types of Discontinuities: Type -1; (Removable type of discontinuities) In case Limit f(x) exists but is not equal to f(c) then the functionis said to have a removable discontinuity
: of200) ]
or discontinuity of the first kind. In this case we can redefine the function such that Limit f(x) = f(c) & X-K;
make it continuous at x= c. Removable type of discontinuity can be further classified as : :ening]
(a)
M I S S I N G P O I N T DISCONTINUITY
: Where Limit f(x) exists finitely but f(a) is not defined. x-»a
n xlf9 x ^ e.g. f(x) = -—Ji—_—L has a missing point discontinuity at x = 1, and (1-x)
ening]
sinx fix)= has a missing point x
discontinuity at x = 0 (b)
ISOLATED POINT DISCONTINUITY
: Where Limit f(x) exists & f(a) also exists but; Limit * f(a). x->a
x—
x 2 -.16 -g- f ( ) — , x 4 & f (4) = 9 has an isolated point discontinuity at x = 4.
e
it of 100]
x =
0 if x e l Similarly f(x) = [x] + [-x] = has an isolated point discontinuity at all x e I. L - l if x e I ning), 3]
Type-2: (Non - Removable type of discontinuities) In case Limit f(x) does not exist then it is not possible to make the function continuous by redefining it. x—>c
ning)] it of 60]
(a)
Such discontinuities are known as non - removable discontinuity or discontinuity of the 2nd kind. Non-removable type of discontinuity can be further classified as: , 1 1 Finite discontinuity e.g. f(x) = x - [x] at all integral x; f(x) =tan — at x = 0 and f(x) = — at x = 0 ( note that f(0+) = 0 ; f(0") = 1)
^<§BansalClasses
Limits, continuity & Differentiability of Functions
1+2*
[5]
rn
(b)
(c)
7t
tanx Infinite discontinuity e.g. f(x) = or g(x) = at x —- and f(x) 2 at x = 4; f(x) = 2 (x-4) x-4 atx = 0.
COSX CL ;
x
d)
e) Oscillatory discontinuity e.g. f(x) = sin J- atx=0. x In all these cases the value off(a) of the function at x= a (point of discontinuity) may or may not exist but Limit does not exist. x-»a
y?
Note: From the adjacent graph note that - fis continuous at x = - 1 - f has isolated discontinuity at x = 1 - f has missing point discontinuity at x = 2 - f has non removable (finite type) discontinuity at the origin.
a) b) i) x 1 2 Nature of discontinuity
«)
In case of dis-continuity of the second kind the non-negative difference between the value ofthe RHL at X=c & LHL at X = c is called THE JUMP O F DISCONTINUITY. A function having afinitenumber of jumps
in a given interval I is called a PIECE WISE interval. 5.
6.
CONTINUOUS or SECTIONALLY CONTINUOUS
All Polynomials, Trigonometrical functions, exponential & Logarithmic functions are continuous in their ») domains. I. If f & g are two functions that are continuous at x= c then the functions defined by: FJ (x) = f(x) ± g(x) ; F2(X) = K f(x), K any real number ; F3(x) = f(x).g(x) are also continuous at x= c. Further, if g (c) is not zero, then F4(x) =
7.
function in this i)
f(x)
is also continuous at x=c.
The intermediate value theorem: Supposefix)is continuous on an interval I, and a and b are any two points ofI. Then ify0 is a number between f(a) and f(b), their exists a number c between a and b such that f(c)=y0The function f, being continuous on [a,b) takes on every value between f(a) and fib)
NOTE VERY CAREFULLY THAT :
(a)
If f(x) is continuous & g(x) is discontinuous at x = a then the product function
(b)
Iff(x) and g(x) both are discontinuous at x=a then the product function <|)(x)=f(x)- g(x) is not necessarily be discontinuous atx= a. e.g. f(x) = -g(x) =
^<§BansalClasses
1 -1
x> 0 x< 0 Limits, continuity & Differentiability of Functions [5]
f(x) =
cosx c)
Point functions are to be treated as discontinuous, eg. f(x) = ^J1 - x + -^x-1 is not continuous at x = 1. il A Continuous function whose domain is closed must have a range also in closed interval.
<0
If fis continuous at x = c & g is continuous at x=f(c) then the composite g[f(x)] is continuous at x = c. eg.fix)=
not exist but
xsinx xsinx will also & g(x) = I x I are continuous at x = 0 , hence the composite (gof) (x) = 2 2 x +2 x +2
x = 0.
be continuous at
CONTINUITY IN A N INTERVAL :
a)
A function fis said to be continuous in (a, b) if fis continuous at each & every point e(a, b).
b) i)
A function fis said to be continuous in a closed interval [a, b] if: f is continuous in the open interval (a, b) &
:«)
f is right continuous at' a' i. e.
f(x)=f(a) = afinitequantity.
fis left continuous at 'b' i.e.
f(x) = f(b) = afinitequantity.
"0 ftheRHLat ber of jumps iction in this,.. J) IUOUS in their!")
Note that a function fwhich is continuous in[a, b] possesses the followiiig properties: If f(a) & f(b) possess opposite signs, then there exists at least one solution ofthe equation f(x) = 0 in the open interval (a, b). If K is any real number between f(a) & f(b), then there exists at least one solution of the equation f(x) = K in the open inetrval (a, b). SINGLE POINT CONTINUITY:
Dntinuous at
Functions which are continuous only at one point are said to exhibit single point continuity e.g. f(x) =f x i f x e Q and g( x ) =[* * . are both continuous only atx = 0. L-xifxgQ LOirxgQ
EXERCISE-I /ncosx 5K.
Let f(x):
VT^-i i
sin4x _
if
j
if x <0
/n(l + tan2x)
Ab) id f(b) :)• g(x) is not Q^r. )t necessarily
x>0
Is it possible to define f(0) to make the function continuous at x=0. If yes what is the value of f(0), if not then indicate the nature of discontinuity. 3
2
Suppose that f(x) = x -3x -4x+ 12 and h(x)
:
f(x) _ —— , x * 3 x-3 then K , x= 3
find all zeros of /(x) JJo)findthe value of K that makes h continuous at x=3 ^j(e)using the value of K found in (b), determine whether h is an even function.
^ <§Bansal Classes
Limits, continuity & Differentiability of Functions
[5]
Q-3.
x2
X2
2
Let yn(x) = x + 7 • ,, 2x2 J 1 + x2 (l+x?)
.+ -,
1 (1+x ) -
2 11 1
and y(x) = Limit y n (x) Discuss the continuity of yjx) (n = 1,2,3 n) and y(x) at x = 0 Draw the graph ofthe function f(x)=x-1 x - x 2 | , -1 < x < 1 & discuss the continuity or discontinuity fin the interval - I < x < 1. i-sinTix
X< 2
l+cos2nx
Letf(x) =
x = j . Determine the value ofp, ifpossible, so that the function is P y/2x-l 2X^TV2 ' X > 2
continuous atx= 1/2. Given the function g (x) = V6-2x and h (x) = 2x2 - 3x + a. Then
6. t
g(x), J b ) If f (x) = - h(x),
M evaluate h (g(2))
x
1+x , 0
t^jrr.
Let f(x) =
^jX8.
Let [x] denote the greatest integer function & f(x) be defined in a neighbourhood of 2 by [x+l]
(exp {(x+2)ln4}) 4 -16 ,x<2 4X-16 f(x) = . l-cos(x-2) ,x>2 (x-2)tan(x-2) Find the values ofA& f(2) in order that f(x) may be continuous at x=2. ktnSx Un
The function f(x) =
if
b+2
if
(ijtanxl^ (l+|cosx|)m if
0
x=f
'
§
Determine the values of'a' & 'b', if fis continuous at x=tc/2. J^IO
Let / ( x )
:
2 1 r x sin—, i f x , ^ 0
0 if x = 0 Use squeeze play theorem to prove that/is continuous at x=0. Q.ll
Let f(x)=x + 2, - 4
Q 12. Let f(x) =
* j ' x>o
^<§BansalClasses
; g
^
=
{(x + ij,/2 ! x > 0 • Discuss the continuity of g(f(x)).
Limits, continuity & Differentiability of Functions
[5]
3cos x n Determine a & b so that f is continuous at x = —. f(x) = a At b(l-sinx) (it-2x)J
if
X
if
X=*§
if
X>f
*
sin(a+l)x+sinx
scontinuity o
4
Determine the values of a, b & c for which the function f (x) =
2
(x+bx^'-x" bx
for
x<()
for x=0 for x>0
is continuous at x = 0.
•
&&
sin 3x+Asin 2x+Bsin x (x* 0) is cont. at x = 0. Find A& B. Also find f(0). Iff(x)=Do not use series expansion or L' Hospital's rule. -Lx-l for 0
of
graph of the function for x e [0,6], Also indicate the nature of discontinuities if any.
yr j^8
If f(x) = x + {-x} + [x], where [x] is the integral part & {x} is the fractional part of x. Discuss the continuity of fin [ - 2 , 2 ]. Find the locus of (a, b) for which the function f (x) = is continuous at x = 1 but discontinuous at x=2.
ax - b for x<1 3x for 1 < x < 2 bx - a for x >2
Prove that the inverse of the discontinuous function y=(1 + x2) sgn x is a continuous function. Ol9
sin2(7t-2x) x n f (x) + h(x) +1 Let g (x) = Lim _ , x ^ 1 and 5g(l) ' \ be a continuous function W = Lim —f 2x + 3x + 3 /n(sec(7i-2x)j at x = 1, find the value of 4 g ( l ) + 2 f ( l ) - h ( l ) . Assume that f (x) and h (x) are continuous at x = 1. If g: [a, b] onto [a, b] is continous show that there is some c e [a, b] such that g (c) = c. 2 + cosx is not defined at x = 0. How should the function be defined at The function f(x) = 3 v x sinx x 4 ^
Q.22 x = 0 to make it continuous atx=0. Use of expansion of trigonometric functions and L' Hospital's rule is not allowed. „sinx „tanx for x > 0 ^ 3 f(x) : a -a tanx-sin x /n(l + x + x,2 ) + / n ( l - x + x ) for x < 0, if fis continuous at x = 0, find 'a' sec x - c o s x r xN cot (x - a) for x a, a ^ 0, a > 0. If g is continuous at x = a then show that now if g (x) = In 2 - v g(e 1 ) = - e .
^Bansal Classes 1101
Limits, continuity & Differentiability of Functions
111]
Q.24 (a)
Let f(x+y) = f(x) + f(y) for all x, y & ifthe function f(x) is continuous at x = 0, then show that f(x) continuous at all x.
(b)
If f(x- y)=f(x)- f(y) for all x, y and f(x) is continuous at x = 1. Prove that f(x) is continuous for all x exce^ atx=0. Given f ( l ) * 0 . J) Givenf(x)=Z i t a n ( J J s e c ^ J ; r , n e N
^.25
r-i
tn |( f ( x ) + t a n | - ) "I (f(x) + t a n ^ -
g(x)=
1 +(f (x) + tan 71 = k for x = —
W
[sin(tanf)]
n
and the domain of g (x) is (0, tc/2).
where [ ] denotes the greatest integer function. f Find the value of k, if possible, so that g (x) is continuous at x=nJ4. Also state the points of discontinui! of g (x) in (0, n/4), if any. , 0 f(x) Let f (x) = x 3 - x2 - 3x - 1 and h (x) = where h is a function such tha,t (a) it is continuous every where except when x = - 1 ,
£
(b) Lim h(x) = oo and (c) Lim h(x) = -p> X—>00
X->-L
2
Find Lim (3h(x) + f(x) - 2g(x>) x-»0
:27
Let f be continuous on the interval [0,1] to R such that f (0) = f (1). Prove that there exists a point c i' such that f(c) = f c + v 2y
Q.28
1 - ax + xa x ^na for x<0 axx2 where a >0. Consider the function g(x) = x x 2 a - x^n2 - x^na - 1 for x>0
9
Without using, L 'Hospital's rule or power series,findthe value of'a' & 'g(O)' so that the function g(x) i continuous at x = 0. (j ^-sm-l(l-{x}2)).sm-l(\-{x}) V2({x}-{x}3)
Let f(x) n
for x * 0
where {x} is the fractional part of
for x = 0
2
Consider another function g(x); such that g(x) = f(x) for x > 0 =2V2 f(x) for x < 0 Discuss the continuity of the functions f(x) & g(x) at x=0. Discuss the continuity of f in [0,2] where f(x) = |4x-5|[x] for x > l . w here [x] is the greates' [cos7tx] for x < l integer not greater than x. Also draw the graph.
[14] Bansal Classes
Limits, continuity & Differentiability of Functions
[15]
,,
.
EXERCISE-II
t+r iowthatf(x)i
^
(OBJECTIVE QUESTIONS)
forallxexce,* 1
State whether True or False.
a).
If f(x) =
for
x*
, then the value which can be given to f(x) at x = j so that the function
becomes continuous every where in (0,n/2) is 1/4. b)
The function f, defined by f(x) =1 — ] — is continuous for real x. + 2"
01 v cK W fix) ' = L' n-)-«^ 1+nsin — 7tx is continuous at x = 1.
2x+l if The function f(x) = -x—1 if x+2 if
yy :>f discontinue
-3
The function defined by f(x)= [T^TF for x * 0 & f(0) = 1 is continuous at x=0. ^im h(x) =
1
/
The function f(x)=2"
21/(1-x)
i f x ^ 1 & f(l) = 1 is not continuous at x = 1.
The functionf(x) = 2xA/(x3 -1) + sVx V(l-x4) + 7x2 V(x-l) + 3x+2 is continuous at x = 1.
'£
There exists a continuous function f: [0,1] [0,10], but there exists no continuous function g: [0, l]-^4(0,10).
istsapointci Q 2.
Fill in the blanks
ftf
Given f(x)=L~ c o s ( c x ) , x ^ 0 & f ( 0 ) = If fis continuous at x = 0, then the value of c is xsinx 2
(h>C The function f(x)=-7-7 has non removable discontinuity at x= / ln|xj = respectively. function g(x) i
& removable discontinuity at x
f i
If f(x) is continuous in [0,1 ] & f(x) = 1 for all rational numbers in [0,1 ] then f
fi.
x + aV2sinx , 0
mal part of yijpf
„. is continuous for 0 < x < n are (e}r 7
y
^
&
If ffx)= ~^ cosx —1 is continuous at x = 4 thenf v '
cot x ~ l
it
4
is the greatesi
[14] Bansal Classes
Limits, continuity & Differentiability of Functions
[15]
Q3,
Indicate the correct alternative(s): The function defined as f(x) = Limit
C0S7tX
'f"51"^ l +x -X 2
1}
rff
2n+1
(A) is discontinuous a t x = 1 becausef(l + ) * f ( l - ) (B) is discontinuous at x = 1 because f(l) is not defined (C) is discontinuous at x = 1 because f(l + ) = f(l")^f(l) (D) is continuous at x = 1 J&f
Let T be a continuous function on R. If f (l/4 n ) =(sinen)e"n2 + - J - J then f(0) is : (A) not unique (C) data sufficient tofindf(0)
(B) 1 (D) data insufficient tofindf(0)
Indicate all correct alternatives if, F(x) = — - 1 , then on the interval [0, TC]
x
(A) tan (f(x)) &
are both continuous
(C) tan (f(x))&f _ 1 (x) are both continuous
Y
(B) tan (f (x)) & —-y are both discontinuous (D) tan (f(x)) is continuous but —j— isnot t (xj
' f i s a continuous function on the real line. Given that X2 + (f(x) - 2) x - V3 • f(x) + 2 V3 - 3 = 0. Then the value off( S ) (A) can not be determined
(B) is 2 ( i - V s )
(C) is zero
(D) is
3)
2(V3 - 2) v '
^J&f
If f(x) = sgn (cos 2 x - 2 sin x + 3), where sgn ( ) is the signum function, then f(x) (A) is continuous over its domain (B) has a missing point discontinuity (C) has isolated point discontinuity (D) has irremovable discontinuity.
^/f)
Let g(x) = tan-1 |x| - cot -1 |x|, f(x) = J-J JJJ {x}, h(x)=jg(f(x))| where (x) denotesfractionalpart and [x] denotes the integral part then which of the following holds good? (A) h is continuous at x = 0 (B) h is discontinuous at x = 0 (C)h(0~)=!
J g T
(D)h(0 + ) = - |
x n - sinx 11 Consider f(x) = Limit T for x > 0, x * 1
^
m=o then (A) f is continuous at x = 1 (B) f has a finite discontinuity atx = 1 (C) f has an infinite or oscillatory discontinuity at x = 1. (D) fhas a removable type of discontinuity at x= 1.
[14] Bansal Classes
Limits, continuity & Differentiability of Functions
[15]
4
Given f ( x ) = ^ - ; e x - 1 sgn (sinx) =0
for x * 0
for x = 0
where {x} is thefractionalpart function; [x] is the step up function and sgn(x) is the signum function ofx then, f(x) (A) is continuous at x = 0 (B) is discontinuous at x = 0 (C) has a removable discontinuity at x = 0 (D) has an irremovable discontinuity a t x - 0 x[x]2 log(1+x) 2 for - 1 < x < 0 i)X
Consider f(x)
/n(ex2 + 2 x tanVx
for 0 < x < 1
where [ * ] & {*} are the greatest integer function & fractional part function respectively, then (A) f(0) = /n2=> fis continuous at x = 0 (B) f(0) = 2 => fis continuous at x = 0 2 (C) f(0) = e => f is continuous at x = 0 (D) f has an irremovable discontinuity at x = 0 Consider
f(x) =
Vl + x - V l - x — x*0 {x}
K - -
g(x) = cos2x - -^f(g(x)) h(x)-- 1 _ f(x)
)nal part and
for x<0 forx =0 forx >0
then, which ofthe following holds good, where {x} denotesfractionalpart function. (A) 'h' is continuous at x = 0 (C) f(g(x)) is an even function x^ jX)
2x — 1 The function f^x) = [x], c o s — — x , where [•] denotes the greatest integer function, is discontinuous at (A) all x (C)nox
[14]
(B) 'h' is discontinuous at x = 0 (D) f(x) is an even function
Bansal Classes
(B) all integer points (D) x which is not an integer
Limits, continuity & Differentiability of Functions
[15]
EXERCISE-III Q.l
rX2 JT3
%
Letf(x) = [x] sin-—— , where [•] denotes the greatest integer function. The domain of fis &the 1. [x + l] [ JEE '96, 2 ] B points of discontinuity of f in the domain are (i Let f(x) be a continuous function defined for l
x = 0. Discuss the continuity ofthe function [e 1 / ( x - 1 ) -2 l/(x-l) +2 '
X^l X = 1
atx= 1.
fe Bansal Classes
[ REE 2001 (Mains), 3 out of 100 ]
Limits, continuity & Differentiability of Functions
[23]
KEY CONCEPTS
(DIFFERENTIABILITY)
THINGS T o R E M E M B E R : IS
&the 1.
Right hand & Left hand Derivatives;
T imit f(a + h ) - f ( a ) [ JEE '96, 2 ] By definition: f ' ( a ) = h " 0 — — u (i) and f (2) =10, [JEE'97,2]
f(a+h)
The right hand derivative of f ' at x = a denoted by f'(a + ) is defined by: , , , a _Limit f(a + h)-f(a) v ; ^ ' t
1
discontinuous (ii)
provided the limit exists & is finite. The left hand derivative: of f at x = a denoted by f'(a + ) is defined by: , , , a . L i m i t+ f ( a - h ) - f ( a ) v ; - h—>o _h Provided the limit exists & is finite. We also write f'(a + ) = f' + (a) & f'(a") = f'_(a). * This geomtrically means that a unique tangent withfiniteslope can be drawn at x = a as shown in the figure. *
ut of200)]
1
continuous at
[REE'99, 6]
lflt exist
(iii)
Derivability & Continuity: (a) If f '(a) exists then f(x) is derivable at x= a => f(x) is continuous at x = a. (b)
If a function f is derivable at x then f is continuous at x. For:r(x)=
Umitf(x
+
h)-f(x)
Also f ( x + h ) - f ( x ) = f ( x out of 100]
Therefore: S ^ x Therefore
^
+ h)
" f ( x ) - , h [ h * 0] h
+hHW] [f (x + h )-f (x)] = 0 =>
f ( x + h } f(x) *=f'(x).o=o h -
f (x+h) = f(x) => fis continuous at x.
Note : If f(x) is derivable for every point ofits domain of definition, then it is continuous in that domain. The Converse ofthe above result is not true: " IF f IS CONTINUOUS AT x , THEN f IS DERIVABLE AT x " IS NOT TRUE. e.g. the functions f(x) = | x | & g(x) = x s i n i x x = 0 but not derivable at x = 0.
& g(0) = 0 are continuous at
N O T E CAREFULLY :
(a)
Let f '+(a) = p & f ' (a) = q where p & q are finite then: (i) p = q => fis derivable at x = a => fis continuous at x = a. (ii) p * q => f is not derivable at x = a. It is very important to note that f may be still continuous at x = a. In short, for a function f: Differentiability => Continuity ; Continuity =£> derivability; Non derivibality =£• discontinuous ; But discontinuity => Non derivability
[14] Bansal Classes
Limits, continuity & Differentiability of Functions
[15]
,vf (b)
If a function fis not differentiable but is continuous at x = a it geometrically implies a sharp corner; x= a.
3.
DERIVABILITY OVER A N INTERVAL:
(i) (ii)
f (x) is said to be derivable over an interval ifit is derivable at each & every point ofthe interval f(x) is sai to be derivable over the closed interval [a, b] if : for the points a and b, f'(a+) & f'(b - ) exist & for any point c such that a < c < b, f '(c+) & f (c - ) exist & are equal.
*
NOTE
:
1.
If f(x) & g(x) are derivable at x = a then the functions f(x) + g(x), f(x) - g(x) , f(x).g(; will also* be derivable at x = a & if g (a) * 0 then the function f(x)/g(x) will also be derivable at x = a. If f(x) is differentiable at x = a & g(x) is not differentiable at x = a, then the product function F(x)=f(: g(x) can still be differentiable at x = a e.g. f(x) = x & g(x) = | x |. If f(x) & g(x) both are not differentiable at x = a then the product function; F(x) = f(x)- g(x) can still be differentiable at x = a e.g. f(x) = | x | & g(x) = | x |. If f(x) & g(x) both are non-deri. at x = a then the sum function F(x) = f(x)+g(x) may be a differential function, e.g. f(x) = | x | & g(x)=-1 x | • If f(x) is derivable at x = a f'(x) is continuous at x=a.
2. 3. 4. 5.
e.g.f(x) =
x 2 sin-L if x ^ 0 0
if x = 0
A surprising result: Suppose that the function f (x) and g (x) defined in the interval (xL, x^ contain the point Xq, and if fis differentiable at x=Xq with f (Xq) = 0 together with g is continuous asx = xQtl the function F (x) = f (x) • g (x) is differentiable at x=x 0 e.g. F (x) = sinx • x2/3 is differentiable atx=0.
EXERCISE-I V^
1
^OfiQ. 3
Discuss the continuity & differentiability ofthe function f(x)=sinx+sin | x | , x s R. Draw a rough ske of the graph of f(x). Examine the continuity and differentiability offix)= | x | + | x - l | + i x—2 | x e R . Also draw the graph of f(x). } Given a function/ (x) defined for all real x, and is such that f ix + h) - f (x) < 6h2 for all real h and x. Show that/(x) is constant. 1
s^jzrfi
Aftinctionfis defined as follows: f(x) = 1+sinx
for
—o6
for
0
2+(x-f) 2 for
f
Discuss the continuity & differentiability at x = 0 & x = %/2. 0T5 6
^ . 7
Examine the origin for continuity & derrivability in the case of the function f defined by f(x) = xtan _1 (l/x) andf(0) = 0. Let f (0) = 0 and f' (0) = 1. For a positive integer k, show that c \ 1 1 1 A Lim — f(x) + f (-] + f = 11 + - + - + +— o 1. x-»o x V 2 3 k y -T-L+i] x Let f(x) = xe ^ ; x ^ 0, f(0) = 0, test the continuity & differentiability at x = 0
[14] Bansal Classes
Limits, continuity & Differentiability of Functions
[15]
a sharp corner a (ft iterval f(x) is sai(^9 jC
g/ip
If f(x)= I x -11. ([x] - [-x]), thenfindf'(1 + ) & f'(!") where [x] denotes greatest integer function. Iff(x) =
ax2 - b if Ixl<1 -R
if
N-1
is derivable at x = 1. Find the values of a & b.
Let f(x) be defined in the interval [-2,2] such that f(x):
-1 x-1
, -2
&
g(x) = f( IX |) + | f(x) |. Test the differentiability of g(x) in ( - 2,2). g(x) , f(x).g(x 2[x] ivableatx=a. 1 where sgn (.) denotes the signum fimction& [.] denotes the greatest sgn r Givenfix)= cos ctionF(x) = f(x v 9^ V, V3x - \x]J) integer function. Discuss the continuity & differentiability of f (x) at x=± 1. J&2 e a differential
Examine for continuity & differentiability the points x = 1 & x = 2, the function f defined by f(x) = f(x) =
~x[x] , 0
, x * 0 & f(0) = - 1 where [x] denotes greatest integer less than or equal to x.
m
Test the differentiability off(x) at x = 0. x^) containin |2x-3|[x] for x>l >us as x=Xq t h e ^ ^ 4 Discuss the continuity & the derivability in [0,2] of f(x): sin-jfor x
w a rough sketc
X 16
The function ax(x-l) + b whenx
whenl
px + qx+2 CO ® (iii)
whenx>3
Find the values of the constants a, b, p, q so that f(x) is continuous for all x f ' ( l ) does not exist f'(x) is continuous at x = 3 l/x _
-l/x
, x * 0 (a > 0) and f(0) = 0 for continuity and existence of l/x , J f f t Examine the function, f(x) =x. a— + a—l/x the derivative at the origin. J^A 8 Discuss the continuity on 0 < x < 1 & differentiability at x = 0 for the function. f(x) = x.sin —.sin — — r wherex*0, x * l / r 7 t & f(0) = f ( l / r a ) = 0, x x.sin L= r 1, 2, 3,
fl8J
^Bansal Classes
Limits, continuity & Differentiability of Functions
-
• [19]
1-x
f(x)
,
(0
x+2 , (l
y = f [f(x)] for 0 < x < 4. 7t
Q.20
if x * 0
cos2x
Consider thefianction,/(x) =
if x = 0
Show that/' (0) exists andfindits value yjo) (c)
Showthat/'
does not exist.
For what values of x, / ' (x) fails to exist.
^ J Z f l 1 Discuss the continuity & the derivability of 'f where f (x) = degree of (ux'+ u2 + 2u - 3) at x = V2. Let/(x) be a function defined on (-a, a) with a > 0. Assume that/(x) is continuous at x = 0 am Lim x->0
J f(x)~ w
X
f(kx) —- = a , where k e (0,1) then compute f' (0+) and f ' (0"), and comment upon tb
differentiability of / a t x= 0. J^23
^"24
Q.25
t
A function f : R -> R satisfies the equation f(x + y) = f(x). f(y) for all x, y in R f(x) * o for any x in R. Let the function be differentiable at x = 0 & f (0) = 2. Show that f (x) = 2f(x) 1 all x in R. Hence determine f(x). Let f(x) be a real valued function not identically zero satisfies the equation, f(x + yn) - f(x) + (f(y))n for all real x & v and f ' (0) > 0 where n (> 1) is an odd natural number. F f(10). Afunctionf: R->RwhereRisasetofrealnumberssatisfiestheequation f(ii±I) = f(x)+f^)+f(0). for a l l x
y in R If t he
function is differentiable at x = 0 then show that
differentiable for all x in R.
EXERCISE-II Fill in the blanks: ^
f
Limit f (3+h2) - f((3-h ) If f(x) is derivable.at x = 3 & f'(3) = 2, then h->o 2h2 Iff(x) = | sinx | & g(x)=x 3 thenf[g(x)] is
&
at x= 0. (State continuity and deriv;
Let f(x) be a function satisfying the condition f(- x) = f(x) for all real x. If f'(0) exists, then its1
^QrzT
For the function f(x) = 1 + el/x '
, the derivative from the right, f (0+) =
& the de
x=0
from the left, f(0~)=_ QCS ^
The number of points at which the function f(x) =* max. {a-x, a+x, b}, be differentiable is .
CO
limits, continuity & Differentiability of Functions
0
Select the correct alternative : (only one is correct) Let f(x) sinx for x 0 & f(0) = 1 then, (A) f(x) is conti. & diff. at x = 0 (C) f(x) is discont. & not diff. at x = O
^
5.7
Givenfix)=
loga(a|[x] + [ - x f
(B) f(x) is conti. & not diff. at x = 0 (D)none
,1: W J
for xl * 0 ; a > 1
3 + aA
where [ ] represents the integral
for x = 0
3) at x=V2. . _ , •us at x — un and (^8"" nment upon the
part function, then: (A) fis continuous but not differentiable at x = 0 (B) fis cont. & diff. at x = 0 (C) the differentiability o f f at x = 0 depends on the value of a (D) f is cont. & diff. at x = 0 and for a = e only, For what triplets of real numbers (a, b, c) with a * 0 the function "x
f(x) =ax2 + bx + c
x
1 x, y in R & :f (x) = 2f(x)for
(A) {(a, l-2a, a) I a s R, a * 0 } (C) {(a, b, c) I a, b, c s R , a + b + c= l }
(B) {(a, l-2a, c) I a, c s R , a ^O } (D) {(a, l-2a,0)| a s R , a * 0 }
al number. Find
A function f defined as f(x)=x[x] for - 1 < x < 3 where [x] defines the greatest integer < x is: (A) conti. at all points in the domain of f but non-derivable at afinitenumber of points (B) discontinuous at all points & hence non-derivable at all points in the domain of f (C) discont. at afinitenumber ofpoints but not derivable at all points in the domain of f (D) discont. & also non-derivable at afinitenumber ofpoints of f.
(^•K) [x] denotes the greatest integer less than or equal to x. Iffix)= [x] [ sin Ttx] in (-1,1) then f(x) is: n show that it is (A) cont. at x = 0 (B) cont. in(-l, 0) (C) differentiable in (-1,1) (D) none A functionfix)= x [ 1 + (1 /3) sin (lnx2)], x * 0.[ ] = integral part f(0) = 0. Then the function: (A) is cont. at x = 0 (B) is monotonic v (C) is derivable at x = 0 (D) can not be defined for x < -1
and derivability)
-x The function f(x) is defined as follows f(x) =xx3-x+l
then its value is
(A) derivable & cont. at x = 0 (C) neither derivable nor cont. at x = 1
.2
if x<0 if 0
(B) derivable at x = 1 but not cont. at x = 1 (D) not derivable at x = 0 but cont at x = 1
& the derivative
x + {x} + xsin{x} for x * 0 where (x) denotes thefractionalpart function, then: 0 for x = 0 (A)'f is cont. & diff. atx = 0 (B) 'f is cont. but not diff. at x = 0 (D) none ofthese (C) 'f is cont. & diff. at x = 2
0 < a < b cannot-Q^
The set of all points where the function f(x) = y ^ y is differentiable is:
£T3
Iffix)=
(A) (-00,00)
[14]BansalClasses
(B) [ 0, 00 )
(C)(-00, 0 ) ^ ( 0 , 00) (D) (0, 00) (E) none
Limits, continuity & Differentiability of Functions
[15]
Select the correct alternative: (More than one are correct) If f(x) = 12x+l | + | X - 21 then f(x) is: (A) cont. at all the points (C) discontinuous at x = - 1 / 2 & x = 2
(B) conti. at x = 2 but not differentiable < (D) not derivable at x = - 1 / 2 & x = 2
*
f(x) = I [x]x | i n - 1 < x < 2 , where [x] is greatest integer < x then f(x) is: (A) cont. atx = 0 (B) discont. x = 0 (C)notdiff. a t x = 2 (D)diff. atx = 2 1.7 f(x) = 1 + x. [cosx] in 0 < x < %/2, where [ ] denotes greatest integer function then, * (A) It is continuous in 0 < x < 7t/2 (B) It is differentiable in 0 < x < TC/2 (C) Its maximum value is 2 (D) It is not differentiable in 0 < x< %/2 Q. 18 f(x) = (Sin-'x)2 Cos(1/x)ifx^0;f(0) = 0,f(x)is: (A) cont. no where in - 1 < x < 1 (B) cont. every where in - 1 < x < 1 (C) differentiable no where in - 1 < x < 1 (D) differentiable everywhere in - 1 < x < JZ.19
jrfS)
^^21
7C % f(x) = | x | + |sinx| i n ( ~ . y j . l t i s : (A) Conti. no where (C) Differentiable no where ,
(B) Conti. every where (D) Differentiable everywhere except at x :
If f(x) = 3(2X+3)2/3 + 2x+3 then, (A) f(x) is cont. but not diff. at x = -3/2 (C) f(x) is cont. at x = 0
(B) f(x) is diff. at x = 0 (D) f(x) is diff. but not cont. at x = - 3/2
Iff(x) = 2+ | sin^xl ,itis: (A) continuous no where (C) differentiable no where in its domain
(B) continuous everywhere in its domain (D) Not differentiable at x = 0
^#22
If f(x) = x2 sin (1/x) ,x*0andf(0) = 0 then, (A) f(x)is continuous a t x = 0 (B)f(x) is derivable atx = 0 (C) f (x) is continuous at x = 0 (D) f'(x) is not derivable at x = 0
yj^h
A function which is continuous & not differentiable at x = 0 is: (A) f(x) = x for x < 0 & f(x) = x2 for x > 0 (B) g(x) = x for x < 0 & g(x) = 2x for x > 0 (C)h(x) = x | x | x s R (D)K(x) = 1 + 1 x | , x e R
9^4
IfSin"1x+ | y | = 2y then y as a function of x is: (A) defined for -1 < x < 1 (B) continuous at x = 0 (C) differentiable for all x
*
Let f(x) = Cosx &H(x)=
(D) such that—= &
Min[f(t)/0
(A) H (x) is cont. & deri. in [0,3 ] (C) H(x) is neither cont. nor deri. at x = %!2
[14] Bansal Classes
*— for-1 < x < 0 3Vl-x*
^
(B) H(x) is cont. but not deri. at x=TC/2 (D) Max. value of H(x) in [0,3 ] is 1
Limits, continuity & Differentiability of Functions [15]
EXERCISE-III ^J^st
Determine the values of x for which the following function fails to be continuous or differentiable fW
1-x , x <1 (1-x) (2-x) , l < x < 2. Justify your answer. 3- x , x> 2
[JEE'97,5]
Let h(x)=min {x, x2 }, for every real number ofx. Then: (A) his cont. for all x (B) his diff. for all x (C) h' (x) = 1, for all x > 1 (D) h is not diff. at two values of x.
v
yfi
Discuss the continuity & differentiability of the function f(x)
:
2 + ,/(l-x2) _2e(1"x)2
[JEE'98,2 ]
|x| < 1 ,\x\>l [REE'98, 6]
The function f(x) = (x2 - 1) | x2 - 3x + 21 + cos (| x |) is NOT differentiable at: (A)-l (B)0 (C)l (D)2 [JEE'99, 2 (out of200)] Q.5
Let f: R —> R be any function. Define g: R (A) onto if fis onto (C) continuous iffis continuous
R by g (x) = |f(x)| for all x. Then g is (B) one one if f is one one (D) differentiable iffis differentiable. [JEE 2000, Screening, 1 out of 3 5 ]
^C^T" Discuss the continuity and differentiability ofthe function, Ixl > 1
f(x) = 1 -ix! ' Q.7 vK
|x|
[REE,2000(3)]
< r |x|<1
[JEE 2001 (Screening)] Let f : R -» R be a function defined by , f (x) = max [ x , x ]. The set of all points where f (x) is NOT differentiable is: (A) { - 1 , 1 } (B) ( - 1 , 0 } (C) {0,1} (P) { - 1 , 0 , 1} 3
The left hand derivative of, f (x) = [ x ] sin (TT x) at x = k, k an integer is: (A) ( - l ) k (k - 1) 71 (B)(-l)*-i(k-l)7t (C)(-l)kk7t
( D ) (— 1 ) k ~ 1 k TT
Which ofthe following functions is differentiable at x = 0 ? (A) cos (! x | ) + | x | (B) cos ( | x j - x (C)sin(!xi) + |x| Q.8
( D ) s i n ( | x | ) - |x|
Let a e R. Prove that a function / : R R is differentiable at a if and only if there is a function g : R -> R which is continuous at a and satisfies /(x) - / ( a ) = #(x) (x - a) for all xeR. [JEE 2001, (mains) 5 out of 100]
fe Bansal Classes
Limits, continuity & Differentiability of Functions
[23]
Q. 9
The domain of the derivative ofthe function ' tan"1 x f(x) = ' (A)R-{0)
J^TO
J ^
if M > 1
(B)R-{1}
is
^
( C ) R - {-1}
Let/: R -> R be such that/(1) = 3 and/'(1) = 6. The Limit (A) 1
1
|(W-i)
if |x| < 1
f
. *• (B)e 1/2
Jx + a if x < 0 ^ | | x - l | if x > 0 =
(C)e 2
( D ) R - {-1,1} [JEE 2002 (Screening), 3] f St
\\
)
V /(1)
equals
(D)e 3 [JEE 2002 (Screening), 3]
Jx + 1 if x < 0 2 g ( ) | ( x - l ) + b if x > 0 x
^
=
Where a and b are non negative real numbers. Determine the composite function gof. If (gof) (x) is continuous for all real x, determine the values of a and b. Further, for these values of a and b, is gof differentiable at x=0? Justify your answer. [JEE 2002,5 out of 60] Q/i2
If a function f: [ -2a, 2a] -» R is an odd function such that f (x) = f (2a - x) for x e [a, 2a] and the left hand derivative atx= a is 0 thenfindthe left hand derivative at x = - a. [JEE 2003, Mains-2 out of 60]
^ ^
Q. 1 3 ^ T h e function given by y= | | x | - l | is differentiable for all real numbers except the points (A) {0,1,-1}
(B)±l
'
(C)l
(D)-l
Q8
'
fl) (b) If f (x) is a continuous and differentiable function and f
I = 0, V n > 1 and n e l , then
(A) f (x) = 0, x e (0, 1] (C) f ' (0) = 0 = f" (0), x e (0, 1]
(B) f (0) = 0, f ' (0) = 0 (D) f (0) = 0 and f 1 (0) need not to be zero [JEE 2005 (Screening), 3 + 3] (c) If |f(x 1 )-f(x 2 )|<(x 1 -x 2 ) 2 , forallXjjXj eR. Find the equation oftangent to the curvey = f(x) at the point (1,2), [JEE 2005 (Mains), 2]
Q 11 Q 14 Q 1 5
Q21,
Q 22, Q 26.
Q l.< Q6./r
[14] Bansal Classes
Limits, continuity & Differentiability of Functions [15]
ANSWER KEY EXERCISE-I 45 Q ^Y 2
Q1.3
Q 3
'
2
Q4
Q 7. (a) TC/2 if a > 0; 0 if a = 0 and -TC/2 if a < 0 Q 8.5050
1 1 2 Q9.a=-;r=-;S= Zf
i
Q 5
VF
Q 6
'
1 ~3
(b) f(x) = | x |
Q10./n2
J
2
P-q ' 2
Q11. does not exist
Q 12.2
£
Q 13. 1
Q 14. — 2
Q 1 5 . — Q 1 6 . — 16V2 tc
Q 17. a = 2 ; limit = 1
Q 18.
nTC
Q21. Domain, x s R, Range, x
Q 19. - - I n 4 e
52
ns I
Q 24. cos 2 a/nCosa + Sin2a /nSina Q27.(/na) n Q 28. - 2
Q 20. 2
Q 22. does not exist
Q 23. 9
Q 25. 8V2(ln3)2 Q29. 0,0
Q 26. - 3, - 3 , - 3 Q.30 4
EXERCISE-II Q 1. e s
Q 2. c = ln2
Q 3. e"'1
Q 4. e
Q 8. e1/2
Q 9. (a,.a2.a3....an)
Q 11. a = c = 1, b = 2
TC2a2+4 Q12.-—416a
8/1,z
Q.5
Q 7. e 1
Q 6. ^ ^
Q l O . f , 71 2 ' 2V2 Q 13.
Q 14.f(x)when | x | > 1 ;g(x)when | x | < 1 ;
- — 4
w henx=
2 3
1 & not defined whenx=-l
1
x
Q 15. 2
Q 16. - — TC
Q 17. a - b
Q 19. 1/2
x x sinx Q21.T(x)= :: - tan 2 -, sinx or t a n , S ( x ) =
l i 2X~2
Q 22. (a) 1 (b) ^2
Q 23. J L tanG
Q 24. a = e2
Q26. j
Q28. (a) 1/2, (b)b
2
Q 20. i
3 i ,linfit=-
s nx
Q 25. e*2
Q 29. (i) a = l , b = - 1 (ii) a = - l , b
EXERCISE-III Q l . e2
Q2.D
Q 3. C
Q 4. C Q 5 . B
Q6./na
Q7. C
Q8. C
Q.9
(!^Bansal Classes
2
1— TC
Limits, continuity & Differentiability of Functions
[25]
CONTINUITY EXERCISE-I
iiS
+
Ql. Q 2. Q 3. Q 4.
f(0 ) = - 2 ; f(0-) = 2 hence f(0) not possible to define (a)-2, 2, 3 (b)K = 5(c) even yn(x) is continuous at x = 0 for all n and y(§) is dicontinuous at x = 0 fis cont. in - 1 < x < 1 Q 5. P not possible.
Q 6. Q 7. Q 8.
(a) 4 - 3V2 + a, (b) a = 3 g(x) = 2+xfor 0 < x < 1, 2 - x f o r 1 < x < 2 , 4 - x f o r 2 < x < 3, g is discontinuous at x = 1 & x = 2 A = 1 ;f(2)= 1/2 Q9.a = 0;b = -1
Q. 11 Q 12. Q 13. Q 15.
f ( f (x) ) is continuous and domain of f ( f (x)) is [-4, ] gof is dis-cont. at x = 0, 1 & -1 a = 1/2, b = 4 Q14. a = -3/2, b * 0, c = 1/2 A = - 4 , B = 5, f(0) = 1 Q 16. discontinuous at x = 1,4 «& 5
Q 17. discontinuous at all integral values in [-2,2] Q 18. locus (a, b) —» x, y is y = x - 3 excluding the points where y = 3 intersects it. Q 20. 5
Q22.
—
60
txv (tan x) if
0
Q 25. k = 0; g(x) = 0
39 Q26. g (x) = 4 (x + 1) and limit = - — Q28. a = - L , g ( 0 ) =
( ^
Q 29. f(0+) = ^ ; f(Q-) = —'•= => f is discont. at x = 0; 2 4V2 + g(0 ) = g(O-) = g(0) = TC/2 =^>g is cont. at x = 0 Q 30. the function fis continuous everywhere in [0,2] except for x = 0, i , 1 & 2.
EXERCISE-II Q 1.
(a) false; (b) false; (c) false; (d) false; (e) false; (f) true; (g) false; (h) true
Q2.
(a)c = ± l ; ( b ) . x ± l , - l & x = 0 ; ( c ) . l ; ( d ) . a = | , b = - ~
Q 3.
(a) D (b). B, C (c). C, D (d). B (e). C (f), A (g). B (h) A (i) D (j) A (k) C
(e). 1/2
EXERCISE-III Q.l
R - [ - l , 0 ) ; discontinuous for all integral values in domain except at zero
Q.2
10
Q.5
Discontinuous at x = 1; f(l + ) = 1 and f(l~) = - 1
[14] Bansal Classes
Q.3 D
Q.4
a=/n| ; b= | ; c=1
Limits, continuity & Differentiability of Functions
[15]
DIFFERENTIABILITY EXERCISE-I Q 1. f(x) is conti. but not derivable at x = 0 Q 2. conti. V x e R, not diff. at x = 0,1 & 2 Q 4. conti. but not diff.at x = 0; diff & conti. at x=Tt/2
Q 5. conti. but not diff. at x = 0
Q 7. fis cont. but not diff. at x = 0
Q 8. f(l + ) = 3 , f ( l ~ ) ~ - l
Q 9. a= 1/2, b = 3/2
Q 10. not derivable at x = 0 & x = 1
Q 11.fis cont. & derivable at x = - 1 but fis neither cont. nor derivable at x= 1 Q12. discontinuous & not derivable at x = 1, continuous but not derivable at x = 2 Q 13. not derivable at x = 0 Q 14. fis conti. at x = 1,3/2 & disconti. at x = 2, fis not diff. at x=1,3/2,2 Q15. (fog)(x) = x+1 for - 2 < x < - 1 , -(x+1) for - 1 < x < 0 & x - 1 for 0 < x < 2. (fog)(x) is cont. at x = -1, (gof)(x) = x+1 for - 1 < x < 1 & 3 - x for 1 < x < 3. (gof)(x) is not differentiable at x= 1 Q 16. a * 1, b = 0, p = — and q = - 1 Q 17. If a e (0, 1) f ' (0+) = - 1 ; f ' (0") = 1 => continuous but not derivable a = 1; f(x) = 0 which is constant => continuous and derivable If a > 1 f ' (0") = - 1 ; f ' (0+) = 1 => continuous but not derivable Q 18. conti. inO
Q.20 (a) f ' (0) = 0, (b) f ' v
3
71
y
Q.21 continuous but not derivable at x= Q.24 f(x) = x
Q.23 f(x) = e2x
1
= — ,v^(c) x= 7n s I 2 ' ' 2n + l 3 v y a Q.22 f ' ( 0 ) = 1-k f(10) = 10
EXERCISE-II Q.l 2 Q.5 2 Q.9 D Q.13 D Q.17 A, B Q.21 B, D Q.25 A, D
Q.2 conti. & diff. Q.6 C Q.10 B Q.14 A Q.18 B, D Q.22 A, B, D
Q.3 0 Q.7 B Q.ll A Q.15 A, B, D Q.19B, D Q.23 A, B, D
Q.4 f'(0 + ) = 0 , f ' ( 0 - ) = l Q.8 A Q.12 D Q.16 A, C Q.20 A, B, C Q.24 A, B, D
EXERCISE-III Ql Q.3 Q.5 Q.7 Q.12
f(x) is conti. & diff. at x = 1; f(x) is not conti. & not diff atx = 2 Q.2 A C, D conti. but not derivable at x = 1, neither cont. nor deri. at x = - 1 Q.4D Q.6 Discont. hence not deri. at x = 1 & - 1 . Cont. & deri. at x = 0 (a)D, (b) A, (c) D Q.9 D Q.10 C Q.ll a = 1; b = 0(gof)'(0) = 0 Q.13 (a) A, (b) B, (c) y - 2 = 0 f'(a~) = 0
[14] Bansal Classes
Limits, continuity & Differentiability of Functions
[15]
YJI
K
J s BAN SAL CLASSES
k..!^
7
.
MATHEMATICS Daily Practice Problems
1 8 T a r g e t SIT JEE 2 0 0 7 CLASS : XII (ABCD) DATE : 28-29/06/2006
TIME: 50 Min each DPR
DATE: 28-29/06/2006 Q.l
tan 9 =
DPP. NO.-25 TIME: 50 Mitt
—j
where 0 e (0,2%), find the possible value of 0.
[2]
2+ —
2 + '--oo Q. 2 Q.3
Find the sum of the solutions of the equation 2e 2 x ~ 5ex + 4 = 0.
[2]
Suppose that x and y are positive numbers for which log9x = log12y = log15(x + y). If the value of - =2 cos 0, where 0 e (0,n/2) find 0.
Q. 4
[3]
Using L Hospitals rule or otherwise, evaluate the following limit: Limit+ Limit x->0
n->«>
l2 (sinx)" ] + [22 (sinx)x ] +
+ [n2 (sinx)x ]
n3
I
where [.] denotes the
greatest integer function. Q.5
1 Consider f ( x ) = - j =
Vb
[4] /b-a
,
.
.
V~T~ S1»2x \ 2
I Va + b t a n
x
, , f o r b > a > 0 & t h e functions g(x)&h(x)
b-a . 1 -j— smx
are defined, such that g(x) = [f(x)] - j-^y-j & h(x) = sgn (f(x» for x e domain o f f , otherwise g(x)=0=h(x) for x <£ domain o f f , where [x] is the greatest integer function of x & {x} is the fractional 7C part of x. Then discuss the continuity of'g' & 'h' at x=— and x = 0 respectively. Q.6 Q.7
[5]
J f ^ d x
[5]
Using substitution only, evaluate: jcosec 3 xdx.
DATE: 30-01/06-07/2006
[5] TIME : 50 Mitt.
Q.l
12 A If sin A = — . Find the value of tan ~ .
Q.2
x y The straight line - + - = 1 cuts the x-axis & the y-axisinA&Brespectively& a straight line perpendicular
[2]
to AB cuts them in P & Q respectively. Find the locus of the point of intersection ofAQ & BP. [2]
Q.3
tanO 1 cot0 If -———-—— = —, find the value of . tan 0 - t a n 30 3 cot0~cot30
1[3] J
Q.4
Q.5
Q.6 Q.7
If a A ABC is formed by the lines 2x + y - 3 = 0 ; x - y + 5 = 0 a n d 3 x - y + l = 0 , then obtain a cubic equation whose roots are the tangent of the interior angles of the triangle. |4] r dx Integrate: J / 2 2 x f~r~ 2 (a - t a n xK/b - tan x
J
[5]
(a>b)
xsinx cosx dx (a cos^ x + b z sin2 x)2
[5]
z
d dy Let — (x2y) - x - 1 where x ? 0 and y = 0 when x = 1. Find the set of values of x for which — dx is positive. IS] S^C 3$S 5J5 sj? ^
^
Sjc
^
TIME; 50Min.
DATE : 03-04/07/2006 Q. 1
Let x = (0.15)20. Find the characteristic and mantissa in the logarithm of x, to the base 10. Assume log in 2 = 0.301 and Iog103 = 0.477. K O l o c / 3 f Z P I ^ o -M^ZPI
Q.2
Two circles of radii R & r are externally tangent. Find the radius ofthe third circle which is between them and touches those circles and their external common tangent in terms of R & r. [2]
Q.3
Let a matrix A be denoted as A=diag. 5 X ,5
Q. 4
then compute the value ofthe integral J( det A)dx.
[3] Using algebraic geometry prove that in an isosceles triangle the sum ofthe distancesfromany point of the base to the lateral sides is constant. (You may assume origin to be the middle point ofthe base of the isosceles triangle) [4]
J'
- x
dx
+ x
Vx + X2 + X3
Q.5
Evaluate:
Q.6
'a3 If the three distinct points, a-1 \
Q.7
,5
[5] a2-3 a-1
f /
b -3 b-1 ' b-1 U3
(
„3
c—1
c — 1 are collinear then
show that abc + 3 (a + b + c) = ab + be + ca.
[5]
Integrate: j\/tanx dx
[5]
3 jVv^otd 2{.l 1 4 -t
&
AX* G
2 vAV
p
MATHEMATICS A
XII (ALL) : I
•• . • . . • M .•: ..::. : . . : i: ".V. '
V:
i •'•
i-ifP
; "••:•;•••• • :
•• '
: . .
..
. •
:
•
1
;
,
V
METHOD OF DIFFERENTIATION
Time Limit: 3 Sitting Each of 70 Minutes duration approx.
Question bank on Method of differentiation There are 72 questions in this question bank. Select the correct alternative : (Only one is correct) Q.l
If g is the inverse of f & f ' (x) = (A) 1 + [g(x)]5
Q.2
£
DfT^o
Q.3
If y = tan"1
(B)
A ix-z O
'
v ^nex
2
then g' (x) :
1
(C) -
5
1 + [g(x)]
(C) 0
& f'(x) = tanx 2 then
Q.4
1
If y = f
(A)
(C)
Q.6
£
1 (5x + 6)
(D) none
(B) sin"1 x
(C) sin-1 Vx
(D) none of these
^2x & f ' (x) = sinx then dx vx + \j
1 + x - x2
1 - x + x2
sm
sin
2 (l + x - x 2 j . f 2 x - l X L sin oi (B) ^ x2 + l j 1+X
x2 + l j
f
2x-JX vx2 + ly
(D) none
(B)
1+a 30
If sin (xy) + cos (xy) = 0 then
(A) x
i| Bansal Classes
T
Z
Let g is the inverse function of f & f ' (x) =
(A) j, Q.7
3x + 4 5x + 6
xJ 1 - x + Vx J l - x 2 l & = —7 + p, then Fp : • ^ J dx 2yjx(l - x)
(A)0
Q.5
(D) - 1
(B) - 2 tan
3tanx2 + 4 tanx 2 z 5tanx + 6
If y = sin
(D) none
dy dx
(A) tanx 3 (C)f
1 + [g(x)f
2fnxthend^ 1 - 6£nx dx
(B) 1
^3x + Ify = f ' v 5x +
1
+ tan-l3 +
y
(A) 2
1+x
.10
. If g(2) = a then g' (2) is equal to
(l + x2)
(C)
.10
1+a
(D)
1 + a10
dy dx (C) -
Q. B. on Method of differentiation
(D)
[9]
Q.8
dy 2x 2 then dx 1+x
If y = sin-1 (A)
Q.9
(B)
The derivative of sec
1
is x =
- 2
(C)
Vs
J_ ' 1 A 2 is w.r.t. - j i c2 at x = — v2x -1. (D) none
(C) 1
(B) 1/4
(A) 4
(D) none
~ 5
/
2
Q.10 If y = P(x), is a polynomial of degree 3, then 2 l^—J ^
(A) P "' (x) + P' (x) Q.ll
Q.12
(B) P " (x). P "' (x)
t ^ Q.15
(D) a constant
r
,w
dy If xP.yi = (x + y)P+c! then ^ is: (B) dependent on p but independent of q (D) independent of p & q both.
g(x). cos^x if x jt 0 " where g(x) is an even function differentiable at x = 0, passing 0 if x = 0
Let f(x) =
through the origin. Then f' (0): (A) is equal to 1 (B) is equal to 0 Q.14
(C) P (x). P "' (x)
(t,!v£(* i v ^ v * " ^
Let f(x) be a quadratic expression which is positive for all real x. If g(x) = f(x) + f' (x) + f" (x), then for any real x, which one is correct. (A) g(x) < 0 (B) g(x) > 0 (C) g(x) = 0 (D) g(x) > 0
(A) independent of p but dependent on q (C) dependent on both p & q
Q.13
. — j j equals :
If y = (A) e
n m
p m
l +x " +x mnp
+
m n
p n
• l +x ~ +x (B) e',mn/p
logsm2xcosx Lim — x-»0 , X log. x COS-
the value equal to
(A) 1
(B)2
2
sin -
(C) is equal to 2 +
then — at em" is equal to: l +x - +x " dx np/m (D) none (C) e m p
Q.17 Let / = ^
n p
2
(C) 4
Q. 16 If f is differentiable in (0,6) & f ' (4) = 5 then Limit (A) 5
(D) does not exist
(B) 5/4
(D) none of these f (4) - f (x 2 ) A,
X
(C) 10
(D) 20
xm (In x)n where m, n e N then:
(A) I is independent of m and n (B) / is independent of m and depends on m (C) I is independent of n and dependent on m (D) / is dependent on both m and n
i| Bansal Classes
Q. B. on Method of differentiation
[ 9]
cosx
Q.18
x 1 Let fix) = 2 sinx x2 2x . Then Limit x -> 0 tanx X 1 (A) 2
Q.19
(B)
cosx sinx Let fix) = cos2x sin2x cos3x sin3x (A) 0
Q.20
(C) - 1
(D) 1
(C) 4
(D) 12
then f '
(B) - 1 2
People living at Mars, instead ofthe usual definition of derivative D f(x), define a new kind of derivative,D*f(x) by the formula f2(x + h)-f2(x) D*f(x)= Limit where f 2 (x) means [fix)]2. If fix) = x /nx then h—>0 h D*f(x)| x = e has the value (A)e
Q.21
(B)2e
(D)none
(C)4e
If f(4) = g(4) = 2 ; f ' (4) = 9 ; g' (4) = 6 then Limit (A) 3V2
VeOO
Vx - 2
ig £ q u a l tQ
.
(D) none
(C) 0
f(x + 3 h ) - f ( x - 2 h ) = If fix) is a differentiable function of x thenLimit h 0 h (C) 0 (D) none & , (A)f'(x) (B) 5f'(x)
Q.22 (H
Q.23
If y = x + ex then (A) ex
Q.24
is : (B) -
(C) -
M
M
(D)
-1
M
2
dy If x2y + y3 = 2 then the value of ~ z atthepoint(l, l)is: dx 3
(A) - 4 Q.25
dy2
(B)
(C)
12
If f(a) = 2, f ' (a) = 1, g(a) = - 1 , g' (a) = 2 then the value of (A) - 5
i| Bansal Classes
(B) 1/5
(C) 5
(D) none Limit 8(x).f(a)-g(a).f(x) x- a (D) none
Q. B. on Method of differentiation
[9]
Q.26 ^
ji
Tf
Q.27
If fis twice differentiable such that f"(x) =-f(x), f'(x) = g(x) h'(x) = [f(x)]2 + [g(x)]2 and h(0) - 2, h(l) = 4 then the equation y = h(x) represents: A (A) a curve of degree 2 ' (C) a straight line with slope 2
(B) a curve passing through the origin (D) a straight line with y intercept equal to - 2.
The derivative ofthe Rmction, f ( x ) = c o s " 1 ( 2 c o s x - 3sinx)|
+ sin -i
j - ^ L (2cosx + 3sinx)j
3 w.r.t. yjIl + x7 at x = —• is :
(A) Q.28
(C)
(B)-
10
(D) 0
Let f(x) be a polynomial in x. Then the second derivative of f(ex), is: (A) fi " (e x)).. ee-x + f ' (ex) (B) f " (e x ). e2x + f ' (e x ). e2x (D) f " (e x ). e 2x + f ' (e x ). ex (C) f " (ex) e 2x
A , , £ ca J ' X •,< t ^ ^' H q ^ c>| f • 1 A,©*' i * ^ * ) ' s
Q.29 The solution set of f ' (x) > g' (x), where f(x) = - (5 2x+1 ) & g(x) = 5X + 4x (In 5) is : 2 Co*H (A) x> 1 (B) 0 < x < 1 (C) x < 0 (D) x > 0 it you Q.30
If y = sin-1
x2-l + sec x2 + l
1
x2 + 1 l I dy , , I x | > 1 then — is equal to : xL —1 1 dx
sjw
„2
(A)
Q.31
Q.32
If y =
(B)
" T X
X
X
X
X
X
a+ b+ a+ b+ a+ b + a _ b (A) (B) ab + 2ay ab + 2 by
(D) 1
oo then ^y = dx ( Q 77 + 2by
(D)
ab + 2ay
Let f (x) be apolynomial function of second degree. If f (1) = f (-1) and a, b, c are in A.P., then f'(a), f'(b) and f'(c) are in (A) G.P. (B)H.P. (C)A.G.P. (D)A.P.
Q. 3 3 If y=sin mx then the value of
£
(C) o
x4 - 1
y yi y2 y 3 y4 y5
(where subscripts of y shows the order of derivatiive) is:
y6 y? y8 (A) independent of x but dependent on m (C) dependent on both m & x
i| Bansal Classes
(B) dependent of x but independent of m (D) independent of m & x.
Q. B. on Method of differentiation
[9]
\ Q.34 If x2 + y2 = R2 (R>0) then k=
W-jT
Jv"
where k in terms of R alone is equal to
(B)4
Q.35
If f&g are differentiable functions such that g'(a)=2& g(a)=b andif fog is an identity function then f' (b) has the value equal to : ; (A) 2/3 (B) 1 (C) 0 (D) 1/2
Q.36
x3 Given f(x) = - — + x2 sin 1.5 a - x sin a. sin 2a - 5 arc sin (a2 - 8a + 17) then: (A) f(x) is not defined at x = sin 8 (C) f ' (x) is not defined at x = sin 8
(B) f ' (sin 8) > 0 (D) f' (sin 8) < 0
Q. 3 7 A function f, defined for all positive real numbers, satisfies the equation fix2)=x3 for every x > 0. Then the value of f ' (4) = (A) 12 (B) 3 (C) 3/2 (D) cannot be determined Q.38
Given: fix) = 4x3 - 6x2 cos 2a + 3x sin2a. sin 6a + Mn (la - a 2 ) then: (A) fix) is not defined at x = 1/2 (C) f ' (x) is not defined at x = 1/2
(B) f ' (1/2) < 0 (D) f ' (1/2) > 0 d2v
Q.39 Jx^fCcJ•
If y = (A + Bx) e
mx
^
Q.40
Q.41 Q.42
£
+ (m - l)" e then (B) e^L \-'t*\
dv - 2m ^
+ m2y is equal to :
(C) v-vv^ e^L
(D)
(B)9
(C) 15
"
"
Let h (x) be differentiable for all x and let/(x) = (kx + ex)h(x) where k is some constant. Ifh(0) = 5, h' (0) = - 2 and f ' (0) = 18 then the value of k is equal to (A) 5 (B)4 (C)3 (D)2.2 f Let e W = In x. If g(x) is the inverse function of fix) then g' (x) equals to : (A) ex
Q.43
x
Suppose/(x) = eax + ebx, where a * b, and that f (x) - 2f (x) - 1 5 / ( x ) = 0 for all x. Then the product ab is equal to
A>25
n^J
2
(B) ex + x
(C) e<* + eX>
(D) e(x + /nx)
dy The equation y2exy = 9e-3-x2 defines y as a differentiable function of x. The value of — for x = - 1 and y = 3 is (A)-y
i| Bansal Classes
(B)-|
(C)3
Q. B. on Method of differentiation
(D)15
[9]
Q.44 v
Q.45 rl&xl
Let f(x) = (xx)X and g(x) = x ^ ' then:
The function f(x) = ex + x, being differentiable and one to one, has a differentiable inverse f '(x). The value of — (f_1) at the point f(Vn2) is dx (A)
Q.46
(B) f ' ( l ) = 2 and g ' ( l ) = 1 (D) f ' ( l ) = 1 and g ' ( l ) = 1
(A) f ' ( l ) = l and g ' ( l ) = 2 (C) f ' ( l ) = l and g ' ( l ) = 0
1 £n2
I f / ( x ) = J°gsinMCQS 3 ^ l°gsin|3x|
COS"
(D) none
(C)
(B)i
for
l x | <
|
x
^
0
v2y for x = 0 /
%
%x
then, the number of points of discontinuity of f in (B)3
(A) 0 Q.47
If y = — — (A)
Q.48
r ? 3 J (C)2
— x ) Vx yja-x + yjx-b
,X + ( a + b ) V(a-x)(x-b)
.(B)
(D)4
dy ^en — wherever it is defined is equal to : dx
?X"(a+b) 2V(a-x)(x-b)
(O-
(a + b) 2V(a-x)(x-b)
(D)
2x + (a + b)
2A/(a-x) (x-b)
d2y dy If y is a function of x then —7 + V 7 - = 0. If x is a function of y then the equation becomes : dx dx ^dx^3 =0 Vdy,
, AX d 2 x dx (A) - r + x 7 = 0 dy dy (C) Q.49
is
d2x
dx \ dy.
2
d2x
'dx^ 2
(D) dy2 - x vdyy
A function f (x) satisfies the condition, f (x) = f ' (x) + f " (x) + f " ' (x) + co where f (x) is a differentiable function indefinitely and dash denotes the order of derivative. If f (0) = 1, then f (x) is: 2 (A) e^ (B) ex (C) e2x (D) e4x cos6x + 6cos4x + 15cos2x+10 cos5x + 5cos3x +10 cosx (A) 2 sinx + cosx (B)-2sinx
Q.50 If y =
i| Bansal Classes
dy dx (C) cos2x ,then
(D) sin2x
Q. B. on Method of differentiation
[9]
Q.51
Q.52
2 cTx " ^dy^ 3 d y H 7 = K then the value of K is equal to dx I f ^ r vdxJ (C)2 (B)-l (A)l
If f(x) = 2sin"1
+ sin-1 (2 A /x(l-x)) where x
e
(D)0
>^
then f' (x) has the value equal to
2 A
(B)zero
( ) Vx(l-x)
CO" if x * 0
Q.53
Let
y = f(x) = 0
if x = 0
Then which of the following can best represent the graph of y = f(x) ? 1(0,1)
(A)""
^
f Q. 5 4 Diffrential coefficient of
n-l x
Q.55
m+nN
i-m
m-n
V
(A) 1
'
(
n+C
w.r.t. x is
V
V
1
/
imn (D)X'
(C)-l
(B) 0
(x+h) f(x) - 2hf(h) • im — is equal to Let f (x) be diffrentiable at x = hthen Lim X h (A) f(h) + 2hf'(h)
(B) 2f(h) + hf'(h)
(C) hf(h) + 2f'(h)
Q.56 If y = at2 + 2bt + c and t = ax2 + 2bx + c, then {; v
' ' 1 ' ' (A) 24 a2 (at + b) f\\vt\V ^e^i^jcnce a arc tan
a-b (A)"
i| Bansal Classes
(B) 24 a (ax + b)2 Vx a
(B)0
(D) hf(h)-2f'(h)
equals
(C) 24 a (at + b)2
(D) 24 a2 (ax + b)
Vx
b arc t a n — has the value equal to b )
(a2 - b 2 ) (Q6a 2u2 b
Q. B. on Method of differentiation
(D)
a 2 - b2 3a 2 b2
[9]
Q.58
rx \ = / ( x ) - / ( y ) for all x,y& Let f(x) be defined for all x > 0 & be continuous. Let f(x) satisfy / yj f(e) = 1. Then : (A) f(x) is bounded
(B) fQ) -> 0 as x -> 0
(C) x.f(x)-> 1 asx-^0
(D) f(x) = /nx
Q. 5 9 Suppose the function /(x) -/(2x) has the derivative 5 at x = 1 and derivative 7 at x = 2. The derivative of the function/(x) -/(4x) at x = 1, has the value equal to t (A) 19 * (B) 9 (C) 17 (D) 14 x4-x2+l dy Q.60 If y = ?= and — = ax + b then the value of a + b is equal to x + V3x + 1 dx ., , J>^
( A ) C O t
571 T
571 (B)cot—
571 (C)tan-
57T (D)xmj
^V
Q.61
Suppose that h (x) =/(x)-g(x) and F(x) = /(g(x)), where/(2) = 3 ; g(2) = 5 ; g'(2) = 4 ; f'(2) = - 2 and f'(5) =11, then (A) F'(2) = 11 h'(2) (B) F'(2) = 22h'(2) (C) F'(2) = 44 h'(2) (D) none
Q.62 Let /(x) = x3 + 8x + 3 which one of the properties of the derivative enables you to conclude that/(x) has an inverse? (A)/(x) is a polynomial of even degree. (B)/(x) is self inverse. (C) domain of / ' (x) is the range of f (x). ( D ) / (x) is always positive. Q.63
Which one ofthe following statements is NOT CORRECT ? (A) The derivative of a diffrentiable periodic function is a periodic function with the same period. (B) If f (x) and g (x) both are defined on the entire number line and are apenodie then the function = f (x). g (x) can not be periodic. (C) Derivative of an even differentiable function is an odd function and derivative of an odd differentiable function is an even function. (D) Every function f (x) can be represented as the sum of an even and an odd function
Select the correct alternatives : (More than one are correct) Q.64
Q.65
dy If y = tan x tan 2x tan 3x then — has the value equal to : dx (A) 3 sec23x tanx tan2x + sec2x tan2x tan3x + 2 sec22x tan3x tanx (B) 2y (cosec 2x + 2 cosec 4x + 3 cosec 6x) (C) 3 sec2 3x - 2 sec2 2x - sec2 x (D) sec2 x + 2 sec2 2x + 3 sec2 3x Ify = e ^ + e " ^ then — equals dx Vx
i| Bansal Classes
s
-Vx
V^
-V^
i
n;—
Q. B. on Method of differentiation
1
p^
[9]
Q.66
2 d If y = x xx then ,y
dx
Q.67
(A)2/nx.x x 2
(B) (2/nx + l).x x
(C) (2 In x + 1). xx2 + 1
(D) x x2+1 - In ex2
Let y= ^x + Jx + Jx + (A) - ± -
x + 2y
V +
(D)
2x + y
dy has the value equal to : dx
2»
1
(B)—
(C) 1 - 2*
2*(l-2')
TO^nj
The functions u = ex sinx; v = ex cosx satisfy the equation: (A)v
du
d2u _ (B) —- = 2 v
dv 2.2 u — = u 2 + v^
d2v (C) —5- = - 2 u dx Q.70
(C) - f1= L4 x=
If 2X + 2y = 2X+ y then ~
(A)-? Q.69
=
(B) - 4 -
2y - 1
Q.68
oo then ^
Let f(x) =
(D) none of these
VxM-1
. x then:
(A) f (10) = 1 (C) domain of f (x) is x > 1
(B) f ' (3/2) = - 1 (D) none
Q. 71 Two functions f & g have first & second derivatives at x = 0 & satisfy the relations, f(0)= ^ - ' f'(0) = 2g'(0) = 4g(0), g"(0) = 5 f"(0) = 6f(0) = 3 then: g(0)
Q.72
(A) if h(x) = ® then h'(0)= ^4 gOO
(B) if k(x) = f(x). g(x) sin x then k' (0) = 2
(C) V ; L™^
(D); none
f'(x)
If y= x ( ' nx) (A) I v( W ^
x
(C)
x lnx
= 2 , then
1
dx
is equal to :
+ 2£nx in ((nx)) '
((In x)2 + 2 In (In x))
i| Bansal Classes
(B) ^ (In x)ln{lnx^ (2 In (In x) + 1) x
(D)
x tnx
(2 In (In x) + 1)
Q. B. on Method of differentiation
[9]
ANSWER a'a
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KEY
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i| Bansal Classes
62 b
Q. B. on Method of differentiation
D 9 zb
[9]
BANSAL CLASSES
TARGET IIT JEE 2007
MATHEMATICS XII (ABCD)
METHOD OF DIFFRENT1ATION AND L' HOSPITAL'S RULE
CONTENTS EXERCISE-I EXERCISE-II EXERCISE-III EXERCISE-IV ANSWER KEY
KEY 1.
CONCEPTS
DEFINITION:
If xandx+h belong to the domain of a function f defined by y = f(x), then l i m i t f(X + h*) — ffxl
h-To .
—^—
EX
IF IT
ISTS
,
called the
IS
.
dy
The derivative of a given function f at a point x = a of its domain is defined as : Limit f(a + h)-f(a) h
"»°
h
, provided the limit exists & is denoted by f'(a).
Note that alternatively, we can define f (a) = 3.
5.
| Z = Limit
= f ( x ) =
|
THEOREMS ON DERIVATIVES : If u and v are derivable function of x, then, (i) v/
— (u+v) = — + —
(iii) v 7
- 7 -
(iv)
— (Jj =
(v)
If y = fiu) & u = g(x) then
dx
dx
dx
(u. v) = u ^
v
d ( 1A
'
(ii)
dx
±
v
—(K u) = K ^ , where K is any constant
v 7
dx
^
v (—) - u (—)
dx
known as "
dx
dx
PRODUCT R U L E "
where v * 0 known as " QUOTIENT
2
T" =
dx
• ~
du dx
RULE "
" CHAIN R U L E "
DERIVATIVE OF STANDARDS FUNCTIONS: (i) D (xn) = n.xn_1; x e R , n e R, x > 0 (iii) D (ax) = ax. In a a > 0
6.
, provided the limit exists.
DERIVATIVE OF f(x) FROM THE FIRST PRINCIPLE /ab INITIO METHOD: Iffix)is a derivable function then, ™
4.
of f at x & is denoted by
Limit f(x + h)-f(x)
x f (x) or ^ . We have therefore, f (x) =
2.
DERIVATIVE
(ii) D(e x ) = ex
(vi) D (sinx) = cosx (ix) D (secx) = secx. tanx
(iv) D (In x) = (v) D (log x) = - log e X X 2 (vii) D (cosx) = - sinx (viii) D = tanx = sec x (x) D (cosecx) = - cosecx. cotx
(xi) D (cotx) = - cosec2x
(xii) D (constant) = 0 where D =
INVERSE FUNCTIONS AND THEIR DERIVATIVES : (a) Theorem: If the inverse functions f & g are defined by y = fix) & x=g(y) & if f'(x) exists & f'(x) * 0 then g'(y) =
(b)
1
dy
. This result can also be written as, if — exists &
Results : (i)
D(sin- 1 x)= .
1
VL-X
^Bansal Classes
, -1
(ii)
D(cos _1 x ) = - ^ L = , - L < x < l
2
M.O.D. andL'HospitalRule
VL-X2
[2]
(iii)
D(tan-1 x)=-
1+x
D (cosec 1 x)=-
(v)
(iv)
, xeR
D (sec-1 x) = [x| V ^ l
-1
(vi)
, |x|>l
D (cot-1 x) =
-1 1+x 2
> X>1
, xeR
Note : In general if y = f(u) then — = f'(u). — . dx
dx
LOGARITHMIC DIFFERENTIATION: Tofindthe derivative of : (i) a function which is the product or quotient of a number of functions (ii)
OR
s:x)
a function ofthe form [f(x)] where f & g are both derivable, it will be found convinient to take the logarithm of the function first & then differentiate. This is called LOGARITHMIC DIFFERENTIATION .
IMPLICIT DIFFERENTIATION:
In answers of dy/dx in the case of implicit functions, both x & y are present.
PARAMETRIC DIFFERENTIATION: If y = f(9) & x = g(0) where 6 is a parameter, then
10.
dyMO
dx / d9
DERIVATIVE OF A FUNCTION W.R.T. ANOTHER FUNCTION: dy
Let y = f(x) ; z = g(x) then ^ =
11.
dx
f'(x)
dy / dx
g'(x)
DERIVATIVES OF ORDER TWO & THREE : Let a function y = f(x) be defined on an open interval (a, b). It's derivative, if it exists on (a, b) is a certain function f'(x) [or (dy/dx) or y ] & is called the first derivative ofy w.r.t. x. If it happens that the first derivative has a derivative on (a, b) then this derivative is called the second derivative ofy w. r. t. x & is denoted by f"(x) or (d2y/dx2) or y". rd
Similarly, the 3 order derivative of y w. r. t. x, if it exists, is defined by
d3y
d
/
n
dy
\
vdx2/
It is also
denoted by f"'(x)ory"'. f(x)
12.
g(x)
h(x)
If F(x) = l(x) m(x) n(x) , where f, g, h, 1, m, n, u, v, w are differentiable functions of x then u(x)
v(x)
w(x)
f'(x) g'(x) h'(x)
F'(x)= l(x) m(x) n(x) u(x)
13.
v(x)
w(x)
f(x)
g(x)
h(x)
l'(x) m'(x) n'(x) u(x)
v(x)
w(x)
+
f(x)
g(x)
h(x)
l(x)
m(x)
n(x)
u'(x) v'(x) w'(x)
L' HOSPITAL'S RULE: Iff(x) & g(x) are functions of x such that : Limit _ n _ Limit Limit Limit g(x) (0 x-ia f(x) = 0 = g(x) OR — 1 m = 00 = _ (ii)
Both f(x) & g(x) are continuous at x = a
(iii)
Both f(x) & g(x) are differentiable at x = a
^Bansal Classes
and
& &
M.O,D.and L 'Hospital Rule
m
(iv)
Both f'(x) & g '(x) are continuous at x = a, Limit f(x) _ Limit f(x) _ Limit f"(x) x^a
14.
g
x-*a
.(x)
g
"(x)
& so on till indeterminant form vanishes.
ANALYSIS AND GRAPHS OF SOME USEFUL FUNCTIONS:
y = f(x) = sin-1
(i)
x^a
g ( x )
Then
2 tan"1 x W<1 1 7t - 2 tan x x>l 1 - (rc + 2 tan" xj x < - 1
N
' 2x 1 + x2
HIGHLIGHTS :
(a)
Domain is x € R & %
range is
(b)
(c)
%
T'T f is continuous for all x but not diff. at x = 1 -1
dx
=
tl/2
(ii)
\D
/1
for x < 1 1+X non existent for |x| = 1 2-r for Ixl > 1 1+x 2
(d)
>y
-1 D\
0
1
I
1 1
-1/2
I in(-1,1) & D in(-00,-1) u (l,oo)
Consider
2
l l + XJ
2tan _1 x if x > 0 1 - 2 tan" x if x < 0
HIGHLIGHTS :
(a) (b)
(c)
Domain is x e R & range is [0,7t) Continuous for all x but not diff. at x = 0 dy dx
y<
D \
7t/2
for x > 0 1+X non existent for x = 0 - -2-sfor x < 0 1+x 2
(d) (iii)
-1
0
|x|
y = f(x) = tan 1 j—^j
HIGHLIGHTS :
f
TZ
1/
%
-1
f is neither continuous nor diff. at x = 1, -1 1x1*1 1- +-x non existent Ixl = 1
dy dx
(d)
I Vx in its domain
^Bansal Classes
(e)
r 0
y
2
(c)
y
It/2
Domain is R - { 1 , - 1 } & range is
(b)
/ T
I in (0, oo) & D in (- oo, 0) 2x
(a)
7t
A
-n/2 '
It is bound for all x
M.O,D.and L 'Hospital Rule
1
m
3 sin 1 xj if
- +
(iv)
1
3
y = f (x) = sin" (3 x - 4 x ) = 3sin x n - 3sin-1x _1
-l
if
4
if
|
HIGHLIGHTS
(a)
Domain is x e [ - 1 , 1 ] & range is
(b)
(C)
(d)
71
71
2'2
Not derivable at Ixl 11 = 2
dx
=
3
if
*6(-W)uft.i)
Continuous everywhere in its domain 3cos_1x-27i
(v)
if
•1
y = f(x) = cos" (4x - 3 x) = 2 7c — 3cos x i f 3cos _ 1 x if
-|
1
3
_1
HIGHLIGHTS :
(a)
Domain is x e [-1,1] & range is [0, 7t]
(b)
Continuous everywhere in its domain but not derivable at x = - , -— 2
(c)
Iin|-i,IJ& Din
(d)
2
£dx- =
U
lf
* e (-W)ufc,i)
GENERAL NOTE :
Concavity in each case is decided by the sign of 2nd derivative as : d2y
^ T > 0 => Concave upwards D = DECREASING
^Bansal Classes
dV dx:2
< 0 => Concave downwards
I = INCREASING
M.O,D.and L 'Hospital Rule
m
EXERCISE-I Q.l /
Find the derivative of the following functions w.r.t. x from thefirstprinciple: cos(Znx), (sinx)cosx, logaC wherea = x x & Cis constant, ^sin Vx and cos"1 (x2).
Vl + x 2 W l - x 2 Q ^ , Differentiate ^ w.r.t. Vl-x 4 . 2 ^ 2 Q.3 (a) Let f (x) = x2 - 4x - 3, x > 2 and let g be the inverse of f. Find the value of g' where f (x) = 2. (b) Let / , g and h are differentiable functions. If /(0) = 1 ;g(0)=2 ; h (0) = 3 and the derivatives of their pair wise products at x = 0 are (f g)'(0) = 6 ; (g h)'(0) = 4 and (hf)'(0) = 5 then compute the value of (fgh)'(O). s.rcsin yo 2 2 Q.4 If v . Provethat x>0. dx (x-y) Q.5
Ify = x +
, prove that —= dx 2
1
x+
x+
Z
1 x+.
— x_ 1
x+
......
x_l
x+.
Q.6
c . \ If x = cosec 9 - sin 9 ; y = cosec" 9 - sin" 9, then show that ( x 2 + 4) dy - n 2 (y 2 + 4) = 0 vdxy
Q.7
Ify = (cosx) /nx + (lnx)x find 6
6
dy dx'
dy = a • (x -y ), prove that dx 3
3
3
Q.8
If V l - x + V l - y
Q.9
Find the derivative with respect to x ofthe function : (lo§cosX s i n x ) Oogsinx C0SX ) _I +
arcsin
=
x 2 11 — y6 y 2 lV1 l - x 6
at X= j .
Q.10 If x = 2cost - cos2t & y = 2sint - sin2t, find the value of (d2y/dx2) when t = (n/2). Q.ll
If y=tm'1-7J= & x = sec-1 2 2 2u Vl-u
-1
, US ' V2 J
f ~ _, -i Vl + sinx + Vl-sinx dy Q.12 TIf y= cot , , find jd-x if XG Vl + s i n x - V l - s i n x
f 1 VV 2 '
u
\
dy prove that 2 — + 1= 0. ,
^ 71 ^
dy + sin 2 tan" IIZi , then find — forx e (-1,1). 1+x
Q.19
Q.14
u
dy I f y = sec4x and x = tan (t), prove that dt
^Bansal Classes
1
16t(l-t 4 ) ( l - 6 t 2 +t 4 ) 2
M. O.D. and L 'Hospital Rule
[6]
Q 15=v
If
(x-a) 4 4 f W = (x-b) (x-c) 4
(x-a) 3 1 (x-a) 4 (x-b) 3 1 then f ' (x) = k. (x-b) 4 (x-c) 3 1 (x-c) 4
Q.16
If [ f (x) ]3 = 3 K x2 - x3 then f " (x) +
Q.17
If y =
nx
(x-a) 2 (x-b) 2 (x-c) 2
1 1 . Find the value of X. 1
= 0. Find the value of n in terms of K.
[f(x)f
+ xVx 2 +1 + ZnVxWx^+T prove that 2y = xy' + In y'. where' denotes the derivative.
1-x2 Q. 18(a) Find the derivative of cos-l when - oo < x < 0, using the substitution x=tan 9. vl + x y
(b) Iff(x)= sin"1
,findf'(x) V x e R , clearly stating the point(s) where f(x) is not derivable.
1+ x
V 'J Also draw the graph of y= f(x) and state its range and monotonic behaviour. r
r Q.1% Ify =
d^y _ bsinx - — - tan — , then show that 2 'a + b 2/ y dx (a + bcosx)2 v
tan
7/
V
Q.20 I f f : R-»R is a function such that f(x) = x3 + x2 f'(l)+xf"(2) + f "'(3) for all x e R, then prove that f(2) = f ( l ) - f ( 0 ) . 1
1
x(x + 1)
Q.21^Ify = x/n[(ax)" + a- ], prove that
O p , Ifflx)
^T
cos(x+x 2 )
sin(x+x 2 )
-cos(x+x2)
sin(x-x2)
cos(x-x2)
sin(x-x2)
sin2x
Q.23
d2y
Let f(x) = x +
0
sin2x
+
dy
= y_1
•
then find f*(x).
2
1
1 2x + 1 2x + 2x + oo Compute the value of /(100) • / ' (100).
Q.24
If (a + bx) e y / x =x, then prove that
Q.25
If y-2 = i + 2V2 cos 2x, prove that
= [x^-y .2 dx
= y(3y:2 +1) (7y2 -1)
EXERCISE-II Q.l
e ex ex dv x If y = e x +e x +x e . Find
dx
Q.2
sina dy = If sin y = x sin (a + y), show that — dx l - 2 x cosa+x 2 '
^Bansal Classes
M. O, D. and L 'Hospital Rule
m
Q.3
If a be a repeated root of a quadratic equation f(x) = 0 & A(x), B(x), C(x) be the polynomials of degree 3, 4 & 5 respectively, then show that
A(x) B(x) A (a) B(a) A'(a) B'(a)
C(x) C(a) C'(a)
is divisible by f(x), where dash
denotes the derivative. Q.4
If y = tan"1
x +x + l
+ tan"1
2
x +3x + 3
+ tan"1 ——2
x +5x+7
l + tan"1 —— 2
+
x +7x + 13
to n terms.
Find dy/dx, expressing your answer in 2 terms. Q.5
cos3x dy If y = arc cos J — . Express explicitly and then show that — = J c q s 2 x V COS X V (l + tan|)2
dy dx
If x = tan— - In
Q.7
d2f = 2 z 3 ^ + z 4 d 2 y2 If x = - and y = f(x), show that : 2 z dx dz dz Prove that if | aj sinx + ajSin 2x + + ansin nx ] < | sinx | for x e R, then
Q.8
\al+2al
Q.9
tanf
+ 3n3 +
Ify = /n x e
find
, sinx>0. si
1 2
Q.6
2
+ cqs4x
Show that —- = - sin y (1 + sin y + cos y).
+ na n | < 1
dy dx'
Q.10 If x 4 + 7 x 2 y 2 + 9 / = 24xy 3 , showthat ^ = dx x g(x),
r
x <0
Q. 11 Let g(x) be a polynomial, of degree one&/(x) be defined by f(x) = V2
Find the continuous function f(x) satisfying f'(l) = f(-l) Q.12 Let f(x) : Q.13
x>0
+ Xy
sinx if x * 0 and f (0) = 1. Define the function f" (x) for all x andfindf" (0) if it exist. x
d 2Jy dy 2 Show that the substitution z = / n tan— changes the equation —— + cot x — + 4y cos ec x = 0 to dx v 2y dx' (d2y/dz2) + 4 y = 0. 3/2
1+
Q.14
Show that R =
d^y dx
1
can be reduced to the form R2/3 =
d^y
2
dx 2
1 2/3
• +
2/3
dy 2
Also show that, if x=a sin2O(l+cos20) & y=acos20 (1- cos20) then the value of R equals to 4a cos39. .2
X„ . X Q- 15
If
y =
dy dx
=
1 +
^
+
(x-x7xx^-x 2 )
+
(x-x 1 )(x X -x X 2 )(x-x 3 ) +
U
P t0 ( n + Dtermsthenprovethat
x. y + +...+ + xn-x x Xj X X2 X X3 X
<§Bansal Classes
M.O.D. andL'HospitalRule
[8]
a+x
b+x c+x Q.16 Let f(x) = £+x m+x n+x . Showthat f "(x) = 0 and that f(x) = f(0) + kx where k denotes p+x q+x r+x the sum of all the co-factors of the elements in f (0). Q.17 If y = logu | cos4x | + | sinx | , where u = sec2x, find
Q.18 Ify=
1 ,/a 2 - b2 - c 2
cos l<
aG - a 2 + b2 + c 2 e Vb2 + c2
dy dx x=-7t/ 6
dy 1 &9 = a + bcosx + csinx;provethat — = —.
sin2nx 2sinx
+ c o s ( 2 n - l ) x = —:— L ,x*K7t,K.eI and deduce from
Q.19 Prove that cosx + cos3x + cos5x +
IN • /O
i\
[(2n+1)sin(2n-1)x-(2n-1)sin(2n+l)x]1
this: sinx +3sin3x+5sin5x +....+ (2n - 1) sin (2n - 1) x = 1
r
4 sin x
.
Q.20 Find a polynomial function /(x) such that /(2x) = / ' (x)/M (x). Q.21 IfY=sX and Z=tX, where all the letters denotes the functions ofx and suffixes denotes the differentiation w.r.t. x then prove that X Y Z X,
Y,
Z,
Y2 Zj
s,
X3 s 2
t,
t2
f x + y\ Q. 22 Let f: R -> ( - %, n) be a derivable function such that f (x) + f (y) = f .1 - xyj Iff(l)=^ & LhmtM=2>
findf(x)_
Q.23 Let fix) be a derivable function at x = 0 & f[
x + y)
f (x) + f (y)
(k e R, k * 0,2). Show that fix)
is either a zero or an odd linear function. Q.24
If f(x+y) = fix)-fiy) for x, y e R&fix) is differentiable everywhere then find fix).
Q.25 Let
f(x + y ) - f ( x )
f(y)-a
2
2
+xy for all real x and y. Iff (x) is differentiable and f'(0) exists for all
real permissible values of'a' and is equal to ^ 5 a - l - a 2
Prove that f (x) is positive for all real x.
EXERCISE-III Evalute the following limits using L'Hospital's Rule or otherwise : 1- x
Ql
Lim x-»0
Q.3
1 Lim 1 •2 x-*0 x 2 sin x
xsin ' x
^Bansal Classes
Q.2 Q.4
L j m xcosx - ln(l+x)
x-»0
im J If Lx— >a Vx - T a
=
-1
M.O,D.and L 'Hospital Rule
find'a'.
m
Q
5
Q.7
~
0
T.
1 + sinx-cosx+ /n(l-x) x™ ^
Q.6
sinx-(sinx)
^ ^ . ( t a n ^ x )
Tim (a + bcosx)x-csinx 5 = 1 X
Determine the values of a, b and c so that
Lim
T.
sinx
. . fsinxV 3x £n\ | + x3
.
^
"2 1-sinx + In (sinx)
x
(x-sinx)
Q.10 Find the value of J{0) so that the function /(x)= 1 x
(l-cosx)
2 , x * 0 is continuous at x = 0 & examine the
e2x - 1
differentiability of f(x) at x = 0. n 11 ^ Q.12
r\
j• Lim
sin(3x ) — /n.cos(2x2-x)
„ t i m a s i n x - b x + cx 2 +x 3 . , „ , , ,. . If "to 1 —o 5 j exists & is finite, find the values of a, b, c & the limit. 2x ,/n(l + x) - 2x + x l-sin
Q.13
Given/(x) =
forx>l
2sin™-i+cos2 ™ j-i
; where h(x) = sin"1 (sgn(x))&
lg(h(x)) forx
Given a real valued function f(x) as follows: x 2 + 2co
f(x) = -
-4
1
for x<0; f(0)= — & f(x) =
sinx - &i(ex cosx) 1
^
forx>0. Test the continuity
and differentiability of/(x) at x=0.
EXER CISE-IV Q.l
Q.2
(a)
Find the values of constants a, b & c so that Lim — — b / n ( l + x) + cxe— _ ^ x->o x sin x
(b)
Find the differential coefficient of the function f(x) = log^ sinx2 + (sinx2)1082" w.r.t. 1 [REE'97, 6 + 6] 2
bx , c Ify= — + — +— + (x-a)(x-b)(x-c) (x-b)(x-c) (x-c)
\
y'.1/ a b c 1, Prove that £ - = - 1 - = - + y xVa-x b - x c-xj [ JEE'98, 8 ]
2
Q.3
'then
If f (x) = ^
^
find t h e d o m a i n 311(1 t h e r a n
and its domain.
<§Bansal Classes
8 e o f f • Show that f is one-one. Alsofindthe function
M.O.D. andL'HospitalRule
[ REE '99, 6 ]
[10]
Q.4(a) If x2 + y 2 = l , t h e n : (A) yy" - 2 (y')2 + 1 = 0 (C) y y " - ( y ' ) 2 - l = 0
(B)
y y " + (y')2
(D)
+ 1 - 0
y y " + 2 (y')
2
+
1 = 0
[JEE 2000, Screening, 1 out of 3 5 ] 2
+ an x n . If | p (x) | < | ex 1 - 1 | for all x > 0 prove that [ JEE 2000 (Mains) 5 out of 100 ]
(b) Suppose p (x) = a0 + a t x + % x +. \al+2a2 + + n an | < 1 . Q.5(a) If In (x + y) = 2xy, theny' (0) = (A)l (B)-l
(C)2
(D)0
[JEE 2004 (Scr.)]
(b) f(x) =
, . -J x + c bsin I 2 .
—- < x <0 2
2
at x = 0
]_
e
;
ax/2_1
0
(D) — n
(b) If P(x) is a polynomial of degree less than or equal to 2 and S is the set of all such polynomials so that P(l) = 1, P(0) = 0 and P'(x) > 0 V x e [0,1], then (A) S = c|) (B) S = {(1 - a)x2 + ax, 0 < a < 2 2 (C) (1 - a)x + ax, a e (0, oo) (D) S = {(1 - a)x2 + ax, 0 < a < l [JEE2005 (Scr.)] (c) I f / ( x - y) =/(x) • g (y) - / ( y ) • g (x) and g (x - y) = g (x) • g (y) +/(x) -/(y) for all x, y e R. If right hand derivative at x = 0 exists for/(x). Find derivative of g (x) at x = 0. [JEE 2005 (Mains), 4] Q.7
1/x Forx> 0, Lim((sinx) + ( l / x f 1 x ) is v x—MD
(A)0
'
(B)-l
(C)l
(D)2 [JEE 2006, 3 (-1)]
<§Bansal Classes
M.O.D. andL'HospitalRule
[11]
ANSWER KEY EXERCISE-I Q 2. Q9.
lWl-x4
Q 3. (a) 1/6; (b) 16 Q 7. Dy = (cosx)1,lnx
32 16 + 7t
Q 18. ( a ) -
8
2
l +x
2
2
;
n n 2'2
(b){0), range
- tanx lnx
l-2x
1 1 Q 1 2 .— - oor r -— - Q13.
3 Q 10. -—
1n2
ln(cosx)
2
(Hx
Q 15. 3
lnx
+ ln(lnx)
Q 1 6 . n = 2K2
Q.22 2(1 + 2x). cos 2(x + x2)
Q.23
100
EXERCISE-II 6 Q 1. — = e ^ V —+eX!lnx +exX x e_1 x x6 [l + elnxj+x®6 eeX dx l l Q4 Q 5 Q9 'l+(x + n ) 2 T ^ ' >' = s i n _ 1 ^ t a n x > ' 1
3
- + ;/ n - x if x < 0 6 2
r
Q.12 f'(x) =
Q 11. f(x) = U+x, Q- 1 7
1 X1 —he lnx x y x inx + x Aix Iny + l x" £nx (1-x-y^na)
dy -V3(12 + ln2) & = ln4
x cos x - sin x . if x Tt 0 if x = 0
if x > 0
Q.20
i ;f"(0) = - ~
4x
1 Q 22. f(x) = 2 tan"1 x Q 24. f(x) = 0 or f(x) = ekx
EXERCISE-III 5 QL-
1 Q2.-
1 Q3.-?
Q 4. a= 1
Q 7. a = 120; b = 60; c = 180 Q 8. 2 Q9-2/5 Q.10 f(0) = 1; differentiable atx= 0, f(0 + ) = -(1/3); f(0") = -(1/3) Q12. a = 6 , b = 6 , c = 0 ; — 40 Q 14. fis cont. but not derivable at x = 0
Q5. -
1
Q 6. 1
Q 11. - 6
Q 13. f is discont. only at x = 0 in (-co, 2)
EXERCISE-IV Q.l
(a) a = 3, b = 12, c = 9 (b) 2^x +' xl [/n2(x) • (sin x2)/n x (2x2 /nx. cot x2 + In (sin x2)) + 2x2 . /nx (cot x2) - In (sin x2)] xln x
Q.3
Domain of f (x) = R - ( - 2, 0}; Range off(x)= R - { - 1/2,1} ; ^ [ f ( x ) ] =
Domain of f" 1 (x) = R - { - 1/2, 1} Q.4 (a) B Q.6 (a) C;(b)B; (c)g'(0) = 0 Q.7 C
<§Bansal Classes
Q.5 (a) A;
M.O.D. andL'HospitalRule
(1
Jx)2
(b) a = 1
[12]
| BAN SAL CLASSES ^
TARGET IIT JEE 2007
PHYSICAL CHEMISTRY XII & XIII
FINAL PRA CTICE PROBLEMS FOR IIT JEE-2007 (With Answers)
ALL THE BEST FOR JEE - 2 0 0 7
Q. 1
Wave number of the second line of Paschen series of hydrogen atom is (RH = 109700 cm ! ) (A) 18750 A (B) 3452 A (C) 7801 A (D) 1542656 A
Q.2
Radiation corresponding to the transition n = 4 to n = 2 in hydrogen atoms falls on a certain metal (work function = 2.0 eY). The maximum kinetic energy of the photoelectrons will be (A) 0.55 eV (B) 2.55 eV (C) 4.45 eY (D) none
Q.3
The maximum number of electrons in a subshell having the same value of spin quantum number is given by (A) 1 + 2 (B) 21 + 1 (C) 2(2/+1) (D) None
Q.4
1 When a graph is plotted with log (Keq) v/s —, then intercept on log (Keq) axis will be AH° ( A )2.303R T^T
Q.5
ASo v(D) ' 2.303R
AS° (Q— R
For the synthesis of NH3 from the Haber process starting with stoichiometric amount of N2 and H2, the attainment of equilibrium is predicted by which curve: N2(g) + 3H2(g) >2NH,(g)
t: (A) Q.6
AH0 v( B7 ) - T T T ^ r 2.303RT
I)'
-NI I,
Nil, t; '
NH3
Hn
(C)lN
ft
2
(D)L
-
N
2
-H,
The reaction A(g) + B(g) ^ C(g) + D(g) is elementry 2nd order reaction opposed by elementry second order reaction. When started with equimolar amounts of A and R, at equilibrium, it is found that the conc. of Ais twice that of C. Specific rate for forward reaction is 2 x 10"3 mol 1 Lsec ! . The specific rate constant for backward reaction is (A) 5.0 x 10"4 M"1 sec"1 (B) 8 x 10"3 M"1 sec"1 (C) 1.5 x 102 M 1 sec"! (D) none of these Question No. 7 to 8 ( 2 questions) A 0.5 L reaction vessel which is equipped with a movable piston is filled completely with a IM aqueous solution of H 2 0 2 .The H 2 0 2 decomposes to H 2 0 and 0 2 (g) in a first order process with half life 10 hrs at 300 K. As gas decomposes, the piston moves up against constant external pressure of 750 mm Hg.
Q.7
What is the net work done by the gas from the start of sixth hour till the end of 10 hrs? (A) 100 J (B) 120 J (C) 130 J (D) 150 J
Q.8
If AH for decomposition of H202(aq.) is X K J/mole, how much heat has been exchanged with surrounding to maintain temperature at 300 K (in KJ) from the start of 6th hour till the end of 10th hour. (A) -0.1035 X (B)-0.2070 X (C)-0.052 X (D) 0.026 X
Q.9
During study of a liquid phase reaction A (aq) > B (aq) + C (aq) the variation in concentration of B with time is given t/min 0 10 20 30 oo conc. (B) mole/L 0 0.1 0.19 0.271 1 The initial rate of reaction was? (A) 1.76 x lo-4 M sec"1 (B) 2.76 x lO^M sec"1 1 (C) 3.86 x iO^M sec(D) 2 xio^Msec- 1
(feBansal Classes
Problems for JEE-2007
[7 ]
Q.IO
A catalyst increases the (A) rate of forward reaction only (B) free energy change in the reaction (C) rates of both forward and reverse reactions (D) equilibrium constant of the reaction
Q.ll
The conductance of a salt solution (AB) measured by two parallel electrodes of area 100 cm2 separated by 10 cm was found to be 0.0001 Q _1 . If volume enclosed between two electrode contain 0.5 mole of salt, what is the molar X conductivity(Scm2mol_1) of salt at same conc. (A) 0.01 (B) 0.02 X 5 10cm (C) 2 x 10" (D) none of these
Y-
400cm2
Q.12
Arrange the following electrolytes in the increasing order of coagulation power for the coagulation of A S 2 S 3 solution. (I) Na 3 P0 4 (II) MgCl2 (III) A1C13 ( A ) I < II < III (B) III = II < I ( C ) I <111 < 1 1 (D) III < I < II
Q.13
If Pd v/s P (where P denotes pressure in atm and d denotes density in gm/L) is plotted for H2 gas d (ideal gas) at a particular temperature. If dP (Pd) (A) 40 K
(B) 400 K
P=8.21atm
(C) 20 K
10, then the temperature will be (D) none
Q.14
A sample of phosphorus that weighs 12.4 gm exerts a pressure 8 atm in a 0.821 litre closed vessel at 527°C. The molecular formula ofthe phosphorus vapour is (A)P 2 (B)P4 (C)P 6 (D)P g
Q.15
For a gas deviation from ideal behaviour is maximum at (A) 0°C and 1.0 atm (B) 100°C and 2.0 atm (C) -10°C and 1.0 atm (D) - 1 0 ° c a n d 4 0 a t m
Q.16
An ideal gaseous mixture of ethane (C2H6) and ethene (C2H4) occupies 28 litre at STP. The mixture reacts completely with 128 gm 0 2 to produce C0 2 and H2Q. Mole fraction at C2H6 in the mixture is (A) 0.6 (B) 0.4 (C) 0.5 (D) 0.8
Q.17
100 cm3 of a solution of an acid (Molar mass = 82) containing 39 gm of the acid per litre were completely neutralized by 95.0 cm3 of aq. NaOH containing 20 gm of NaOH per 500 cm3. The basicity of the acid is (D) data insufficient (A) 1 (B) 2 (C) 3
Q.18
SPM Study the following figure and choose the correct options. i (A) There will be no net moment of any substance across the membrane 0.2 M a q . ;0,15Maq. (B) CaCL will flow towards the NaCl solution. NaCl Solution I CaCl-, Solution (C) NaCl will flow towards the CaCl2 solution. (D) The osmotic pressure of 0.2 M NaCi is lower than the osmotic pressure of 0.15 M CaCl2, assuming complete dissociation of electrolyte.
Q.19
What is [Ag' J in a solution made by dissolving both Ag 2 Cr0 4 and Ag 2 C 2 0 4 until saturation is x 10 ! 2 reached with respect to both salts. K sp (Ag 2 C 2 0 4 ) = 2 x 10 n , K sp (Ag 2 Cr0 4 ) : 4 5 6 (A) 2.80 x 10(B) 7.6 x 10" (C) 6.63 x !0" " (D) 3.52 x
^
(feBansal Classes
Problems for JEE-2007
[7 ]
0,20
A3B2 is a sparingly soluble salt of molar mass M (g moH) and solubility x g lit-1. The ratio ofthe molar concentration of B3~ to the solubility product of the salt is x5
1 M4
1 M4
Q.21
A 1 litre solution containing NH4C1 and NH4OH has hydroxide ion concentration of 10~6 mol/lit. Which of the following hydroxides could be precipitated when the solution is added to 1 litre solution of 0.1 M metal ions i Ag(OH) (K sp = 5 x lCr3) II Ca(OH)2 (K sp = 8 x l (r 6 ) 11 ffl Mg(OH)2 (K sp = 3 x 1CT ) IV Fe(OH)2 (K sp = 8 x 10"16) (A) I, II, IV " (B) IV (C) III and IV (D) II, III, IV
Q.22
From separate solutions of four sodium salts NaW, NaX, NaY and NaZ had pH 7.0, 9.0, 10.0 and 11.0 respectively. When each solution was 0.1 M, the strongest acid is: (A) HW (B) HX (C) HY (D) HZ
Q.23
AG for the conversion of 2 mol of C6H6(f) at 80°C (normal boiling point) to vapour at the same temperature and a pressure of 0.2 atm is (A) -9.44 Kcal/mol (B) -2.27 Kcal/nioi (C) -1.135 Kcal/moi (D) zero
Q.24
A certain ideal gas has C v m = a + bT, where a = 25.0 J/(mol. K) and b = 0.03 J (mol.K2), Let 2 mole of this gas go from 300 K and 2.0 litre volume to 600 K and 4.0 litre. AS,,.,. gdS is (A) 32.08 J/K (B)-1-2 08 J/K (C) 64.17 J/K (D) None
Q.25
What is ArG (KJ/mole) for synthesis of ammonia at 298 K at following sets of partial pressure: N2(g) + 3H2(g) ^ 2NH 3 (g); ArG° = -33 KJ/mole. [Take R = 8.3 J/K mole, log2 = 0.3; log3 = 0.48]
Q.26
Gas
N2
H,
NH3
Pressure (atm)
1
3
0.02
(A)+ 6.5 (B) - 6.5 (C) + 60.5 (D)-60.5 Give the correct order of initials T (true) or F (false) for following statements. (I) Two electrons in the same atom can have the same set of three quantum numbers (II) The lowest energy configuration for an atom with electrons in a set of degenerate orbital s is that having the maximum number of unpaired electrons with the same spin. (III)
The value of (PV) / (RT) at critical state is 0.325
Tr 8 The ratio of Critical Constants -—- is ——— pr Rxb c (A) TTFF (B) FTFT (C)FFTT (IV)
Q.27
(D) TTTF
Give the correct order of initials T (true) or F (false) for following statements. (i) The ratio of the radii of the first three Bohr orbit of hydrogen atom is 1 : 8 : 27 (ii) (iii)
A
m (molar conductivity) for a strong electrolyte decreases as the electrolyte concentraton increases. The frequency of a green light is 6 x io 14 Hz, then its wavelength is 500 nm
(iv)
The mass of an equivalent of H 3 P0 4 when it combines with NaOH using the equation H3PG4 + 2NaOH > Na2HP04 + 2H 2 0 is 49 g. ( A)FFTT (B) FTFT * (C) TFTF (D)FTTT
^Bansa I Classes
Problems for JEE-2007
[4]
Q.28
Give the correct order of initials T (true) or F (false) for following statements I. An azeotropic solution of two liquids has a boiling point lower then either of them when it shows negative deviation from Raoult's Law. II. The fraction of volume occupied by atoms in a body centered cubic unit cell is 0.74. III. In the face-centered cubic unit cell of closest packed atoms, the radius of atoms in terms V'3a of edge length (a) of unit cell is —— IV.
If the anions (Y) form hexagonal closest packing and cations (X) occupy only 2/3 octahedral voids in it, then the general formula of the compound is X0Y3(present in 2 : 3). (A)FFFT (B)FFTT (C) TFFT ' (D) TTFF
Q.29
Which of the following statement(s) is/are correct: Statement (a) : For the reaction Br, + QH~
> B r + BrOT + H2G, The equivalent weight of
3M Br2 is - y where M = Molecular weight. Statement (b) : For an aqueous solution to be neutral it must have pH = 7 Statement (c) : For the reaction 1 1 — N2 +
NO
equilibrium constant is KL
and 2NO ^ N 2 + (X equilibrium constant is K2 then K2. KL = 1 Statement (d) : If the partial orders are equal to corresponding coefficients in the balanced reaction, the reaction must be elementary (simple) (A) Statement a (B) Statement b, d (C) Statement a. b, d (D) All Q.30
Give the correct order of initials T (true) or F (false) for following statements. I. If the equilibrium constants for the reaction Br, ^ 2Br at 500 K and 700 K are 10"10 and 10~5 respectively, then reaction is exothermic. II. When a solution prepared by mixing 90 ml of HQ solution (pH = 3), 10 mi solution of NaOH (pH = 11) and 60 ml water will be 3.1. III.
In the nuclear reaction p 5 U
»
7
Pb. the total number of a and (3 particles lost would
bell. An aqueous solution of an alcohol (B P 56°C) in water (B.P. 100°C) has vapour pressure more than that of water. (A)TFTT (B)FFFT (C) FFTF (D)FFTT
IV.
(feBansal Classes
Problems for JEE-2007
[7 ]
ANSWER Q.l _1
V = - = Rt Q.2
A
Sol.
En = -
13.6
3
1_
2 _
y = 7801 cm"1
52
eV ; E2 = -
13.6
;
=
13.6
eV/atom
AE = E4 - E2 = 2.55 eV Absorbed energy = work function of metal + K.E. 2.55 = 2 + K.E. K.E. = 0.55 eV Ans. Q.3
Q.4 Q.5 Q.6 Sol. Q.7 Sol.
B D A B k = k
k K
c
/ e£
i
b
eq
At the end of 5 hours A
A
°
at the end of 10th hours =
\ , 2
2~
An
V2-1
A 0r
Aq = 0 . 2 0 7 ^ ~ 2 ~ V2 => 0.2071 mole/L amount decayed in 0.5 L = 0.2071/2 \,2
1
I
0.2071" f 0.2071^ moles of 0 2 formed = ~ ( mole 4 2 j W = -PAV = -nRT = - (0.2071/4) x 8.314 x 300 = -129.14 Joule Q.8
A
Sol.
Heat exchanged (q )
Q.10 Q.ll
A C B
Sol.
G*
Q.9
/ a
10 100
^0.2071
x AH = -0.1035 X kJ/mole
0.1 ; G = 0.0001 S ; V = 100 x 10= 1000 cm3 = 1 litre K = G G * = 0. x 0 . 0 0 0 1 = 10- 5
KXIOO
(0. l x 0 . 0 0 0 1 ) x 1 0 0 0
M
0.5
(feBansal Classes
= 0.02 Scm2 moH Ans.
Problems for JEE-2007
[7]
Q.12 Sol.
As2S3 is negativley charged sol., so coagulation power Al3+ > Mg 2+ > Na+
Q.13 Sol.
A PM = dRt f
M
\
d(Pd) _ 2PM dP ~ RT
V RT J T = 40 K Q.14 Q.15 Q.16 Sol.
10
2x8.21x2 0.0821xT
B D B C2H6 + 3.5 0 2 -> 2C0 2 + 3H 2 0 ; C2H4 + 3 0 , -> 2C0 2 + 21^0 Let vol. of ethane is x 22.4 x 4 = 3.5 x + 3 ( 2 8 - x ) 89.6 = 3.5 x + 84 - 3x x = 11.2 litrE at const. T & PV a n Mole fraction of C 2 H 6 in mixture
Q.17 Sol.
11.2
28
xlOO = 0.4
B meq" + acid = meqn of base Acid: 3.9 gm acid present in 100 cm3 Base: 20 gm NaOH present in 500 cm3 NIV,=N 2 V 2 39 (—) InJ
Q.18 Sol.
= 10
f 20 x 1000 = 95 x I xlOO V40500
D 7tj = ij x CjRT ; 7t2 = i2 x C2RT = 2 x 0.2RT; =3xO."l5RT
n=2
[only solvent (H 2 0) molecules can passed through SPM]
h
D
Sol.
Ag 2 Cr0 4 (s) ^ 2Ag+ (aq) + CrO 2 " (aq) 2x + 2y
x
+
Ag 2 C 2 0 4 (s) ^ 2Ag (aq) + C 2 0 2 - (aq) 2y + 2x
y
sp, x 2x10" x —— = - = _ ;t=> - =0.1 K so y 2xl0 y K
2 x 10"11 = (2x + 2y)2.y = (2.2y)2.y 2 x 10~n = 4.84 y3 y = 1.6 x 10"4 x = 0.16 x 10"14 Total (Ag+) = 2x + 2y = (2 x 0.16 + 2 x 1.6) x 10^ = 3.52 x 10"4
(feBansal Classes
Problems for JEE-2007
Ans.
[7]
Q.20 Sol.
C Solubility
S = x/M 3A 2 " (aq) + 2 B 3 " (aq)
A3B2(S) ^
3S ,
2S „
Ksp=(3S)W
=»
[A3-]
.
JOBS' ; L - J - ^
2S
Q.21
B
Sol.
when 1 litre each are mixed [OH"] = 10"6 M (Buffer solution) Mn+ = 0.05 M for M2+ = Q = [0.5][10-6]2 = 5 x 10~14 Q > K s p only for Fe2+ only Fe(OH)2 is ppt.
Q.22
A
Q.23
B
Sol.
C6H6(/) -> C6H6(g) At 1 atm, AG = 0 we know for reversible processes, dG = VmdP - SdT dT = 0, dG = VmdP for a reaction, d(AG) = n(AVm)dP (for n moles) P
P
2
=»
1
M4
^
(Gibbs equation)
2
Jd(AG) - n j(AV m )dP; Pi Pi n = 2 and, AVm = V
P t = 1 atm and P 2 = 0.2 atm
- V
g
?
AG0.2 atm - AG, a t m = n J =>
1
R T
1
g
=
H P
^
~
^
AG
1 atm =
0
0.2
AG0 2 atm - 2 x RT In — = - 2.27 Kcal/mol
Q.24
C
Sol.
, r nC v dT jds = J — +
r dT „ f ( a + bT)dT j nR — = 2j — +
„fRJW n r (25 + 0.03T)dT 2J - d V = 2j i — ' — + 2R^n2
= 2[25 ^n2 + 0.03 [T2 - T J ] + 2Rin2 => 2[25 x 0.693 + 0.03 x 300] + 2 x 8.314 x 0.693 = 52.65 + 11.52 AS = 64.17 J/k Q.25
Sol.
D 0
AG = AG + RT In
P2 NH3
Pn 2 xPH 2 AG = -33000+ 8.314 x 2.303 log Q.26
A
Q.27
D
(feBansal Classes
Q.28
A
(0.02)
f - = - 60.5 kJ/mole 1x3 Q.29 A Q.30 D Problems for JEE-2007
[7 ]
1
S BANSAL CLASSES Target IIT JEE 2007
MATHEMATICS Daily Practice
Problems
CLASS : XII (ABCD)
Q.>
DPP ON PROBABILITY DPR NO.- 1 st After 1 Lecture 100 cards are numbered from 1 to 100. The probability that the randomly chosen card has a digit 5 is: (A) 0.01 (B) 0.09 (P^0.19 (D) 0.18
Q.2
A quadratic equation is chosen from the set of all the quadratic equations which are unchanged by squaring their roots. The chance that the chosen equation has equal roots, is: (A) 1/2 (B) 1/3 (C) 1/4 (D) 2/3
•Q.3 i
If the letters ofthe word "MISSISSIPPI" are written down at random in a row, the probability that no two S's occur together is : (A) 1/3 (B) 7/33 (C) 6/13 • (D) 5/7
Q.4
A sample space consists of 3 sample points with associated probabilities given as 2p, p 2 ,4p - 1 then (A)p=VU-3
(B)VT0-3
(C) 1/4
(D)none
Q.5
A committee of 5 is to be chosen from a group of 9 people. The probability that a certain married couple will either serve together or not at all is: (A) 1/2 (B) 5/9 (C) 4/9 (D) 2/3
Qj)
There are only two women among 20 persons taking part in a pleasure trip. The 20 persons are divided into two groups, each group consisting of 10 persons. Then the probability that the two women wi 11 be in the same group is: (A) 9/19 (B) 9/38 (C) 9/35 (D) none
Q.7
A bag contain 5 white, 7 black, and 4 red balls, find the chance that three balls drawn at random are all white.
Q. 8
If four coins are tossed, find the chance that there should be two heads and two tails.
Q.9
Thirteen persons take their places at a round table, show that it is five to one against two particular persons sitting together.
Q.10
In shuffling a pack of cards, four are accidentally dropped,findthe chance that the missing cards should • be one from each suit.
^ Bansal Classes
[1]
Q.ll
A has 3 shares in a lottery containing 3 prizes and 9 blanks, B has 2 shares in a lottery containing 2 prizes ancl|6 blanks. Compare their chances of success.
Q.12
There are three works, one consisting of 3 volumes, one of 4 and the otlrer of one volume. They are placed 3 on a shelf at random, prove that the chance that volumes of the same works are all together is
.
l
Q.13 Q.14
The letter forming the word Clifton are placed at random in a row. What is the chance that the two vowels come together? Three bolts and three nuts are put in a box. If two parts are chosen at random, find the probability that one is a bolt and one is a nut.
Q.15
There are 'm' rupees and 'n ten nP s, placed at random in a line. Find the chance of the extreme coins being both ten nP's.
Q.16
A fair die is tossed. If the number is odd, find the probability that it is prime.
Q.17
Three fair coins are tossed. If both heads and tails appear, determine the probability that exactly one head appears.
Q.18
3 boys and 3 girls sit in a row. Find the probability that (i) the 3 girls sit together, (ii) the boys are girls sit in alternative seats.
Q.19
A coin is biased so that heads is three times as likely to appear as tails. Find P (H) and P (T).
Q.20
In a hand at "whist" what is the chance that the 4 kings are held by a specified player?
'If I
MATHEMATICS
J | BANSAL CLASSES ^ T a r g e t IIT JEE 2 0 0 7
Daily Practice
CLASS : XII (ABCD) Q. 1
DPP ON PROBABILITY After 2nd Lecture Given two independent events A, B such that P (A) = 0.3,P (B) = 0.6. Determine (i) P (A and B) (ii) P (A and not B) (iii) P (not A and B) (iv) P (neither A nor B) (v) P (A or B)
Problems DPR NO.- 2
Q.2
A card is drawn at random from a well shuffled deck of cards. Find the probability that the card is a (i) king or a red card (ii) club or a diamond (iii) king or a queen (iv) king or an ace (v) spade or a club (vi) neither a heart nor a king.
Q.3
A coin is tossed and a die is thrown. Find the probability that the outcome will be a head or a number greater than 4.
Q.4
Let A and B be events such that P(A) = 4/5. P(B)= 1/3, P(A/B) = 1/6, then (a) P(A n B); (b) P(A u B) ; (c) P(B/A) ; (d) Are A and B independent? 1 1 1 If A and B are two events such that P (A) = - , P (B) = - and P (A and B) = - , find
Q.5
(i) P (A or B),
|
(ii) P (not A and not B)
A 5 digit number is formed by using the digits 0,1,2,3,4 & 5 without repetition. The probability that the number is divisible by 6 is: (A) 8% (B) 17% (C) 18% (D) 36% Q.7
An experiment results in four possible out comes S p S2, S3 & S4 with probabilities p,, p2, p3 & p4 respectively. Which one of the following probability assignment is possbile. [Assume S, S, S3 S4 are mutually exclusive] (A) Pj = 0.25 , p", = 0.35, p3 = 0.10 , p4 = 0.05 (B)p, = 0.40. p2 = -0.20 , p3 = 0.60 , p4 = 0.20 (C) p, = 0.30 , p~ = 0.60, p3 = 0.10 , p4 = 0.10 (D) p, = 0.20, p2 = 0.30 , p3 = 0.40, p4 = 0.10
Q.8
In throwing 3 dice, the probability that atleast 2 of the three numbers obtained are same is (A) 1/2 (B) 1/3 (C) 4/9 (D) none
' (<5.9 /• There are 4 defective items in a lot consisting of 10 items. From this lot we select 5 items at random. The probability that there will be 2 defective items among them is 1 ~2 Q.10
( B )
2 5
5 2l
( D )
10 2l
From a pack of 52 playing cards, face cards and tens are removed and kept aside then a card is drawn at random from the ramaining cards. If A : The event that the card drawn is an ace 11: The event that the card drawn is a heart S : The event that the card drawn is a spade then which ofthe following holds ? (A) 9 P(A) = 4 P(H) (B) P(S) = 4P (A n H) (C) 3 P(H) = 4 P(A u S) (D) P(II) = 12 P(A n S)
^ Bansal Classes
113]
Q.ll
6 married couples are standing in a room. If 4 people are chosen at random, then the chancc thai exactly one married couple is among the 4 is : 16
8
(A) 55 Q.12
17
(B) 55
(C) 55
(D)
24
55
The chance that a 13 card combination from a pack of 52 playing cards is dealt to a player in a game of bridge, in which 9 cards are of the same suit, is 13 a . i3p ,39 r /ii. 13 r - 3 9 r r -39r (A) 117; (B) 52^ (C) JT^ (D) none 13
13
13
Q.13
If two of the 64 squares are chosen at random on a chess board, the probability that they have a side in common is: (A) 1/9 (B) 1/18 (C) 2/7 * (D) none
Q.14
Two red counters, three green counters and 4 blue counters are placed in a row in random order. The probability that no two blue counters are adjacent is 1 7 5 ' (A) (B)— (C)— (D) none
tnf P^" ' Q.15
The probabilities that a student will receive A, B, C or D grade are 0.40,0.35,0.15 and 0.10 respectively. Find the probability that a student will receive (ilj not an A grade (ii) B or C grade (iii) at most C grade
Q.16
In a single throw of three dice, determine the probability ofgetting (i) a total of 5 (ii)atotalofatmost5 (iii) a total ofat least 5.
Q.17
A die is thrown once. If E is the event "the number appearing is a multiple of 3" and F is the event "the number appearing is even",findthe probability of the event "E and F". Are the events E and F independent?
Q.18
In the two dice experiment, if E is the event of getting the sum of number on dice as 11 and F is the event Qj|"getting a number other than 5 on the first die, find P (E and F). Are E and F independent events?
Q.19
A natural number x is randomly selected from the set of first 100 natural numbers. Find the probability 100 that it satisfies the inequality. x+ >50
Q.20
3 students A and B and C are in a swimming race. A and B have the same probability of winning and each is twice as likely to win as C. Find the probability that B or C wins. Assume no two reach the winning point simultaneously. A box contains 7 tickets, numbered from 1 to 7 inclusive. If 3 tickets are drawn from the box, one at a time, determine the probability that they are alternatively either odd-even-odd or even-odd-even.
•
Q.21 ^
x
Q.22
5 different marbles are placed in 5 different boxes randomly. Find the probability that exactly two boxes remain empty. Given each box can hold any number of marbles.
Q.23
South African cricket captain lost the toss of a coin 13 times out of 14. The chance of this happening was 7 (A)^I
Q.24
1 (B)^]j
13 (C)^4
13 (D)^y
There are ten prizes, five A's, three B's and two C's, placed in identical sealed envelopes for the top ten contestants in a mathematics contest. The prizes are awarded by allowing winners to select an envelope at randomfromthose remaining.; When the 8th contestant goes to select the prize, the probability that the remaining three prizes are one A, one B and one C, is (A) 1/4 (B) 1/3 (C) 1/12 (D) 1/10 »
(i§
Bansal Classes
[4]
MATHEMATICS
I5 BAN SAL CLASSES t a r g e t IIT JEE 2 0 0 7 CLASS: XII (ABCD)
Daily Practice
Problems
DPP ON PROBABILITY
DPR NO.- 3
rd
After 3 Lecture Whenever horses a, b, c race together, their respective probabilities of winning the race are 0.3,0.5 and 0.2 respectively. If they race three times the probability that "the same horse wins all the three races" and the probablity that a, b, c each wins one race, are respectively
Q.l
(A)
9_
50
50
(B)
16 100' 100
(C)
12 15
50 ' 50
(D)
10 _8_
50
50
Q.2
Let A & B be two events. Suppose P(A) = 0.4, P(B) = p & P(A u B) = 0.7. The value of p for which A & B are independent is : (A) 1/3 (B) 1/4 (C) 1/2 (D) 1/5
Q.3
A & B are two independent events such that P (A ) = 0.7, P (B ) = a & P(A u B ) = 0.8, then, a = (A) 5/7
(B) 2/7
(C) 1
(D) none
Q.4
A pair of numbers is picked up randomly (without replacement) from the set {1,2,3,5,7,11.12,13,17,19}. The probability that the number 11 was picked given that the sum of the numbers was even, is nearly: T2"/?^ (A) 0.1 (B) 0.125 (C) 0.24 f (D) 0.18 A 3
Q.5
For a biased die the probabilities for the diffferent faces to turn up are given below: Faces: 1 2 3 4 5 6 Probabilities: 0.10 0.32 0.21 j 0.15 0.05 0.17 The die is tossed & you are told that either face one or face two has turned up. Then the probability that it is face one is: (A) 1/6 (B) 1/10 (C) 5/49 (D) 5/21 A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the determinant chosen has the value non negative is : (A) 3/16 (B) 6/16 (C) 10/16 (D) 13/16
Q.7 r
Q.8
Q.9
15 coupons are numbered 1,2,3, ,15 respectively. 7 coupons are selected at random one at atime with replacement. The probability that the largest number appearing on a selected coupon is 9 is: f9y (A) 16.,
(B)
V15y
(C)
(D)
97-87
15'
A card is drawn & replaced in an ordinary pack of 52 playing cards. Minimum number of times must a card be drawn so that there is atleast an even chance of drawing a heart, is (A) 2 , (B) 3 (C) 4 (D) more than four ^s. r An electrical system has open-closed switches S,, S2 and S3 as shown. S 3
The switches operate independently of one another and the current will flow from A to B either if S, is 1 closed or if both S, and S3 are closed. If P(S,) = P(S2) = P(S3) = , find the probability that the circuit will work. (i§ Bansal Classes
[5]
Q.IO (i) (ii) Q.ll Q.12
A certain team wins with probability 0.7, loses with probability 0.2 and ties with probability 0.1. The team plays three games. Find the probability V 3 ( ( o - 1 ) l C o * 0 / ' c t^ "S that the team wins at least two of the games, but lose none, that the team wins at least one game. ! 0') j — V An integer is chosen at randomfromthe first 200 positive integers. Find the probability that the integer is divisible by 6 or 8. A clerk was asked to mail four report cards to four students. He addresses four envelops that unfortunately paid no attention to which report card be put in which envelope. What is the probability that exactly one ofthe students received his (or her) own card?
Q 13 Find the probability of at most two tails or at least two heads in a toss of three coins. Q.14
What is the probability that in a group of (i) 2 people, both will have the same date of birth. (ii) 3 people, at least 2 will have the same date of birth. Assume the year to be ordinarry consisting of365 days.
11
Ql 15 The probability that a person will get an electric contract is — and the probability that he will not get 4 2 plumbing contract is —. If the probability of getting at least one contract is —, what is the probability that he will get both ? ive horses compete in a race. John picks two horses at random and bets on them. Find the probability that John picked the winner. Assume dead heal. Q.17
Two cubes have their faces painted either red or blue. The first cube has five red faces and one blue face. When the two cubes are rolled simultaneously, the probability that the two top faces show the same colour is 1/2. Number of red faces on the second cube, is (A)l (B)2 (C) 3 (D)4
Q.18
AH and W appear for an interview for two vaccancies for the same post. P(H)= 1/7; P(W) = 1/5. Find the probability ofthe events (a) Both are selected (b) only one of them is selected (c) none is selected.
Q.19
A bag contains 6R, 4W and 8B balls. If 3 balls are drawn at random determine the probability of the event (a) all 3 are red; (b) all 3 are black; (c) 2 are white and 1 is red; (d) at least 1 is red; (e) 1 of each colour are drawn (f) the balls are drawn in the order of red, white, blue. The'l^dds that a book will be favourably reviewed by three independent critics are 5 to 2,4 to 3, and 3 to 4 respectively. What is the probability that of the three reviews a majority will be favourable? In a purse are 10 coins, all five nP's except one which is a rupee, in another are ten coins all five nP's. Nine coins are takenfromthe former purse and put into the latter, and then nine coins are takenfromthe latter and put into the former. Find the chance that the rupee is still in the first purse.
\
|
'
Q.22
A, B, C in order cut a pack of cards, replacing them after each cut, on condition that the first who cuts a spade shall win a prize. Find their respective chances, 'if
Q.23
A and B in order drawfroma purse containing 3 rupees and 4 nP's, find their respective chances of first drawing a rupee, the coins once drawn not being replaced.
^ Bansal Classes
113]
MATHEMATICS
J | BANSAL CLASSES 1 8 T a r g e t IIT JEE 2007
Daily Practice
Problems
CLASS: XII (ABCD) Q. 1
D P P ON P R O B A B I L I T Y DPR NO.- 4 t h After 4 Lecture There are n different gift coupons, each of which can occupy N(N > n) different envelopes, with the same probability 1/N P j: The probability that there will be one gift coupon in each ofn definite envelopes out ofN given envelopes P2: The probability that there will be one gift coupon in each of n arbitrary envelopes out of N given envelopes Consider the following statements N! n t 1 n r2 W ' l " Nn 2 N (N - n ) ! n! N! (iv)P2=Nn(N_n)! (v)p,= ^ r Now, which of the following is true (A) Only (i) (B) (ii) and (iii)
Q.2
(C) (ii) and (iv)
(D) (iii) and (v)
The probability that an automobile will be stolen and found wifhing one week is 0.0006. The probability that an automobile will be stolen is 0.0015. The probability that a stolen automobile will be found in one week is (A) 0.3 (B) 0.4 (C) 0.5 (D)0.6 One bag contains 3 white & 2 black balls, and another contains 2 white & 3 black balls. Aball is drawn from the second bag & placed in thefirst,then a ball is drawnfromthefirstbag & placed in the second. When the pair of the operations is repeated, the probability that thefirstbag will contain 5 white balls is: (A) 1/25 (B) 1/125 (C) 1/225: (D) 2/15
j
A child throws 2 fair dice. If the numbers showing are unequal, he adds them together to gjlit his final score. On the other hand, if the numbers showing are equal, he throws 2 more dice & adds all 4 numbers showing to get hisfinalscore. The probability that hisfinalscore is 6 is: 145 146 147 148 ( A ) B ) D n% < f^ < >T296 Q.5
A person draws a card from a pack of 52 cards, replaces it & shuffles the pack. He continue^ doing this till he draws a spade. The probability that he will fail exactly thefirsttwo times is: H' (A) 1/64 (B) 9/64 (C) 36/64 (D) 60/64
Q. 6
Indicate the correct order sequence in respect of the following : I. If the probability that a computer will fail during thefirsthour of operation is 0.01, then if we turn on 100 computers, exactly one will fail in thefirsthour of operation. II. A man has ten keys only one of whichfitsthe lock. He tries them in a door one by one discarding the one he hasliied. The probability thatfifthkey fits the lock is 1/10. III. Given the events A and B in a sample space. If P(A) = 1, then A and B are independent. IV. When a fair six sided die is tossed on a table top, the bottom face can not be seen. The probability that the product of the numbers on thefivefaces that can be seen is divisible by 6 is one. (A) FTFT (B)FTTT (C)TFTF (D)TFFF
(i§ Bansal Classes
[7]
Q.7
An unbaised cubic die marked with 1,2,2,3,3,3 is rolled 3 times. The probability of getting a total score of 4 or 6 is 50 60 16 (D)none (C) (B) (A) 216 216 216
Q.8
lA bag contains 3 R & 3 G balls and a person draws out 3 at random. He then drops 3 blue balls into the bag & again draws out 3 at random. The chance that the 3 later balls being all of different colours is (A) 15% (B) 20% (C) 27% (D)40%
Q.9
A biased coin with probability P, 0 < P < 1, of heads is tossed until a head appears for thefirsttime. Ifthe probability that the number of tosses required is even is 2/5 then the value of P is (A) 1/4 (B) 1/6 (C) 1/3 (D) 1/2 %
Q.10
If a, b e N then the probability that a2 + b2 is divisible by 5 is (A)
Q.ll
25
(B)
7
11
17
18
In an examination, one hundredI candidates took paper in Physics and Chemistry. Twentyfivecandidates failed in Physics only. Twenty candidates failed in chemistry only. Fifteen failed in both Physics and Chemistry. A candidate is selected at random. The probability that he failed either in Physics or in Chemistry but not in both is (A)
20
3 (B)7
2
(Q-
(D)
U_ 20
Q.12
In a certain game A's skill is to be B's as 3 to 2,findthe chance of A winning 3 games at least out of 5.
Q.13
In each ofa set of games it is 2 to 1 in favour of the winner of the previous game.What is the chance that the player who wins thefirstgame shall wins three at least of the next four?
Q.14 A coin is tossed n times, what is the chance that the head will present itself an odd number of times'? • hn') f Q.15 Afairdieistossedrepeatidly.Awinsifitis 1 or 2 on two consecutive tosses and B wins ifit is 3, 4, 5 or 6 on two consecutive tosses. The probability that A wins if the die is tossed indefinitely, is
K W
(A)
1
£
2 CD) 5
Q.16
Counters marked 1,2,3 are placed in a bag, and one is withdrawn and replaced. The operation being repeated three times, what is the chance of obtaining a total of 6?
Q.17 (a) (b)
A normal coin is continued tossing unless a head is obtained for thefirsttime. Find the probability that number of tosses needed are at most 3. number of tosses are even.
<§ Bansal Classes
18]
mm M R i
Q.18
A purse contains 2 six sided dice. One is a normal fair die, while the other has 2 ones, 2 threes, and 2 fives. Adie is picked up and rolled. Because of some secret magnetic attraction ofthe unfair die, there is 75% chance of picking the unfair die and a 25% chance of picking a fair die. The die is rolled and shows up the face 3. The probability that a fair die was picked up, 1 (D) (A) 7 (B) \ (C) -6 24
Q.19
Before a race the chance of three runners, A, B, C were estimated to be proportional to 5,3,2, but during the race A meets with an accident which reduces nis chance to 1/3. What are the respective chance of B and C now?
^
Q.20 A fair coin is tossed a large number of times. Assuming the tosses are independent which one ofthe following statement, is True? (A) Once the number of flips is large enough, the number of heads will always be exactly half of the total number of tosses. For example, after 10,000 tosses one should have exactly 5,000 heads. (B) The proportion of heads will be about 1/2 and this proportion will tend to get closer to 1/2 as the number of tosses inreases (C) As the number of tosses increases, any long run of heads will be balanced by a corresponding run of tails so that the overall proportion of heads is exactly 1/2 (D) All ofthe above Q.21
A and B each throw simultaneously a pair of dice. Find the probability that they obtain the same score.
Q.22 A is one of the 6 horses entered for a race, and is to be ridden by one of two jockeys B or C. It is 2 to 1 that B rides A, in which case all the horses are equally likely to win; if C rides A, his chance is trebled, what are the odds against his winning? i Direction for Q.23 to Q.25 Let S and T are two events defined on a sample space with probabilities P(S) = 0.5, P(T) = 0.69, P(S/T) = 0.5 Q.23
Q.24
Events S and T are: (A) mutually exclusive (C) mutually exclusive and independent
(B) independent (D) neither mutually exclusive nor independent
The value of P(S and T) (A) 0.3450 (B) 0.2500
(C) 0.6900
(D) 0.350
(C) 0.8450
(D)0
Qr25 The value of P(S or T) (A) 0.6900 (B) 1.19
^ Bansal Classes
• If '
I
113]
J a BANSAL CLASJSES Target IIT JEE 2007
MATHEMATICS Daily Practice
Problems
CLASS: XII (ABCD) Q. 1
DPP ON PROBABILITY DPP. NO.- 5 th After 5 Lecture The first 12 letters ofthe english alphabets are written down at random. The probability that there are 4 letters between A & B is : (A) 7/33 (B) 12/33 (C) 14/33 (D) 7/66
Q.2
Events A and C are independent. If the probabilities relating A, B and C are P (A) = 1/5; P (B) = 1/6 ; P (A n C) = 1/20 ; P (B u|C) = 3/8 then (A) events B and C are independent (B) events B and C are mutually exclusive (C) events B and C are neither independent nor mutually exclusive (D) events B and C are equiprobable
Q. 3
Assume that the birth of a boy or girl to a couple to be equally likely, mutually exclusive, exhaustive and independent of the other children in the family. For a couple having 6 children, the probability that their "three oldest are boys" is 20
*>64
(B)
64
(C)
2 64
(D)
64
Q.4
A and B play a game. A is to throw a die first, and is to win if he throws 6, If he fails B is to throw, and to win if he throws 6 or 5. If he fails, A is to throw again and to win with 6 or 5 or 4, and so on,findthe chance of each player.
Q. 5
Box A contains 3 red and 2 blue marbles while box B contains 2 red and 8 blue marbles. A fair coin is tessed. If the coin turns up heads, a marble is drawnfromA, if it turns up tails, a marble is drawn from bag B. The probability that a red marble is chosen, is 2
(A)
(B)-
3
(C)-
(D)
1
Q.6
A examination consists of 8 questions in each of which one of the 5 alternatives is die correct one. On the assumption that a candidate who has done no preparatory work chooses for each question any one of thefivealternatives with equal probability, the probability that he gets more than one correct answer is equal to: (A) (0.8)8 (B) 3 (0.8)8 (C) 1 - (0.8)8 (D) 1 - 3 (0.8)8
Q. 7
The germination of seeds is estimated by a probability of 0.6. The probability that out of 11 sown seeds exactly 5 or 6 will spring is: (A)
Q. 8
C5. 6 510
(B)
'C 6 (3 5 2 5 )
(C) n c 5 V6
(D) none of these
The probability of obtaining more tails than heads in 6 tosses of a fair coins is: (A) 2/64 (B) 22/64 (C) 21/64 . (D) none
^ Bansal Classes
113]
Q.9
An instrument consists of two units. Each unit must function for the instrument to operate. The reliability of thefirstunit is 0.9 & that of the second unit is 0.8. The instrument is tested & fails. The probability that "only thefirstunit failed & the second unit is sound" is: j (A) 1/7 (B) 2/7 (C) 3/7 (D) 4/7
1 Q.10
LotAeonsistsof3G and 2D articles. Lot B consists of4G and ID article. A new lot C is formed by taking 3 articles from A and 2 from B. The probability that an article chosen at random from C is defective, is (A)
Q.ll
1
(C)
(B)
(D)none
25
A die is weighted so that the probability of different faces to turn up is as given: Number Probability
1 0.2
2 0.1
3 0.1
4 0.3
5 0.1
.
6 0.2
If P (A / B) = pj and P (B / C) = p2 and P (C / A) = p3 then the values of pj, p2, p3 respectively are Take the events A, B & C as A= {1,2, 3}, B = {2, 3, 5} and C = {2,4, 6} (A) Q.12
3
3' 6
4
(C)
3' 6
(D)
2 M 3' 6' 4
If mn coins have been distributed into m purses, n into each find (1) the' chance that two specified coins will be found in the same purse, and (2) what the chance becomes when r purses have been examined and found not to contain either of the specified coins.
Q.13 A box has four dice in it. Three of them are fair dice but the fourth one has the numberfiveon all of its faces. A die is chosen at random from the box and is rolled three times and shows up the facefiveon all the three occassions. The chance that the die chosen was a rigged die, is (A)
Q.14
216 (B)
217
215 219
(C)
216
219
(D)none
On a Saturday night 20% of all drivers in U.S.A. are under the influence of alcohol. The probability that a driver under the influence of alcohol will have an accident is 0.001. The probability that a sober driver will have an accident is 0.0001. If a car on a Saturday night smashed into a tree, the probability that the driver was under the influence of alcohol, is (A) 3/7 (B)4/7 (C) 5/7 (D) 6/7 ;
J
Direction for Q.15 to Q.17 (3 Questions)
A JEE aspirant estimates that she will be successful with an 80 percent chance if she studies 10 hours per day, with a 60 percent cTiance if she studies 7 hours per day and with a 40 percent chance if she studies 4 hours per day. She further believes that she will study 10 hours, 7 hours and 4 hours per day with probabilities 0.1,0.2 and 0.7, respectively Q.15
The chance she will be successful, is (A) 0.28 (B) 0.38
(!§ Bansal Classes
(C) 0.48
(D) 0.58
Q.16
Given that she is successful, the chance she studied for 4 hours, is 6
(A)-
(C)
12
(D)
12
Q.17
Given that she does not achieve success, the chance she studied for 4 hour, is 21_ 20 19 (D) (A) ii (C) (B) 26 26 26 26
Q.18
There are four balls in a bag, but it is not known of what colour they are; one ball is drawn at random and found to be white. Find the charlce that all the balls are white. Assume all number of white ball in the bag to be equally likely.
Q.19
A letter is known to have come eitherfromLondon or Clifton. On the postmark only the two consecutive letters ON are legible. What is the chance that it camefromLondon? -fupp Ajt^e / too *** qH Q.20 A purse contains n coins of unknown value, a coin drawn at random is found to be a rupee, what is the chance that is it the only rupee hi the purse? Assume all numbers of rupee coins in the purse is equally likely. Q.21
One of a pack of 52 cards has been lost, from the remainder of the pack two cards are drawn and are 'found to be spades, find the chance that the missing card is a spade.
Q. 2 2 A, B are two inaccurate arithmeticians whose chance of solving a given question correctly are (1/8) and (1/12) respectively. They solve a problem and obtained the same result. If it is 1000 to 1 against their making the same mistake, find the chance that the result is correct. Q.23 (a) (b) (c) (d) (e) (f)
We conduct an experiment where we roll a die 5 times. The order in which the number read out is important. What is the total number of possible outcomes of this experiment? What is the probability that exactly 3 times a "2" appears in the sequence (say event E)? What is the probability that the face 2 appears at least twice (say event F)? Which of the following are true : E c F , F c E ? Compute the probabilities : P(E n F), P(E/F), P(F/E) |( Are the events E and F independent?
7>
(!§ Bansal Classes
[121
MATHEMATICS
BANSAL CLASSES
Daily Practice Problems
T a r g e t I1T JEE 2 0 0 7
DPP O N PROBABILITY
CLASS: XII (ABCD)
DPP. NO.- 6
A f t e r 6th L e c t u r e A bowl has 6 red marbles and 3 green marbles. The probability that a blind folded person will draw a red marble on the second draw from the bowl without replacing the marblefromthefirstdraw, is
Q.l
(B)
(A)
i
(C)
1
(D)
8
Q.2
5 out of 6 persons who usually work in an office prefer coffee in the mid morning, the other always drink tea. This morning of the usual 6, only 3 are present. The probability that one of them drinks tea is: (A) 1/2 (B) 1/12 (C) 25/72 (D) 5/72
Q.3
Pal's gardner is not dependable, the probability that he will forget to water therosebushis2/3.The rose bush is in questionable condition. Any how if watered, the probability of its withering is 1/2 & if not watered then the probability of its withering is 3/4. Pal went out of station & after returning he finds that rose bush has withered. What is the probability that the gardner did not water the rose bush.
Q.4
The probability that a radar will detect an object in one cycle is p. The probability that the object will be detected in n cycles is: (A) 1 - p" (B) 1 - (1 - p)" (C) p" (D) p(l - p)""1
Q.5
Nine cards are labelled 0,1,2,3,4, 5,6,7,8. Two cards are drawn at random and put on atable in a successive order, and then the resulting number is read, say, 07 (seven), 14 (fourteen) and so on. The probability that the number is even, is (A)
Q. 6
(D)
Two cards are drawn from a well shuffled pack of 52 playihg cards one by one. If A: the event that the second card drawn is an ace and B : the event that the first card drawn is an ace card, then which of the following is true? (A)P(A)=~P(B)=^
(B) P (A)
(C) P (A) =
(D) P (A);
; P (B) = ~
(l + 3p) (1-p) Q.7
(C)j
(B)
If
-
> -
16 221
; P ( B )
_ 4 "?I
(l-2p) &
— ~ — are the probabilities of three mutually exclusive events defined on a
sample space S, then the true set of all values of p is 1 I (A) 3' 2 (B) (C) 4 ' 3
,1
; P (B) = ^
Classes
(D) 4' 2
Probability [43
Q.8
A lot contains 5 0 defective & 50 non defective bulbs. Two bulbs are drawn at random, one at a time, _ It with replacement. The events A, B, C are defined as : jp 1 A= {the first bulb is defective}; B = { the second bulb is non defective} C = {the two bulbs are both defective or both non defective} Determine whether (i) A,B,C are pair wise independent (ii) A,B,C are independent
Q.9
An Urn contains'm' white and 'n' black balls. All the balls except for one ball, are drawn from it. The probability that the last ball remaining in the Urn is white, is
(A)
m m+ n
(B)
n m+n
1 (C) (m + n)!
Q
mn (D) (m + n)!
Q, 10 A Urn contains'm' white and 'n' black balls. Balls are drawn one by one till all the balls are drawn. Probability that the second drawn ball is white, is (A)
n(m + n - l ) (B) (m + n)(m + n -1)
m m+ n
m(m-l) (C) (m + n)(m + n - l ) Q.ll
mn (D) (m + n)(m + n -1)
Mr. Dupont is a professional wine taster. When given a French wine, he will identify it with probability 0.9 correctly as French, and will mistake it for a Californian wine with probability 0.1. When given a Californian wine, he will identify it with probability 0.8 correctly as Californian, and will mistake it for a French wine with probability 0.2. Suppose that Mr. Dupont is given ten unlabelled glasses of wine, three with French and seven with Californian wines. He randomly picks a glass, tries the wine, and solemnly says: "French". The probability that the wine he tasted was Californian, is nearly equal to (A) 0.14 (B) 0.24 (C) 0.34 (D)0.44
Q.12 Let A, B & C be 3 arbitrary events defined on a sample space'S' and if, P(A) + P(B) + P(C) = p,, P(A n B) + P(B n C) + P(C n A) = p2 & P(A n B n C) = p3, then the probability that exactly one of the three events occurs is given by: (A) p, - p2 + p3 (B) P l - p2 + 2p3 (C) p, - 2p2 + p3 (D) p, - 2p2 + 3p3 Q.13
Three numbers are chosen at random without replacement from {1,2,3, the minimum ofthe chosen numbers is 3 or their maximum is 7 is (A)-
(B)
1
(C)
(D)
,10}. The probability that _n 40
Q.14 A biased coin which comes up heads three times as often as tails is tossed. If it shows heads, a chip is drawn from urn-I which contains 2 white chips and 5 red chips. If the coin comes up tails, a chip is drawn from urn-II which contains 7 white and 4 red chips. Given that a red chip was drawn, what is the probability that the coin came up heads? Q.15
In a college, four percent of the men and one percent of the women are taller than 6 feet. Further 60 percent of the students are women. If a randomly selected person is taller than 6 feet, find the probability that the student is a women.
Probability [1
Q.16
If at least one child in a family with 3 children is a boy then the probability that 2 ofthe children are boys, is 3 1 1 (D)? (C) (A) (B)
Q.17
The probabilities of events, A n B, A, B & A u B are respectively in A.P. with probability of second term equal to the common difference. Therefore the events Aand B are (A) compatible (B) independent (C) such that one of them must occur (D) such that one is twice as likely as the other
Q.18
From an urn containing six balls, 3 white and 3 black ones, a p erson selects at random an even number of balls (all the different ways of drawing an even number of balls are considered equally probable, irrespective of their number). Then the probability that there will be the same number of black and white balls among them (B)
(A)
11
15
U_ 30
(C)
(D)
Q. 19 One purse contains 6 copper coins and 1 silver coin; a second purse contains 4 copper coins. Five coins are drawn from thefirstpurse and put into the second, and then 2 coins are drawnfromthe second and put into thefirst.The probability that the silver coin is in the second purse is (A) Q.20
1
(B)
(C)
5 9
(D)
7 persons are stopped on the road at random and asked about their birthdays. Ifthe probability that 3 of K them are born on Wednesday, 2 on Thursday and the remaining 2 on Sunday is ^ , then K is equal to (A) 15
(B) 30
(C) 105
(D)210
Q.21
Two buses A and B are scheduled to arrive at a town central bus station at noon. The probability that bus A will be late is 1/5. The probability that bus B will be late is 7/25. The probability that the bus B is late given that bus A is late is 9/10. Then the probabilities (i) neither bus will be late on a particular day and (ii) bus A is late given that bus B is late, are respectively (A) 2/25 and 12/28 (B) 18/25 and 22/28 (C) 7/10 and 18/28 (D) 12/25 and 2/28
Q.22
A box contains a normal coin and a doubly headed coin. A coin selected at random and tossed twice, fell headwise on both the occasions. The probability that the drawn coin is a doubly headed coin is (A)
Q.23
(B)
8
4 (D)?
(C)
Abox contains 5 red and 4 white marbles. Two marbles are drawn successively from the box without replacement and the second drawn marble drawn is found to be white. Probability that thefirstmarble is also while is (A)
Bansal Classes
(B)|
(C)
1
[15]
r Q.24
A and B in order draw a marble from bag containing 5 white and 1 red marbles with the condition that ' Q. whosoever draws the red marble first, wins the game. Marble once drawn by them are not replaced into the bag. Then their respective chances of winning are 2
1
3 B
2
2
3
&
< >5 5
1 1
In a maths paper there are 3 sections A, B & C. Section A is compulsory. Out of sections B & C a student has to attempt any one. Passing in the paper means passing in A & passing in B or C. The probability ofthe student passing in A, B & C are p, q & 1/2 respectively. If the probability that the student is successful is 1/2 then: (A) p = q = l
(B) p = q =
1
(C) p = l , q = 0
(D) p = l , q
1
Q.26 Abox contains 100 tickets numbered 1,2,3,.... ,100. Two tickets are chosen at random. It is given that the maximum number on the two chosen tickets is not more than 10. The minimum number on them is 5, 1 with probability
(A)
1
(B)
2 11
(C)
3 19
(D)none
Q.27
Sixteen players Sj, s 2 , , s16 play in a tournament. They are divided into eight pairs at random. From each pair a winner is decided on the basis of a game played between the two players of the pair. Assume l 'ir that all the players are of equal strength. The probability that "exactly one of the two players s, & s2 is among the eight winners" is 4
(A)Q.28
7 (B)-
(C)
8_ 15
(D)
15
The number 'a' is randomly selected from the set {0,1,2,3, 98,99}. The number 'b' is selected a b from the same set. Probability that the number 3 + 7 has a digit equal to 8 at the units place, is (A)
16
(C)
16
(D)
16
Q.29
We are given two urns as follows: Urn Acontains 5 red marbles, 3 white marbles and 8 blue marbles. Urn B contains 3 red marbles and 5 white marbles A fair dice is tossed if 3 or 6 appears, a marble is chosenfromB, otherwise a marble is chosenfromA. Find the probability that (i) a red marble is chosen, (ii) a white marble is chosen, (iii) a blue marble is chosen. (Use Tree Diagram)
Q.30
We are given two Urns as follows: Urn Acontains 5 red marbles, 3 white marbles. Urn B contains 1 red marbles and 2 white marbles. A fair die is tossed, if a 3 or 6 appears, a marble is drawn from B and put into A and then a marble is drawn from A; otherwise, a marble is drawnfromA and put into B and then a marble is drawn from B. (Use Tree Diagram) (i) What is the probability that both marbles are red? (ii) What is the probability that both marbles are white?
Probability [
on that ed into
Q.31
Two boys Aand B find the jumble of n ropes lying on thefloor.Each takes hold of one loose end. Ifthe 1 ] . probability that they are both holding the same rope is —— then the number of ropes is equal to 1u1 (B) 100 (D) 50 (A) 101 (C) 51 Direction for Q.32 to Q.35 (4 Questions) Read the passage given below carefully before attempting these questions.
& Ca C. The tat the
A standard deck of playing cards has 52 cards. There are four suit (clubs, diamonds, hearts and spades), each of which has thirteen numbered cards (2, ,9,10, Jack, Queen, King, Ace) In a game of card, each card is worth an amount of points. Each numbered card is worth its number (e.g. a 5 is worth 5 points); the Jack, Queen and King are each worth 10 points; and the Ace is either worth your choice of either 1 point or 11 points. The object of the game is to have more points in your set of cards than your opponent without going over 21. Any set of cards with sum greater than 21 automatically loses.
that m is 5.
Here's how the game played. You and your opponent are each dealt two cards. Usually the first card for each player is dealt face down, and the second.card for each player is dealt face up. After the initial cards are dealt, the first player has the option of asking for another card or not taking any cards. The first player can keep asking for more cards until either he or she goes over 21, in which case the player loses, or stops at some number less than or equal to 21. When the first player stops at some number less than or equal to 21, the second player then can take more cards until matching or exceeding the first player's number without going over 21, in which case the second player wins, or until going over 21, in which case the first player wins. We are going to simplify the game a little and assume that all cards are dealt face up, so that all cards are visible. Assume your opponent is dealt cards and plays first.
From sume s2 is Q.32
The chance that the second card will be a heart and a Jack, is (A)
.-cted
52
(B)
13 52
(C)
17 52
(D)
52
Q. 3 3 The chance that the first card will be a heart or a Jack, is 13 (A) ^ Q.34
16 (B) n
(D)none
Given that the first card is a Jack, the chance that it will be the heart, is (B)
Q.35
17 CO 52
4 li
:
o»
5
Your opponent is dealt a King and a 10, and you are dealt a Queen and a 9. Being smart, your opponent does not take any more cards and stays at 20. The chance that you will win if you are allowed to take as many cards as you need, is ^
97 564
,(B)
25 282
(C)
15 188
(D)i
i f>i
>
Bansal Classes
[17]
Atari .awrv More than one alternative are correct: Q.3 6 If A& Bare two events such that P(B)* 1,BC denotes the event complementry to B, then v
> \
I
j _ p(B)
(B) P (A n B) > P(A) + P(B) - 1
Q>
(C) P(A)>
(D) P(A/BC) +P(A C /B C ) = 1 >
Q.3 7 A bag initially contains one red & two blue balls. An experiment consisting of selecting a ball at random, Q.16 noting its colour & replacing it together with an additional ball of the"feame colour. If three such trials are made, then: (A) probability that atleast one blue ball is drawn is 0.9 (B) probability that exactly one blue ball is drawn is 0.2 Q.l (C) probability that all the drawn balls are red given that all the drawn balls are of same colour is 0.2 (D) probability that atleast one red ball is drawn is 0.6. Q.3 Q.38 Two real numbers, x & y are selected at random. Given that 0 < x < 1 ; 0 < y < 1. Let A be the event Q.7 that y2 < x ; B be the event that x2 < y, then : 1 Q.l (A) P (A n B) = (B) A & B are exhaustive events (C) A & B are mutually exclusive
(D) A & B are independent events.
Q.l
Q. 3 9 For any two events A & B defined on a sample space , (A) P (A/B) >
P(A)
p ^ B ) ~ 1 , P ( B ) ^ 0 is always true
(B)P (AKB) = P (A) - P ( A n B ) (C) P (A u B) = 1 - P (Ac). P (Bc), if A& B are independent lj(D) P (A u B) = 1 - P (Ac). P (Bc), if A & B are disjoint Q.40
If E, and E2 are two events such that P(E,) = 1/4, P(E2/E,) =1/2 and P(E,/ E2) = 1/4 (A) then E, and E2 are independent (B) E, and E2 are exhaustive (C) E2 is twice as likely to occur as E, (D) Probabilities of the events Ej n E 2 , E, and E2 are in G.P.
Q.l Q-
Q Q Q
Q.41 Let 0 < P(A) < 1 , 0 < P(B) <1 & P(A u B) = P(A) + P(B) - P(A). P(B), then : (A) P(B/A) = P(B) - P(A) (B) P(AC u B c ) = P(AC) + P(BC) c C C (C) P((A u B) ) = P(A ). P(B ) (D) P(A/B) = P(A) Q .42 If M & N are independent events such that 0 < P(M) < 1 & 0 < P(N) < 1, then : (A) M & N are mutually exclusive (B) M & N are independent (C) M & N are independent
Classes
(D) P(M/N) +P(M/N) = 1
Probability [18
ANSWER KEY - X C
Q.l
3
Q.8
8
Q.16 ire
2
Q.2
A
Q.IO
2197 Q.ll 20825
Q.17
3
Q.3
1
Q.4
B
952 to 715 1
Q.18
A
Q.5
Q-13
Tj
Q.6
Q'f
4
3 5
A
J_ 56
Q.7
n(n-l) (m + n)(m + n - l )
Q 15
"
4
1 ^
Q.19
Q.20
4' 4
c„-48cr 521 "13
D P P - 2
at
Q.l
(i)0.18,(ii)0.12,(iii)0.42,(iv)0.28, (v)0.72
Q.3
2
Q.7
Q.4
3 D
(a) 1/18, (b) 43/90, (c) 5/18, (d) NO Q.5 Q.9
Q.8 Q.15
Q.14 C
D
Q.10
(i) 0.6, (ii) 0.5, (iii) 0.25
Q.17P(E) = -j , P ( F ) 4 , P(E and F) = g;Yes
Q
1
9
55 loo " =
11 20
Q.l
A
Q.2
Q.8
B
Q.9
Q.14
(0 3 6 5
Q.18
^ 10
Q-2'
C
Q.20
3 -
Q.3
B
5 Q.10 8 364x363 , ( i i ) l - (365)2
2 24 Q.19 7 ' 35 Q.22
7 1 2 2 1 . 9 Q.2(i) — ,(ii),!-,(iii) — ,(iv) —, (v) - , (vi) —
(a)
A Q.16
D P P r ' 3 Q.4 C Q.5
17 105
15
Q.ll 4
To
16 12 9 37 ' 37 ' 37
Q
'
2 3
B
Q.3
C
Q.8
C
Q.9
C
Q.10
A
Q.ll
Q.21
4H Bansal Classes
Q.7
D
I 4
Q.12
1 3
Q.13
7 8
Q.17
C 4 2 Q.20 17' 51
(b)i
Q.17 Q.22
13 to 5
209 343
35 • .-•<
Q.2
B
D
(
B
Q.20
Q.6
22 13
Q.l
7 27 73 648
D
5 7 3 149 (b) (C) (d) 204' 102' 68' 204
P I * - ^ Q.4 D Q.5
Q.16
Q.13 B
30 Q.18P(E)=^,P(F)== - , P ( E n F ) = — ; Not independent 36 12 2 Q.24 A . Q.23 A Q.21 Q.22 25
O
Q.15 B
C
Q.6
Q.ll A Q.12 A 1 5 53 (i) — , (ii) — , (in) —
(i) 0.49; (ii) 0.973
Q-
5 3 (i) - , (ii) -
A
B
Q.6
1
B
Q.7
Q.14
Q.12
2133 Q.13 3125
4 9
Q.18
A
Q.19
B=
Q.23 B
Q.24
A
2
5
;
C
=
Q.25
B 1 2 4 15 C [19]
I>
I ' P - 5
Q.l
D
Q.2
A
Q.3
D
Q.4
Q.7
A
Q.8
B
Q.9
B
Q.10
155 169 ;B: 324 ' " ' 324 C Q.ll D
Q.13
C
Q-12
n-1
( 1. )mn-1 ^ 7 . ( 2 ) mn - rn - 1
Q.17 D
Q.18
Q.23 (a)
65
(e)
2
Q.19
(b)
n — 17
10-5
P(E n F) =P(E); P(E/F) =
Q.5
B
Q.6
D
C
Q.15
C
Q.16
B
Q.21
n_ 50
A:
Q.14 2
Q.20
n(n + l)
(c)
1-2
(d)
EcF
; P(F/E) = 1
(f)
No
P(E)
Q.22
13 14
A
D P P - 6 Q.2
A
Q.3
3
Q.4
B
Q.5
Q.l
A
Q.8
(i) A,B>C are pairwise independent (ii) A,B,C are n^i'hdependent.
Q.ll
C
Q.12 D
Q.13
D
Q.14
Q.18 B
Q.19 C
Q.20
B
Q.21
4
165 193 C
Q.15 Q.22
n
D
i i Q.25
D
Q 30
(i)
'
Q.26
A
61 371 (U) 216 ' 1296
Q.27
C
Q.31
C
A, B, C
<13 Bansal Classes
Q.40
B
Q.7
Q.9
A
Q.10 A
Q.16
A
Q.17 D
Q.23
A
Q.24 D
>
Q.28 D
Q.29
Q.32 D
Q.33
More than one alternative are correct: Q.36 A, B, C, D Q.37 A, B, C, D Q.39
Q.6
A, C, D
Q.38
A, B
Q.41
C,D
\
1
1
(i)-;(ii)-;(iii)-
B
Q.34
C
Q.35 D
Q.42
B, C, D
[20]
BANSAL CLASSES
TARGET IIT JEE 2007
MATHEMATICS XII (ABCD)
PROBABILITY
CONTENTS KEY CONCEPTS EXERCISE-I EXERCISE-II EXERCISE-III ANSWER KEY
KEY
CONCEPTS
THINGS T O REMEMBER : RESULT
- 1
(i)
SAMPLE-SPACE
(ii)
EVENT
(iii)
COMPLEMENT
: The set of all possible outcomes of an experiment is called the
SAMPLE-SPACE(S).
: A sub set of sample-space is called an EVENT. OF AN EVENT A
: The set of all out comes which are in S but not in A is called
t h e COMPLEMENT O F T H E EVENT A DENOTED BY A OR A C .
(iv)
: If A & B are two given events then AnB is called is denoted by AnB or AB or A & B . COMPOUND EVENT
(v)
MUTUALLY
(vi)
EQUALLY LIKELY EVENTS
COMPOUND EVENT
and
: Two events are said to be MUTUALLY EXCLUSIVE (or disjoint or incompatible) if the occurence of one precludes (niles out) the simultaneous occurence ofthe other. If A & B are two mutually exclusive events then P (A & B) = 0. EXCLUSIVE
EVENTS
: Events are said to be EQUALLY LIKELY when each event is as likely to occur
as any other event. (vii)
EXIIA USTIVE EVENTS
: Events A,B,C L are said to be EXHAUSTIVE EVENTS if no event outside this set can result as an outcome of an experiment. For example, if A & B are two events defined on a sample space S, then A & B are exhaustive A u B = S=> P (Au B) = 1 .
(viii)
CLASSICAL DEF. OF PROBABILITY
: If n represents the total number of equally likely, mutually exclusive and exhaustive outcomes of an experiment and m of them are favourable to the happening of the event A, then the probability of happening of the event A is given by P(A) = m/n . Note : (1) 0 < P(A) < 1 (2) (3)
P(A) + P( A) = 1, Where A = Not A . — X If x cases are favourable to A & y cases are favourable to A then P(A) = - and (x + y) P( A) = —-— We say that O D D S I N FAVOUR O F A are x: y & odds against A are y: x
Comparative study of Equally likely, Mutually Exclusive and Exhaustive events. Experiment 1. Throwing ofa die 2. A ball is drawn from an urn containing 2W, 3Rand 4G balls 3. Throwing a pair of dice •
4. From a well shuffled pack of cards a card is drawn 5. From a well shuffled pack of cards a card is drawn
^Bansal Classes
Events A : throwing an odd face {1, 3, 5} B : throwing a composite face {4,. 6} E, : getting a W ball E,: getting a R ball E3: getting a G ball A : throwing a doublet {11,22,33,44,55,66} B : throwing a total of 10 or more {46,64,55,56,65,66} E, : getting a heart E, : getting a spade E,: getting a diamond E4: getting a club A = getting a heart B = getting a face card
Probability
Exhaustive
No
M/E Yes
No
Yes
Yes
Yes
No
No
Yes
Yes
Yes
No
No
No
E/L
No
[2]
RESULT -
(i)
2
AUB = A+ B = A or B denotes occurence of at least A or B. For 2 events A & B : (See fig.l) P(AuB) = P(A) + P(B) - P(AnB) = P(A. B) + P( A .B) + P(A.B) = 1 - P( A . B )
(ii)
Opposite of " atleast A or B" is NIETHER A NOR B i.e. A + B = 1-(A or B) = A n B Note that P(A+B) + P( A n B ) = 1.
(iii)
If A & B are mutually exclusive then P(AuB) = P(A) + P(B).
(iv)
For any two events A & B, P(exactly one of A, B occurs) = P(AnB) + P(BnA)
(v) (vi) (vii)
= P(A) + P(B) - 2P(AnB)
= P ( A u B ) - P ( A n B ) = p(A c uB c )-P(A c nB c ) If A & B are any two events P(AnB) = P(A).P(B/A) = P(B).P(A/B), Where P(B/A) means conditional probability of B given A & P(A/B) means conditional probability of A given B. (This can be easily seen from the figure) D E MORGAN'S LAW : - If A & B are two subsets of a universal set U, then (a) (AuB)c = AcnB° & (b) (AnB)c = AcuBc A u (BnC) = (AuB) n (AuC)
RESULT -
& A n (BuC) = (AnB) u (AnC)
3
For any three events A,B and C we have (See Fig. 2) (i)
P(A or B or C) = P(A) + P(B) + P(C) - P(AnB) - P(BnC)P(CnA) + P(AnBnC)
(ii)
P (at least two of A, B,C occur) = P(BnC) + P(CnA) + P(AnB) - 2P(AnBnC)
(iii)
P(exactly two of A,B,C occur) = P(BnC) + P(CnA) + P(AnB) - 3P(AnBnC)
(iv)
P(exactly one of A,B,C occurs) = P(A) + P(B) + P(C) - 2P(BnC) - 2P(CnA) - 2P(AnB)+3P(AnBnC)
Fig. 2
NOTE :
If three events A, B and C are pair wise mutually exclusive then they must be mutually exclusive, i.e P(AnB) = P(BnC) = P(CnA) = 0 w P(AnBnC) = 0. However the converse of this is not true.
Classes
Probability
[3]
RESULT -
4
: Two events A& Bare said to be independent if occurence or non occurence of one does not effect the probability ofthe occurence or non occurence of other. If the occurence of one event affects the probability ofthe occurence ofthe other event then the events are said to be DEPENDENT or CONTINGENT . For two independent events A and B : P(AnB) = P(A). P(B). Often this is taken as the definition of independent events. Three events A, B & C are independent if & only if all the following conditions hold; P(AnB) = P(A). P(B) ; P(BnC) = P(B). P(C) P(CnA) = P(C). P(A) & P(AnBnC) = P(A). P(B). P(C) i.e. they must be pairwise as well as mutually independent . Similarly for n events A t , A^, A^, An to be independent, the number of these conditions is equal to nc02+ "c,.5+ + ncn = 2n - n - 1. The probability of getting exactly r success in n independent trials is given by P(r) = nCr pr qn"r where : p = probabi lity of success in a single trial . q = probability of failure in a single trial, note : p + q = 1. Note : Independent events are not in general mutually exclusive & vice versa Mutually exclusiveness can be used when the events are taken from the same experiment & independence can be used when the events are taken from different experiments.
INDEPENDENT EVENTS
(i) (ii)
(iii)
RESULT - 5 :
BAYE'S THEOREM
OR
T O T A L PROBABILITY T H E O R E M
:
If an event A can occur only with one of the nmutually exclusive and exhaustive events B,, B2,.... Bn & the probabilities P(A/B.|), P(A/B,)....... P(A/Bn) are known then, P(B/A)=
'':""'tA " ' £ P(B,).P(A/B,) i=l
PROOF:
The events A occurs with one of the n mutually exclusive & exhaustive events B , , B 2 , B 3 , A = AB, + AB., + AB, + + AB 1 2 3 n
BN
n
P(A) = P(AB,) + P(AB2) +.......+ P(ABn) = X P(AB.) N O T E : A = event what we have ; B. = event what we want ; B2',, B3', B n are alternative event. Now, P(AB.) = P(A) . P(B/A) = P(B.) . P(A/B.) P(B,/A)
\ \ \B\
/ //Bn-l, /
B "
/
_ P(Bi).P(A/Bi) = P(Bi).P(A/Bi) P(A) ~ " I P(ABi)
P(Bi/A) =
/
WflL /B
P(B|) . P(A/Bi) XP(B i ).P(A/B j ) Fig. 3
Classes
Probability
[447]
RESULT -
6
If p, and p, are the probabilities of speaking the truth of two indenpendent witnesses A and B then P (their combined statement is true) =
JVP2 Pi P 2 + ( 1 - P i ) G - P 2 )
In this case it has been assumed that we have no knowledge ofthe event except the statement made by AandB. However if p is the probability of the happening of the event before their statement then P (their combined statement is true) =
PP]_P2 PPiP2+0-P)(1-Pi)(1-P2)
Here it has been assumed that the statement given by all the independent witnesses can be given in two ways only, so that if all the witnesses tell falsehoods they agree in telling the same falsehood. If this is not the case and c is the chance of their coincidence testimony then the Pr. that the statement is true = Pp ! p ? Pr. that the statement is false = (l-p).c (1—p|)(l—p2) However chance of coincidence testimony is taken only if the joint statement is not contradicted by any witness. RESULT -
7
(i)
A PROBABILITY DISTRIBUTION spells out how a total probability of 1 is distributed over several values of a random variable.
(ii)
Mean of any probability distribution of a random variable is given by : \
p.
X-
n = y p " ' = Z Pi xi (iii)
Variance of a random variable is given by, a 2 = X (x - p)2. p I p. x2. - (i2
(iv)
(since2P; = l)
(Note that SD =
)
The probability distribution for a binomial variate 'X' is given by ; P (X= r)=nC p1 qn~r where all symbols have the same meaning as given in result 4. The recurrence formula p ( r + 1) = n - r H , is very helpful for quickly computing P(r)
r+1
q
P(l), P(2). P(3) etc. if P(0) is known . (v)
Mean of BPD = np
; variance of BPD = npq .
(vi)
If p represents a persons chance of success in any venture and 'M' the sum of money which he will receive in case of success, then his expectations or probable value = pM expectations = pM
RESULT-8
(i) (ii)
: GEOMETRICAL APPLICATIONS
:
The following statements are axiomatic : If a point is taken at random on a given staright line AB, the chance that it falls on a particular segment PQ of the line is PQ/AB . If a point is taken at random on the area S which includes an area cr, the chance that the point falls on a is a/S .
Classes
Probability
[5]
Q.l
EXERCISE-I Let a die be weighted so that the probability of a number appearing when the die is tossed is proportional to that number. Find the probability that, (i) An even or a prime number appears (ii) An odd prime number appears (iii) An even composite number appears (iv) An odd composite number appears.
Q. 2
Numbers are selected at random, one at a time,fromthe two digit numbers 00,01, 02, ,99 with replacement. An event E occurs if & only if the product ofthe two digits of a selected number is 18. If four numbers are selected, find the probability that the event E occurs at least 3 times.
Q.3
In a box, there are 8 alphabets cards with the letters : S, S, A, A,A, H, H, H. Find the probability that the word 'ASH' will form if: the three cards are drawn one by one & placed on the table in the same order that they are drawn. the three cards are drawn simultaneously.
(i) (ii) Q.4
There are 2 groups of subjects one of which consists of 5 science subjects & 3 engg. subjects & other consists of 3 science & 5 engg. subjects. An unbiased die is cast. If the number 3 or 5 turns up a subject is selected at random from first group, otherwise the subject is selected from 2nd group . Find the probability that an engg. subject is selected.
Q.5
A pair of fair dice is tossed. Find the probability that the maximum of the two numbers is greater than 4.
Q.6
In a bui Iding programme the event that all the materials will be delivered at the correct time is M, and the event that the building programme will be completed on time is F . Given that P (M) = 0.8 and P (M n F) = 0.65, find P (F/M). If P (F) = 0.7,findthe probability that the building programme will be completed on time if all the materials are not delivered at the correct time.
Q.7
In a given race, the odds in favour of four horses A, B, C & D are 1 : 3,1 :4,1: 5 and 1 : 6 respectively. Assuming that a dead heat is impossible,findthe chance that one of them wins the race.
Q.8
A covered basket of flowers has some lilies and roses. In search of rose, Sweety and Shweta alternately pick up a flowerfromthe basket but puts it back if it is not a rose. Sweety is 3 times more likely to be the first one to pick a rose. If sweety begin this 'rose hunt' and if there are 60 lilies in the basket,findthe number of roses in the basket.
Q.9
Least number of times must a fair die be tossed in order to have a probability of at least 91/216, of getting atleast one six.
Q.10
Suppose the probability for A to win a game against B is 0.4. If A has an option of playing either a "BEST OF THREE GAMES" or a "BEST OF 5 GAMES" match against B, which option should A choose so that the probability of his winning the match is higher? (No game ends in a draw).
^ ^
Q.ll
A room has three electric lamps . From a collection of 10 electric bulbs of which 6 are good 3 are selected at random & put in the lamps. Find the probability that the room is lighted.
Q.12
A bomber wants to destroy a bridge. Two bombs are sufficient to destroy it. If four bombs are dropped, what is the probability that it is destroyed, if the chance of a bomb hitting the target is 0.4.
^Bansal
Classes
Probability
[6]
Q.13 The chance of one event happening is the square ofthe chance of a 2nd event, but odds against the first are the cubes of the odds against the 2 nd . Find the chances of each, (assume that both events are neither sure nor impossible). Q.14 A box contains 5 radio tubes of which 2 are defective. The tubes are tested one after the other until the 2 defective tubes are discovered. Find the probability that the process stopped on the (i) Second test; (ii) Third test. If the process stopped on the third test, find the probability that the first tube is non defective. Q.15 Anand plays with Karpov 3 games of chess. The probability that he wins a game is 0.5, looses with probability 0.3 and ties with probability 0.2. If he plays 3 games then find the probability that he wins atleast two games. Q.16 An aircraft gun can take a maximum of four shots at an enemy's plane moving awayfromit. The probability of hitting the plane at first, second, third & fourth shots are 0.4, 0.3,0.2 & 0.1 respectively. What is the probability that the gun hits the plane. Q.17 In a batch of 10 articles, 4 articles are defective. 6 articles are taken from the batch for inspection. If more than 2 articles in this batch are defective, the whole batch is rejected Find the probability that the batch will be rejected. Q. 18 Given P(AuB) = 5/6 ; P(AB) = 1/3 ; P( B) = 1/2. Determine P(A) & P(B). Hence show that the events A & B are independent. Q.19 One hundred management students who read at least one of the three business magazines are surveyed to.study the readership pattern. It is found that 80 read Business India, 50 read Business world and 30 read Business Today. Five students read all the three magazines. A student was selected randomly. Find the probability that he reads exactly two magazines. Q.20 An author writes a good book with a probability of 112. If it is good it is published with a probability of 2/3. If it is not, it is published with a probability of 1 /4. Fi nd the probability that he will get atleast one book published if he writes two. Q.23
3 students {A,B,C} tackle a puzzle together and offers a solution upon which majority ofthe 3 agrees. Probability ofA solving the puzzle correctlyisp. Probability of B solving the puzzle correctly is also p. C is a dumb student who randomly supports the solution of either A or B. There is one more student D, whose probability of solving the puzzle correctly is once again, p. Out of the 3 member team {A, B, C} and one member team {D}, Which one is more likely to solve the puzzle correctly.
Q.22 A uniform unbised die is constructed in the shape of a regular tetrahedron with faces numbered 2,2,3 and 4 and the score is takenfromthe face on which the die lands. If two such dice are thrown together, find the probability of scoring. (i) exactly 6 on each of 3 successive throws. (ii) more than 4 on at least one of the three successive throws. Q.23 Two cards are drawn from a well shuffled pack of 52 cards. Find the probability that one of them is a red card & the other is a queen.
Classes
Probability
[450]
Q.24 A cube with all six faces coloured is cut into 64 cubical blocks of the same size which are thoroughly mixed. Find the probability that the 2 randomly chosen blocks have 2 coloured faces each. Q. 2 5 Consider the following events for a fami ly with children A = {of both sexes} ; B = {at most one boy} In which of the following (are/is) the events A and B are independent, (a) if a family has 3 children (b)ifa family has 2 children v ' A s s u m e that the birth of a boy or a girl is equally likely mutually exclusive and exhaustive. Q.26 A player tosses an unbiased coin and is to score two points for every head turned up and one point for f every tail turned up. If Pn denotes the probability that his score is exactly n points, prove that V ^ i P P n- n-. = 2 (Pn-2-Pn-l) ^>3 Also compute P, and P2 and hence deduce the pr that he scores exactly 4. Q.27 Each of the 'n' passengers sitting in a bus may get down from it at the next stop with probability p. Moreover, at the next stop either no passenger or exactly one passenger boards the bus. The probability of no passenger boarding the bus at the next stop being pQ. Find the probability that when the bus continues on its way after the stop, there will again be 'n' passengers in the bus. Q . 2 8 The difference between the mean & variance of a Binomial Variate1X' is unity & the difference of their square is 11. Find the proba^'ity distribution of'X'. Q.29 An examination consists of 8 questions in each of which the candidate must say which one of the 5 alternatives is correct one. Assuming that the student has not prepared earlier chooses for each of the question any one of 5 answers with equal probability. (i) prove that the probability that he gets more than one correct answer is (58 - 3 x 48) / 5 8 . (ii) find the probability that he gets correct answers to six or more questions. (iii) find the standard deviation ofthis distribution. Q.30
Two bad eggs are accidently mixed with ten good ones. Three eggs are drawn at random without replacement, from this lot. Compute mean & S.D. for the number of bad eggs drawn.
EXERCISE-II Q. 1
The probabilities that three men hit a target are. respectively, 0.3,0.5 and 0.4. Each fires once at the target. (As usual, assume that the three events that each hits the target are independent) (a) Find the probability that they all: (i) hit the target; (ii) miss the target (b) Find the probability' that the target is hit: (i) at least once, (ii) exactly once. (c) If only one hits the target, what is the probability' that it was thefirstman?
Q.2
Let A & B be two events defined on a sample space . Given P(A) = 0.4 ; P(B) = 0.80 and P (A/B) =0.10. Then find ;
Q. 3
(i)P(AuB) &
-p[(AnB) u (AnB)].
Three shots arefiredindependently at a target in succession. The probabilities that the target is hit in the first shot is 1/2, in the second 2/3 and in the third shot is 3/4. In case of exactly one hit, the probability of destroying the target is 1/3 and in the case of exactly two hits, 7/1.1 and in the case of three hits is 1.0. Find the probability of destroying the target in three shots.
Classes
Probability
[8]
Q.4
In a game of chance each player throws two unbiased dice and scores the difference between the larger and smaller number which arise. Two players compete and one or the other wins if and only if he scores atleast 4 more than his opponent. Find the probability that neither player wins.
Q.5
A certain drug, manufactured by a Company is tested chemically for its toxic nature. Let the event "THE DRUG IS TOXIC" be denoted byH & the event " T H E CHEMICAL TEST REVEALS THAT THE DRUG IS TOXIC" be denoted by S. Let P(H) = a, P(S / H) = P(S / H) = l - a . Then show that the probability that the drug is not toxic given that the chemical test reveals that it is toxic, is freefrom'a'.
Q.6
A plane is landing. If the weather is favourable, the pilot landing the plane can see the runway. In this case the probability of a safe landing isp r Ifthere is a low cloud ceiling, the pilot has to make a blind landing by instalments. The reliability (the probability of failurefreefunctioning) of the instruments needed for a blind landing is P. Ifthe blind landing instruments function normally, the plane makes a safe landing with the same probability P] as in the case of a visual landing. Ifthe blind landing instruments fail, then the pilot may make a safe landing with probability p 2 < p 1 - Compute the probability of a safe landing if it is known that in K percent of the cases there is a low cloud ceiling. Also find the probability that the pilot used the blind landing instrument, if the plane landed safely.
Q.7
A train consists of n carriages, each of which may have a defect with probability p. All the carriages are inspected, independently of one another, by two inspectors; the first detects defects (if any) with probability Pj, & the second with probability p 2 . If none of the carriages is found to have a defect, the train departs . Find the probability of the event; " THE TRAIN DEPARTS WITH ATLEAST ONE DEFECTIVE CARRIAGE " .
Q.8
A is a set containing n distinct elements. A non-zero subset P ofA is chosen at random. The set A is reconstructed by replacing the elements of P. A non-zero subset Qof A is again chosen at random. Find the probability that P & Q have no common elements.
Q.9
In a multiple choice question there are five alternative answers of which one or more than one is correct. A candidate will get marks on the question only if he ticks the correct answers. The candidate ticks the answers at random. If the probability of the candidate getting marks on the question is to be greater than or equal to 1 /3 find the least number of chances he should be allowed.
Q.10 n people are asked a question successively in a random order & exactly 2 of the n people know the answer: (a) If n > 5, find the probability that thefirstfour of those asked do not know the answer. (b) Show that the probability that the rUl person asked is the first person to know the answer is : 2(n-r) n(n-l) Q.ll (i) (ii) (iii)
if 1 < r < n
A box contains three coins two of them are fair and one two - headed. A coin i s selected at random and tossed. If the head appears the coin is tossed again, if a tail appears, then another coin is selected from the remaining coins and tossed. Find the probability that head appears twice. If the same coin is tossed twice,findthe probability that it is two headed coin. Find the probability that tail appears twice.
Classes
Probability
[9]
Q.12 The ratio of the number of trucks along a highway, on which a petrol pump is located, to the number of cars running along the same highway is 3 :2. It is known that an average of one truck in thirty trucks and two cars in fifty cars stop at the petrol pump to be filled up with the fuel. If a vehicle stops at the petrol pump to be filled up with the fuel, find the probability that it is a car. Q.13 Abatch of fifty radio sets was purchased from three different companies A, B and C. Eighteen of them were manufactured by A, twenty of them by B and the rest were manufactured by C. The companies Aand C produce excellent quality radio sets with probability equal to 0.9; B produces the same with the probability equal to 0.6. What is the probability of the event that the excellent quality radio set chosen at random is manufactured by the company B? Q.14 The contents of three urns 1,2 & 3 are as follows : 1 W, 2 R, 3B balls 2 W, 3 R. IB balls 3 W, 1R, 2B balls An urn is chosen at random & from it two balls are drawn at random & are found to be " 1 RED nd & 1 WHITE ". Find the probability that they came from the 2 urn. Q.15 A slip of paper is given to a person "A" who marks it with either a (+)ve or a (-)ve sign, the probability of his writing a(+)ve sign being 1/3. "A" passes the slip to "B" who may leave it alone or change the sign before passing it to "C", Similarly "C" passes on the slip to "D" & "D" passes on the slip to Refree, who finds a plus sign on the slip. If it is known that B, C & D each change the sign with a probability of 2/3, then find the probability that "A" originally wrote a (+)ve sign. Q.16 There are 6 red balls & 8 green balls in a bag. 5 balls are drawn out at random & placed hi a red box; the remaining 9 balls are put in a green box. What is the probability that the number ofred balls in the green box plus the number of green balls in the red box is not a prime number? Q.17 Two cards are randomly drawnfroma well shuffled pack of 52 playing cards, without replacement. Let x be the first number and y be the second number. Suppose that Ace is denoted by the number 1; Jack is denoted by the number 11; Queen is denoted by the number 12; King is denoted by the number 13. Find the probability that x and y satisfy log3(x + y) - log3x - log3y +1=0. Q. 18(a) Two natural numbers x & y are chosen at random. Find the probability that x2 + y2 is divisible by 10. (b) Two numbers x&y are chosen at random from the set {1,2,3,4,....3n}. Find the probability that x2 - y2 is divisible by 3 . (c) If two whole numbers x and y are randomly selected, then find the probability that x 3 +y 3 is divisible by 8. a2 Q.19 A hunter's chance of shooting an animal at a distance r is — (r>a). He fires when r = 2a &
if he misses he reloads & fires when r = 3a, 4a escapes. Find the odds against the hunter.
If he misses at a distance 'na', the animal
Q.20 An unbiased normal coin is tossed ' n' times. Let : E, : event that both Heads and Tails are present in ' n' tosses. E2 : event that the coin shows up Heads atmost once. Find the value o f ' n' for which Ej & E2 are independent.
Classes
Probability
[10]
n+2 Q.21 A coin is tossed (m+n) times (m>n). Show that the probability of at least m consecutive heads is Q.22 There are two lots of identical articles with different amount of standard and defective articles. There are N articles in the first lot, n of which are defective and M articles in the second lot, m of which are defective. K articles are selectedfromthefirstlot and L articles from the second and a new lot results. Find the probability that an article selected at randomfromthe new lot is defective. Q.23 An instrument is being tested, upon each trial the instrument fails with probability p. After thefirstfailure the instrument is repaired and after the second failure it is considered to be unfit for operation. Find the probability that the instrument is rejected exactly in the kth trial. Q.24(a) Prove that if A, B & C are random events in a sample space & A, B, Carepairwise independent and Ais independent of (BuC) then A, B&C are mutually independent. (b) An event A is known to be independent of the events B, BuC & BnC . Showthat it is also independent of C. Q.25 In a knockout tournament 2n equally skilled players; S,, S2,
S n are participating. In each
round players are divided in pairs at random & winnerfromeach pair moves in the next round. If S2 reaches the semifinal thenfindthe probability that S, wins the tournament.
EXERCISE-III Q.l
Ifp&q are chosen randomly from the set {1,2,3,4,5,6,7, 8,9,10} with replacement. Determine the probability that the roots of the equation x2 + px + q = 0 are real. [ JEE '97, 5 ]
Q. 2
There is 3 0% chance that it rains on any particular day. What is the probability that there is at least one rainy day within a period of 7 - days ? Given that there is at least one rainy day, what is the probability that there are at least two rainy days ? [ REE '97,6 ]
Q.3 (i)
Select the correct alternative(s). [ JEE '98,6 x 2= 12 ] 7 white balls & 3 black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals: (A) 1/2 ' (B) 7/15 (C) 2/15 (D) 1/3
(ii)
If from each of the 3 boxes containing 3 white & 1 black, 2 white & 2 black, 1 white & 3 black balls, one ball is drawn at random, then the probability that 2 white & 1 black ball will be drawn is: (A) 13/32 (B) 1/4 (C) 1/32 (D) 3/16
(iii)
If E & F are the complementary events of events E & F respectively & if 0 < P (F) < 1, then: (A) P (E | F) + P(E |F) = 1
(B) P(E|F) + P(E| F ) = l
(C) P(E |F) + P(E| F ) = l
(D) P(E| F) + P ( E I F ) = l
Probability
[11]
(iv)
There are 4 machines & it is known that exactly 2 of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified . Then the probability that only 2 tests are needed is: (A) 1/3 (B) 1/6 (C) 1/2 (D) 1/4
(v)
If E & F are events with P(E) < P(F) & P(E n F) > 0, then : (A) occurrence of E => occurrence of F (B) occurrence of F => occurrence of E (C) non-occurrence of E => non-occurrence of F (D) none of the above implications holds.
(vi)
A fair coin is tossed repeatedly. If tail appears on first four tosses, then the probability of head appearing on fifth toss equals : (A) 1/2 (B) 1/32 (C) 31/32 (D) 1/5
Q.4
3 players A, B&Ctossacoin cyclically in that order (that is A, B, C,A, B, C, A, B, ) till a head shows . Let p be the probability that the coin shows a head. Let a, p & y be respectively the probabilities that A, B and C gets the first head. Prove that (3 = (1 - p)a. Determine a, (3 & y (in terms of p). [ JEE '98, 8 ]
Q. 5
Each co-efficient in the equation ax2+bx + c = 0 is determined by throwing an ordinary die. Find the probability that the equation will have equal roots. [ REE '98,6 ]
Q.6(a) If the integers m and n are chosen at random between 1 and 100, then the probability that a number of the form 7m + 7" is divisible by 5 equals
7
K
<
C
>I
(b) The probability that a student passes in Mathematics, Physics and Chemistry are m, p and c respectively. Of these subjects, the student has a 75% chance of passing in at least one, a 50% chance of passing in at least two, and a 40% chance of passing in exactly two, which of the following relations are true? 19
(A) p + m + c = —
27
(B) p + m + c = —
1
(C)pmc=—
1
(D)pmc=-
(c) Eight players P,, P7, P3, Pg play a knock-out tournament. It is known that whenever the players P; and P • play, the player P; will win if i < j. Assuming that the players are paired at random in each round, what is the probability that the player P4 reaches the final. [ JEE '99, 2 + 3 + 10 (out of 200)] Q.7
Four cards are drawn from a pack of 5 2 playing cards. Find the probability (correct upto two places of decimals) of drawing exactly one pair. [REE'99,6]
Q.8
A coin has probability ' p' of showing head when tossed. It is tossed 'n' times. Let pn denote the probability that no two (or more) consecutive heads occur. Prove that, p, = 1 , p2 = 1 - p2 & pn = (1 - p) p n _! + p (1 - p) p n _ 2 , for all n > 3. [ JEE' 2000 (Mains), 5 ]
Probability
[12]
Q.9
A and B are two independent events. The probability that both occur simultaneously is 1 /6 and the probability that neither occurs is 1/3. Find the probabilities of occurance ofthe events A and B separately. [REE'2000(Mains), 3]
Q.10 Two cards are drawn at random from a pack of playing cards. Find the probability that one card is a heart and the other is an ace. [ REE' 2001 (Mains), 3 ] Q. 11 (a) An urn contains'm' white and 'n' black balls. Aball is drawn at random and is put back into the urn along with K additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. What is the probability that the ball drawn now is white. (b) An unbiased die, with faces numbered 1,2,3,4,5,6 is thrown n times and the list of n numbers showing up is noted. What is the probability that among the numbers 1,2,3,4,5,6, only three numbers appear in the list. [JEE' 2001 (Mains), 5 + 5 ] Q.12 A box contains N coins, m of which are fair and the rest are biased. The probability of getting a head when a fair coin is tossed is 1/2, while it is 2/3 when a biased coin is tossed. A coin is drawnfromthe box at random and is tossed twice. Thefirsttime it shows head and the second time it shows tail. What is the probability that the coin drawn is fair? [ JEE' 2002 (mains)] Q. 13(a) Aperson takes three tests in succession. The probability of his passing the first test is p, that of his passing each successive test is p or p/2 according as he passes or fails in the preceding one. He gets selected provided he passes at least two tests. Determine the probability that the person is selected. (b) In a combat, A targets B, and both B and C target A. The probabilities ofA, B, C hitting their targets are 2/3,1/2 and 1/3 respectively. They shoot simultaneously and Ais hit. Find the probability that B hits his target whereas C does not. [JEE' 2003, Mains-2 + 2 out of 60] Q. 14(a) Three distinct numbers are selectedfromfirst100 natural numbers. The probability that all the three numbers are divisible by 2 and 3 is 4
(A)5
4
(B ) 5 5
4
(Q-
4
CD) —
(b) If A and B are independent events, prove that P (A u B) • P (A' n B') < P (C), where C is an event defined that exactly one of A or B occurs. (c) A bag contains 12 red balls and 6 white balls. Six balls are drawn one by one without replacement of which atleast 4 balls are white. Find the probability that in the next two draws exactly one white ball is drawn (leave the answer in terms ofnCr). [JEE 2004,3 + 2 + 4]
Probability
[13]
Q. 15(a) A six faced fair dice is thrown until 1 comes, then the probability that 1 comes in even number of trials is (A) 5/11 (B) 5/6 (C) 6/11 (D) 1/6 [JEE 2005 (Scr)] 13 2 1 (b) A person goes to office either by car, scooter, bus or train probability of which being —,—,— and — 2 14 1 respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is —, —, — and — respectively. Given that he reached office in time, then what is the probability that he travelled by a car. [JEE 2005 (Mains), 2] Comprehension (3 questions) There are n urns each containing n +1 balls such that the Ith urn contains i white balls and (n +1 - i) red balls. Let u, be the event of selecting r111 urn, i= 1,2,3, ,n and w denotes the event of getting a white ball. Q. 16(a) If P(Uj) oc i where i = 1,2,3, (A) 1
, n then Lim P(w) is equal to
(B) 2/3
n-»oo
(C) 3/4
(D) 1/4
(b) If P(Uj) = c, where c is a constant then P(un/w) is equal to 2 (A) v ' n+1
1 (B) v ' n+1
n t ' n+1
v(C)
v(D)
1 2
(c) If n is even and E denotes the event of choosing even numbered urn (P(u.) = —), then the value of ' n P(w/E), is n+2
n+2
n
1 [JEE 2006,5 marks each]
Classes
Probability
[14]
ANSWER KEY EXERCISE-I Q 1.
70 8 10 (i) f y (ii) y x (iii) y (iv) 0
Q 2. 97/(25)4
Q 4. 13/24
Q 5. 5/9
Q 7. 319/420
Q8. 120 29 Q 11. —
Q 10. best of 3 games
Q 3. (i) 3/56 (ii) 9/28 Q 6. P(F/M) = % 16 ; P (F/m) = 4\ Q 9. 3 328 1 1 Q 12. — Q 13. - , -
Q 14. (i) 1/10, (ii) 3/10, (iii) 2/3
Q 15. 1/2
Q 17.19/42
Q 18. P(A) = 2/3, P(B) = 1/2
Q 20. 407/576
Q 21. Both are equally likely 24
Q 23. 101/1326
C Q24. " ^ T
or
Q 16. 0.6976 Q 19. 1/2
125 63 Q 22. (i) 773- ; (ii) 77 16 o4
23
Q25. Independent in (a) and not independent in (b)
Q 26. P, = 1/2 , P2 = 3/4
Q 27. (1 - p ) - 1 . [ p0 (1 - p) + np(l -p 0 )]
Q 28.
481 4V2 Q 29. - 5 - , 5 5
Q30. mean = 0.5
EXERCISE-II Q 1.
(a) 6%, 21%; (b) 79%, 44%, (c) 9/44 « 20.45%
Q 4. 74/81
Q6.
Q7. l - [ l - p ( l - p i ) ( l - p 2 ) ] n
Q 8.
Q10- ^^f
Q1L 1/2 1/2 1/12
Q.13
Q 14. 6/11
Bansal Classes
Q 3. |
Q5. p ( H / s ) = l / 2
K K . K ^ m . /1 r.x ]! ;P(H /x x /A) 1 a :\ P(E) = (I- — -)p,+—[Pp,+(1-P)p 2 2
~
Q 2. (i) 0.82, (ii) 0.76
4 [1UPUP l + ( 1 " P ) P 2 ] p1 + A [ P P 1 + ( 1 _ P ) P 2 ] 100
(3n-2n+1+1 )/(4n-2n+1+1)
' '
Probability
Q 9. 11
Q12
-1
Q 15. 13/41
[15]
Q 16. 213/1001
Q.17
~
Q 19. n+1: n-1
Q
22
Q 18. (a) 9/50 (b)
(c)
^
Q 20.n=3
KnM+LmN ' MN(K + L)
3 23
2
Q -
P d-P)"
2
Q
25
'
EXERCISE-III Q.2 [1 - (7/10)7 - 7Cj (3/10) (7/10)6]/1 - (7/10)7
Q.l 31/50
Q.3 (i) B (ii) A (iii) - A, D Q.4 a =
^—j , l-(l-p)3
Q.6 (a) A
Q.10
P
(b) B, C
1 — 26
(iv) A(v) D
(vi) A
= - l t J > ) ! L3 , y = i h P ) ! P _3 l-(l-p) l-(l-p)
(c) 4/35
Q.ll "
Q.5
Q.7 0.31
6
m
(a)—;(b) m+n
Q.9 | & |
C 3 (3"-3.2 n +3)Z 6"
Q.12 ^
or
J& -
9m m + 8N
12C 6q 10c 2Q +12 Q 6Q 11c
2
Q.13 (a)p (2-p) ; (b) 1/2
1 Q.14 ( a ) D , ( c ) 12/-i 31124 6p 2V
Q.15 (a) A, (b) ^
5/216
Classes
Q.16
2
1
4
+
12 p 6p H
1
+
5
,
1
12 p 6p 6
(a) B, (b)A, (c)B
Probability
[16]
BANSAL CLASSES IMATHEMATICS I TARGET IIT JEE 2007 XI (PQRS & J)
^ffteAiny
oj/ou
QI/ouv
Q&/
QUESTION BANK ON STRAIGHT LINE COMPLEX NUMBER SOLUTIONS OF TRIANGLE SEQUENCE & PROGRESSION
Time Limit: 5 Sitting Each of 75 Minutes duration approx. This Question Bank will be discussed just after the Deepawali vacation.
Question bank on Straight line, Sequence & Progression, Solutions of Triangle &XompIex Number. There are 125 questions in this question bank. Select the correct alternative : (Only one is correct)
Q. 1
A
B
'
r
1
If in a triangle ABC, b cos2 — + a cos2 — = - c then a, b, c are : (A) in A.P.
(B) in GP.
(C) in H.P.
(D)
None
Q.2 ^ On the portion ofthe straight line, x + 2y = 4 intercepted between the axes, a square is constnicted on the side ofthe line away from the origin. Then tire point ofintersection of its diagonals has co-ordinates (A) ( 2 . 3 ) (B) (3,2) (C) (3.3) (D) (2,2) 3.3
If A is the area and 2s the sum ofthe 3 sides of a triangle, then ! (A) A <
3-4
, W3
If (73 4 i)'
(B) A
(D) None
2W (a + ib), then a2 + b2 is equal to : (B) 2
(A) 1
(C) 3
(D) 4
from the point M (2, - 3) is : (C) 2
The sum of n terms ofthe series. 2
(A) 2 n i n
2n + in
2
(B)n + 2n
(D)
oo
1
f 2n i +1 2n + .v 1 +3 +5 + is: 2n - I v 2n - 1 / i i (C) m + n (D) none of these
).7
II ' A (3, 4). B (5, - 2) & C are 3 points such th^t AC = BC & the area of A ABC = 10, then the co-ordinates ofC are : (A) ( 7 , - 2 ) or ( 1 , 0 ) (B) (-7.2) or (0,1) ( C ) ( - 7 . - 2 ) or (-1.0) (D) (7,2) or (1,0)
).8
If —is purely imaginary then (A)
).9
(D) 0
(B) b (c + a)
(C) a (b + c)
(D) c2
I f the point B is symmetric to the point A (4, -1) with respect to the bisector of thefirstquadrant, then 1 the lenuth AB is : (A) 3 ^2
'.11
2
c c In any triangle ABC, (a + b)2 sin2— + (a - b)2cos2 — (A) c (a + b)
i.IO
(B)
z, + z, is equal to : Z, - z , (C) 3
(B) 4^2
(C) 5 h
(D) none
In a triangle ABC, CM and CM are the lengths of the altitude and median to the base AB. If a = 10, b = 26, c = 32 then length (HM) (A) 5 (B) 7 (C) 9 (D) none
(S? Bansal Classes
Q. B. on St. line, sequence & progression,complex No.
[2]
Q.12 Number of points denoting the complex number Z on the complex plane and satisfying simultaneously 7 the system of equations 0 < argZ <7t/4 and | Z - 6 i | = 5is /yy/ (A) 0 (B) 1! (C) 2 (D) more than 2 Q.13 A stick of length 10 units rests against the tloor & a wall ofa room . If the stick begins to slide on the floor then the locus of its middle point is : (A) x2 + y2 = 2.5 (B) x2 + y2 = 25 (C) x2 + y2 = 100 (D) none Q. 14 If a, b, c are in 1 LP., then a. a - c. a - b are in : (A) A.P. (B) G.P. (C)H.P.
(D) none of these
Q. 15 Through a given point P (a, b) a straight line is drawn to meet the axes at Q & R. If the parallelogram Sy-yf OQSR is completed then the equation ofthe locus of S is (given'O' is the origin): a b a ib h . (B) ^ = (A) - + -i (C) t + - = 2 (I)) 2 x y • y x x V! v x 1 — iz ll Q. 16 11 z = x+ iy & to = then | to | = 1 implies that, in the complex plane : z- i (B) z lies on the real axis (A) z lies on the imaginary axis (D) none (C) z lies on the unit circle C
I
Q. 17 In a triangle ABC. CD is the bisector ol the angle C. 11 cos — has the value ~ and /(CD) - 6, then 2 j •i — + j has the value equal to (A)
(C)
(B)
(I)) none
Q. 1JL Two mutually perpendicular straight lines through the origin from an isosceles triangle with the line 2x + y = 5 . Then the area of the triangle is : > (A) 5 (B) 3 (C) 5/2 (D) 1 Q. 19 If the median ofa triangle ABC through A is perpendicular to AB then — --- has the value equal to tan Ii n- > A 5 (A) (B) 2 (C) - 2 (D) Q.20 The region represented by inequalities Arg Z < — ; | Z | < 2; lm(z) > 1 in the Argand diagram is given by: a
2-
\
1 7V>0°
(S? Bansal Classes
\ , 2
\
1
(D)
TV1"'
1
2
Q. B. on St. line, sequence & progression,complex No.
[462]
Q.21
Distance between the two lines represented by the line pair, x2 - 4xy + 4y2 +.x - 2y - 6 = 0 is : • "(A)
V5
(B) V5
(C) 2 4
(D) none
Q.22
If three positive numbers a. b, c are in H.P. thene'n + u-2b+c) simplifies to 2 (A) (a - c) (B) zero (C) ( a - c) (D) 1
Q.23
The distance between the two parallel lines is 1 unit. A point 'A' is chosen to lie between the lines at a distance'd' from one of them . TriangleABC is equilateral with B on one line and C on the other parallel line. The length ofthe side ofthe equilateral triangle is (A) | V d 2 + d + l
Q.24
(B)
~
3
d+
'
(C) 2 V d 2 - d
+
l
(D) V d 2 _ d + i
Let /. be a complex number, then the fey ion represented by the inequality |z + 2| < | z 4' 4 [ is given by: (A) Re(z)>-3 (B) Im(z) < - 3 (C) Re (z) < - 3 & Im (z) > - 3 (D) Re (z) < - 4 & Im (z) > - 4
Q.25 .With usual notations, in a triangle ABC, a cos(B - C) + b cos(C - A) + c cos( A - B) is equal to (A)
Q.26
abc F"
abc
4abc
i
abc
1
The co-ordinates ofthe points A, B, C are (- 4. 0), (0, 2) & (-13, 2) respectively . The point of intersection ofthe line which bisects the angle CAB internally and the line joining C to the middle point of AB is: ( A )
(_|,i)
m
[_!.!!)
m
( l , J i )
1 (D)
Q.27
With usual notations in a triangle ABC, (11, )• (112) • (113) has the value equal to (A) R2r (B) 2R2r (C)4R2r (D) 16R2r
Q.28
If co be a complex cube root of unity, then the value of —
,
1+2©
(A) 0
(B) 1
(C) - 1
— + -—-— is :
1 + co
2 +co
(D) none
Q.29
The points A(a, 0), B(0, b), C(c, 0) & D(0, d) are such that ac = bd & a, b, c, d are all non-zero. The the points: (A) form a parallelogram (B) do not lie on a circle (C) form a trapezium (D) areconcyclic
Q.30
Ifthe roots of the cubic x3 - px2 + qx - r = 0 arc in G.P. then (A) q3 = p3r (B)p3 = q3r (C) pq = r
Q.31
(D)pr = q
If A, is the area ofthe triangle with the vertices (0,0); (atanO, bcotO); (asinO, bcosO); A2isthe area of the triangle with the vertices (a, b); (a sec2 0 , b cosec2 0); (a + a sin2 0 , b + b cos2 0) and A3 is the area ofthe triangle with the vertices (0,0); (atanO, -bcotO); (asinO, bcosO) then the values of 0 for which A,,A2, A3 areinGP. is: (A) 0 (B) tt/2 (C) 0 , 0 e R (D) none
(S? Bansal Classes
Q. B. on St. line, sequence & progression,complex No.
[4]
Q.32 -kQ.33
If
-7 - 1 . then the expression z"
(A) 1
+z
(B) - 1
is equal to : '
r
(C) i
i (D)-i
In a A ABC, ifthe median, bisector and altitude drawn from the vertex A divide the angle at the vertex into four equal parts then the angles of the A ABC are : 2i it i n n k rt .nr n n jit it (A)T.7.~ T T 1 <0 J.-J-j W - . - . j
Q. 34 The acute angle between two straight lines passing through the point M(- 6.-8) and the points in which the line segment 2x + y + 10 = 0 enclosed between the co-ordinate axes is divided in the-ratio 1 :2 :2 in the directionfromthe point of its intersection with the x - axis to the point of intersection with the y-axis is: (A) ft/3 (B) TI/4 (C) N / 6 (D) TI/12 Q.35
With usual notation in a AABC
—
+
V r, equal to: (A) 1
(B) 16
,• 1 1 — -t— - , , , where K has the value r, j vr, \\J a"b c"
-
r2j \r2
(C) 64
(D) 128
< . i Q.36 If P the affix ofz in the Argand diagram & P moves so that - ' is always purilv imaginary, then the z-1 locus ofz is : V I (A) circle centre , , radius -J= (B) circle centre ^ 1, - - j • radiu^ -|= (C) circle centre (2,2) and radius 1/2
(D) none of these
I
(|).37 In an A.P. with first tenn 'a' and the common difference d (a. d * 0). the ratio' p' ofthe sum ofthe first a n terms to sum ofn terms succeeding them does not depend on n. Then the ratio — and the ratio' p'. respectively are 11 (A)-,-
1 (B)2, -
( C )
11 2'3
( D )1
2'2
„ _„ . . . . , _ a cos A + bcosB + ccosC . Q.38 In a AABC, the value of is equal to : a + b+ c r
(A) -
R
(B)-
_
R
(C)7
CD)
2r
-
Q.39 Ifthe straight lines joining the origin and the points of intersection ofthe curve 5x2 + 12xy - 6y2 + 4x - 2y + 3 = 0 and x + ky - 1 = 0 are equally inclined to the co-ordinate axes then the value of k : (A) is equal to 1 (B) is equal to - 1 (C) is equal to 2 (D) does not exist in the set of real numbers , (S? Bansal Classes
Q. B. on St. line, sequence & progression,complex
No.
[5]
Q .40
Q.41
If co is one ofthe imaginary cube root of unity then the value of tire expression, (1 + 2co + 2or)lt) + (2 + co + 2co2)10 + (2 + 2co + to2)10 is: (A) 0 -(B) 1 '(C) co (D) co2 (r, +r,)(r, + r,)(r, + r,) w dth u s u a l n o t a t b n in a A ABC, if R = k — where k has the value equal to r, r2 + r2 r3 + r3 r, (A) 1
Q.42
Q.43
(B) 2
'
(C) 1/4
(D) 4
I fa, b, c are in CP., then the equations, ax2 + 2bx + c = 0 & dx2 + 2ex + f = 0 have a common root, i f d c r - -,-arcin: , .. a bb c' c > (A) A.P. (B) G.P. • (C) HP. (D) none i j I In a triangle ABC, AI) is the altitude from A. Given b > c , angle C = 23° & ! AD = ? h c ; 1
Iv - c'
then angle B = (A)157°
(B)113°
(C)147°
(D)nonc
Q.44
If ot& [5 are imaginary cube roots of unity then a n +P" is equal to : n it (A) 2cos-2— (B) cos + (C)2isin— (D) isin 3 3 3
Q.45
A pair of perpendicular straight lines is drawn through the origin forming with the line 2x t 3y 6 an isosceles triangle right angled at the origin. The equation to the line pair is: (A) 5x2 - 24xy - 5y2 0 I (B) 5x2 - 26xy - 5y2 = 0 (C) 5x2 i 24xy - 5y2 0 ' (D) 5x2 + 26xy - 5y2 = 0
Q.46
In a A ABC, cos 3A eos 3B + cos 3C = 1 then : (A) A ABC is right angled (B) A ABC is acute angled (C) A ABC is obtuse angled (D) nothing definite can be said about the nature of the A.
Q.47
If the sum ofthe roots ofthe quadratic equation, ax2 + bx + c = 0 is equal to sum ofthe squares of their , . a b c reciprocals, then —, —, — are in : c a b (A) A.P. (B) GP.
(C)H.P.
(D) none
71 Q.48. The locus of z, for arg z = - — is 2n (A) same as the locus of z for arg z = —
, 7 i (B) same as the locus ofz for arg z = ~
(C) the part ofthe straight line V3 x + y =0 with (y < 0, x > 0) (D) the part ofthe straight line V3x + y =0with(y>0,x<0)
(S? Bansal Classes
Q. B. on St. line, sequence & progression,complex No.
[6]
Q.49
If the incirclc ofthe A ABC touches its sides respectively at L. M and N and if x. v, z be the circumradii of the triangles MIN. NIL and LIM where I is the incentre then the product xy/, is equal to : (A) R r
(B) rR2
(C) - Rr2
(D) - rR2
If the line y = mx bisects the angle between the lines ax2 + 2hxy + by2 = 0 then m is a root of the quadratic equation: (A) hx2 + ( a - b ) x - h = 0 (B) x2 + h (a - b) x - 1 = 0 (C) (a - b) x2 + hx - (a - b) = 0 (D) (a - b) x2 - hx - (a - b) = 0 I Q.5 1 AD, BE and CF are the perpendiculars from the angular points of a A ABC upon the opposite sides. The perimeters ofthe A DEF and A ABC are in the ratio : Q.50
(A)
£i
.
where r is the in radius and R> is the circum radius ofthe A ABC f j Q.52
Q.53
i
!
(COS0 - isinO)4
(sinQ +• icosO)5 (A) cosO-isinO
(B) cos 90 - i sin 90
(C) sin 90 - i cos 90
(D) sin 0 - i cos 0
Letan, n e N is an A.P. with common'difference 'd'&all whose terms are non-zero. If n approaches infinity, then the sum —-— + —!— + a,a->
a T
1 — will approach i+i
i
(A);a jd Q.54
Q.55
(D)a.d
(B) ajd
Points A & B are in the first quadrant; point'O'is the origin . If the slope of OA is 1. slope of OB is 7 and OA = OB, then the slope of AB is : (A) - 1/5 (B) - 1/4 (C) - 1/3 (D) - 1/2 B C In a A ABC if b + c = 3a then cot— • cot— has the value equal to : (A) 4
Q.56
2
(B) 3
(C) 2
(D) 1
Expressed in the form r(cos0 + isin0), - 2 + 2i becomes : (A) 2x/2 cos] — + i sin -— 4; V4 (C) ?V2 cos|
3ti ) . . f. 3K + 1 sin 4/ V 4
3 3n f *!] r f i + ism — — (B) 2V2 cos V V 4J 4 •
•
71 . . 7t (D) 72 cos| -— + tsin 4J K4
Q.57 If P (1,2), Q(4,6), R(5,7)& S (a, b) are the vertices of a parallelogram PQRS, then: (A) a = 2, b = 4 (B) a = 3, b = 4 (C) a = 2, b = 3 (D) a = 3, b = 5
(S? Bansal Classes
Q. B. on St. line, sequence & progression,complex No.
[7]
Q.58
Let f„ g, h be the lengths ofthe perpendiculars from the circumcentre ofthe A ABC on the sides a. b and c respectively .If — + — + —.= X ——- then the value of X is : f g h • tgh (A) 1/4
Q.59
(B) 1/2
(C) 1
(D) 2
The sum to infinite number of terms of series (A/2 + l| + 1 +(V2 - l) + ~> yj 2
-
oo = 4 + 371
( O - U . 3-2V2
(D)
(C) 2nn ± ~ 3
(D) 2n n ± ~ 6
^ ™: 3 -l- 21sinx . . . . Q.60 11 is purely imaginary then x = 1 - 2isinx (A) nn !
1
6
I
(B) nu ± -7 3
1
Q.61
. I II the orthocentre anjd circumcentre of a triangle ABC be at equal distances from the side BC and lie on the same side of BC then tanB lanC has the value equal to: (A) 3
Q.62
(C)-3
(D)- ~
P is a point inside the triangl? ABC. Lines arc drawn through P, parallel to the sides of the triangle. The three resulting triangles with the vertex at P have areas 4,9 and 49 sq. units. The area ofthe triangleABC is (A) 2^3
Q.63
(B)j
(B) 12
(D) 144
(C) 24
The product ofthe distances ofthe incentre from the angular points ofa AABC is : (A) 4 R2 r
(B) 4 Rr2
(C)
(a b c) R
(D)
(abc)s R
Q.64
The number of solutions ofthe equation z2 + z = 0 where z is a complex number, is: (A) 4 (B) 3 (C) 2 (D) 1
Q.65
If Sn denotes the sum ofthefirstn tenns of a GP., with thefirstterm and the common ratio both positive, then (A) S„ , S 2 n , S3n form a G.P. ( B ) Sn ' S2n • " Sn , S 3n , -S 2n form a G.P. (C) S2n - Sn , S3n - S2n , S3n - Sn form a GP. (D) S2n-Sn , S 3 -S 2 n , S 3 -S n form a GP.
Q.66 ;
In a AABC if b = a (V3 — 1) (A) 15° (B) 45°
ar,
d ZC = 30° then the measure of the angle A is (C) 75° (D) 105°
VII Q.67
A is a point on either of two lines y + V3 I x I
4
=
2 at a distance of j j units from their point of intersection.
The co-ordinates of the foot of perpendicular from A on the bisector of the angle between them are (D) (0,4) (A) (B) (0,0) (C) .S (S? Bansal Classes
Q. B. on St. line, sequence & progression,complex No.
[8]
Q.68
If s n = 1 + i"2 + T[ + H> + (A) zero lorn odd (C) positive for n even
+ i -2" ; n e N and i =V^T then 'Sn is (B) zero ofneven - (D) negative for n odd 5P
Q.69
In a AABC, a = a, = 2 , b = a-,, c = a3 such that ap+, = where (A) r, =p r=2 1,2 then
(B)r, = 2r,
'
.
ap
5P
v
(C) r2 = 2r,
1 P
(D) r2.= 3r
Q.70
In a triangle AI3C. side AB has the equation 2 x + 3 y = 29 and the side AC has the equation . x + 2y = 16. If the mid-point of BC is (5,6) then the equation of BC is : (A) x - y = - 1 (B)5x-2y=*13 (C) x + y = 11 (D)3x-4y = - 9
Q.71
Consider an A.P. a, , a-, „ a, a, + a6 + a 7 is (A)-8 (B)5
such that a3 + a- + a8 = 11 and a^ + an = -2. then the value of j j (C)7 (D) 9
Q.72 The solution set ofthe equation, z2 + (3 + 2i)z - 7 + 17i = 0 where z is a complex number expressed in the form of a + bi is (A) 2 - 3i; - 5 + i (B)2 + 3 i ; - 5 + i (C) 2 - 3i; 5 - i (D)none Q.73 Three lines x + 2y + 3 = 0 ; x + 2y - 7,= 0 and 2x - y - 4 = 0 form the three sides of two squares. The equation to the fourth side of each square is ( A ) 2 x - y + 14 = 0 & 2x - y + 6 = 0 ( B ) 2 x - y + 14 = 0 & 2 x - y - 6 = 0 (C) 2x - y - 14 = 0 & 2x - y - 6 = 0 (D) 2x - y - 14 = 0 & 2x - y + 6 = 0 '1 1 1 Q.74 If'()' is the circumcentre ofthe A ABCj" and R,, R, and R3 are the radii ofthe circumcircles of triangles OBC, OCA and OAB respectively then — + — + — has the value equal to: j R, R2 R3 (A)
abc
2R
,
v(B)
R1
4A
(C) —T R
abc
K(D) J
A ^—7 4R2
Q.75 The circumcentre ofthe triangle formed by the lines, xy + 2x + 2y + 4 = 0 and x + y + 2 = 0 is: (A) ( - 2 , - 2 ) (B) ( - 1 , - 1 ) (C) (0,0) (D) ( - 1 , - 2 ) Q.76
The principal value ofthe arg(z) and | z | of the complex number flljO z = 1 + cos
. . flliO + 1 sin ^ ^ " J
' , ' T11TTt c o s Isn 2n In (C) ~ , 2 cos —
ar
e respectively : ,
< B » -In l ? - 2 c o s l In ? .71 n (D)- - , - 2 c o s —
Q.77 The medians ofa A ABC are 9 cm, 12 cm and 15 cm respectively. Then the area of the triangle is (A) 96 sq cm (B)84sqcm (C)72sqcm (D)60sqcm
Bansal Classes
Q. B. on St. line, sequence & progression,complex
No.
[9]
Q.78
ABC is an isosceles triangle'.*Ifthe co-ordinates ofthe base are (1.3) and (- 2. 7), then co-ordinates of vertex A can be : t (A) ( 4 . 5 ) •
Q.79 A circle of radius r is inscribed in a square. The mid points ofsides of the square have been connected by line segment and a new square resulted. The sides ofthe resulting square were also connected by segments so that a new square was obtained and so on, then the radius ofthe circle inscribed in the nlh square is 1-11" 1 2 r (A) 3.80
3-3n " (B) 2
(C)
+ f n
(l + sina)
(D)
_ 5-3 n O" 2
—
4
a +
—
2
(B)
(l-sina) ,
n
a
4
2
(D) 72 (l + sina) ,
The equatiohs ofthe straight line PL, PM are 13x + 4y = 8 and 19x - 3y = 17 respectively, then the length ofthe perpendiculars from any point on the line PQ to the lines PL, PM arc in the ratio [where coordinates of Q s(5,3)] t :
(A)1:V2 82
(C)
n
2
Let z = 1 -,sin a + i cos a where a e (0, n/2), then the modulus and the principal value ofthe argument ofz are respectively: j (A) ^ ( l - s i n a )
3.81
r
2
(B) 1 :2
(C) 2 : 1
;
(0)^2:1
1
I
If x, y and z are the distances of incentre from the vertices ofthe triangle ABC respectively then ja be !xyz" is M equal to (A) n t a n
A
^ A (B)2>ly
^ A (C) Z t a n Y
A (D)Z sm -
83 The line x + y = p meets the axis of x&y at A&B respectively .A triangle APQ is inscribed in the triangle OAB, O being the origin, with right angle at Q . P and Q lie respectively on OB and AB . Ifthe area ofthe triangle APQ is 3/8lh ofthe area of the triangle OAB, then (A) 2 84
(B) 2/3
(C) 1/3
is equal to : (D) 3
The root ofthe equation z5 + z4 + z3 + z2 + z + 1 =0 having the least positive argument is : n n (A) cos — + 1 sin —
7t 71 (B) cos — +1 sin —
71 71 (C) cos — +1 sin —
(D) cos — + i sin -
6
4
(S? Bansal Classes
6
4
Q. B. on St. line, sequence & progression,complex No.
[10]
Q.85
la a AABC, a semicircle is inscribed, whose diameter lies on the side c. Then the radius ofthe semicircle 7 is ' 2A 2A (A)— (B)— a+b a+b-c Where A is the area ofthe triansjle ABC.
Q.8 6 , Q.87
If in triangle ABC , A = (1. 10). circumcentre s
(D)
Given am+n = A ; ain = B as the terms ofthe GP. a, . a2 . a, following holds?
(C) am = a,
c
j - §) and orthocentre =
co-ordinates of mid-point of side opposite to A is : (A) (1,-11/3) (B)(l-5) (C) (1,-3)
(V ' 4) lhcn llic
(D) (1,6) then for A ^ 0 which ofthe
(B) ^ =2VA"B"
(A)a=VAB
Q.88
2A (O—
A\
in m n - mn nun
„ (A) (D)a„=a, vB,
in" 11 n-11
A point 'z' moves on the curve | z - 4 - 3 i | = 2 in an argand plane. The maximum and minimum values of | z | are : (A) 2, 1 (B) 6, 5 (C) 4, 3 (D) 7.3
Q.89 The points with the co-ordinates (2a, 3a), (3 b. 2b) & (c.c) are collinear : (A) for no value of a, b, c (B) for all values of a, b, c i (C) if a, - . b are in II.P. (D) if a. - c. b arc in H.P. 5
^ „, Q.90
Q.91
5
... II in a A ABC.
cos A cosB cosC ; = —— = lien the triangle is a b c ' (A) right angled (B) isosceles (C) equilateral
(D)obtuse
If x,, y, are the roots of x2 + 8 x - 20 = 0, x 2 , y2 are the roots of 4 x2 + 32 x - 57 = 0 and x 3 , y3 are the roots of 9 x2 + 72 x - 1 1 2 = 0 , then the points, (x,, y,), (x2, y2) & (x3, y3) (A) are collinear (B) form an equilateral triangle (C) form a right angled isosceles triangle (D) are concyclic
Q.92
If o) is an imaginary cube root of unity, then the value of, (p + q)1 + (P (J) + q O)2)3 -I- (p G)2 -I- q oi)3 is : 3 (A) p + q3 (B) 3 (p3 + q3) (C) 3 (p3 + q3) - p q (p + q) (D) 3 (p3 + q3) + p q (p + q) i Q.93 If cos A + cosB + 2cosC = 2 then the sides of the AABC are in (A) A.P. (B)G.P (C)H.P. ' (D) none Q.94 'friangle formed by the lines x + y = 0 , x - y = 0 and /x + my = 1. If 1 and m vary subject to the condition / 2 + m2 = 1 then the locus of its circumcentre is (A) (x2 - y2)2 = x2 + y2 (B) (x2 + y2)2 = (x2 - y2) (C) (x2 + y2) = 4x2 y2 (D) (x2 - y2)2 = (x2 + y2)2 (S? Bansal Classes
Q. B. on St. line, sequence & progression,complex No.
[11]
Q.95
Ifthe sum of the first 11 terms ofan arithmetical progression equals that ofthe first 19 terms, then the sum of itsfirst30 terms, is • (A) equal to 0 (B) equal to - 1 (C) equal to 1 (D) non unique
Q 96
If z = (3 + 7i) (p + iq) where p.qe I - {0}, is purely imaginary then minimum value of |zp is 3364 (C)—— (D) 3364 j Two rays emanate from the point Aand form an angle of 43° with one another. Lines L,,L, and L, (no two of which are parallel) each form an isosceles triangle witli the original rays. The largest angle ofthe triangle formed by lines L,, L-, and L, is : (A) 127° ~ " (B) 129° (C) 133° (D) 137° (A) 0
Q.97
Q.98
(B) 58
If in a A ABC.cosAcosB -t sinA sinB' ,sin2C — 1 then, the statement which isiincorrect, is (A) A ABC is isosceles but not right angled (B) AABC is acute angled (C) A ABC is right angled
Q.99
(D) least angle ofthe triangle is —
The co-ordinates ofthree points A(-4,0); B(2,1) and C(3,1) determine the vertices of an equilateral trapezium ABCD . The co-ordinates ofthe vertex D are : (A) (6,0) (B) (- 3, 0) (C) (-5,0) (D) (9, 0)
Q 100 If in a triangle ABC p. q, r are the altitudes from the vcrticcs A, B, C to the opposite sides, then which ofthe following does not hold good? , (A) (EP) ( v i j =(XA) [ v i j
(B) ( I p ) ( E a ) = | J z ^ ^ V
(C) (Z p) (Z pq) ( n a) = (Z a) (Z ab) (fl p)
(D)
0 (1 1 s i n :- + - - V p q rJ V p) (
where R is the circum-radius of A ABC Q.101 The image ofthe pair of lines represented by ax2 + 2h xy + by2 = 0 by the linemirrory = 0 is (A) ax2 - 2h xy - by2 = 0 (B) bx2 - 2h xy + ay2 = 0 2 2 (C) bx + 2h xy + ay = 0 (D) ax2 - 2h xy + by2 = 0 Q.l 02 Let C be a circle with centre P() and AB be a diameter of C. Suppose P, is the mid point ofthe line segment P()B, P, is the mid point ofthe line segment P,B and soon. Let C,, C-,, C3, be circles with diameters P()Pp P,P2, P2P3 respectively. Suppose the circles C p C2, C3, are all shaded. The ratio of the area ofthe unshaded portion of C to that ofthe original circle C is (A)8:9 (B)9:10 (C)10:11 (O) 11:12 Q. 103 The set of values of 'b' for which the origin and the point (1,1) lie on the same side ofthe straight line, a2x + a by + 1 = 0 V a G R, b > 0 are : (A) b 6 (2, 4) (B) b e (0, 2) (C) b g [0, 2] (D) (2, 00) Q. 104 The product of the arithmetic mean of the lengths ofthe sides of a triangle and harmonic mean ofthe lengths of the altitudes ofthe triangle is equal to: (A) A (B) 2 A (C) 3 A (D) 4 A [ where A is the area of the triangle ABC ] (S? Bansal Classes
Q. B. on St. line, sequence & progression,complex No.
[12]
Q. 105 The co-ordinates ofthe vertices P. Q, R & S of square PQRS inscribed in the triangle ABC with vertices As(0.0). B = (3.0) & C = (2, 1) given that two of its vertices P Q are on the side AB are respectively ,A)
0 U
U - V u
(C) (1,0)
.0
f > U ' 3 ' 27
S
1 1 M 4 4 4J . 12
V' 2
4
Q.l 06 If in a triangle sin A: sin C = sin (A-B) : sin (B-C) then a 2 : b 2 : c2 (A) are in A.P. (B)areinGP (C) are in H.P. (D) none of these Q.107 The line 2x + 3y= 12 meets the x-axis at A and they-axis at B .The line through (5,5) perpendicular toAB meets the x-axis, y- axis & the line AB at C, D, E respectively. IfO is the origin, then the area of the ojrEB is : , i f 2j0 23 26 5752 (A) — sq. units (B) — sq. units (C) — sq. units (D) sq. units
Q. 108 If in a triangle ABC (A)
n?
2 cos A a
(B)
8
cos B ^ cos C 3 b + —— + = — + — then the value of the angle A is : b c be ca t n 71 ( C ) -jr (D)~
Q. 109 Let As (3.2) and B s (5,1). ABP is an equilateral triangle is constructed on the side'ofAB remote from the origin then the orthocentre of triangle ABP is ' (A) 4-—73. -) ?- - 7 3
73, — + 73 (B) 4 + — i i
,0,14-175.2-^
(D) 4 + — 73, — + - 73 z j
Q. 110 With usual notations in a triangle ABC, if r, = 2r, = 2r3 then (A) 4a = 3b (B) 3a = 2b (C) 4b = 3a
\
(D)2a = 3b
Q. 111 The vertex ofa right angle of a right angled triangle lies on the straight line 2x + y - 10 = 0 and the two other vertices, at points (2, -3) and (4,1) then the area of triangle in sq. units is (B)3
(A)7io
(C)
33
CD) 11
(
a 2 + b2 + c2 ^ A B C Q.l 12 In a AABC . sin~ siny sin~ simplifies to v sinA' sinB sinC (A) 2A
(B) A
(C)
(D)
where A is the area of the triangle (S? Bansal Classes
Q. B. on St. line, sequence & progression,complex No.
[13]
Q.l 13 GivenA = (1, 1) and AB is any line through it cutting the x-axis in B. IfAC is perpendicular to AR and meets the y-axis in C, then the equation of locus of mid- point P of BC is (A) x + y =j 1 (B) x + y = 2 (C)x + y = 2xy (t>)2x + 2y=l Q. 114 Area ofa triangle inscribed in a circle of radius 4 if the measures of its angles are in the ratio 5 : 4 : 3 is (A) 4 (V3-V2)
(B) 4(73 + 72)
(C) 4(3-75")
(0)4(3 + 73)
^.l 15 ABCD is a rhombus. Its diagonals AC & BD intersect at the point M&satisfy BD = 2AC. Ifthe points D & M represent the complex numbers 1 + i and 2 - i respectively, then A represents the complex numbers are (A) w 1 (C)
3-
r
3 —1
v
(B) 3 + 11 ,
2 ,
i • 1 + —i
3. —1
v
2 ,
(D)none
). 116 The number of possible straight lines. passing through (2,3) and forming a triangle with coordinate axes, whose area is 12 sq. units, is (A) one
(B) two
(C) three
(D)four
•elect the correct alternative : (More than one are correct)
). 117 All the points lying inside the triangle formed by the points (1. 3), (5.6) & (- 1.2) satisfy (A) 3x + 2y > 0 (B) 2x + y + I > 0 (C) 2x + 3y - 12 > 0 (D) - 2x + 11 > 0 .). 118 Two vertices ofthe AABC are at the points A(- 1. - 1) and B(4, 5) and the third vertex lines on the straight line y 5(x - 3). Ifthe area ofthe A is 19/2 then the possible co-ordinates ofthe vertex C are: 1 (A) (5. 10) ..(B) (3.0) (C) (2,-5) (D) (5.4) ) 119 Which ofthe following expressions are not the trigonometric formsofany complex number? (A) - 3
(C) 2
cos
sm
[
, < 1 sm-.
4
- -t 1 cos
.1
7t
(B) 2 I V
4
4
. .
71
- ism— 4
71 2n (D) cos ~ + i sin —
3
'.120 If one vertex of an equilateral triangle of side 'a' lies at the origin and the other lies on the line x - 73 y ~ 0 then the co-ordinates ofthe third vertex are : (A) (0, a) , x
(B)
y
^73a v 2
a") 2y
f 73j1 a a
(C) (0, - a) x
( D )
y
x
9
' 9
y
.121 II - t- - --- 1 is a line through the intersection of — + — = 1 & — + — = 1 and the lengths ofthe c c d a b b a perpendiculars drawn from the origin to these lines are equal in lengths then (A) — 4 V = V ^ "T a" b" c" d' 1 1 1 1 (C) - + - = - + a b e d (S? Bansal Classes
a'
b"
c
a"
(D) none
Q. B. on St. line, sequence & progression,complex No.
[14]
2 in 2 m' Q.l 22 ' The bisectors of angle between the st. lines. v-b=— (x-a)& v - b = -^-—(x-a)are I-nr " I - m'" (A) (y - b) (m + m') + (x - a) (1 - m m') = 0 (B) (y - b) (m + m') - (x - a) (1 - m m') = 0 (C) (y - b) (1 - mm') + (x - a) (m + m') = 0 (D) (y - b) (1 - m m') - (x - a) (m + m') = 0 Q. 123 In the quadratic equation x2 + (p + iq) x + 3i = 0. p & q are real. If the sum ofthe squares ofthe roots is 8 then : (A) p = 3. q = - 1 (B) p = 3. q = 1 (C) p = 43, q = - 1 (D)p - 3 , q = l Q.l24 The x - co-ordinates of the vertices of a square of unit area are the roots of the equation x2 - 3 | x | + 2 = 0 and the y - co-ordinates of the vertices are the roots of the equation y2 - 3y + 2 = 0 then the possible vertices ofthe square is/are : (A) (1. 1). (2. 1). (2. 2). (1.2) (B) (-1. 1), (- 2. 1). (-2. 2). (-1. 2) (C) (2, 1), (1, - 1). (1. 2), (2, 2) (D) (-2, 1). (- 1,-1). (- 1, 2|. (-2. 2) Q. 125 Consider the equation y - y, = m (x - x,). If m & x, arefixedand different lines are drawn for different values of y,, then: (A) the lines will pass through afixedpoint (B) there will be a set of parallel lines (C) all the lines intersect die line x = x, (D) all the lines will be parallel to the line y = x,. I i 1
t '
i i
(S? Bansal Classes
i
I
I i
Q. B. on St. line, sequence & progression,complex No.
[15]
[9ll
'Illrf '°N xsidwoD'uoissBjSojd y douanbds 'vui] 75" no 7/ ()
sdssoj^)
jvsuvfjlj
Select the correct alternative : (Only one is correct)
Q.l Q.6 Q.ll Q.16 Q.21 Q.26 Q.31 Q.3 6 Q.41 Q.46 Q.51 Q.56 Q.61 Q.66 Q.71 Q.76 Q.81 Q.8 6 Q.91 Q.% Q.lOl Q. 106 Q.l 11 Q.ll 6
D A C B B 1) I) A C C C B A D c B I) A A 1) D A B C
.
Q.2 C Q.7 D Q.12 D Q.17 A Q.22 1 A Q. 27 1D Q.3 2 B Q.3 7 (C Q.42 A' Q.47 C Q.52 D Q.57 C Q.62 D Q.67 B Q.72 A Q.77 C Q.82 B Q.87 A Q.92 B Q.97 B Q.l 02 D Q.l07 B Q.l 12 B
Q.3 Q.8 Q.13 Q.18 Q.23 Q.28 Q.3 3 Q.3 8 Q.4 3 Q.48 Q.53 Q.5 8 Q.63 Q.68 Q.73 Q.7 8 Q.8 3 Q.88 Q.9 3 Q.98 Q.l 03 Q.l 08 Q.l 13
A A B A B A C A B C A A B B I) 1) 1) I1 1) A C B 1) A
Q.4 Q.9 Q. 14 Q.19 Q.24 Q.29 Q.34 Q.3 9 Q.44 Q.49 Q.54 Q.59 Q.64 Q.6 9 Q.74 Q.79 Q.84 O 89 Q.94 Q.9 9 Q.l 04 Q.l 09 Q.l 14
I) I) C C A 1) B B A C 1) D C 1) c A 1) 1) A I) B 1) 1)
Q.5 B Q.10 C Q.15, A Q.20 B Q.2 5 A Q.30 A Q35 C Q.40 A Q.45 A Q.50 A Q.55 C Q.60 B Q.65 B Q.70 C Q-75 B Q.80 A Q.8 5 A Q.90 C Q.9 5 A Q. 100 B Q.l 05 1) Q.l 10 "C Q.l 15 A
Select the correct alternative : (More than one are correct)
Q.l 17 A. B. D Q.121 A. C Q.l25
Q.l 18 A, B Q.l22 A, D
Q.l 19 A, B. C. D Q.l23 B.C
B.C
A3>l H3AVSNV
Q.l20 A. B. C. D Q.l24 A. B
BANSAL CLASSES TARGET IITJEE 2007
MATHEMATICS XII (ABCD)
APPLICATION OF DERIVATIVE i
N P E X
TANGENT & NORMAL KEY CONCEPT EXERCISE-I EXERCISE-II EXERCISE—III
:
Page-2 Page -3 Page -5 Page -6
!
MONOTQNQCITY KEY CONCEPT EXERCISE-I EXERCISE-II EXERCISE-III
Page Page Page Page
-7 -8 -10 -11
Page Page Page Page
-13 -16 -18 -20
MAXIMA - MINIMA KEY CONCEPT EXERCISE-I EXERCISE-II EXERCISE-III ANSWER KEY
.
1
Page-22
TANGENT & NORMAL THINGS TO REMEMBER: I The value of the derivative at P (Xj, y{) gives the slope of the tangent to the curve at P. Symbolically = Slope of tangent at iyi P(x1y1) = m(say). Jx
n
Equation of tangent at (x p y^ is; dy
HI
(X-XJ).
*iyi Equation of normal at(x1,y1) is;
Length of I Subnormal
dy dx xi yi NOTE: 1. The point P (Xj, y,) will satisfy the equation of the curve & the equation of tangent & normal line. 2. If the tangent at any point P on the curve is parallel to the axis of x then dy/dx = 0 at the point P. If the tangent at any point on the curve is parallel to the axis of y, then dy/dx = oo or dx/dy=0. 3. If the tangent at any point on the curve is equally inclined to both the axes then dy/dx=± 1. 4. If the tangent at any point makes equal intercept on the coordinate axes then dy/dx=-1. 5. Tangent to a curve at the point P (Xj, y^ can be drawn even through dy/dx at P does not exist, 6. e.g. x = 0 is a tangent to y = x2/3 at (0,0). 7. If a curve passing through the origin be given by a rational integral algebraic equation, the equation ofthe tangent (or tangents) at the origin is obtained by equating to zero the terms ofthe lowest degree in the equation, e.g. If the equation of a curve be x2 - y2 + x3 + 3 x2 y - y3 = 0, the tangents at the origin are given by x2 - y2 = 0 i.e. x + y = 0 and x - y = 0. IV
Angle of intersection between two curves is defined as the angle between the 2 tangents drawn to the 2 curves at their point of intersection. If the angle between two curves is 90° eveiy where then they are called ORTHOGONAL curves.
i f V+ [W ^
VI
yr ^ (a) Length of the tangent (PT) =
(b) Length of Subtangent (MT) = - p —
(c) Length of Normal (FN) = y, ^l + [f'(x,)f
(d) Length of Subnormal (MN) = y, f11 (x^
f(Xj)
DIFFERENTIALS:
The differential of a function is equal to its derivative multiplied by the differential of the independent variable. Thus if, y=tan x then dy = sec2 x dx. In general dy = f ' (x) d x. Note that: d (c) = 0 where 'c' is a constant. d(u + v - w) = du + d v - d w d(uv) = udv + v d u Note
1.
2.
For the independent variable 'x', increment A x and differential d x are equal but this is not the case with the dependent variable 'y' i.e. Ay * dy. dy The relation dy = f' (x) d x can be written as — = f1' (x); thus the quotient of the differentials of'y' and 'x' is equal to the derivative of'y' w.r.t. 'x'.
%BansaIClasses
dx
Application of Derivative
[81
EXERCISE-III Q. 1
Find the equations of the tangents drawn to the curve y2 - 2x3 - 4y + 8 = 0 from the point (1,2).
Q.2
Find the point of intersection of the tangents drawn to the curve x2y = 1 - y at the points where it is intersected by the curve xy = 1 - y.
Q. 3
Find all the lines that pass through the point (1,1) and are tangent to the curve represented parametrically as x = 2t -1 2 and y - 1 +12.
Q. 4
In the curve xayb=Ka+b, prove that the portion of the tangent intercepted between the coordinate axes is divided at its point of contact into segments which are in a constant ratio. (All the constants being positive).
Q.5
A straight line is drawn through the origin and parallel to the tangent to a curve 2
„2
{ r a + ^a2 - y 2 = In at an arbitary point M. Show that the locus of the point P of
intersection of the straight line through the origin & the straight line parallel to the x-axis & passing through the point M is x2 + y2 = a2. Q. 6
n - x2 a a + Va r-z Prove that the segment of the tangent to the curve y=~ In < 0 ~ V a - x contained between 2 a - V a - x2 the y-axis & the point of tangency has a constant length.
Q. 7
A function is defined parametrically by the equations
. 1 ift*0 •"
2t +12 sin-
Q. 8 Q. 9
1
- _sint 2 if t * 0 t f(t) = x = 1 0 if t = 0 "*™m>m o if t = 0 Find the equation of the tangent and normal at the point for t = 0 if exist. Find all the tangents to the curve y = cos (x + y), - 2n < x < 2n, that are parallel to the line x + 2y = 0. (a) (b)
Find the value of n so that the subnormal at any point on the curve xy n =a" +1 may be constant, Show that in the curve y=a. In (x2 - a2), sum of the length of tangent & subtangent varies as the product of the coordinates of the point of contact.
Q. 10 Prove that the segment of the normal to the curve x = 2a sin t + a sin t cos 2 t; y = - a cos3t contained between the co-ordinate axes is equal to 2a. Q.ll
Show that the normals to the curve x = a(cost + t sin t) ; y = a(sint-t cost) arctangent linesto the circle x2 + y2 = a2.
Q. 12 The chord of the parabola y = - a2x2 + 5 ax - 4 touches the curve y=—— at the point x = 2 and is 1—x bisected by that point. Find 'a'. Q. 13 If the tangent at the point (xl5 y,) to the curve x3 + y3 = a3 (a * 0) meets the curve again in (x2, y2) then x v showthat—+— = - 1 . x
i
yi
Q. 1'4 Determine a differentiable function y=f (x) which satisfies f 1 (x) = [f(x)]2 and f(0)=- —. Find also the equation of the tangent at the point where the curve crosses the y-axis.
(!§Bansal Classes
Application of Derivative
[20J
Q. 15 If Pj & p2 be the lengths of the perpendiculars from the origin on the tangent & normal respectively at anyJ ypoint v(x,Jy)J on a curve, then show that
p^lxsinT-ycos^l v ,T,
. ', p 2 =|xcos F + ysin v F|
above case, the curve be x2/3 + y2/3 = a2/3 then show that:
dy where tan ¥ = — . If in the dx
4 p } 2 + p22 = a2.
Q. 16 The curve y=ax 3 + bx 2 +cx + 5, touches the x - axis at P ( - 2,0) & cuts the y-axis at a point Q where its gradient is 3. Find a, b, c. Q. 17 The tangent at a variable point P ofthe curve y=x 2 - x3 meets it again at Q. Show that the locus of the middle point of PQ is y = 1 - 9x + 28x2 - 28x3. Q.18
Show that the distance from the origin of the normal at any point of the curve
0
9
0
0"
x = ae e sin—+ 2cos— &y = ae e c o s — 2 sin— is twice the distance of the tangent at the point 2 2. 2 2 from the origin. Q. 19 Show that the condition that the curves x273 + y2/3 = c2/3 & (x2/a2) + (y2/b2) = 1 may touch if c = a + b. Q.20 The graph of a certain function/contains the point (0,2) and has the property that for each number 'p' the line tangent to y = / (x) at (p, /(p)) intersect the x-axis at p + 2. Find/(x). Q.21 A curve is given by the equations x = at2 & y = at3. A variable pair of perpendicular lines through the origin 'O' meet the curve at P & Q. Show that the locus of the point of intersection of the tangents at P & Q is 4y2 = 3ax - a2. x2 y2 = Q.22(a) Show that the curves ~2—TT + T i — a +Kj b +Kj
1&
x2 + y2 = ~ i — 1 — 1 intersect orthogonally. a + K2 b + &2
2 X v2
x2 v2
(b) Find the condition that the curves— + — = 1 & — - + — = 1 may cut orthogonally. a b a b Q.23
Show that the angle between the tangent at any point A' of the curve /n (x2 + y2) = C tan 1xx and the line joining A to the origin is independent of the position ofAon the curve.
i i2
Q.24 For the curve x2/3 + y2/3 = a2/3, show that | z | + 3p2 = a2 where z = x + i y & p is the length of the perpendicular from (0,0) to the tangent at (x, y) on the curve. Q.25 Aand B are points of the parabolay=x2. The tangents at Aand B meet at C. The median of the triangle ABC from C has length'm' units. Find the area of the triangle in terms of'm'.
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EXERCISE-II Q. 1
RATE MEASURE AND APPROXIMATIONS Water is being poured on to a cylindrical vessel at the rate of 1 m3/min. If the vessel has a circular base of radius 3 m, find the rate at which the level of water is rising in the vessel.
Q. 2
A man 1.5 m tall walks awayfroma lamp post 4.5 m high at the rate of 4 km/hr. (i) how fast is the farther end of the shadow moving on the pavement ? (ii) how fast is his shadow lengthening ?
Q. 3
A particle moves along the curve 6 y=x 3 + 2. Find the points on the curve at which the y coordinate is changing 8 times as fast as the x coordinate.
Q.4
An inverted cone has a depth of 10 cm & a base of radius 5 cm. Water is poured into it at the rate of 1.5 cm3/min. Find the rate at which level of water in the cone is rising, when the depth of water is 4 cm.
Q.5
A water tank has the shape of a right circular cone with its vertex down. Its altitude is 10 cm and the radius of the base is 15 cm. Water leaks out of the bottom at a constant rate of leu. cm/sec. Water is poured into the tank at a constant rate of C cu. cm/sec. Compute C so that the water level will be rising at the rate of 4 cm/sec at the instant when the water is 2 cm deep.
Q.6
Sand is pouringfroma pipe at the rate of 12 cc/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always l/6th of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm.
Q.7
An open Can of oil is accidently dropped into a lake; assume the oil spreads over the surface as a circular disc ofuniform thickness whose radius increases steadily at the rate of 10 cm/sec. At the moment when the radius is 1 meter, the thickness of the oil slick is decreasing at the rate of 4 mm/sec, how fast is it decreasing when the radius is 2 meters.
Q. 8
Water is dripping outfroma conical funnel of semi vertical angle TC/4, at the uniform rate of 2 cm3/sec through a tiny hole at the vertex at the bottom. When the slant height of the water is 4 cm, find the rate of decrease of the slant height of the water.
Q. 9
An air force plane is ascending vertically at the rate of 100 km/h. If the radius of the earth is R Km, how fast the area of the earth, visiblefromthe plane increasing at 3min after it started ascending. Take visible a r e a A = — — Where h is the height of the plane in kms above the earth. R+h
Q. 10 A variable A ABC in the xy plane has its orthocentre at vertex 'B', a fixed vertex 'A' at the origin and the 7x2 third vertex 'C' restricted to lie on the parabola y = H——. The point B starts at the point (0,1) at time 36 t = 0 and moves upward along the y axis at a constant velocity of 2 cm/sec. How fast is the area of the triangle increasing when t = ^ sec. Q. 11 A circular ink blot grows at the rate of 2 cm2 per second. Find the rate at which the radius is increasing 6 22 after 2 — seconds. Use it =—. 11 7
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Q. 12 Water is flowing out at the rate of 6 m3/minfroma reservoir shaped like a hemispherical bowl of radius % 2 R = 13 m. The volume of water in the hemispherical bowl is given by V = (3R - y) when the (a) (b)
water is y meter deep. Find At what rate is the water level changing when the water is 8 m deep. At what rate is the radius of the water surface changing when the water is 8 m deep.
Q. 13 If in a triangle ABC, the side 'c' and the angle 'C' remain constant, while the remaining elements are changed slightly, show that
cos A
^ — = 0. cosB
Q. 14 At time t > 0, the volume of a sphere is increasing at a rate proportional to the reciprocal of its radius. At t = 0, the radius of the sphere is 1 unit and at t= 15 the radius is 2 units. (a) Find the radius of the sphere as a function of timet. (b) At what time t will the volume of the sphere be 27 times its volume at t = 0. Q. 15 Use differentials to a approximate the values of; (a)\25.2 and(b) V26. EXERCISED! Q. 1
Find the acute angles between the curves y = | x2 -11 and y = | x2 - 3 ! at their point of intersection. [REE '98,6]
Q.2
Find the equation of the straight line which is tangent at one point and normal at another point of the curve, x — 3t 2 , y — 2t3. [ REE 2000 (Mains) 5 out of 100 ] 3tt
Q. 3
If the normal to the curve, y=f(x) at the point (3,4) makes an angle — with the positive x-axis. Then f'(3) (A)-l
(B)-|
(C)j
(D)l [JEE 2000 (Scr.) 1 out of 35 ]
Q.4
The point(s) on the curve y3 + 3x2 = 12y where the tangent is vertical, is(are) c 4 f A \ ' TT ^ 4 (A) 2) (B) { ^ J , l j (C) (0,0) (D) . 2J [JEE 2002 (Scr.), 3]
Q.5
Tangent to the curve y=x 2 + 6 at a point P (1,7) touches the circle x2 + y2 + 16x+ 12y+c = 0atapoint Q. Then the coordinates of Q are (A) (-6,-11) (B) (-9,-13) (C) ( - 10, - 15) (D)(-6,-7) [JEE 2005 (Scr.), 3]
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MONOTONOCITY (Significance of the sign of thefirstorder derivative) DEFINITIONS: 1. A function f (x) is called an Increasing Function at a point x=a if in a sufficiently small neighbourhood around x=a we have
f (a + h)
> f (a) and
f (a - h)
< f (a)
Similarly decreasing if
f (a + h)
< f (a) and
f(a-h)
> f (a)
increasing;
disregards whether f is non derivable or even discontinuous at x = a
| decreasing.
A differentiable function is called increasing in an interval (a, b) if it is increasing at every point within the interval (but not necessarily at the end points). A function decreasing in an interval (a, b) is similarly defined. A function which in a given interval is increasing or decreasing is called "Monotonic" in that interval. Tests for increasing and decreasing of a function at a point: If the derivative f'(x) is positive at a point x = a, then the function f (x) at this point is increasing. If it is negative, then the function is decreasing. Even if f (a) is not defined, f can still be increasing or decreasing.
2.
3. 4.
f'<0
O
f'<0 c
0 increasing at x = 0
decreasing at x = c
Note : If f'(a) = 0, then for x = athe function may be still increasing or it may be decreasing as shown. It has to be identified by a seperate rule. e.g. f (x) = x3 is increasing at every point. Note that, dy/dx = 3 x2. yt
5.
J'<0
N
/
f'>01 f'>0
O
x=a
•f'(a)=0
f'(a)=0
sf'<0
O
x=a
—»x
Tests for Increasing & Decreasing of a function in an interval: SUFFICIENCY TEST : If the derivative function f'(x) in an interval (a, b) is every where positive, then the function f(x) in this interval is Increasing; If f'(x) is every where negative, then f (x) is Decreasing.
General Note: (1) If a function is invertible it has to be either increasing or decreasing. If a function is continuous the intervals in which it rises and falls may be separated by points at which its (2) derivative fails to exist. If f is increasing in [a, b] and is continuous then f (b) is the greatest and f (c) is the least value of f in [a, (3) b]. Similarly if f is decreasing in [a, b] then f (a) is the greatest value and f (b) is the least value. 6.
(a) (i)
00
(iii)
ROLLE'S THEOREM: Let f(x) be a function of x subject to the following conditions: f(x) is a continuous function of x in the closed interval of a < x < b. f' (x) exists for every point in the open interval a < x < b. f(a) = f(b). Then there exists at least one point x = c such that a < c < b where f' (c) = 0. Note that if f is not continuous in closed [a, b] then it may lead to the adjacent graph where all the 3 conditions of Rolles will be valid but the assertion will not be true in (a, b).
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(b) (i) (ii) (iii)
LMVT THEOREM: Let f(x) be a function of x subject to the following conditions: f(x) is a continuous function of x in the closed interval of a < x < b. f' (x) exists for every point in the open interval a < x < b, f(a)*f(b). Then there exists at least one point x = c such that a < c < b where f' (c) =
'——^ b- a Geometrically, the slope of the secant line joining the curve at x = a & x = b is equal to the slope of the tangent line drawn to the curve atx = c. Note the following: "a.
Rolles theorem is a special case of LMVT since
b- a Note : Now [f (b) - f (a)] is the change in the function f as x changes from a to b so that [f (b) - f (a)] / (b - a) is the average rate ofchange of the function over the interval [a, b]. Also f'(c) is the actual rate of change of the function for x = c. Thus, the theorem states that the average rate of change of a function over an interval is also the actual rate of change ofthe function at some point of the interval. In particular, for instance, the average velocity of a particle over an interval of time is equal to the velocity at some instant belonging to the interval. This interpretation of the theorem justifies the name "Mean Value" for the theorem. (c)
APPLICATION O F ROLLES THEOREM FOR ISOLATING T H E R E A L ROOTS O F A N EQUATION f ( x ) = 0
(i) (ii) (iii)
Suppose a & b are two real numbers such that; f(x) & its first derivative f' (x) are continuous for a < x < b. f(a) & f(b) have opposite signs. f' (x) is different from zero for all values of x between a & b. Then there is one & only one real root of the equation f(x) = 0 between a & b.
EXER CISE-I Q. 1
Find the intervals ofmonotonocity for the following functions & represent your solution set on the number line. (a) f(x) = 2. e x2 "4x (b) f(x) = ex/x Also plot the graphs in each case.
(c) f(x) = x2 e~x
(d) f (x) = 2x2 - In \ x |
1
Q.2
Find the intervals in which f (x) = cosx-;, cos3x is a decreasing function.
Q. 3
Find the intervals of monotonocity ofthe function (a) f(x) = sinx-cosx inx e[0,27i] (b)
Q.4
Q.5
g(x) = 2 sinx + cos2x in(0
Show that, x3 - 3x2 - 9 x + 20 is positive for all values of x > 4.
3
2
Let f (x) = x - x + x + 1 and g(x) =
r max{f(t):0
3-x Discuss the conti. & differentiability of g(x) in the interval (0,2). Q.6, ^
,0
Find the set of all values of the parameter 'a' forwhich the function, f(x) = sin 2x - 8(a +1 )sin x + (4a2 + 8a -14)x increases for all x e R and has no critical points for all x e R.
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Application of Derivative
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Q.7
Find the greatest & the least values of the following functions in the given interval ifthey exist. (a)/(x) = sin
Q.8
1
(b) y = xx in (0,oo) (c) y = x5 - 5x4+ 5x3 +1 in [-1,2]
/—— - / n x i n
»
2
V*+l
oJb\ AV>\ » r •
Find the values of'a' for which the function f(x)=sinx - a sin2x - — sin3x+2ax increases throughout the number line.
Q.9
e Prove that f (x) = j(9cos 2 (2/nt)-25cos(2/nt) + 17)dt is always an increasing function of x,VxeR
V-0
x3 + (a -1) x2 + 2x +1 is monotonic increasing for every x e R then find the range of
Q.IO Iff(x) =
\ values of'a'.
y
Q. 11 Find the set of values of 'a' for which the function, f(x) = {
V 2 1 - 4a - a2 a+1
x3 + 5x + V7 is increasing at every point of its domain.
Q.12 Find the intervals in which the function f (x) = 3 cos4 x + 10 cos3 x + 6 cos2 x - 3 , 0 < x < 7 i ; i s monotonically increasing or decreasing. Q.13 Find the range of values of'a'for which the function f(x)=x 3 + (2a+3)x2 + 3(2a+ l)x+5 is monotonic in R. Hence find the set of values of'a' for which f (x) in invertible. Q.14 Find the value of x> 1 for which the function
^ crn O
I
F ( x ) - Jf-/n
t
1
Q.15 \
/ ^
2
I 32 J
dt is increasing and decreasing.
Find all the values of the parameter 'a' for which the function; f(x) = Sax - a sin 6x - 7x - sin 5x increases & has no critical points for all x € R.
Q.16 Iff(x) = 2ex - ae~x+(2a +1 )x - 3 monotonically increases for every x e R then find the range of values
Q.17
Construct the graph of the function f (x) = -
x2 - 9 and comment upon the following --x+x+3 x—1
(a) Range ofthe function, (b) Intervals of monotonocity, (c) Point(s) where f is continuous but not dififrentiable, (d) Point(s) where f fails to be continuous and nature of discontinuity. (e) Gradient of the curve where f crosses the axis of y. Q.18 Prove that, x2 - 1 > 2x In x > 4(x - 1 ) - 2 In x for x > 1. , * f 3n } ) CUP Q.19 Prove that tan2x + 6 In secx + 2cos x + 4 > 6 sec x for x e — , 2 n \ . ' ~ * U' J Y^tesU^Q.20 If ax2 + (b/x) > c for all positive x where a > 0 & b > 0 then show that 27ab2 > 4c3. fa^ Q.21
fr
'\
::
. >1
If 0 < x < 1 prove that y = x In x - (x2/2) + (1/2) is a function such that d2y/dx2 > 0. Deduce that x/nx>(x 2 /2)-(l/2).
4§Bansal Classes
Application of Derivative * 0 » »
[9]
Q.22 Prove that 0 < x. sirix - (1/2) sin2x < (1/2) (71 - 1 ) for 0 < x < tc/2. Q.23
Show that x2 > (1 + x) [/n(l + x)]2 Vx>0.
Q.24 Find the setofvalues of x for which the inequality /n(l +x)> x/(l +x) is valid. Q.2^ If b> a,findthe minimum value of |(x-a) 3 |+ | ( x - b ) 3 | , x € R.
EXERCISE-II Q. 1 .2
Verify Rolles throrem for f(x) = (x - a)m (x - b)n on [a, b]; m, n being positive integer. Let f: [a, b] R be continuous on [a, b] and differentiable on (a, b). Iff (a) < f (b), then show that f ' (c) > 0 for some c e (a, b).
Q.3
Let f (x) = 4x3 - 3x2 - 2x +1, use Rolle's theorem to prove that there exist c, 0< c <1 such that f(c) = 0.
Q. 4
Using LMVT prove that: (a) tan x > x in
V
i sin x < x
for x > 0
Q.5 Prove that if / i s differentiable on [a, b] and if f (a) = f (b) = 0 then for any real a there is an x e (a, b) X ^ N such that a f ( x ) + f'(x).= 0. 3 x=0 r 2 Q.6 For what value of a, m and b does the function f (x) = L - x +3x + a 0 < x < l mx+b l
Let f, g be differentiable on R and suppose that f(0) = g (0) and f ' (x) < g' (x) for all x > 0. Show that f (x) < g (x) for all x > 0.
Q.9
Let f be continuous on [a, b] and differentiable on (a, b). If f (a) = a and f (b) = b. show that there exist distinct c p c2 in (a, b) such that f ' (Cj) + f'(c 2 ) = 2.
Q. 10. Let f (x) and g (x) be differentiable functions such that f' (x) g (x) * f (x) g' (x) for any real x. Showthat between any two real solutions of f (x) = 0, there is at least one real solution of g (x) = 0. Q. 11 Let f defined on [0,1 ] be a twice differentiable function such that, | f" (x) | < 1 for all x e [0,1 ] If f (0) = f (1), then showthat, | f' (x) | < 1 for all x e [0,1] Q. 12 f (x) and g (x) are differentiable functions for 0 < x < 2 such that f (0) = 5, g (0) = 0, f (2) = 8, g (2) = 1. Show that there exists a number c satisfying 0 < c < 2 and f ' (c) - 3 g' (c). Q.13 If f,
f'(c) *'(c) = 0 ^'(c)
Q.14 Show that exactly two real values of x satisfy the equation x 2 =x sinx + cos x. Q.15 Let a > 0 and/ be continuous in [-a, a]. Suppose that / (x) exists a n d / (x) < 1 for all x € (-a, a). If /(a) = a and/(- a) = - a, show that f (0) = 0. Q. 16 Let a, b, c be three real number such that a < b < c, f (x) is continuous in [a, c] and differentiable in (a, c). Also f 1 (x) is strictly increasing in (a, c). Prove that (c - b) f (a) + (b - a) f (c) > (c - a) f (b) % Bans aIClasses
Application of Derivative
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Q.17 Use the mean value theorem to prove,
X—1
l
X
Q.18 Use mean value theorem to evaluate, Lhn (Vx + l -Vx)_ Q.19 Using L.M. V.T. or otherwise prove that difference of square root of two consecutive natural numbers greater than N2 is less than 1 . Q.20 Prove the inequality ex > (1 + x) using LMVT for all x e Rq and use it to determine which of the two numbers e" and ne is greater.
EXERCISE-III 3
Q.l
Letf(x) =
xe* x<0 , , ' ; where 'a' is a positive constant. Find the interval in which f'(x) is x + ax - x , x > 0
increasing. Q.2
Q.3
Iff(x)=
[ JEE '96,3 ] X
sin x
&g(x)=
X
tan x
, where 0 < x < 1, then in this interval:
(A) both f(x) & g (x) are increasing functions (B) both f(x) & g (x) are decreasing functions (C) f (x) is an increasing function (D) g (x) is an increasing function [ JEE '97 (Scr), 2 ] dg Let a+b = 4, where a < 2 and let g (x) be a differentiable function. If > 0 for all x, prove that dx a
b
o
o
j g(x) dx + J g(x) dx increases as (b - a) increases.
[JEE '97,5]
Q.4(a) Let h(x) = f(x) - (f(x))2 + (f(x))3 for every real number x. Then: (A) h is increasing whenever f is increasing (B) h is increasing whenever f is decreasing (C) h is decreasing whenever f is decreasing (D) nothing can be said in general. x2- 1 (b) f(x) = —r—, for every real number x, then the minimum value of f: x +1 (A) does not exist because f is unbounded (B) is not attained even though f is bounded (C) is equal to 1 (D) is equal to - 1 . [JEE '98,2 + 2 ] Q.5(a) Foralfx e (0,1): (A) ex < 1 + x (B) loge(l +x)
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Q.6(a) If /(x) = xex(1 ~x), then f(x) is
( (A) increasing on
M ~ >l J
(B) decreasing on R
4 ' (C) increasing on R (D) decreasing on 3 (b) Let - 1 < p < 1. Show that the equation 4x - 3x - p = 0 has a unique root in the interval identify*. Q.7
and
[ JEE 2001,1 + 5 ]
The length of a longest interval in which the function 3 sinx - 4sin3x is increasing, is 7t 7X 3TC (A) — (B) — (C)y '((D) D) n7i [JEE 2002 (Screening), 3]
Q.8(a) Using the relation 2(1 - cosx) < x 2 , x ^ 0 or otherwise, prove that sin (tanx) > x, V x« 0'i (b) Let f: [0,4] -» R be a differentiable function. (i) Show that there exist a, b € [0,4], (f (4))2 - (f (0))2 = 8 f'(a) f (b) (ii) Show that there exist a, P with 0 < a < p < 2 such that 4 | f(t) dt = 2 (a f (a 2 ) + p f (P 2 )) [JEE 2003 (Mains), 4 + 4 out of 60]
o
Q.9(a) Let f (x):
x°7nx,x > 0 0,
x=0
(A)-2
. Rolle's theorem is applicable to f for x e [0,1], if a =
(B)-l
(C)0
1
(D)-
f(x2)-f(x) . (b) If/is a strictly increasing function, then Lim — i s equal to T.
(A) 0
(B) 1 101
(C)-l
(D)2
[JEE 2004 (Scr)]
100
Q.10 If p (x) = 5 lx - 2323x - 45x +1035, using Rolle's theorem, prove that at least one root of p(x) lies between (451/100,46). [JEE 2004,2 out of 60] Q. 11 (a) If/(x) is a twice differentiable function and given that f(l) = 1, f(2) = 4, f(3) = 9, then (A) f" (x) = 2, for V x e (1, 3) (B) f" (x) = f ' (x) = 2, for some x e (2, 3) (C) f" (x) = 3, for V x € (2, 3) (D) f" (x) = 2, for some x e (1,3) [JEE 2005 (Scr), 3] (b) f (x) is differentiable function and g (x) is a double differentiable function such that | f (x) | < 1 and f '(x) = g (x). Iff^O) +g2(0) = 9. Prove that there exists some c e (-3,3) such that g (c) • g"(c)<0. [JEE 2005 (Mains), 6]
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Application of Derivative
[12]
MAXIMA - MINIMA
FUNCTIONS OF A SINGLE VARIABLE HOW MAXIMA & MINIMA ARE CLASSIFIED 1.
relative maximum A function f(x) is said to have a maximum absolute maximum No greater value of f. No greater value of f. at x = a if f(a) is greater than every other near by relative maximum Also a relative maximum No greater value of f. value assumed by f(x) in the immediate A nearby y=f(x) neighbourhood of x=a. Symbolically f(a) > f(a + h) >x=a gives maxima for f(a) > f(a-h) relative minimum R No smaller value of f. a sufficiently small positive h. near by absolute minimum Similarly, a function f(x) is said to have a No smaller value of f. Also a relative minimum minimum value at x=b if f(b) is least than every other value assumed by f(x) in the immediate x=a x=b neighbourhood at x=b. Symbolically if
f(b) < f(b + h)
x=b gives minima for a sufficiently small positive h. p/1 \ C f1 1N » i f(b) < f(b-h) Note that: (i) the maximum & minimum values of a function are also known as local/relative maxima or local/relative minima as these are the greatest & least values ofthe function relative to some neighbourhood of the point in question. the term 'externum' or (extremal) or 'turning value' is used both for maximum or a minimum value, (ii) a maximum (minimum) value of a function may not be the greatest (least) value in a finite interval, (iii) (iv) a function can have several maximum & minimum values & a minimum value may even be greater than a maximum value. maximum & minimum values of a continuous function occur alternately & between two consecutive (v) maximum values there is a minimum value & vice versa. w
(0 (ii) (iii) (iv)
A NECESSARY CONDITION FOR MAXIMUM & MINIMUM : If f(x) is a maximum or minimum at x = c & iff' (c) exists then f' (c) = 0. Note : The set of values of x for which f' (x) = 0 are often called as stationary points or critical points. The rate of change of function is zero at a stationary point. In case f' (c) does not exist f(c) may be a maximum or a minimum & in this case left hand and right hand derivatives are of opposite signs. The greatest (global maxima) and the least (global minima) values of a function fin an interval [a, b] are f(a) or f(b) or are given by the values of x for which f' (x) = 0. dy Critical points are those where — = 0, if it exists, or it fails to exist either by virtue of a vertical tangent dx or by virtue of a geometrical sharp corner but not because of discontinuity of function. SUFFICIENT CONDITION FOR EXTREME VALUES :
f'(c-h) > 0 => x - c is a point of local maxima, where f' (c)=0. h is a sufficiently f'(c+h) < 0 small positive . f'(c-h) < 0 • x - c i s aFpoint of local minima, where f'(c) = 0. quantity Similarly v f'(c+h) > 0_ ' Note : Iff' (x) does not change sign i.e. has the same sign in a certain complete neighbourhood of c, then f(x) is either strictly increasing or decreasing throughout this neighbourhood implying that f(c) is not an extreme value of f.
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Application of Derivative
[81
4. (a) (b) 5.
USE OF SECOND ORDER DERIVATIVE IN ASCERTAINING THE MAXIMA OR MINIMA: f(c) is a minimum value of the function f, if f' (c) = 0 & f" (c) > 0. f(c) is a maximum value of the function f, f' (c) = 0 & f" (c) < 0. Note : iff" (c) = 0 then the test fails. Revert back to thefirstorder derivative check for ascertaning the maxima or minima. SUMMARY-WORKING
RULE:
FIRST:
When possible, draw a figure to illustrate the problem & label those parts that are important in the problem. Constants & variables should be clearly distinguished. SECOND:
Write an equation for the quantity that is to be maximised or minimised. If this quantity is denoted by 'y', it must be expressed in terms of a single independent variable x. his may require some algebraic manipulations. THIRD:
If y = f (x) is a quantity to be maximum or minimum, find those values of x for which dy/dx = f'(x) = 0. FOURTH:
Test each values of x for which f'(x) = 0 to determine whether it provides a maximum or minimum or neither. The usual tests are: (a) If dVdx 2 is positive when dy/dx = 0 => y is minimum. If dVdx 2 is negative when dy/dx = 0=>yis maximum. If d^/dx 2 = 0 when dy/dx = 0, the test fails. (b)
dy
positive
If— is zero dx
negative
for
x < x0
for x = xn for
• a maximum occurs at x = xQ.
x > x0
But if dy/dx changes signfromnegative to zero to positive as x advances through x0 there is a minimum. If dy/dx does not change sign, neither a maximum nor a minimum. Such points are called INFLECTION POINTS. FIFTH:
If the function y = f (x) is defined for only a limited range of values a < x < b then examine x = a& x = b for possible extreme values. SIXTH:
Ifthe derivative fails to exist at some point, examine this point as possible maximum or minimum. Important Note: Given a fixed point A(xl5 yt) and a moving point P(x, f (x)) on the curve y=f(x). Then AP will be maximum or minimum if it is normal to the curve at P. If the sum of two positive numbers x and y is constant than their product is maximum ifthey are equal,i.e. x + y = c , x > 0 , y > 0 , t h e n xy= ^ [(x + y ) 2 - ( x - y ) 2 ] 6.
If the product of two positive numbers is constant then their sum is least if they are equal, i.e. (x + y)2 = ( x - y)2 + 4xy
USEFUL FORMULAE OF MENSURATION TO
riP
REMEMBER:
Volume of a cuboid = /bh. Surface area of a cuboid = 2 (7b+bh+h/). Volume of a prism = area of the base x height. Lateral surface of a prism=perimeter of the base x height. Total surface of a prism = lateral surface + 2 area of the base (Note that lateral surfaces of a prism are all rectangles).
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Application of Derivative
[81
Volume of a pyramid=- area of the base x height. Curved surface of a pyramid=- (perimeter of the base) x slant height. (Note that slant surfaces of a pyramid are triangles).
<3=
Volume of a c o n e = i A . 3
Curved surface of a cylinder = 2 it rh. Total surface of a cylinder = 2 it rh + 2 it r2. cr-
Volume of a sphere=— tci3. Surface area of a sphere = 4 tt r2. Area of a circular s e c t o r r 2 8 , when 0 is in radians. 2
7.
SIGNIFICANCE O F T H E SIGN O F 2 N D ORDER DERIVATIVE A N D POINTS O F INFLECTION : nd
The sign of the 2 order derivative determines the concavity of the curve. Such points such as C & E on the graph where the concavity of the curve changes are called the points of inflection. From the graph wefindthat if:
dV
>0 => concave upwards
(i)
dx
(ii)
d2y —7 < 0 => concave downwards, dx
2
d2y At the point of inflection wefindthat —y = 0 & dx
dV changes sign. 2
dx
d2y Inflection points can also occur if—-y fails to exist. For example, consider the graph of the function dx defined as, ,3/5
for x e ( - oo, 1) f ( x ) = [ 2 - x2 for x e (1, oo) Note that the graph exhibits two critical points one is a point of local maximum & the other a point of inflection.
faBansal Classes
Application of Derivative
V A
1
\2-x2 V2 "
[15]
EXERCISE-III Q.l
A cubic f(x) vanishes at x = - 2 & has relative minimum/maximum at x = - l and
l If Jf (x) -l
find
the cubic/(x).
3
X
Q.2
Investigate for maxima & minima for the function, f(x)= j [2 (t — 1) (t - 2)3 + 3 (t - l) 2 (t - 2)2] dt
Q.3
Find the maximum & minimum value for the function;
i
(a) Q.4
y = x + sin2x, 0 < x < 2 7 i
(b)
y = 2 c o s 2 x - cos4x, 0
Suppose f(x) is real valued polynomial function of degree 6 satisfying the following conditions; (a) f has minimum value at x = 0 and 2 (b) f has maximum value at x = 1
(c)
f(x) x
for all x, Limit I / n 0 X 1
!
i 0
Q
1
= 2.
i
Determine f(x). Q. 5
Find the maximum perimeter of a triangle on a given base4 a' and having the given vertical angle a.
Q. 6
The length of three sides of a trapezium are equal, each being 10 cms. Find the maximum area of such a trapezium.
Q. 7
The plan view of a swimming pool consists of a semicircle ofradius r attached to a rectangle of length '2^ and width's'. If the surface area A of the pool isfixed,for what value of'r' and V the perimeter 'P! of the pool is minimum.
Q. 8
For a given curved surface of a right circular cone when the volume is maximum, prove that the semi vertical angle is sin-1
Q.9
V3
.
Of all the lines tangent to the graph of the curve y -
6 2~77 >findthe equations of the tangent lines of x +5
minimum and maximum slope. Q.IO A statue 4 metres high sits on a column 5.6 metres high. How farfromthe column must a man, whose eye level is 1.6 metresfromthe ground, stand in order to have the most favourable view of statue. Q. 11 By the post office regulations, the combined length & girth of a parcel must not exceed 3 metre. Find the volume ofthe biggest cylindrical (right circular) packet that can be sent by the parcel post. Q. 12 A running track of440ft.is to be laid out enclosing a footballfield,the shape of which is a rectangle with semi circle at each end. Ifthe area ofthe rectangular portion is to be maximum,findthe length of its sides. 22
Use : n « — .
(!§Bansal Classes
Application of Derivative
[20J
Q.13 A window of fixed perimeter (including the base of the arch) is in the form of a rectangle surmounted by a semicircle. The semicircular portion is fitted with coloured glass while the rectangular part isfittedwith clean glass. The clear glass transmits three times as much light per square meter as the coloured glass does. What is the ratio of the sides of the rectangle so that the window transmits the maximum light? Q.14 A closed rectangular box with a square base is to be made to contain 1000 cubic feet. The cost of the material per square foot for the bottom is 15 paise, for the top 25 paise and for the sides 20 paise. The labour charges for making the box are Rs. 3/-. Find the dimensions of the box when the cost is minimum. Q.15 Find the area of the largest rectangle with lower base on the x-axis & upper vertices on the curve y = 12-x 2 . Q.16 A trapezium ABCD is inscribed into a semicircle of radius / so that the base AD of the trapezium is a diameter and the vertices B & C lie on the circumference. Find the base angle 0 of the trapezium ABCD which has the greatest perimeter. 3X. 4" b
Q.l?
If y =————— has a turning value at (2,-1) find a & b and show that the turning value is a
maximum Q.18 Prove that among all triangles with a given perimeter, the equilateral triangle has the maximum area. Q.19 A sheet of poster has its area 18 m2. The margin at the top & bottom are 75 cms and at the sides 50 cms. What are the dimensions of the poster if the area of the printed space is maximum?
x2 y 2
Q.20 A perpendicular is drawnfromthe centre to a tangent to an ellipse h—j - 1 . Find the greatest value a b of the intercept between the point of contact and the foot of the perpendicular. X
Q.21 Consider the function, F (x) = j(t 2 -1) dt, x e R. (a) (b) (c) (d) (e)
-l
Find the x and y intercept of F if they exist. Derivatives F' (x) and F" (x). The intervals on which F is an increasing and the invervals on which F is decreasing. Relative maximum and minimum points. Any inflection point.
Q.22 A beam of rectangular cross section must be sawnfroma round log of diameter d. What should the width x and height y of the cross section be for the beam to offer the greatest resistance (a) to compression; (b) to bending. Assume that the compressive strength of a beam is proportional to the area of the cross section and the bending strength is proportional to the product of the width of section by the square of its height Q. 2 3 What are the dimensions of the rectangular plot of the greatest area which can be laid out within a triangle of base 36ft.& altitude 12ft? Assume that one side of the rectangle lies on the base of the triangle. Q .24 Theflowerbed is to be in the shape of a circular sector of radius r & central angle 0. If the area is fixed & perimeter is minimum,findr and 0. Q.25 The circle x2 + y 2 = 1 cuts the x-axis at P&Q. Another circle with centre at Q and varable radius intersects the first circle at R above the x-axis & the line segment PQ at S. Find the maximum area of the triangle QSR.
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Application of Derivative
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EXERCISE-II 3
Q.1
The mass of a cell culture at time t is given by, M (t) =
(a)
Find Lim M(t) and Lim M(t)
(b)
dM 1 ,,„ , Show that — - - M(3 - M)
(c)
Find the maximum rate of growth of M and also the vlaue of t at which occurs.
Q. 2
Find the cosine of the angle at the vertex of an isosceles triangle having the greatest area for the given constant length I of the median drawn to its lateral side.
Q. 3
From a fixed point A on the circumference of a circle of radius 'a1, let the perpendicular AY fall on the tangent at a point P on the circle, prove that the greatest area which the AAPY can have
t->-00
~zr l + 4e
t-»00
2
• , ra is3V3 — sq. units. Q. 4
Given two points A ( - 2,0) & B (0,4) and a line y = x. Find the co-ordinates of a point M on this line so that the perimeter of the A AMB is least.
Q. 5
A given quantity of metal is to be casted into a half cylinder i.e. with a rectangular base and semicircular ends. Show that in order that total surface area may be minimum, the ratio of the height of the cylinder to the diameter of the semi circular ends is n/(n+2).
Q. 6
Depending on the values of p e R,findthe value of'a' for which the equation x3 + 2 px 2 +p = ahas three distinct real roots.
Q.7
Show that for each a > 0 the function e~ax. xa2 has a maximum value say F (a), and that F (x) has a minimum value, e~e/2.
Q.8
Let f (x) = sin3 x + X sin2x, -n/2
Q. 9
For what real values of 'a' and 'b' are all the extremum values of the function, f(x) = a2x3 + ax2 - x + b negative and the maximum is at the point x0 = - 1 . - Vx/nx
Q.IO Consider the function /(x) =
L
whenx>0
0
for x = 0
(a) (b)
Find whether/ is continuous at x = 0 or not. Find the minima and maxima if they exist.
(c)
Does/' (0) ? Find Lim / ' (x).
(d)
Find the inflection points of the graph of y=f (x)..
x-> 0
Q. 11 Consider the function y =/(x) = In (1 + sin x) with -2n
feBansal Classes
Application of Derivative
[18]
Q. 12 Arightcircular cone is to be circumscribed about a sphere of a given radius. Find the ratio of the altitude of the cone to the radius of the sphere, if the cone is of least possible volume. Q.13 Find the point on the curve 4 x2 + a2y2 = 4 a 2 ,4 < a2 < 8 that is farthestfromthe point (0, - 2). 3 5 Q.14 Find the set of value of m for the cubic x3 - — x2 + — = log1/(4 (m) has 3 distinct solutions. Q.15 Let A (p2, - p), B (q2, q), C (r2, - r) be the vertices of the triangle ABC. A parallelogram AFDE is drawn with vertices D, E & F on the line segments BC, CA & AB respectively. Using calculus, show that
1
maximum area of such a parallelogram is: — (p + q) (q + r) (p - r). T
Q.16 A cylinder is obtained by revolving a rectangle about the x - axis, the base of the rectangle lying on the x - axis and the entire rectangle lying in the region between the curve x y = & the x - axis. Find the maximum possible volume of the cylinder. Q.17 For what values of' a' does the function f (x) = x3 + 3 (a - 7) x2 + 3 (a2 - 9) x - 1 have a positive point ofmaximum Q.18 Among all regular triangular prism with volume V,findthe prism with the least sum of lengths ofall edges. How long is the side of the base of that prism? Q.19 A segment of a line with its extremities on AB and AC bisects a triangle ABC with sides a, b, c into two equal areas. Find the length of the shortest segment. Q.20 What is the radius of the smallest circular disk large enough to cover every acute isosceles triangle of a given perimeter L? Q.21 Find the magnitude of the vertex angle ' a ' of an isosceles triangle of the given area 'A' such that the radius 'r' of the circle inscribed into the triangle is the maximum. Q.22 Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed
is6rv3. Q.23 The function f (x) defined for all real numbers x has the following properties (i) f (0) = 0, f (2) = 2 and f 1 (x) = k(2x - x2)e_x for some constant k > 0. Find (a) the intervals on which/is increasing and decreasing and any local maximum or minimum values. (b) the intervals on which the graph/is concave down and concave up. (c) Hie function f (x) and plot its graph. Q.24 Find the minimum value of j sin x + cos x + tan x + cot x + sec x + cosec x | for all real x. Q.25 Use calculus to prove the inequality, sin x >
2x
n - in 0 < x < — •
71
2
x2 You mav use the inequality to prove that, cos x< 1 - — 71
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TE in 0 < x < ~
Application of Derivative
2
[81
EXERCISE-III Q. 1
A conical vessel is to be prepared out of a circular sheet of gold of unit radius. How much sectorial area is to be removedfromthe sheet so that the vessel has maximum volume. [ REE '97,6 ]
Q. 2(a) The number of values of x where the function f(x) = cos x + cos (V2x) attains its maximum is: (A)0
(B) 1
(C) 2
(D) infinite
(b) Suppose f(x) is a function satisfying the following conditions: (i)
f(0) = 2, f(l)= 1
(ii)
5 fhas a minimum value at x = - and
2ax 2ax-l 2ax + b + l b b+1 -1 2(ax + b) 2ax + 2b +1 2ax + b Where a, b are some constants. Determine the constants a, b & the function f(x). [JEE '98,2 + 8] (w)
for all x f' (x) =
Find the points on the curve ax2 + 2bxy + ay2 = c ; c > b > a > 0, whose distance from the origin is minimum [REE'98,6]
Q.3
X
The function f(x) = j t (el - 1) (t - 1) (t - 2)3 (t - 3)5 dt has a local minimum at x =
Q.4
(A) 0
' (B) 1
(C)2
Q. 5
(D)3 [JEE'99 (Screening), 3] Find the co-ordinates of all the points P on the ellipse (x2/a2) + (y2/b2) = 1 for which the area of the triangle PON is maximum, where O denotes the origin and N the foot ofthe perpendicularfromO to the tangent at P. [JEE '99,10 out of 200]
Q.6
Find the normals to the ellipse (x2/9) + (y2/4) = 1 which are farthestfromits centre.
Q.7
Find the point on the straight line, y=2 x +11 which is nearest to the circle, 16 (x2 + y2) + 32 x - 8 y - 50 = 0. [REE 2000 Mains, 3 out of 100]
[REE '99,6]
r Ixl for 0 < |x| < 2 Let f (x) = [ j f x = Q . Then at x = 0,'f' has :
Q.8
(A) a local maximum (C) a local minimum Q. 9
(B) no local maximum (D) no extremum. [ JEE 2000 Screening, 1 out of 35 ]
Find the area of the right angled triangle of least area that can be drawn so as to circumscribe a rectangle of sides 'a' and 'b', therightangles of the triangle coinciding with one of the angles of the rectangle. [ REE 2001 Mains, 5 out of 100 ]
Q. 10(a) Let f(x) = (1 + b2)x2 + 2bx + 1 and let m(b) be the minimum value of f(x). As b varies, the range of m (b) is (A) [0,1]
(D) (0,1]
(C)
(b) The maximum value of (cos ctj) • (cos a 2 ) (cos a n ), under the restrictions iz O < ctj, a 2 , a n <— and cot otj • cot a 2 cot a n = 1 is
(A)^
ffl>£
(O^
(!§Bansal Classes
Application of Derivative
[20J
Q. 11 (a) If a,, aj, , an are positive real numbers whose product is a fixed number e, the minimum value of a, + aj + a3 + + an l + 2an is (A)n(2e)1/n (B)(n+l)e1/n (C)2ne1/n (D)(n+l)(2e)1/n [JEE 2002 Screening] (b) A straight line L with negative slope passes through the point (8,2) and cuts the positive coordinates axes at points P and Q. Find the absolute minimum value of OP + OQ, as L varies, where O is the origin. [ JEE 2002 Mains, 5 out of 60] Q. 12(a) Find a point on the curve x2 + 2y2 = 6 whose distancefromthe line x + y=7, is minimum. [JEE-03, Mains-2 out of 60] (b) For a circle x2 + y2 = r2, find the value of'r' for which the area enclosed by the tangents drawnfromthe point P(6,8) to the circle and the chord of contact is maximum. [JEE-03, Mains-2 out of 60] Q. 13(a) Let f (x) = x3 + bx2 + cx + d, 0 < b2 < c. Then f (A) is bounded (B) has a local maxima (C) has a local minima (D) is strictly increasing . , . _ 3x-(x + l) w (b) Prove that sin x + 2x > - VX G n
[JEE 2004 (Scr.)]
. (Justify the inequality, if any used). [JEE 2004,4 out of 60]
Q.14 If P(x) be a polynomial of degree 3 satisfying P(-l) = 10, P( 1) = - 6 and P(x) has maximum at x = - 1 and P'(x) has minima at x = 1. Find the distance between the local maximum and local minimum of the curve. [JEE 2005 (Mains), 4 out of 60] Q.15(a) If/(x) is cubic polynomial which has local maximum at x = - 1. If / ( 2 ) = 18,/(1) = - 1 and / ( x ) has local maxima at x = 0, then (A) the distance between (-1,2) and (a,/(a)), where x = a is the point of local minima is 2^5 • (B)/(x) is increasing for x e [1, 2-/5 ] (C)/(x) has local minima at x = 1 (D) the value o f / 0 ) - 5 ex 0 < x <1 x x_1 1 < x < 2 andg(x) = Jf(t)dt ,x e [1,3]theng(x)has (b)/(x) = 2 - e x-e 2
[JEE 2006,5marks each]
(c) If/(x) is twice differentiable function such that /(a) = 0,/(b) = 2,/(c) = - l,/(d) = 2,/(e) = 0, where a < b < c < d < e , thenfindthe minimum number of zeros of g(x) = (f' (x))2 + f (x).f" (x) in the interval [a, e]. [JEE 2006,6]
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A N S W E R T A N G E N T
K E Y
&
N O R M A . L
EXERCISE-I Q.l 2V3 x - y = 2 ( V 3 - l ) or 2V3 x + y = 2(V3 + l)
Q.2 (0,1)
Q.3
x= 1 whent= l,m->co; 5 x - 4 y = 1 i f t * l,m = 1/3]
Q.7
T : x - 2 y = 0 ; N:2x + y = 0
Q.9 (a) n = - 2 Q.20
1
Q.14
Q.12a=l
2ex
Q.8 x + 2y = 7t/2 & x + 2y = -37r/2 x+2
; x-4y = 2
Q.16
a = - 1/2 ; b - - 3 / 4 ; c = 3
Q.23 0 = tan_1 —
Q.22 (b) a - b = a' - b'
Q.25
mVm V2
EXERCISE-II Q.2 (i) 6km/h (ii) 2 km/hr Q.3 (4,11) & (-4,-31/3) Q.5 1 + 36 7t cu. cm/sec Q.6 1/48 71 cm/s Q.7 0.05 cm/sec
Q.l 1/9 n m/min Q.4 3/8 n cm/min
V2 4ti
Q.9 2007tr 5 /(r + 5) 2 km 2 /h
Q.8 — cm/s Q.12
(a)-
24ti
m/min., (b)-
5 m/min. 288u
Q.IO
66
Q.ll -7 cm/sec. 4
Q.14 (a) r = (1 +1)1/4, (b) t = 80 Q.15 (a) 5.02, (b)
80
27
EXERCISE-III Q.l 0 = tan -1
4V2
Q.2 V2 x + y - 2 V2 = 0 or ^ x - y - 2 V 2 = 0 M O N O
TO
Q.3 D N O
Q.5
Q.4D
D
C I T Y
EXERCISE-I Q.l
(a) I in (2,oo) & Din ( - 0 0 , 2 ) (b) I in (1,00) & D in ( - 00, 0) u (0,1) (c) I in(0,2) & D in(-00,0) u(2,oo) (d) I for x > -J- or - - < x < 0 & D for x < - - or 0 < x < w 2
Q.2 Q.3
2
71
2
371
2
7t
xe 2n7c + — ,2mn u 2mt—,2n7i uf2mr + 71, 2mt + — I; n e l I 4' 4 4 I 4 (a) I in [0, 3ti/4) u (7tt/4 , 2 tt] & D in (3tt/4 , 7 7t/4) (b) I in [0 , ti/6) u (tc/2 , 5tx/6) u (3tc/2 ,2 71] & D in (tt/6 , tt/2) u (5ti/6, 3 ti/2)]
J
Q.5
continuous but not diff. at x= 1
Q.7
(a)(7r/6)+(l/2)/n3,(7i/3)-(l/2)/n3, (b) least value is equal to (l/e)1/e, no greatest value, (c) 2 &-10
Q.8
[l,oo)
Q.10
Q.6
a < - (2+V5 j or a > V?
a e (-oo,-3]u[l,oo)
Q.ll
[ - 7 , - 1 ) u [2, 3]
Q.12 increasing in x e (tt/2 ,27t/3) & decreasing in [0 , n/2) u (27t/3, 7t] Q.13
0
% Bans aIClasses
Q.14
t in (3,00) and I in (1,3)
Q.15
(6,00)
Application of Derivative
Q.16
a>0
[81
r 5 ^ Q.17 (a) ( - oo, 0] ; (b) t in 1 a n d I in (-oo,l)u -,00 -{-3} ; (c) x = - ;
(d) removable discont. at x=-3 (missing point) and non removable discont. at x = 1 (infinite type) (e)-2 Q.24. (-1, 0) u (0, oo) Q.25 (b-a) 3 /4
EXERCISE-II Q.l
c = -m'° + n a which lies between a & b m+n
Q.6
Q.7
y = - 5x - 9 and y = 5x + 11
Q.18 0
a = 3, b = 4 and m = 1
EXERCISE-1II Q.l ( - 2/a, a/3)
Q.2 C
Q.6 (a) A, (b) cos(j cos"1 p
Q.4(a) A,C;(b) D Q.7 A
Q.5 (a)B;(b)D; (c)C
Q.9
(a) D ; (b) C
Q.ll (a)D
MAXIMA. - MINIMA EXERCISE-I Q.l
f (x) = x3 + x2 - x + 2 Q.2
Q.3
(a) Max at x = 2 n, Max value = 2 IT, Min. at x = 0, Min value = 0 (b) Max at x-n/6 & also at x = 5 n/6 and Max value = 3/2, Min at x = 7t/2, Min value = - 3
Q.4 f(x)= | x6 - y x5 + 2x 4
Q.7
r=
2A 'ti + 4 ' S
Q.IO 4^2 m Q.12 110', 70* Q.15 32 sq. units
2A Vn + 4
max.at x = l ; f(l) = 0, min.at x = 7/5; f(7/5) = -108/3125
Q.5 Pmax = a | 1 + coseca 2J
Q.6 75 73 sq. units
Q.9 3x + 4y - 9 = 0 ; 3x - 4y + 9 = 0
Q.ll l/n cum Q.13 6/(6 + 71) Q.16 0 = 60°
Q.14 side 10', height 10' Q.17 a = 1, b = 0
Q.19 width 2 V3 m, length 3 V3 m Q.20 | a — b j Q.21 (a) (-1, 0), (0, 5/6); (b) F ' (x) = (x2 - x), F " (x) = 2x - 1, (c) increasing (- oo, 0) u (1, oo), decreasing (0,1); (d) (0,5/6); (1,2/3); (e) x = 1/2 Q.22(a)x = y = ^ , ( b ) x = - ^ , y = ^ | d
Q.23 6' x 18'
Q.24 r = Va , 0 = 2 radians
Q 25
% Bans aI Classes
'
173
Application of Derivative
[81
EXERCISE-III Q.l
(a) 0, 3, (c) - , t = In 4
Q.2 cos A - 0.8
Q.4 (0,0) Q.8 -3/2
Q.6 p < a < y ^ + p if p > 0 ; ^ - + p < a < p i f p < 0 1 f Q.9 a = - ~ and b e -oo,— or a= 1,be(-oo,- 1)
Q.IO (a) f is continuous at x = 0; (b) - — ; (c) does not exist, does not exist; (d) pt. of inflection x = 1 6 Q.ll
71 3it (a) x = - 27t, - 7t, 0,it, 2n, (b) no inflection point, (c) maxima at x = — and - — and no'minima, 37C
7t
(d) x = — and x = - —, (e) - n in 2 f
Q.12 4
Q.13 (0,2) & max. distance = 4
Q.17 ( - oo, - 3) u (3 ,29/7) Q.20 L/4
Q.18 H = x = Q.21
Q.14 m e
\_
r
Q.16
32' 16.
'4VY/3
Q.19
vV3y
* 4
(c+a-b)(a + b - c )
71
-
Q.23 (a) increasing in (0,2) and decreasing in (-oo, 0) u (2, oo), local min. value = 0 and local max. value = 2 (b) concave up for ( - oo, 2 - V2) u (2 + V2, oo) and concave down in (2 - V2), (2 + V2) (c)f(x)=|e2x-x2
Q.24
2V2-I
EXERCISE-IJI sq. units
Q.l *
Q.3
Q.2 (a) B, (b) a = - j ; b = 4
& i2(a + b) ' y2(a+b)J
Q.5
± -f
Q.8
A
V2(a+b)'
4
; f(x)
y 2(a+b),
4
Q.4 (a) B, D,
Q.6 ± V 3 x ± V 2 y = V 5 Q.9 2ab Q.IO (a) D; (b) A
Q.13 (a)D Q.14
(x2 - 5x + 8)
Q.ll (a) A; (b) 18
Q.7 (-9/2,2) Q.12
(a) (2,1); (b) 5
4^65 Q-15 (a) B, C; (b) A, B, (c) 6 solutions
(!§Bansal Classes
Application of Derivative
[20J
BANSAL CLASSES TARGET IIT JEE 2007
MATHEMATICS XII (ABCD)
BOOLEAN ALGEBRA
CONTENTS KEY- CONCEPTS SUGGESTED EXERCISE ANSWER-KEY
KEY
CONCEPTS
Mathematical logic is the science of reasoning. It is a process by which we arrive at a conclusion from known statements or assertions with the use of valid assumption which is known as Laws of Logic. 1.
Basic Concepts: A statement is a sentence which is either true of false but not both simultaneously.
2.
Truth value of statement: If a statement is true, we say that its truth value is TRUE or T and if it is false we say that its truth value is FALSE or F.
3.
Compound statements: A statement is said to be simple, if it cannot be broken down into two or more sentences. New statements that can be formed by combining two or more simple statements are called sub-statements or component statements of the compound statements.
4.
The compound statement S consisting of sub statements p, q, r, is denoted by S (p,q, r,..). A fundamental property of a compound statements is that its truth value is completely determined by the truth value of each of its sub statements together with the way in which they are connected to form the compound statement.
5.
Basic logical connectives: The words which combine simple statements to form compound statements are called connectives. To define a set of connectives with definite meanings in the language of logic is called object language. Three basic connectives (logical) are
Object language (i) (ii) (iii) 6.
English word
Conjuction Disconjuction negation
Symbol
and or not
A v ~
Conditional and Bi-conditional stetements: " If p then q", such statements are called conditional statements and are denoted by p -» q read as 'p implies q'. " p if and only if q " such statements are called bi-conditional statements and are denoted by p o q . Table (to be remembered) S.No. Connective
1. 2. 3. 4. 5.
and or not If then If then only if
tH.BansatClasses
Nature ofthe compound statement formed by the Connective. Conjuction Disjunction negation implication or conditional equivalence or biconditional
Symbol Symbol^ ; form
Boolean Algebra
V
pA q pv q
~
~P
A
— >
<->
p -> q
p^q
Negation
(~P) V ( ~ q )
(~p) A (~ q) p) = p p
A
(~ q)
[ p A ( ~ q ) ] v [ q A ( ~ p ) ]
[2]
Truth table: Let S(p, q, r) be a compound statement consisting of sub statements p, q, r,.... etc. A simple concise way to show the relationship between the truth table of S and the truth values of its substatements p, q, r,... etc. is by means of a table called the truth table for the statement S. Example: (1) Construct the truth table for [ p A (~ p) ] P ~ P [PA P)] T F F F T F (2) (i)
Construct truth table for (i) p A q p q pAq T^ T* * j^ T F F F T F F F F
and (ii) p v q (ii) p q
p v q
1
T F F
F T F
T T F
Tautologies and contradiction: A statement is said to be tautology if it is true for all logical possibilities. Analogously a statement is said to be a contradiction if it is false for all logical possibilities. A straight forward method to determine whether a given statement is tautology (or contradiction) is to construct its truth table. We denote tautology by T and contradiction by 'C'. Logical equivalence: Two statements S, (p, q, r,...) and S2 (p, q, r,....) are said to be logically equivalent, or simply equivalent if they have the same truth values for all logical possibilities and denoted by S, (p, q, r,...) = S2 (p, q, r,....) Note: (1) (2) (3) (4) (5)
p -> q = ( ~ p ) v q ~ ( p -> q) s p A (--q) p -> q s ( ~ q) -> p) p <-> q - (p -> q) A (q->p) ~ (p q) = ( p A - •q) V (~p
A
q)
(Conditional statement) (Negation of Conditional statement) (Contrapositive of Conditional statement) (Bicond statement) (Negation of Biconditional statement)
Duality: Two compound statements S, and S2 are said to be duals of each other if one can be obainedfromother by replacing A by v and v by A . The connections A and v are called duals of each other. Algebra of statements: 1. Idempotent laws : If p is any statement then (a) p v p = p (b) p A p = p 2. Associative laws : If p, q and r are any three statements, then (a) p v (q v r) = (p v q) v r (b) p A (q A r) = (p A q) A r 3. Commutative laws: If p and q are two statements, then (a) p v q s q v p (b) p A q = q A p 4. Distributive Laws: If p, q and r are three statements then, (a) p v (q A r) = (p v q) A (p v r) (b) p A (q v r) S (p A q) v (p A r)
tH.Bans at Classes
Boolean Algebra
[3]
5. 6.
7.
Identity laws: If p is any statement, t is tautology and c is contradiction, then (a) p v t = t (b) p A t = p (c) p v c = p (d) p A c = c Complement laws: If t is tautology, c is a contradiction and p is any statement, then (a)p v (~ P ) = t (b)p A (~ P ). = c (c) ~ t = c (d) ~ c = t Involution law: If p and q be any two statements then ~ p) = p
Definition: A non empty set B together with two operations generally denoted by '+' and '.'is said to be a Boolean Algebra if the following axioms hold: (I)
(II)
(III)
For all x, y e B (a) x+y e B (b) x, y e B
(Closure property for +) (Closure property for.)
For all x, y e B (a) x+y=y+x (b) x . y = y. x
(Commutative law of+) (Commutative law of.)
For all x, y and z in B (a) (x + y) + z = x + (y + z) (b) (x . y). z = x. (y. z)
(Associative law of +) (Associative law of.)
(IV)
(a) (b)
(Distributive law of + over.) (Distributive law of. over +)
(V)
There exists elements denoted by 0 and 1 in B. Such that for all x e B . (a) x+ 0= x (0 is identity for +) (b) x. 1 = x (1 is identity for.)
(VI)
For each x e B, there exists an element denoted by x', called the component or negation of x, in B such that (a) x + x' = 1 (b) x . x' = 0 (Complement law) Here'+' and '.' are not ordinary addition and multiplication. These are simply operations.
*
x + (y. z) = (x + y).(x + z) x . (y + z) = (x. y) + (x . z)
Important points: 1.
Boolean algebra is designated as (B,'+','.',"', 0,1) in order to emphasise it six parts, namely set B, the two binary operations '+' and '.', the complement operation''' and the two special elements 0 and 1. The symbols 0 and 1 not necessarily represent the numbers zero and one but elements are called zero elements and unit elements.
2.
For the set S of all logical statements, the operations v and A play the roles of'+'and'.'respectively. The tautology t and the contradiction C play the roles of 1 and 0, and the operationplays the role ? 11
3.
For P(A), the set of all subsets of a set A, the operations u and n play the roles of '+' and'.', A and (j) play the roles of 1 and 0, and complementation plays the role of " ' .
tH.BansatClasses
Boolean Algebra
[4]
Principal of duality: The dual of any statement in a boolean algebra B is the statement obtained by interchanging the elements 0 and 1 in the original statement. Concept of duality as defined in mathematical logic is same here, the only difference between the two is of notations as '+' is used for v ,'.' is used for A , 0 is used for contradiction c and 1 used for the tautology!:. Principle of Duality: Dual of any theorem in Boolean algebra is also a theorem. Theorem 1 :In a Boolean algebra 0 and the unit element 1 are unique. Theorem 2: Let B be a Boolean algebra. Then for any x and y in b we have, (a) x + x = x (a') x. x = x (b) x + 1 = 1 (b') x . 0 = 0 (c) x + (x.y) = x (c) x. (x + y) = x (d) 0' = 1 (d')l' = 0 (e)(x')' = x (f)(x + y)' = x',y'
(f')(x.y)' = x'+y'
Note that a', b', c', d' and f ' are duals of a, b , c, d and f. Arguments and their validity: An argument is the assertion that statement S, follows from other statements, S,, S 2 , S 3 , ...Sn We denote the argument by (S,, S 2 , S 3 ,.... ,S n ; S). The statement S is called the comclusion and the statements S,, S 2 ,...., Sn are called hypothesis. An argument consisting of the hypothesis S,, S 2 ,..., Sn and conclusion S is said to be valid if S is true whenever all S., 1 ' 2,...., ' S' nare true. Application of Boolean algebra to switching circuits: switch P —9
switch q
0
-switch P
0
, switch q Lamp /
R
"
Switches p and q are in parallel.
Switches p and q are in series. Switches p closed closed open open
q
closed open closed open
Lamp State on off off off
Switches P closed closed open open
q
closed open closed open
Lamp State on on on off
If we replace the words 'closed' and 'on' by the word 'True (or T)' and words 'open' and 'off by the word 'False (or F)' then the tables becomes truth table for logical experiment p A q and p v q respectively. In the language of logic we use symbols 1 and 0 to represent T and F.
tH.BansatClasses
Boolean Algebra
[5]
Switches p 1 1 0 1
q 1 0 1 1
Lamp State 1 0 0 1
Switches p 1
Switches p and q are in series, i.e. p A q
q 1
Lamp State 1
1
0
1
1 1
1 1
1 1
Switches p and q are in parallel, i.e. p v q .
These type of tables are called input/output tables with input as all possible values in bits of the switches p, q etc. and output as the corresponding values in bits of their outcome. We define the logical operations '+' and '.' on the set of bits {0,1} by Operation 1+ 1 = 1 1+0 = 1 0+1 = 1 0+0=0
OR (+) T v T=T T v F=T F v T=T Fv F=F
Operation AND (.) 1.1 = 1 T A T=T 1.0 = 0 T A F=F 0.1=0 F A T=F 0.0 = 0 FA F=F
The NOT operation ',' on the set {0, 1} is defined by 0'= 1 as ~ F = T and 1' = 0 as ~T = F. Definition 2: Let {B, + , . , ' , 0 , 1 , } be a Boolean algebra and Xj, x 2 , in Xj, x2, ,...,xn are definedrecursivley as follows: (I) 0,1, Xj, x,,..., xn are all Boolean expression. (II) If x and y are Boolean expressions, then (a) x' (b) x + y(orx v y) (c) x . y (or x A y)
, xn are in B. Then Boolean expressions
are also Boolean expressions. Note: We denote a Boolean expression X in x,, x 2 ,...., xn by X( x,, x 2 ,...., xn) Definition 3: Let X(x, , x 2 , ...., xn) be a boolean expression in , x 2 , ...,xn . Then a function f of the form f(x,,x 7 ,...,x ) = X(x , X2, ..., X] ) is called a Boolean function. Three basic gates : Gates are circuits constructed using solid state devices, which are capable of switching voltage levels ( bits 0 and 1). The input / output tables for these gates are similar to the truth tables of conjuction, discunjuction and negation respectively with T = 1 and F = 0. Definition: An AND gate is a Boolean function defined by f (x p x?) = x, .x2 Xj, x2 e {0,1} In DUt 1 1 0 0
tH.BansatClasses
x2 1 0 1 0
Boolean Algebra
Output x1 . x 2 1 0 0 0
[6]
Definition: An OR gate is a Boolean function defined by f (x,, x2) = x, + x2 x,, x2 e {0,1} In DUt
Output
Xl
X2
+ x2
1 1 0 0
1 0 1 0
1 1 1 0
Definition: An NOT gate is a Boolean function defined by f(x) = x' Input
X 0 1
NOT
x-r
OR
XR-
x,, x2 e {0,1}
Output
X' 1
0
NOT
(2)
(1)
In the circuit (1): In a circuit if the output S is uniquely defined for each combination of inputs x,, x2and x 3 . Such a circuit is said to be combinatorial circuit or combinatorial circuit. In the circuit (2): We observe that if x, = 1 and x2 = 0 then the inputs to the AND gate are 1 and 0 and so the output of the AND gate is 0 . Which is the input to the NOT gate which yields output 1 i.e. S = 1. But the diagram states that x2 = S i.e. 0 = 1 a contradiction. Therefore we conclude that output S is not uniquely defined. Such a circuit is not a combinational circuit. Definition: Two combinatorial circuits are equivalent if their input/output tables are identical.
OR
x,-
NOT
In DUt
Xl 1 1 0 0
tH.BansatClasses
s,
In DUt
Output x2 1 0 1 0
Si 0 0 0 1
Xt 1 1 0 0
Boolean Algebra
x2 1 0 1 0
Output S2 0 0 0 1
[7]
SUGGESTED EXERCISE Ql.
Let B = { {1} , {2}, {1,2 } ,(j>}. Show that (B , u , n , ' , (j), {1,2 }) is a Boolean Algebra.
Q2.
Let L be set of all logical statements. Define operations '+', '.' and " ' b y p + q = p v q ; p . q = p A q ; p' = ~p for all p, q E L where v , A has usual meaning in mathematical logic. Show that (L, + , . , ' , C , t) is a Boolean Algebra.
Q3.
Let B be a Boolean algebra. Then, for any x and y in B, prove (i) (x + y). (x + 1) = x + x . y + y (ii) x = 0 if and only if y = x.y' + x'.y + x'.y for all y "(iii) x + x. ( y + l ) = x
Q4.
Construct truth tables for the following (i) [p v (~p) A q)] -> q (ii) p A (q r) (iii) [p A r) -» (q v r)
Q 5.
Which ofthe following are equivalent ? (i) p q ; (~ p) -> (~ q) (ii) p ^ q ; (p A q) v ( ~ P A (~ q) )
Q6.
Examine the validity ofthe folloiwng arguments: (i) Sj : p -> q ; S 2 : q ^ p ; S : p v q (ii) S , : [ p A (~ q) ] —» r ; S2 : p v q ; S 3 : q p
Q7.
Q8.
Construct an input output table for each of the following Boolean Algebra functions: (i) f (x,, x 2 , x3) = ((x,. x2') + x 3 ). x,' (ii) f (x,, x2) - (x,. x2') + x2 ' Write the Boolean expression for the following input/output table. Show that it is a Boolean function and also draw its arrow diagram. Input
1 1 1 1 0 0 0 0
Q 9.
; S:r
x2 1 1 0 0 1 1 0 0
Output
1
S 1 0 0 1 0 0 1
0
0
x3
1 0 1 0 1
0
Find the combinatorial circuit corresponding to the following Boolean expressions: (i) (ii) (iii)
x.+(x1'.x2) {x, + (x 2 '. X3) } + X3 { x ; + ( x 2 ' . x 3 ) } + x,
tH.BansatClasses
Boolean Algebra
[8]
Q10.
Prove Demorgans Laws for any elements a, b in a Boolean algebra, i.e. prove (i) (a + by = a' b' (ii) (ab)' = a' + V
Qll.
Simplify (i) { [ (a' A b')' v c] A (a v c)} ' (ii) (x A y) v [ (x v y ' ) A y]' where B is a Boolean algebra
a, b e B x,yeB
Q12.
Draw the circuit which realizes the function a A [(b v d1) v (c' A (a v d v c')] A b
Q13.
Write the Boolean expression corresponding to the following switching circuit. Use laws of Boolean algebra to simplify the circuit. Construct the network for the simplified circuit.
.
^. q
q
rq'
Q14.
If A, B, C represent three switches in an on position and A', B' and C' represent the switches in an off position, then construct a network for the polynomial ABC + AB'C + A'B'C. Using the laws of the Booean algebra, show that above polynomial is equivalent to C(A+ B') and construct an equivalent switching circuit.
Q15.
Simplify the combinational circuit:
OR not>ON0T>O-
Q16. For each x in a Boolean Algebra B, prove that x + y = lj x.y = of
tH.BansatClasses
y = x'
Boolean Algebra
[9]
ANSWER KEY
Q4.
(i)
p
q
~P
~p A q
T T F F
T F I F
F F T T
F F T F
P T T T
q
r T F T F T F T F
T T F F T T F F
T F F F F
(ii)
(iii)
Q5.
q
P T T T T F F F F
No,Yes
Q6. Xl 0 0 0 0 1 1 1 1
©
x2
1 1 0 0
1 0 1 0
tH.BansatClasses
T
1
F T F
A
q) p v (~p
T T T F
A
q) -> q
T F T T
P A(q->r) T F T T F F F F
T F T T T F T T
p
A
(~ r) F T F T F F F F
qv r T T T F T T T F
(PA ~r) - > ( q v r ) T T T F T T T T
(i) invalid; (ii) invalid X3 0 1 0 1 0 1 0 1
*2 0 0 1 1 0 0 1 1
Xl
~r F T F T F T F T
r T F T F
T T F F T T F F
pv (~p
X
x'2 1 1 0 0 1 1 0 0
X-|.X'2 0 0 0 0 1 1 0 0
(x 1 .x' 2 ) + x 3
0 1 0 1 1 1 0 1
1X2
(xvx2)'
( x v x 2 )' + x 2
1 0 0 0
0 1 1 1
1 1 1 1
Boolean Algebra
X'l 1 1 1 1 0 0 0 0
( ( x 1 , x ' 2 ) + x 3 ) . xS
0 1 0 1 0 0 0 0
[10]
Q8.
f (x) = x, • x2 • x3 + x, • x2' • x3' + x,' • x2' •
Arrow Diagram
Q9.
(i)
(ii)
(iii)
Qll.
(i) (a' A c');(ii) 1 b
V
Q12.
V -
Q13. pq + r ( r ' + q)(p' + r q ' ) Simplified form: pq + qr q
r
Q14.
B
C
A
B'
c"
A'
B'
c"
xr Q15.
X,
T
A B* \ x ; + x2
- N0j>O—
tH.BansatClasses
Boolean Algebra
[11]
BANSAL CLASSES
TARGET IIT JEE 2007
MATHEMATICS XII (ABCD) :. : :.:. "
:
v.. :
-
VECTORS
&
3-D
CONTENTS KEY- CONCEPTS EXERCISE-I EXERCISE-II EXERCISE-III EXERCISE-IV ANSWER KEY
:
KEY 1.
CONCEPTS
DEFINITIONS: A VECTOR
may be described as a quantity having both magnitude & direction. A vector is generally
represented by a directed line segment, say AB . A is called the initial point & B is called the terminal point. The magnitude of vector AB is expressed by i AB ! • a vector of zero magnitude i.e.which has the same initial & terminal point, is called a ZERO VECTOR. It is denoted by O. ZERO VECTOR
UNIT VECTOR
a vector of unit magnitude in direction of a vector a is called unit vector along a and is
denoted by a symbolically a=— • two vectors are said to be equal if they have the same magnitude, direction & represent the same physical quantity. EQUAL VECTORS
two vectors are said to be collinear if their directed line segments are parallel disregards to their direction. Collinear vectors are also called PARALLEL VECTORS. If they have the same direction they are named as like vectors otherwise unlike vectors. COLLINEAR VECTORS
Simbolically, two non zero vectors a and b are collinear if and only if, a=Kb , where K e R COPLANAR VECTORS a given number of vectors are called coplanar if their line segments are all parallel to the same plane. Note that "Two VECTORS A R E ALWAYS COPLANAR".
let O be a fixed origin, then the position vector of a point P is the vector OP • If a & b & position vectors of two point A and B, then ,
POSITION VECTOR
AB = b - a = pvofB - pvofA. 2.
VECTOR ADDITION
:
If two vectors a & b are represented by OA & OB , then their sum a + b is a vector represented by OC , where OC is the diagonal of the parallelogram OACB. a + b = b + a (commutative) a + 0=a = 0 + a 3.
MULTIPLICATION
(a + b) + c = a + (b + c) (associativity) a +(-a)=0 = (-a)+a
OF VECTOR
BY SCALARS
:
If a is a vector & m is a scalar, then m a is a vector parallel to a whose modulus is | m | times that of a. This multiplication is called SCALAR MULTIPLICATION. If a & b are vectors & m, n are scalars, then: m ( a ) = (a)m = ma m ( n a ) = n ( m a ) = (mn )a ( m + n ) a = ma + na 4.
SECTION FORMULA
m(a + b) = ma + mb :
If a & b are the position vectors of two points A & B then the p. v. of a point which divides AB in the t -r= na + mb . Note , , , p. v. of. mid ., point . x oi AB = a + b . ratio m: n is given by: m+n 2
^Bansal
Classes
Probability
[2]
DIRECTION
COSINES
:
Let a = aji + a 2 j + a 3 k the angles which this vector makes with the +ve directions OX,OY & OZ are called DIRECTION ANGLES & their cosines are called the DIRECTION COSINES . ao 2 cos a = 1 -! i r 5> cos p=~(3=7^> cos r = 1 J r • Note that, cos2 a + cos213 + cos2 F = 1 a aI la Figure B (B VECTOR EQUATION OF A LINE : Parametric vector equation of a line passing through two point a
a
A(a) & B(b) is given by, f = a + t(b - a) wheretisaparameter. If the line passes through the point A(a) & is parallel to the vector b then its equation is, r = a +1 b Note that the equations of the bisectors of the angles between the lines r = a + A . b & r = a + p c i s : r = 5 +1 (b + c) & r = a + p ( c - b ) . TEST OF COLLINEARITY
:
Three points A,B,C with position vectors a,b,c respectively are collinear, if& only if there exist scalars x, y, z not all zero simultaneously such that ; xa + yb + zc = 0, where x + y + z = 0.
8.
SCALAR PRODUCT
OF TWO
VECTORS
:
a.b = la! cos 9(0 < 9 < rt), note that if 9 is acute then a.b > 0 I |2 7 "
& if 9 is obtuse then a.b < 0 — —
a.a=|aj =a ,a.b=b.a (commutative)
a . (b + c) = a.b + a.c (distributive)
a.b = 0<=>a±b
(a*0 b*0)
A
A
A
A
A
A
A
i.i = j.j = k.k = 1 ; projection of a on b = ^ . jbi
A
A
A
A
A
i.j = j.k = k.i = 0
Note: That vector component of a along b
=
(-a • xy b b and perpendicular to b = a 2 v b j a.b
the angle d between a & b is given by cos<|) A A A if a = a1i + a 2 j + a 3 k & |aj= Y a i 2 Note
+ a
22
+ a
j
/
0<<|><7t
b ^ b, 2 + b 2 2 + b ;
_i Maximum value of a . b = [ a I I b |
(ii)
Minimum values of a . b = a . b = - | a | Classes
V o
— — • b = bji+ b 2 j + b3k then a.b = a,bj + a2b2 + a3b3
(i)
a •b
lb Probability
[3]
(iii)
Any vector a can be written as , a = (a . ij i + (a . jj j + (a . kj k .
(iv)
a b A vector in the direction of the bisector of the angle between the two vectors a&b is pr + pr. Hence l l |b| a
bisector of the angle between the two vectors g&b is X (a + bj, where X e R+. Bisector ofthe exterior angle between a&b is X (a - b j , X e R+ . 9.
VECTOR PRODUCT OF TWO VECTORS : If a&b are two vectors & 9 is the angle between them then a x b = |a
(0
sin 9n
where n is the unit vector perpendicular to both a&b such that a , b & n forms a right handed screw system . Lagranges Identity: for any two vectors a & b;(axb)2 =|a|~|b| -(a.b) 2 =
(ii) (iii)
a.a a.b a.b b.b
Formulation of vector product in terms of scalar product:
or
The vector product a x b is the vector c , such that (i) | c | = yja2b2 -(a-b) 2
(ii) c • a = 0; 5 • b =0 and
(iii) a, b, c form a right handed system (iv) ®°axb = 0 o a & b are parallel (collinear) (a * o, b ^ 0) i.e. a = Kb , where K is a scalar, axb
bxa
( n o t commutative)
(ma) x b = a x (mb) = m(a x b) ax(b + c) = (axb) + (axc)
where m is a scalar . (distributive)
ix j = k, jxk = i, kxi = j A A A i j k yv A /v — ~ t If a = aji + a 2 j + a 3 k & b = bji + b 2 j + b3k then axb a l a 2 a 3 bj b2 b3
^ ixi = j x j = kxk = 0
(v)
(vi)
Geometrically axb
area of the parallelogram whose two adjacent sides are
represented by a & b . (vii)
Unit vector perpendicular to the plane of a & b is n = ±
axb axb
(a x b j A vector of magnitude 'r' & perpendicular to the palne of a & b is ± y — axb If 9 is the angle between a&b then sin 9 =
Classes
axb lal b
Probability
[4]
(viii)
Vector area If a ,b & c are the pv's of 3 points A, B & C then the vector area of triangle ABC = 1 axb + bxc + cxa The points A, B & C are collinear if axb + bxc + cxa = 0 Area of any quadrilateral whose diagonal vectors are d & d , is given by
10.
1
dj xd2
SHORTEST DISTANCE BETWEEN TWO LINES : If two lines in space intersect at a point, then obviously the shortest distance between them is zero. Lines which do not intersect & are also not parallel are called SKEW LINES. For Skew lines the direction of the shortest distance would be perpendicular to both the lines. The magnitude of the shortest distance vector would be equal to that of the projection of AB along the direction of the line of shortest distance, LM is parallel to p x q
1.
i.e.
AB . (pxq)
( b - a ) . (pxq)
pxq
pxq
LM = Projection of AB on LM
Projection of AB on pxq
The two lines directed along p & q will intersect only if shortest distance = 0 i.e (b-a).(pxq) = 0 i.e. (b - aj lies in the plane containing p & q .=> (b-aj p q = 0 .
2.
bx(a2 - a , ) If two lines are given by r, = a, + Kb & r2 = a, + Kb i.e. they are parallel then, d =
11.
SCALAR TRIPLE PRODUCT / BOX PRODUCT / MIXED PRODUCT :
or
The scalar triple product of three vectors a, b & c is defined as : axb.c=|a|b |c| s in0 cos
A A A A A /V A A A — * b, b2 b3 If a = aji + a 2 j + a 3 k ; b = bji + b 2 j + b3k & c = Cji + c 2 j + c3k then [a b c] = C[ c 2 c 3
In general , if a = a, 1 + a2m + a3n ; b = b j + b2m + b3n & c = c, 1 + c2m + c3n a
then |abc
l
a
2
a
3
t>i b2 b3 Cj c2 c3
Trnnj ; where 1, rh & n are non coplanar vectors .
If a , b , c are coplanar <=> [a b c] = 0 .
Classes
Probability
[5]
Scalar product of three vectors, two of which are equal or parallel is 0 i.e. [abc] = 0, Note : If a , b, c are non - coplanar then [a b c] > 0 for right handed system & [a b c] < 0 for left handed system. or
[i j k] = 1
[Kabc] = K[abc]
®=
[(a + b) c d] = [a c d] + [b c d]
The volume of the tetrahedron OABC with O as origin & the pv's of A, B and C being a , b & c respectively is given by V = — [a b c] 6
cir
"
The positon vector of the centroid of a tetrahedron if the pv's of its angular vertices are a , b, c & d are given by ~ [a + b + c + d]. Note that this is also the point of concurrency of the lines joining the vertices to the centroids ofthe opposite faces and is also called the centre of the tetrahedron. In case the tetrahedron is regular it is equidistantfromthe vertices and the four faces ofthe tetrahedron. Remember that
*12.
a-b
b-c
&
0
c-a
a+b b+c c+a = 2 a b c
VECTOR TRIPLE PRODUCT : Let a , b , c be any three vectors, then the expression a x ( b x c ) is a vector & is called a vector triple product. ax(bxc)
GEOMETRICAL INTERPRETATION OF
Consider the expression a x (bx c) which itself is a vector, since it is a cross product of two vectors a & (b x c). Now a x ( b x c ) is a vector perpendicular to the plane containing a & (b x c) but b x c is a vector perpendicular to the plane b & c , therefore a x (b x c) is a vector lies in the plane of b&c
and perpendicular to a . Hence we can express ax ( b x c ) in terms of b & c
i.e. a* x (b x c) = xb + yc where x & y are scalars . a x (b x c) = (a . c)b - (a . b)c
( a x b ) x c = ( a . c ) b - ( b . c)a
(a x b) x c # a x (b x c) 13.
LINEAR COMBINATIONS / Linearly Independence and Dependence of Vectors : Given a finite set of vectors a, b, c, combination of a, b, c,
(a)
for any x, y, z
FUNDAMENTALTHEOREM I N PLANE
coplanar with a,b
then the vector r = xa + yb + zc +
iscalleda linear
e R. We have the following results :
: Let a,b be nonzero, non collinear vectors. Then any vector r
can be expressed uniquely as a linear combination of a,b
i.e. There exist some unique x,y e R such that xa + yb=r . (b)
FUNDAMENTAL THEOREM I N SPACE
: Let a ,b ,c be non-zero, non-coplanar vectors in space. Then
any vector r, can be uniquily expressed as a linear combination of a ,b ,c i.e. There exist some unique x,y e R such that xa + y b + z c = r .
Classes
Probability
[6]
(c)
If x, ,x 2 ,
x n are n non zero vectors, & k,, k2,
kjXj + k 2 x 2 +
k n x n = 0 => k, = 0,k2 = 0
kn are n scalars & if the linear combination k n = 0 then we say that vectors x ,,x 2 ,
xn
a r e LINEARLY INDEPENDENT VECTORS .
(d)
If x j ,x 2 ,
x n are not LINEARLY INDEPENDENT then they are said to be LINEARLY
vectors . i.e. if k , x , + k 2 x 2 + x,,x 2 ,
DEPENDENT
+ k n x n = 0 & if there exists at least one k * 0 then
x n are said to be LINEARLY DEPENDENT .
Note : If a = 3i + 2j + 5k then a is expressed as a LINEAR COMBINATION of vectors i, j, k • Also, a,
I5 js K
form a linearly dependent set of vectors. In general, every set of four vectors is a linearly dependent system. i, j, k are LINEARLY LNDEPENDENT set of vectors. For KJI+K2j + K3k = 0
^
= > Kj = 0 = K 2 = K 3 .
Two vectors a & b are linearly dependent => a is parallel to b i.e. axb= 0 => linear dependence of a & b • Conversely if axb^O then a & b are linearly independent . If three vectors a ,b ,c are linearly dependent, then they are coplanar i.e. [a, b, c] = 0, conversely, if [a, b, c] * 0 , then the vectors are linearly independent.
14.
COPLANARITY OF VECTORS : Four points A, B, C, D with position vectors a, b, c, d respectively are coplanar if and only if there exist scalars x, y, z, w not all zero simultaneously such that xa+ yb+zc+wd=0
15.
where, x + y + z + w=0.
RECIPROCAL SYSTEM OF VECTORS : If a, b, c & a',b',c' are two sets of non coplanar vectors such that a.a'=b .b -c.c'=l two systems are called Reciprocal System of vectors. XT ^ Note:
then the
. u- 2xa , axb a = r - •, ; b'= _ , ; c'jSbcj
abcj
|abcj
16.
EQUATION OF A PLANE :
(a)
The equation (r - r0 >.n — 0 represents a plane containing the point with p.v. r0 where n is a
(b)
vector normal to the plane. ?.n=d is the general equation of a plane. Angle between the 2 planes is the angle between 2 normals drawn to the planes and the angle between a line and a plane is the compliment of the angle between the line and the normal to the plane.
17.
APPLICATION OF VECTORS :
(a)
Work done against a constant force F over a
\
displacement s is defined as W =F.s (b)
L,
j a
The tangential velocity V of a body moving in a circle is given by V = w x r where r is the pv of the
—=
i p
/ V
point R
Classes
Probability
[482]
(c)
The moment of F about 'O' is defined as M = f x Fwherer is the pv of P wrt ' 0 \ The direction of M is along the normal to the plane OPN such that r , F & M right handed system.
(d)
Moment ofthe couple = ( r , - r 2 ) x F
where
form a
h & ?2 are pv's of the
point ofthe application of the forces F & - F. 3 -D COORDINATE GEOMETRY USEFUL RESULTS A
General:
(1)
Distance (d) between two points (x,, y,, z,) and (x 2 , y 2 , z2)
(2)
d=
yl(x2-xl)2+(y2-y1)2+(z2-zrf
Section Fomula m m2x1 + m 1 x 2 2Yl + m i Y2 m!+m 2 ' : m]+m 2 (For external division take -ve sign)
z=
A
(X|,y,, z,)
m
i
P(x, y, z) —!- m,
—B (x2, y? j z.)
+nii z 2 mj +m 2
m2Zj
Direction Cosine and direction ratio's of a line (3) (a)
Direction cosine of a line has the same meaning as d.c's ofa vector. Any three numbers a, b, c proportional to the direction cosines are called the direction ratios i.e. I _ m _ n _ 1
a b c _ Va2+b2+c2 same sign either +ve or -ve should be taken through out. note that d.r's of a line joining x,, y,, z, and x 2 , y 2 , z2 are proportional to x2 - x, , y, - y, and z2 - z, (b)
If 9 is the angle between the two lines whose d.c's are /,, m,, n, and l 2 , m 2 , n2 cos9 = /, /2 + m ; m2 + n ; n2 hence if lines are perpendicular then /, l2 + m, m2 + nj n2 =0 U n = m i = ni if lines are parallel then l2 m 2 n 2 (x„ y,,z,) h ml ni note that if three lines are coplanar then
h
m2
n
h
m3
n3
2
=0
(4)
Projection of the join of two points on a line with d.c's /, m, n are / (x2 - x,) + m(y2 - y,) + n(z2 - z,)
B (i) (ii)
PLANE General equation of degree one in x, y, z i.e. ax + by + cz + d = 0 represents a plane. Equation of a plane passing through (x,, y,, z,) is a (x - x,) + b (y - y,) + c (z - z,) = 0 where a, b, c are the direction ratios of the normal to the p l a n e . Equation of a plane if its intercepts on the co-ordinate axes are xl, y,, z, is x y z h— +— =1 x i Yi z i
(iii)
Bansal
Classes
Vectors
y2 .zb)
d.c's I, m, n
[8/
(iv)
(v)
Equation of a plane if the length of the perpendicular from the origin on the jJane is p and d.c's ofthe perpendicular as /, m,, n is /x + my + nz = p Parallel and perpendicular planes - Two planes aj x + bj y + CjZ + d} = 0 and a,x + b2y + c2z + d2 = 0 are perpendicular if a} a7 + bj b, + Cj c2 = 0 a,
parallel if coincident if (vi)
b, _ c,
a2"b2"c2
and d2
Angle between a plane and a line is the compliment of the angle between the normal to the plane and the .. » Line : r = a + A,b then Une.lt pi a n e : f . n = d
cos(90-9)= sinG =
bl.lnl"
where 6 is the angle between the line and normal to the plane. (vii)
'line
b.n 9 plane
/
Length of the perpendicular from a point (Xj, y,, z^ to a plane ax + by + cz + d = 0 is axj + byj + czj + d , ,2 . c2 yj/ a2 + b +
(viii)
Distance between two parallel planes ax + by + cz + dj = 0 and ax + by + cz + d2 = 0 is d
l"d2 V a + b 2 +c 2 2
(ix)
Planes bisecting the angle between two planes a,x + b,y + CjZ + dj = 0 and a2 + b2y + c2z + d2 = 0 is given by a1x + b1y + c1z + d1 Vai + b f + c f
+
a 2 x + b2y + c2z + d 2
2 ^ a12 +Tb?L> +2c? T Of these two bisecting planes, one bisects the acute and the other obtuse angle between the given planes.
(X) C (i)
Equation of a plane through the intersection of two planes P, and P2 is given by P( + XP2 = 0 STRAIGHT LINE IN SPACE Equation of a line through A (Xj, y t , z,) and having direction cosines I ,m, n are x - x 1 = y - y 1 = z-zi I m n and the lines through (x,, y, ,z,) and (x2, y2 ,z2) X-XI z-z. y-yi X
(ii) (iii)
2~ X 1 Y2~y\ Z 2~ Z 1 Intersection of two planes ajX + b,y + c t z + d, = 0 and a2x + b2y + c2z + d2 = 0 together represent the unsymmetrical form of the straight line. x-xj _ y-yj _ z-zj is General equation of the plane containing the line / m n A (x - x^ + B(y - y}) + c (z - z ) = 0 where Al + bm + cn = 0 .
&Bansal Classes
Vectors
[9]
LINE OF GREATEST SLOPE AB is the line of intersection of G-plane and H is the horizontal plane. Line of greatest slope on a given plane, drawn through a given point on the plane, is the line through the point 'P' perpendicular to the line of intersetion of the given plane with any horizontal plane.
EXERCISE-I Q.l
If
a&b
are non collinear
vectors
such that,
p = (x + 4y)a + (2x + y + l)b
&
q = ( y - 2 x + 2)a + ( 2 x - 3 y - l ) b .find x&ysuchthat 3p = 2q. Q.2
(a) (b)
Show that the points a - 2b + 3c;2a + 3b - 4 c & - 7 b + 10c are collinear. Prove that the points A=(1,2,3), B (3,4,7), C (-3 ,-2,-5) are collinear &findthe ratio in which B divides AC.
Q.3
Points X & Y are taken on the sides QR & RS, respectively of a parallelogram PQRS, so that QX = 4XR
Q.4
& RY = 4 YS. The line XY cuts the line PR at Z. Prove that PZ = — PR. {25 J Find out whether the following pairs of lines are parallel, non-parallel & intersecting, or non-parallel & non-intersecting.
(2\~\
(i)
i- = i + j + 2k + A. (3i - 2j + 4k) . . . \ . . r 2 =2i + j + 3k + p. (-6i + 4 j - 8 k )
(n)
i = i - j + 3k + X (i - j + k) / - h = 2i + 4j + 6k + p (2i + j + 3k)
r, = i + k + X (i + 3j + 4k) r2 =2i + 3j + p (4i - j + k) Q.5
Let OACB be paralelogram with O at the origin & OC a diagonal. Let D be the mid point of OA. Using vector method prove that BD & CO intersect in the same ratio. Determine this ratio.
Q.6
Aline EF drawn parallel to the base BC of a AABC meets AB & AC in F & E respectively. BE & CF meet in L. Use vectors to show that AL bisects BC.
Q.7
'O'is the origin of vectors and A is a fixed point on the circle of radius'a'with centre O. The vector OA is denoted by a. A variable point 'P' lies on the tangent at A & OP = r. Showthat a.r =|a| .Hence if P = (x,y) & A s (x19y,) deduce the equation of tangent at Ato this circle.
Q.8
Q.9
(a)
By vector method prove that the quadrilateral whose diagonals bisect each other at right angles isarhombous.
(b)
By vector method prove that the right bisectors of the sides of a triangle are concurrent.
The resultant of two vectors a&b is perpendicular to a. If |b| = V2|a| show that the resultant of 2a & b is perpendicular to b.
s Q-10 and a, b,De(l c and-1.2) d are the positionIf vectors of the A=(x, 3x); C = (2z, y) respectively. | a | = 2sl3.; (aApoints b)= (a'c); (a^)y,= z); \ aBn d=(a(y,j -) i 2z, obtuse, then find3x, x, y,- z.
Bansal Classes
Vectors
[10]
Q.ll
If r and s are non zero constant vectors and the scalar b is chosen such that | r + b s | is minimum, then show that the value of jbsj2 +1 r + bs |2 is equal to | f | 2 .
Q.12 Use vectors to prove that the diagonals of a trapezium having equal non parallel sides are equal & conversely. Q. 13 (a) Find a unit vector a which makes an angle (rc/4) with axis of z & is such that a+i + j is a unit vector. L ( af _a bl (b) Prove that 2 1 b IJ I?'
'b J
Q.14 Given four non zero vectors a, b, c and d. The vectors a, b & c are coplanar but not collinear pair by A
A
A
_A
pair and vector d is not coplanar with vectors a ,b & c and (ab) = (be) = —, (da) = a, (db) = P then prove that ( d c) = cos-1 (cos P - cos a ) . Q.15 (a) (b)
Use vectors to find the acute angle between the diagonals ofa cube. Prove cosine & proj ection rule in a triangle by using dot product.
Q.16 In the plane of a triangle ABC, squares ACXY, BC WZ are described, in the order given, externally to the triangle on AC & BC respectively. Given that CX = b, CA = a , CW = x, CB = y • Prove that a.y + x.b = 0. Deduce that AW.BX=0. Q.17 A A OAB is right angled at O ; squares OALM & OBPQ are constructed on the sides OA and OB externally. Show that the lines AP & BL intersect on the altitude through 'O'. Q.18
Given that u = i - 2 j + 3k ;v = 2i + j + 4k ;w = i + 3j + 3k and (u R -10)i + (v • R - 20) j + (w • R - 20)k = 0. Find the unknown vector R .
Q.19 IfO is origin ofreference, point A( a) ; B(b);C(c) ; D(a + b);E(b + c);F(c + a);G(a + b + c) where /\ /V — > /V /\ A /V /V a = aji + a 2 j + a 3 k ; b = b1i + b 2 j + b3k and c = c } i+ c 2 j + c3k then prove that these points are vertices of a cube having length of its edge equal to unity provided the matrix. a, a 2 a 3 b
i
C,
b
2
C2
b
3
C3^
is orghogonal. Also find the length XY such that X is the point of intersection of CM and
GP; Y is the point of intersection of OQ and DN where P, Q, M, N are respectively the midpoint of sides CF, BD, GF and OB. Q.20 (a) (b)
If a + b + c = 0 , showthat axb=bxc=cxa . Deduce the Sine rule fora AABC. IfA, B, C, D are any 4 points in space, prove that ABx CD + BCx AD+ CAx BD =4 (area of triangle ABC).
Q.21 (a) (b)
Determine vector of magnitude 9 which is perpendicular to both the vectors: 4 i - j + 3k & - 2 i + j - 2 k fn sq. units. A triangle has vertices (1,1,1);(2,2,2),(1, l,y) and has the area equal to csc v4 j Find the value of y.
Probability
[486]
Q.22 The internal bisectors of the angles of a triangle ABC meet the opposite sides in D, E, F; use vectors to prove that the area of the triangle DEF is given by (2abc) A (a + b) (b + c) (c + a)
w h e r e A is t h e a r e a o f t h e triangle
"
Q.23 If a ,b ,c ,d arc position vectors of the vertices of a cyclic quadrilateral ABCD prove that: axb + bxd + dxa
bxc + cxd + dxb
(b-a).(d-a)
(b-c).(d-c)
=0
Q.24 The length ofthe edge ofthe regular tetrahedron D-ABC is'a'. Point E and Fare taken on the edges AD and BD respectively such that E divides DA and F divides BD in the ratio 2:1 each. Thenfindthe area of triangle CEF. Q.25 If the point R(r) is on the line, which is parallel to the vector, a i + b j + ck (where a, b, c * 0) and passing through the point S (s), then prove that, ? x (a i + b j + c kj = s * (a i + b j -f ck]. Further if, T (t) is a point outside the given line then show that the distance of the line from the point (t-s).(cj-bk)" + (t-s).(ak-ci)
T (t) is given by,
+ ( t - s ) . (b i - a jj
/a 2 + b2 + c2
EXERCISE-II Q.l
_ 1 A(a) ; B(b); C(c) are the vertices ofthe triangle ABC such that a = - ( 2 i - r - 7 k ) ; b = 3r + j - 4 k ; c = 22i -11 j - 9r • A vector p = 2j - k is such that (? + p) is parallel to i and (f - 2i) is parallel to
Q.2
Q.3
p. Show that there exists a point D(d) on the line AB with d = 2 t i + ( l - 2 t ) j + (t-4)k .Also find the shortest distance CfromAB. The position vectors of the points A, B, C are respectively (1,1,1); (1, -1,2); (0,2, -1). Find a unit vector parallel to the plane determined by ABC & perpendicular to the vector (1,0,1). Let
(a,-a)2 (b,-a)2 (c,-a)z
(a,-b)2 (b,-b)2 (Cj-b) z
(a,-c) (b, - c ) = 0 and if the vectors a = i + aj + a 2 k; (3 = i + bj + b 2 k; ( C l -C) 2
y = j + c j + c2^ are non coplanar, show that the vectors dj = i +a,j + af k;p, = i + b,j + b, k and y, = i + Cj j + c 2 k are coplaner. Q. 4 (i)
Given non zero number x,, x2, x 3 ; y,, y2, y3 and z,, z2 and z3 such that Xj > 0 and y; < 0 for all i = 1,2, 3. Can the given numbers satisfy
©
x]x2+y1y2+z1z2=0 1 x 2 x3 y y =0 and • x 2 x 3 +y 2 y 3 +z 2 z 3 =0 3 Yi 2 z Z Z l 2 3 IfP = (x, , x2, x 3 ); Q (y ], y2, y3) and O (0,0,0) can the triangle POQ be a right angled triangle? X
Classes
Probability
[12]
Q.5
The pv's of the four angular points of a tetrahedron are: A (j + 2k) ; B (3 i + kj ; C (4i + 3j + 6kj & D(2i+3j + 2k).Find: (i) the perpendicular distancefromA to the line BC. (ii) the volume of the tetrahedron ABCD. (iii) the perpendicular distance from D to the plane ABC. (iv) the shortest distance between the lines AB & CD.
Q.6
-»
The length ofan edge ofa cube ABCDA.B^jDj is equal to unity. ApointE taken on the edge AA, is 1 — . Apoint F is taken on the edge BC such that BF = ~ . If Oj is the centre of
—
such that
AE the cube, indtl le shortest distance of the vertex B,fromthe plane of the A OjEF. Q.7
Thevector OP = i + 2j + 2k turns through a right angle, passing through the positive x-axis on the way. Find the vector in its new position.
Q. 8
Find the point R in which the line AB cuts the plane CDE where a = i + 2j + k, b= 2i + j + 2k, c - - 4 j + 4k, d = 2 i - 2 j + 2k & e = 4i + j + 2k-
Q.9
If a = aji + a 2 j + a 3 k ; b = b,i + b 2 j + b3k and c = c,i + c 2 j + c3k then show that the value of the a-i a-j a-k scalar triple product [ na + b nb + c nc + a] is (n 3 +1) b-i b-j b-k c i c-j c-k
Q.10 Find the scalars a & p if Sx(bxc) + (a.b)b = (4-2p-sina)b + (p2 - l)c & (c.c)5 = c while b & c are non zero non collinear vectors. Q.ll
Ifthe vectors b,c,d are not coplanar, then prove that the vector (a x b) x (c x d) + (a x c) x (d x b) + (a x d) x (b x c) is parallel to a .
Q.12
a , b, c are non-coplanar unit vectors. The angle between b & c is a, between c & a is P and between a & b is y . If A (a cosa), B (b cosp), C (c cosy), then show that in AABC, a x | b x c)|
|b x (c x a)j
Jc x ^a x b |
sinA
sinB
sinC
n
Q.13
]~[ ja x (b x c)| sina cosp cosy n,)
where
bxc . _ cxa „ axb i " 1 7 — ' n2 - p—^ & n, = c x a bx c axb
Given that a,b,p,q are four vectors such that a + b = pp,b.q = 0& ( b ) 2 = l , where p is a scalar then prove that |(a.q )p - (p.q )a |= |p.q |.
Q.14
Show that a = px(qxr) ; b = q x ( r x p ) & c = r x ( p x q ) represents the sides of a triangle. Further prove that a unit vector perpendicular to the plane of this triangle is n,tan(pAq) + n2tan(qAr) + ri3tan(rAp) where a, b, c, p, q are non zero vectors and jn, tan(pAq) + n2 tan(qAr) + n3 tan(r Ap)| ,, perpendicular , & „ n, - = pxq . ; n- = jz—zr qxr & „ n. - rxp no two ofr- -p,q,r are mutually 2 3 x rx Pi qxr P q
Classes
Probability
[13]
Q.15 Let p, q, ? be three mutually perpendicular vectors of the same magnitude. If a vector x satisfies the equation p x ((* - q)x p) + q x ((* - f )x q) + r x ((x - p)x f ) = 0. Then find the vector x in terms of q, r .
P,
Q.16 Let a = cxi + 2 j - 3 k , b = i + 2 a j - 2 k andc = 2i - a ] + k. Find the value(s) ofa, if any, such that 0a x bj x ^b x cj! x (c x a) =0. Find the vector product when a = 0. Q.17 Prove the result (Lagrange's identity) (p x q) • (r x s) =
p.r p.s & use it to prove the following. Let q.r q.s
(ab)denote the plane formed by the lines a,b. If (ab) is perpendicular to (cd) and (ac) is perpendicular to (bd) prove that (ad) is perpendicular to (bc). Q.18
b +(b
' ? 5 ~? p ( b x 5 ) • P(P" + a )
(a)
If px + (xxg) = b;(p*0) prove that x =
(b)
Solve the following equation for the vector p ; pxa + (p.bjc = bxc where a, b, c are non zero non coplanar vectors and a is neither perpendicular to b nor to c , hence show that pxa +
abc a.c
is perpendicular to b - c.
Q.19 Find a vector v which is coplanar with the vectors i + j - 2 k & i - 2 j + k and is orthogonal to the vector-2i + j + k. Itis given that the proj ection ofv along the vector i - j + k is equal to 6 73 . Q.20
Consider the non zero vectors a, b, c & d such that no three of which are coplanar then prove that b5d + c|abdj =
a cd + d a b c
. Hence prove that a, b, c & d represent the position vectors of
the vertices of a plane quadrilateral if and only if Q.21
bed + abd acd
+ abc
=1 .
The base vectors a1,a2,a3 are given in terms of base vectors b p b 2 ,b 3 as, aj = 2b, +3b 2 - b 3 ; a 2 = b, - 2b2 + 2b3 & a 3 = -2b, + b2 - 2b3 . If F = 3b, - b2 + 2b 3 , then express F in terms of a,,a 2 & a 3 .
Q.22
If A (a) ; B(bj & C(c) are three non collinear points , then for any point P(p) in the plane of the AABC , prove that;
(i) (ii)
abc
axb + bxc + cxa
)
The vector v perpendicular to the plane of the triangle ABC drawn from the origin 'O' is given by = ±
abc (axb + bxc + cxa) where A is the vector area ofthe triangle ABC. 4A
Classes
Probability
[14]
Q.23
Given the points P (1,1, -1), Q (1,2,0) and R (-2,2,2). Find (a) (b)
PQxPR Equation of the plane in (i) scalar dot product form (ii) parametric form (iii) cartesian form (iv) if the plane through PQR cuts the coordinate axes at A, B, C then the area ofthe AABC
Q.24 Let a,b & c be non coplanar unit vectors, equally inclined to one another at an angle 9. If axb + bxc = pa + qb + rc . Find scalars p , q & r in terms of9. Q.25
Solve the simultaneous vector equations for the vectors x and y. x+cxy=a
and
y + c x x=b where c is a non zero vector.
EXERCISE-III Q.l
Find the angle between the two straight lines whose direction cosines /, m, n are given by 2/ + 2 m - n = 0 and mn + n/ + /m = 0.
Q. 2
If two straight line having direction cosines I, m, n satisfy al + bm + cn = 0 and f m n + g n / + h / m = 0 f g h are perpendicular, then show that — + — + — =0. a b c Pisanypointon theplane/x+my+nz=p.ApointQtakenonthelineOP (where O is the origin) such that OP. OQ = p2. Show that the locus of Q is p( lx + my + nz) = x2 + y2 + z2.
Q.3 Q.4
Find the equation of the plane through the points (2,2,1), (1, -2,3) and parallel to the x-axis.
Q.5
Through a point P (f, g, h), a plane is drawn at right angles to OP where 'O' is the origin, to meet the r5 coordinate axes in A, B, C. Prove that the area of the triangleABC is 2f gh w ^ e r e OP = r-
Q. 6
The plane lx + my = 0 is rotated about its line of intersection with the plane z-0 through an angle 9. Prove that the equation to the plane in new position is lx + my + zVl2 + m2 tan 9 = 0
Q. 7
Find the equations of the straight line passing through the point (1,2,3) to intersect the straight line x + l = 2 ( y - 2 ) = z + 4 and parallel to the plane x + 5y + 4z = 9.
Q.8
Find the equations of the two lines through the origin which intersect the line angle of -n .
Q.9
A variable plane is at a constant distance pfromthe origin and meets the coordinate axes in points A, B and C respectively. Through these points, planes are drawn parallel to the coordinates planes. Find the locus of their point of intersection. x+2
Q.10 Find the distance of the point P (- 2,3, - 4)fromthe line
2y+ 3
x-3 y-3 z ^ = ——=y at an
3z + 4
= ——— = —-— measured parallel to
the plane 4x + 12y- 3z + 1 = 0.
Classes
Probability
[490]
Q.ll
Find the equation to the line passing through the point (1, -2, -3) and parallel to the line 2x + 3y - 3z + 2 = 0 = 3x - 4y + 2z ~ 4.
Q.12 Find the equation of the line passing through the point (4, -14,4) and intersecting the line of intersection ofthe planes: 3x + 2 y - z = 5 and x - 2 y - 2 z = - l at right angles. Q.13 Let P = (1,0,- 1); Q = (l, 1, l)andR = (2,1,3) are three points. (a) Find the area of the triangle having P, Q and R as its vertices. (b) Give the equation of the plane through P, Q and R in the form ax + by + cz = 1. (c) Where does the plane in part (b) intersect the y-axis. (d) Give parametric equations for the line through R that is perpendicular to the plane in part (b). Q.14 Find the point where the line of intersection of the planes x - 2y + z = 1 and x + 2y - 2z = 5, intersects the plane 2x + 2y + z + 6 = 0. Q.15 Feet ofthe perpendicular drawnfromthe point P (2,3, -5) on the axes of coordinates are A, B and C. Find the equation of the plane passing through their feet and the area of AABC. Q.16 Find the equations to the line which can be drawn from the point (2, -1,3) perpendicular to the lines x—1 y-2 z-3 x-4 y z+3 . , — =— = — and — = y = — at right angles. x—1 Q.17 Find the equation ofthe plane containing the straight line — =
y+2
z =-
and perpendicular to the
plane x - y + z + 2 = 0.
x+l y-p z+2 x y-1 z+ 7 . , Q.18 Find the value of p so that the lines —-= —— and 1— = —— = —— are in the same D = —— Z* I plane. For this value of p, find the coordinates of their point of intersection and the equation ofthe plane containing them. Q. 19 Find the equations to the line of greatest slope through the point (7,2, -1) in the plane x - 2y + 3z = 0 assuming that the axes are so placed that the plane 2x + 3y - 4z = 0 is horizontal. Q.20 Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the area of triangles ABC, ACD and ADB be denoted by x, y and z sq. units respectively. Find the area of the triangle BCD. Q.21
The position vectors of the four angular points of a tetrahedron OABC are (0,0,0); (0,0,2); (0, 4, 0) and (6,0,0) respectively. Apoint P inside the tetrahedron is at the same distance 'r'fromthe four plane faces of the tetrahedron. Find the value of V. x + 6 y + 10 z + 14 _ = —-— = —-— is the hypotenuse of an isosceles right angled triangle whose opposite 5 3 8 vertex is (7,2,4). Find the equation of the remaining sides.
Q.22 The line
Q.23 Find the foot and hence the length of the perpendicular from the point (5, 7, 3) to the line x-15 y - 2 9 5-z —-— = —~— = - r - . Alsofindthe equation ofthe plane in which the perpendicular and the given 3 8 5 straight line lie. x—1 y-2 z+3 Q.24 Find the equation of the line which is reflection of the line ——=——=—— in the plane 3 x - 3 y + 10z = 26. x - l y z . x - 3 _ y _ z-2 Q. 2 5 Find the equation ofthe plane containing the line = — - — and parallel to the line ^ 4 • Find also the S.D. between the two lines.
Classes
Probability
[16]
EXERCISE-IV —>
—»
—»
Q.l (a) Let OA=a, 0B=l()a+2b and OC=b where O.A&C are non-collinear points. Let p denote the area of the quadrilateral OABC, and let q denote the area of the parallelogram with OA and OC as adjacent sides. If p = kq, then k = . (b) If A, B&C are vectors such that |B| = |C| , Prove that ; ( a + B)X(a+C)1 x (BXC).(B+C) = 0
Q.2(a) Vectors
x,y&z
[ JEE' 97,2 + 5 ]
each of magnitude -J2, make angles of 60° with each other. If
xx(yxz) = a,yx(zxx) = b and xxy = c thenfmd x, y and z intermsof a, b and c(b) The position vectors of the points P&Qare 5i + 7 j - 2 k and - 3 i + 3 j + 6 k respectively. The vector A = 3 i - j + k passes through the point P & the vector B = - 3 i + 2 j + 4 k passes through the point Q. A third vector 2i + 7 j - 5 k intersects vectors A&B. Find the position vectors of the points of intersection. [ REE' 97, 6 + 6 ] Q. 3 (a) Select the correct alternative(s) A A A A A A A A A f— (i) I f a = i + j + k,b = 4 i + 3 j + 4k and c = i + «j + Bk are linearly dependent vectors & |cj =V3 , then: (A) a = 1, p = - 1 (B) a = 1, p = ±1 (C)
Which ofthe following expressions are meaningful ? (A)u.(vxw) (B) (u. v ) . w (C) (u. v) w
(D)ux(v.w)
(b) Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezeum lies on the line passing through the mid-points ofthe parallel sides. (You may assume that the trapezeum is not a parallelogram.) (c) For any two vectors u & v, prove that (i) (u.v)2 +|uxv| 2 = ju|2|v|2 &
(ii) (l + |u| 2 )(l+|v| 2 ) = (1-U.v)2 + |U + v+(Uxv)| 2 [ JEE ' 9 8 , 2 + 2 + 2 + 8 + 8]
Q.4(a) If x x y = a, y x z = b, x.b = y, x.y = l and y.z = l thenfmd x,y &z intermsof a, b and y • >
A A A
^
A
A
A
(b) Vectors AB = 3i - j + k & CD = - 3 i + 2 j + 4k are not coplanar. The position vectors of points A and C are 6i + 7 j + 4k and - 9 j + 2k respectively. Find the position vectors of a point P on the line AB & a point Q on the line CD such that PQ is perpendicular to AB and CD both. [ REE' 98 , 6 + 6 ] Q.5 (a) Let a = 2i + j - 2 k & b = i + j. If c is a vector such that a - c = | c | , ] c - a | = 2^2 and the angle between (axb) and c is30°, then (axb)xc = (A) 2/3 Bansal Classes
(B) 3/2
(C) 2 Vectors
(D) 3
[17]
(b) Let g=2i+j+k, b = i + 2 j - k and a unit vector c be coplanar. If 5 is perpendicular to a, then c (A) - j . ( - ] + k)
(B) j . (-i - 1 - k)
(C) J L (i - 2 j )
(D) ^ (i - j - k)
(c) Let a & b be two non-collinear unit vectors. If u = a - (a . bj b & v = a x b, then |v| is: (A) |u|
(B) |u| + |u.a|
(C) |u| + |u . bj
(D) u + C . (a + b)
(d) Let u & v be unit vectors. If w is a vector such that w + (w x u) = v , then prove that | (u x v). w | < ^ and the equality holds if and only if u is perpendicular to v . [ JEE '99, 2 + 2 + 3 + 10, out of 200 ] Q.6(a) Air arc AC ofa circle subtends a right angle at the centre O. The point B divides the arc in the ratio 1 : 2. If OA = a & OB = b , then calculate OC in terms of a & b . (b) If a , b, c are non-coplanar vectors and d is a unit vector, then find the value of, (a.d)(bx5) + (b.d)(cx5) + (5.d)(Sxb) independent of d . Q.7 (a)
[REE'99,6 + 6]
Select the correct alternative:
(i) If the vectors a , b & c form the sides BC, C A & AB respectively of a triangle ABC, then (A) a . b + b . c + c . a = 0
(B)axb = b x c = c x a
( C ) a . b = b . c = c.a
(D) a x b + b x c + c x a
=
0
(ii) Let the vectors a , b, c & d be such that (a x b )x (c x d) = 0 .Let P, & P2 be planes determined by the pairs of vectors a , b & c,d respectively. Then the angle between P, and P2 is: (A) 0 (B) 7 i / 4 (C) 7T/3 (4) tx/2 (iii) If a , b & c are unit coplanar vectors, then the scalar triple product 2a-b (A) 0 (b)
2b-c
2c-a
(B) 1
(C) - V3 (D) 73 [ JEE ,2000 (Screening) 1 + 1 + 1 out of 35 ]
Let ABC and PQR be any two triangles in the same plane. Assume that the perpendicularsfromthe points A, B, C to the sides QR, RP, PQ respectively are concurrent. Using vector methods or otherwise, prove that the perpendiculars from P, Q, R to BC, CA, AB respectively are also concurrent. [ JEE '2000 (Mains) 10 out of 100 ]
Q.8 (i)
If a = i + j - k , b — - i + 2j + 2k & c = - i + 2 j - k , find a unit vector normal to the vectors
(ii)
a + b and b - c. Given that vectors a & b are perpendicular to each other, find vector 6 in terms of a & b satisfying the equations, 6 . 5 = 0 , 6 • b = 1 and J o , a , b = 1
Classes
Probability
[18]
(iii)
a , b & c are three unit vectors such that a x (^b x cj = - (b + cj . Find angle between vectors a & b given that vectors b & c are non-parallel.
(iv)
A particle is placed at a corner P of a cube of side 1 meter. Forces of magnitudes 2,3 and 5 kg weight act on the particle along the diagonals of the faces passing through the point P. Find the moment of these forces about the corner opposite to P . [ REE '2000 (Mains) 3 + 3 + 3 + 3 out of 100 ]
Q.9(a) The diagonals ofa parallelogram are given by vectors 2i + 3 j - 6 k and 3i - 4 j - k • Determine its sides and also the area, (b) Find the value of X such that a, b, c are all non-zero and (~4i + 5 j ja + (3i - 3 j + k)b + (i + j + 3k)c - X (ai + bj + ck)
[ REE '2001 (Mains) 3 + 3]
Q,10(a) Find the vector r which is perpendicular to g = i - 2 j + 5k and b=2i + 3 j - k and r • (2i + j + k) +8 = 0. A
A
A
A
A
A
(b) Two vertices of a triangle are at - i + 3 j and 2i + 5 j and its orthocentre is at i + 2 j. Find the position vector of third vertex. [ REE '2001 (Mains) 3 + 3] Q. 11 (a) If a, b and c are unit vectors, then a - b (A) 4 (B) 9 (b) Let a = i - k , b = xi + j + ( l - x)k (A) only x (B)onlyy
+ |c-a| 2 does NOT exceed (C) 8
+ b-c
a n d c = yi +
Q. 12(a) Show by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices. (b) Find 3-dimensional vectors v,, v 2 , v3 satisfying v, • v, = 4,
v, • v2 = -2, v, • v3 = 6, v2 • v2 = 2, v2 • v3 = -5, v3 • v3 = 29.
(c) Let A(t) = f, (t)i + f 2 (t)j and B(t) = g, (t)i + g2 (t)j, t e [0,1], where f,, f2, g,, g2 are continuous functions. If
A(t) and B(t)are nonzero vectors for all t and A(0) = 2 i + 3 j ,
A(l) = 6i + 2j, B(0)= 3i + 2j and B(l) = 2i + 6j, then showthat A(t) and B(t) are parallel for some t. [ JEE '2001 (Mains) 5 + 5 + 5 out of 100 ] Q.l 3(a) If a and b are two unit vectors such that a +2b and 5 a - 4 b are perpendicular to each other then the angle between a and b is (A) 45°
(I) (C)cos-1 -
(B) 60° A A A
_
A
2
(D)cos , _
A
(b) Let V = 2i + j - k and W = i + 3k. If u is a unit vector, then the maximum value of the scalar triple product [U V W is (A)-l
Classes
(B) 7l0 + V6
(C)V59 Probability
(D) V60 [JEE 2002(Screening), 3 + 3] [19]
Q.14 Let V be the volume of the parallelopiped formed by the vectors a =a,i + a 2 j + a 3 k, b =bji + b2 j+b 3 k, c =c j + c2 j+c 3 k. If a , b , c , where r = 1, 2, 3, are non-negative real 3
numbers and £ (ar + br + c r ) = 3L, show that V < L3.
[JEE 2002(Mains), 5]
Q.15 If a = i + aj +k , b = j +ak , c = ai +k, then find the value of 'a' for which volume of parallelopiped formed by three vectors as coterminous edges, is minimum, is 1 1 1 (C)± (D)none (A)^ (B)--JJ V3 [JEE 2003(Scr.), 3] Q.16 (i) Find the equation ofthe plane passing through the points (2,1,0),(5,0, l)and(4,1,1). (ii)
If P is the point (2,1,6) then find the point Q such that PQ is perpendicular to the plane in (i) and the mid point of PQ lies on it. [ JEE 2003,4 out of 60]
Q.17 If u , v , w are three non-coplanar unit vectors and a, P, y are the angles between u and v , v andw , w and u respectively and x , y , z are unit vectors along the bisectors of the angles r_ _ 2 «a l1r1r I-\2l 7 . 22PP sec2 — a, P, y respectively. Prove that [xxy y xz z x x j =—[u v wj —c sec — — sec sec — - sec 2
16
2 2.
L2
[JEE 2003,4 out of 60] x - 1 y+1 z - 1 x-3 y-k z Q. 18(a) If the lines — y = - y - = and —— = —— = y intersect, then k = (A) |
(B)|
(C)0
(D)-l
(b) A unit vector in the plane of the vectors 2i + j + k, i - j + k and orthogonal to 5i + 2 j + 6k A
A
6i-5k
3j —k
A
A
2i-5k
A
A
A
2i + j - 2 k
(c) If a = i + j + k , a-b = 1 and 5xb = j-k,then b = (A) i
(B) i - j + k
(C) 2j — k
Q.l9(a) Let a, b, c, d are four distinct vectors satisfying axb = cxd
(D) 2i [JEE 2004 (screening)] and axc = bxd. Show that
a-b + c - d ^ a - c + b-d. (b) T is a parallelopiped in which A, B, C and D are vertices of one face. And the face just above it has corresponding vertices A', B', C', D'. T is now compressed to S with face ABCD remaining same and A', B', C', D' shifted to A., B., C„ D. in S. The volume of parallelopiped S is reduced to 90% of T. Prove that locus ofA. is a plane. (c) Let P be the plane passing through (1,1,1) and parallel to the lines L, and L2 having direction ratios 1,0, -1 and -1,1,0 respectively. IfA, B and C are the points at which P intersects the coordinate axes, find the volume of the tetrahedron whose vertices are A, B, C and the origin. [JEE 2004,2 + 2+ 2outof60]
Classes
Probability
[20]
Q.20(a) If a, b, c are three non-zero, non-coplanar vectors and b, = b - - ^ - a , b, = b + - - a , 2 Ia| | a |2 - c, =c
c-a - b-c - - c-a c Ta + —r-b, c 2 = 2 2 Ia| |c| '' | a| then the set of orthogonal vectors is (A) (a, bj, c 3 )
(B)(5,bj,c 2 )
bj-c— = |bj| 2 ''
3
^ c-a _ b-c r c - — T2 a + — r2b .bc ,4 =c |c| |c|
(C) (a, b p c j
c-a _ |c|
2
b-c — ^ 2- br 1. |b|
(D) (a, b 2 , c j
(b) A variable plane at a distance of 1 unit from the origin cuts the co-ordinate axes at A, B and C. If the centroid D (x, y, z) of triangleABC satisfies the relation - y + —j + ~y = k, then the value of k is x y z (A) 3 (B) 1 (C) 1/3 (D) 9 [JEE 2005 (Screening), 3] (c) Find the equation of the plane containing the line 2x-y + z - 3 = 0,3x+y + z = 5 and at a distance of 1 ^ from the point (2,1,-1). (d) Incident ray is along the unit vector v and the reflected ray is along the vN
unit vector w. The normal is along unit vector a outwards. Express w in terms of a and v.
mmmmMfmmuum [ JEE 2005 (Mains), 2 + 4 out of 60 ]
Q.21(a) A plane passes through (1,-2, 1) and is perpendicular to two planes 2x - 2y + z = 0 and x - y + 2z = 4. The distance of the plane from the point (1,2,2) is (A)0 (B)l (C) 4 1 (D) 2 V2 (b) Let a = i + 2 j + k , b = i - j + k and c = i + j - k • A vector in the plane of a and b whose projection - . 1 . on c is —p=, is V3 (A) 4 i - j + 4k
(B) 3i + j - 3 k
(C) 2 i + j - 2 k
(D) 4 i + j - 4 k
[JEE 2006,3 marks each] (c) Let A be vector parallel to line of intersection of planes P j and P9 through origin. P, is parallel to the A
/V
/\
/V
a
a
a
a
vectors 2 j + 3 k and 4 j - 3 k and P2 is parallel to j - k and 3 i + 3 j , then the angle between vector A and 21 + j - 2 k is ,. s n (A) 2 4
(C)
n 6
(D)
3n T [JEE 2006,5]
Bansal Classes
Vectors
[21]
(d) Match the following (i) Two rays in the first quadrant x + y = | a | and ax - y = 1 intersects each other in the interval a £ (a0, co), the value of a0 is (ii) Point (a, (3, y) lies on the plane x + y + z = 2. Let
a
= ai + P j + y k ,
k x ( k x a ) = 0,
0 1 2 J(y 2 -l)dy J(l-y )dy + (iii) l 0 (iv) If sinA sinB sinC + cos A cosB = 1, then the value of sin C =
then y =
v2i2
(D)l
j
b+c
[JEE 2006,6]
, cos02 = — , cosG3 = ° a+c a+b
9 0-, 9 0, then tan — + tan — = 2 2 (iii) A line is perpendicular to x + 2y + 2z = 0 and passes through (0,1,0). The perpendicular distance of this line from the origin is
1 + x dx
(A)0
(ii) Sides a, b, c of a triangle ABC are in A.P. and cos9, =
(B) 4/3 i (C) j V l - x d x + o
<5?
(e) Match the following / 11 x = t , then tan t = (i) Xtan--1 i=l
(A) 2
(B)l
V5 3 (D) 2/3 (C)
Probability
[JEE 2006,6]
[22]
ANSWER KEY EXERCISE-I Q. 1
x = 2 , y = -1
Q.2
Q.4 Q.5
(i) parallel (ii) the lines intersect at the point p.v.-2? + 2] (iii) lines are skew 2:1 Q.7 xxj + y y ^ a 2 Q.10 x = 2,y = - 2 , z = -
Q.13
—1 1 1 (a)yi-~j+-^k
Q.15
Q.19
-s/lT —
• ~ - \ ,,. „ ^ „, (a)± 3 ( i - 22jj -— 2 k2k), ) , (b)y (b) y= =3 3orory = - 1 Q.24
^ ,, Q.21
(b) externally in the ratio 1:3
1 (a)arccos-
Q.18
- i + 2j + 5k 5a? ^sq.units
EXERCISE-II +
Q.l
2Jl7
Q-2
Q.5
(i) ^
(ii) 6 (iii) f VlO (iv) S
Q.8
p.v.ofR = r = 3i+3k
Q-4 Q.6
~
Q.10 -a=n7t + ( ~ 1 )
7t
2
N0
'
N 0
Q.7
,nel&p = l Q.15
p ( p + q + r)
Q.16 a = 2/3 ; i f a = 0 then vector product is-60(2i + k Q.18
L
! Q.21
abc
(b) p =
r _ a + cxb
+
b.c b b.blc) U r - U
(a.c)(a.b)
(a.b)
Q.19
/ - r\ 9-j +k
(a.b) J
F = 25j + 5a2 + 3a3
Q.23 (a) 2 i - 3 j + 3k, (b)(i)-4,(ii) f = i + j - k + M j + k) + p(-3i + j + 3k),(iii)-4,(iv) „ „. Q.24 p = -
1 2cos0 1 ; q= ; r= - , V l + 2cos0 V1 + 2COS0 yl + 2cos0 1 2cos9 p= -r —— ; q ~ — F — = ; r : VT+TcosO" ' / l + 2cos0 '
or
Q.25 x=
1 ^/l + 2cos0_
a + (c.a)c + b x c b + (c.b)c + a x c —^ , y= _ 1+c 1 + 2c
EXERCISE-III Q.l
0 = 90°
vQ . 8
- = - = —
Q.II
Q.4
x y z x or — 1 2 - 1 - 1 x-1 y +2 z+ 3 = = 13 17
Classes
y + 2z = 4 y
z
= - = —
1
-2
Q.7
x-1 2
v0 . 9
~T 22
Q- 12
y-2 2
z-3 -3
1 . 11 . 1 1 + _ 222 + _ 222 = " T2 x y zz pP" x - 4 y+ 14 z - 4 3 10 Probability
vQ . 1 0
17 2
—-
[23]
3 2x 2y z / 0 , ^" , 0 A ; (d) x = 2t + 2 ; y = 2t + 1 and z = - 1 + 3 Q.13 ( a ) - ; (b) y + y - - = l ; (c) 2 / Q.14 ( 1 , - 2 , - 4 ) Q.17
x y z 19 - + j + — = l , A r e a = y sq.units
Q.15
2x + 3y + z + 4 = 0
Q.18
p = 3, (2,1,-3); x + y + z = 0
Q 20
V(x 2 +y 2 +z 2 )
-
Q.22
x-7_y-2_z-4
x-2 y+1 z-3 Q.16 — = — = - y
|
y+1
z-7
x-7_y-2_z-4 x-4
Q.23
Q.21
(9,13,15); 14 ; 9 x - 4 y - z = 14
Q.25 x - 2y + 2z - 1 = 0; 2 units
Q.24
EXERCISE-IV
Q.l (a) 6 Q.2 (a) x = a x c ; y ^ b x c
; z = b + a x c or b x c - a
(b) (2, 8, - 3) ; (0, 1, 2)
Q.3 (a) (i) D (ii) C (iii) A, C axb - axb —-—a x —— Q.4 (a) x = — axbSY*
;
'
_ r axb y=—;
y
axb r axb —^— + b x — -^LZ=.-Lfa x b Y
Q.5 (a) B (b) A (c) A, C
Q.6
(b) P = (3,8,3)& Q = (-3,-7,6)
(a) 5 = -V3a + 2b
(b)
abc
Q.7
(a) (i) B (ii) A (iii) A
Q.8
(i) ± i ; ( i i ) 4 + ^ i r ; b (axb)
Q.9
(a) y(5i - j-7k), y ( - i + 7j-5k); | Vl 274 sq. units (b)A, = 0, X = -2± V29
«
( " i ) JT ;
(iv) |M| - V 7
5 a 17 a Q.10 (a) r = -13i + l l j + 7k; ( b ) - i + y j Q.ll
(a) B (b) C
Q.12
(b) v, = 2i, v2 = - i ± j, v3 = 3i ± 2 j ± 4 k
Q.13 (a) B ; (b) C Q.15 D Q.16 (i)x + y - 2 z = 3 ; (ii) (6,5,-2) Q.18 (a)B, (b) B, (c)A Q.19 (c) 9/2 cubic units Q.20 (a) B, (b) D ; (c) 2 x - y + z - 3 = 0 and 62x + 29y + 19z- 105 = 0, (d) w = v - 2 ( a • v) a Q.21 (a) D, (b) A, (c) B, D, (d) (i) -> D, (ii) ->A, (iii) -> B, C, (iv) D, (e) (i) B, (ii) D, (iii)->C
Classes
Probability
[24]