MATHS
ASSIGNMENTS Exercise - 01
CBSE FLASH BACK 1.
x (x 2 + 1 ) ( x + 2) + 1 = 0 .
(i )
Solve for real x :
(ii)
Solve : 5x + 1 + 5 2 – x = 5 3 + 1 .
(iii)
Solve : x2/3 + x 1/3 – 2 = 0
6
6
6
......
2.
Evaluate :
.
3.
A number num ber ex ceed s its posi tive squa s quare re root by 12. Find F ind the th e num ber .
4.
A two digit num ber is four tim es the sum and three t hree t ime s the pr oduc t of it s digi ts. Find F ind th e num ber.
5.
Rectangular field is 16 metres long and 10 meters wide. There is a path of equal width all around it, having an area of 120 sq. metres. Find the width of the path.
6.
A journey jou rney of 192 km k m take t akes s 2 hou rs le ss by a f ast train tra in then th en by a slow s low train t rain.. If the averag a verag e speed spe ed of slow train be 16 km/hr. less than that of fast train. Find the average speed of each train.
7.
If the roots of the equation, equation, a (b – c) x2 + b (c – a) x + c (a – b) = 0 are equal, show that 2 1 1 = + . b a c
8.
If the roots of the equation , x2 – 8 x + a2 – 6 a = 0 are real and distinct. Then find all possible values of ‘ a ’ .
9.
If the roots of the equation, x2 + 2 c x + a b = 0 are real and unequal. unequal. Prove that the the roots of 2 2 2 2 x – 2 (a + b) x + a + b + 2 c = 0 are non-real complex quantities.
10.
Show that the roots of the equation, equation, x2 + p x + q = 0 are rational rational if p = k
q , k
where p , q , k are rational. rational. 11.
If the roots of the equation, x2 – x + m = 0 differ by 1, then prove prove that 2 = 4 m + 1 .
12.
If the ratio of the roots of the equation , a x 2 + b x + c = 0 is ‘ x ‘ , then prove that
b2 (r 1)2 = . ac r
29
MATHS
Exercise - 02
OBJECTIVE 1.
2.
4.
)
1, 9 5
x 1 x
The number of real roots of
(C)
4 , 1 5
3
+ x +
(D) do d oe s no t e xi s t
( B) 2
[ 1 , 1]
(C) 4
( B) x
[ 0 , 2]
If S is the set of all real x such that
(A)
, 3 2
(B)
3 , 1 2 4
(C) x 2x 2x
3
(D)
5 , 1 9
( D) N o ne o f t h es e
1 = 0 , is : x
If x2 2 x + sin2 = 0 , then : (A) x
6.
( B)
If x = 2 + 22/3 + 2 1/3 , then the value value of x3 – 6 x 2 + 6 x is : (A ) 3 ( B) 2 (C) 1
(A ) 0 5.
(C) eq equals to –5
If a1 , a 2 , a 3 (a 1 > 0) are in G.P. with common ratio r , r , then the value of r , r , for which the inequality 9 a1 + 5 a3 > 14 a2 holds , can not lie in the interval : (A) [1 ,
3.
x + 6 = 0.
Sum of the real roots of the equation , x2 + 5 (A) eq eq u al s t o 5 (B) eq eq u a l s t o 1 0
1
3 x2 x (C)
( D) 6
[2, 2 ]
(D) No None of these
is positive , then S contains :
1 , 1 4 2
(D) None of these
7.
The sum of the roots of a equation is 2 and sum of their cubes is 98, then the equation is : (A) x 2 + 2 x + 15 = 0 (B) x2 + 15 x + 2 = 0 (C) 15 x 2 2 x + 15 = 0 (D) x 2 2 x 15 = 0
8.
If x2 – 4x + log 1/2 a = 0 does not have have two distinct real roots, roots , then maximum value of ‘ a ‘ is : (A)
1 4
(B)
1 16
(C) –
1 4
(D) None of these
9.
If p and q are the roots of the equation, x2 + px + q = 0 then : (A ) p = 1 ( B) p = 1 o r 0 (C) p = 2
(D) p = 2 or 0
10.
If the roots roots of the equation, x2 – p x + q = 0 differ by unity, then : 2 (A) p = 1 – 4 q (B) p 2 = 1 + 4 q (C) q2 = 1 – 4 p
(D) q 2 = 1 + 4 p
11.
The value of the biquadratic expression, x 4 8 x 3 + 18 x 2 8 x + 2 when x = 2 + (A ) 1
12.
( B) 2
(C) 0
3 is : ( D) N o ne o f t h es e
If , are the roots of the equation equation , 2 x 2 + 4 x 5 = 0 , the equation whose roots are the reciprocals of 2 – 3 and 2 – 3 is : (A) x 2 + 10 x 11 11 = 0 (B) x2 + 10 x + 11 = 0 (C) 11 x 2 + 10 10 x + 1 = 0 (D) 11 x 2 10 x + 1 = 0
30
MATHS 13.
If
and are the roots of the equation , 2 + 2 , 2 + 2 is :
(A) 4 x 2 + 49 x + 118 = 0 (C) 4 x 2 – 49 x – 118 = 0 14.
(B) 4 x 2 – 49 x + 118 = 0 (D) x 2 – 49 x + 118 = 0
If the roots of the equation, A x 2 + B x + C = 0 are , and the roots of the equation, x 2 + px + q = 0 are 2 , 2 , then value of ‘ p ’ will be : (A)
15.
2 x 2 – 3 x – 6 = 0, then the equation whose roots are
B2
2 A C A
2
(B)
2 A C B 2 A
2
(C)
B2
4 A C
(D) None of these
A 2
If both the roots of the equation , (3 a + 1) x2 (2 a + 3 b) x + 3 = 0 are infinite then : (A) a = ; b = 0 (B) a = 0 ; b = (C) a =
1 2 ; b= 3 9
(D) a =
;
b=
16.
If the equation (a – 5) x2 + 2 (a – 10) x + a + 10 = 0 has real roots of the same sign, then : (A) a > 10 (B) – 5 < a < 5 (C) a < – 10 and 5 < a 6 (D) None of these
17.
The value of ‘p’ for which the sum of the square of the roots of 2 x 2 – 2 (p – 2) x – p – 1 = 0 is least, is : (A) 1
18.
(B)
3 2
(C) 2
If b > a , the equation (x a) (x b) + 1 = 0 , has : (A) must be in (a , b) (C) one root in (– , a) and other in (b , )
(D) – 1
(B) must be in [a , b] (D) None of these
19.
The equations , a x 2 + b x + a = 0 and x3 – 2 x 2 + 2 x – 1 = 0 have two roots in common. Then a + b must be equal to : (A) 1 (B) – 1 (C) 0 (D) None of these
20.
The equations , x3 + 5 x 2 + p x + q = 0 and x3 + 7 x 2 + p x + r = 0 have two roots in common. If the third root of each equation is represented by x1 and x2 respectively , then the ordered pair (x 1 , x2) is (A) ( 5 , 7) (B) (1 , 1) (C) ( 1 , 1) (D) (5 , 7)
31
MATHS
Exercise - 03
SUBJECTIVE 1.
Solve th e follow ing fo r real values of x : (a) (b) (c) (d) (e) (f)
2.
3 x 2 – 4 x + 2 = 5 x – 4
x + 4 x + 2 + 2 x + 5 = 0 (x + 3) x + 2 + 2 x + 3 + 1 = 0 (x – 1) x – 4 x + 3 + 2 x + 3 x – 5 = 0 x + 1 + x – x – 2 = 0 2 – 2 – 1 = 2 + 1 2
2
2
3
2
x+2
x+1
x+1
Solve the following equations / inequations for real x :
log2
x
(a)
3 . 8 2 = 6. x
x
log x 6
(b)
3
(c)
(e)
(f)
1 1 < x 1 log4 ( x log4 x 2
log
x
(d)
log 1/2 x + log3 x > 1.
2 log 2 (x 2 – x – 2) 1
6
log5 ( x 2
3)
x 1 > 0 2 x
4 x 11)2 log11 ( x 2 4 x 11)3 2 5x
3 x2
0
Solve for a2 – b = 1.
3.
x 2 15
a b
(g)
a b
(a)
If
(b)
If
(c)
If , be the roots of the equation , 2 (x 2 – x ) + 2 x + 3 = 0 and values of for which and are connected by the relation ,
x
+ a
If
= 2 a
(h)
+ a
b
x 2 15
= 2 a
be a root of the equation , 4 x2 +2 x 1 = 0 then prove that 4 3 – 3 is the other root , are the roots of the equation , x2 2 x + 3 = 0 obtain the equation whose roots are 3 – 3 2 + 5 – 2 , 3 – 2 + + 5 .
+ = (d)
b
x
, are
show that
4 , then find the quadratic equation whose roots are 3
the roots of a x 2 + b x + c = 0 and
,
are the roots of
, –
21 2
1 , 2 and
be the two
22 1
are the roots of a x 2 + b x + c = 0
1 1 b b b b 2 a a x + x + c c =
0.
32
MATHS 4.
(a)
If the ratio of the roots of , x 2 + n x + n = 0 is p : q , then prove that
p + q (b)
q – p
n
= 0 , where , n , p , q
1 x p
If the roots of the equation
+
R + .
1 x q
=
1 are equal in magnitude but r
opposite in sign, show that p + q = 2r and that the product of the roots is equal to –
5.
6.
7.
1 2 p + q 2 . 2
(c)
Show that if p , q , r and s are real numbers and p r = 2 (q + s) , then at least one of the equations , x2 + p x + q = 0 , x 2 + r x + s = 0 has real roots .
(d)
If x1 , x 2 be the roots of the equation x 2 3 x + A = 0 and x3 , x 4 be those of the equation x 2 12 x + B = 0 and x1 , x 2 , x 3 , x 4 are in G.P. Find A and B.
Show that the function , z = 2 x 2 + 2 x y + y 2 2 x + 2 y + 2 is not smaller than
Find the range of values of ‘ a ‘ , such that f (x) =
(a)
(b)
Find the least value of ,
2
6 x2 5 x2
x, y
R.
2 (a 1) x 9 a 4 is always negative . x 2 8 x 32
2
x 1 cannot x2 1 x R . x
Prove that the function y = values smaller than 1/2 for
ax
3 .
22 x 21 18 x 17
have values greater than 3/2 and
for all real values of ‘ x ‘ , using the theory of
quadratic equations. (c)
8.
Find the minimum value of the expression , 2 . log10 x log x 0.01 ; where x > 1 .
3 <
2
ax 2 x2 x 1
x
(d)
Find the values of ‘ a ’ for which
< 2 is valid for all real x .
(a)
If the quadratic equation , x2 + b x + a c = 0 and x2 + c x + a b = 0 have a common root , prove that the equation containing their other roots is x2 + a x + b c = 0 , where a 0 .
(b)
If a x 2 + 2 b x + c = 0 and a1 x 2 + 2 b1 x + c 1 = 0 have a common root and
b c a , , a1 b1 c1
are in A.P. , show that a1 , b 1 and c 1 are in G.P. (c)
9.
If the equations a x 2 + 2b x + c = 0 and a1 x 2 + 2 b1 x + c1 = 0 with rational coefficients have one and only one root in common then prove that b2 a c and b12 a1c 1 will be both perfect squares .
x 2 (a 5) x + 4 a = 0 (a
R)
be a quadratic equation. Find the value of ‘ a ’ for which
(a)
both roots are real and distinct
(b)
both roots are equal
(c)
roots are not real
(d)
roots are opposite in sign
(e)
roots are equal in magnitude but opposite in sign
(f)
both roots are positive
(h)
atleast one root is positive
(i)
one root is smaller than 2, the other root is greater than 2
(g)
both roots are negative
33
MATHS
10.
(j)
both roots are greater than 2
(l)
exactly one of the roots lie in the interval (1, 2)
(m)
both roots lie in the interval (1, 2)
(o)
one root is greater than 2, the other roots is smaller than 1
(p)
atleast one root is greater than 2.
(a)
If , are the two distinct roots of x2 + 2(k 3) . x + 9 = 0, then find the values of k such that , (– 6 , 1) .
(b)
For what real values of ‘ a ’ the equation a x 2 + x + a 1 = 0 posses two distinct real roots
and satisfying the 11.
inequality
13.
14.
(n)
1
1
atleast one root lie in the interval (1, 2)
> 1 .
If is a root of a x 2 + bx + c = 0 , is a root of a x 2 + bx + c = 0, where 0 < < , show that the equation a x 2 + 2bx + 2c = 0 has a root satisfying 0 < < < .
(b)
If the quadratic equation ax 2 + bx + c = 0 has real roots, of opposite signs in the interval
3) < 0
b c – > 0 2a 4a
(a)
If (x 3 a) (x a
(b)
Find all numbers ‘ a ‘ for each for which the least value of quadratic trinomial 4 x 2 4 a x + a2 2 a + 2 on the interval 0 x 2 is equal to 3 .
for all x
[1 , 3] . Find ‘ a ‘ .
(a)
2 2 Find the set of values of ‘ p ‘ for which the equation , p . 2cos x + p . 2 cos x – 2 = 0 has real roots.
(b)
Solve the equation , 9 – x – 2 – 4 . 3 – x – 2 – a = 0 for every real number a.
(a)
The quadratic equation x 2 + px + q = 0 where p and q are integers has rational roots. Prove that the roots are all integral.
(b)
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.
(c)
If a , b , c
I and
f ()
15.
both roots are smaller than 2
(a)
( 2 , 2) then prove that , 1 +
12.
(k)
a x 2 + b x + c = 0 has an irrational root . Prove that 1 q
2
, where
Q =
p q
and f (x) = a x 2 + b x + c .
(d)
Let a , b and c be integers with a > 1 and let p be a prime number. Show that if ax 2 + bx +c is equal to ‘p’ for two distinct integral values of x, then it can’t be equal to ‘2p’ for any integral value of x .
(a)
How many roots does the equation ,
(b)
Find the value of a < 0 for which the inequalities , 2 a x < 3 a – x and x –
x2
1 –
1 x2
5 3
= x posses ? Find them .
x a
>
6 a
have
solutions in common .
34
MATHS
Exercise - 04
OBJECTIVE R ,
a 0 and the quadratic equation a x2 b x + 1 = 0 has im aginary roots then a + b + 1 is (A) positive (B) negative (C) zero (D) depends on the sign b
1.
If a , b
2.
If c > 0 and 4 a + c < 2 b , then a x 2 – b x + c = 0 has a root in the interval : (A) (0 , 2) (B) (2 , 4) (C) (0 , 1) (D) (– 2 , 0)
3.
If the equation ,
(A) 4.
(2 n + 1 )
3
a
– 4 x + 13 = sin
x
has a solution , then ‘ a ‘ is equal to : (n I)
(B) 3 (4 n + 1 )
2
(C) 3 (1 + 4 n)
2
(D) None of these
R
(B) a
7 , 6 6
(C) a
5 , 6 6
(D) None of these
The solution set of the inequation , log1/2 (2 x+2 –4 x) – 2 is : (A)
6.
2
If the product of the roots of the equation , 2 x 2 + a x + 4 sin a = 0 is 1, then roots will be imaginary, if (A) a
5.
x
, 2
13
(B)
, 2
13
(C) ( – , 2 )
(D) None of these
The set of real value(s) of ‘ p ‘ for which the equation, 2 x + 3 + 2 x 3 = p x + 6 has more than two solutions is : (A) [0 , 4 ) (B) ( 4 , 4 ) (C) R {4 , 4 , 0 } (D) { 0 }
7.
The least value of the expression , x2 + 4 y2 + 3 z2 – 2 x – 12 y – 6 z + 14 is : (A) 0 (B) 1 (C) no least value (D) None of these
8.
If both roots of the quadratic equation (2 x) (x + 1) = p are distinct and positive then ‘p ‘ must lie in the interval : (A) p > 2
9.
(B) 2 < p <
9 4
(C) p <
2
(D) –
< p <
The value of ‘ p ‘ for which both the roots of the quadratic equation , 4 x 2 20 px + (25 p2 + 15p 66) are less than 2 lies in : (A)
4 , 2 5
(B) (2 ,
)
(C)
1 , 4 5
(D) ( – , – 1)
10.
The number of values of ‘ k ‘ for which the equation , x2 3 x + k = 0 has two real and distinct roots lying in the interval (0 , 1) are : (A) 0 (B) 2 (C) 3 (D) infinitely many
11.
If both the roots of the equation , x 2 – 2 a x + a 2 + a – 3 = 0 are less than 3 , then : (A) a < 2 (B) 2 a 3 (C) 3 < a 4 (D) a > 4
12.
The equation ,
a ( x b) ( x c ) b ( x c ) ( x a) c ( x a) ( x b) + + = x is satisfied by : (a b) (a c ) (b c ) (b a ) ( c a) ( c b )
(A) no value of x (C) exactly three values of x
(B) exactly two values of x (D) all values of x
35
MATHS n
13.
The constant term of the quadratic expression , (A) – 1
(B) 0
x k 1
1 x as k k 1 1
(C) 1
is :
(D) None of these
14.
If , , are the roots of the equation , x 3 + P 0x 2 + P 1x + P2 = 0, then (1 – equal to (A) (1 + P1) 2 – (P 0 + P2) 2 (B) (1 + P1) 2 + (P 0 + P2) 2 (C) (1 – P1)2 – (P 0 – P2) 2 (D) None of these
15.
If
,
n
2) (1 – 2) (1 – 2) is
are the roots of the quadratic equation, 6 x 2 – 6 x + 1 = 0 , then
1 2
(a + b + c 2 + d 3) +
1 2
12 d 6 c 4 b a 12 (C) 12 a + 6 b + 4 c + 9 d
(a + b + c 2 + d 3)
=
1 (12 a + 6 b + 4 c + 3 d) 12 (D) None of these
(A)
(B)
16.
Number of positive integers ‘ n ‘ for which n 2 + 96 is a perfect square is : (A) 4 (B) 8 (C) 12 (D) infinite
17.
Consider the equation x 2 + x – n = 0, where ‘ n ‘ is an integer lying between 1 to 100. Total number of different values of ‘ n ’ so that the equation has integral roots, is : (A) 6 (B) 4 (C) 9 (D) None of these
18.
Let p (x) = 0 be a polynomial equation of least possible degree, with rational coefficients , having 7 + 3 49 as one of its roots . Then the product of all the roots of p(x) = 0 is : (A) 7 (B) 49 (C) 56 (D) 63 3
19.
20.
If , are the roots of x2 a x + b = 0 and (A) V n + 1 = a V n + b Vn 1 (C) Vn + 1 = a Vn b V n 1
n + n = V n , then
(B) Vn + 1 = a V n + a Vn 1 (D) V n + 1 = a Vn 1 + b Vn
The inequalities , y (– 1) – 4 , y (1) 0 and y (3) 5 are known to hold for y = ax 2 + b x + c then the least value of ‘ a ’ is : (A)
1 4
(B)
1 3
(C)
1 4
(D)
1 8
36
MATHS
Exercise - 05 SUBJECTIVE 1.
2.
3.
Find the values of a for which the equation x4 + (1 2 a) x 2 + a2 1 = 0 (a)
has no solutions
(b)
has one solution
(c)
has two solutions
(d)
has three solutions.
(e)
has four distinct real solutions
Find all real values of ‘ a ‘ for which the equation x4 + (a 1) x 3 + x 2 + (a 1) x + 1 = 0 possesses at least two distinct negative roots .
Find the real values of ‘m’ for which the equation ,
x 2 1 x 2 – (m – 3)
x 1 x 2 + m = 0
has real
roots ? 4.
Find all values of ‘a’ f or which the equation, (x2 + x + 2) 2 (a 3) (x2 + x + 2) (x2 + x + 1) + (a 4) (x2 + x + 1)2 = 0 has at least one real roots.
5.
If the equation x (x + 1) (x + a) (x + 1 + a) = a2 has four real roots , prove that a
6.
,
5
2
Prove that the minimum value of ,
5
2, (a
5
2
x ) (b c ) (c x )
5 2 ,
, x > – c is
a
c
b
c
2
a > c and b > c .
7.
For what real value of ‘a’ do the roots of the equation x2 2 x a2 + 1 = 0 lie between the roots of the equation x2 2 (a + 1) x + a (a 1) = 0 ?
8.
A quadr atic trinomial f (x) = ax 2 + bx + c is such that the equation f (x) = x has no real roots . Prove that in this case the equation f (f (x) ) = x has no real roots either .
9.
Find all real values of ‘ a ‘ for each of which the equation ,
x a
(x2 + (1 + 2 a2) x + 2 a2) = 0
has
only two distinct roots. Write the roots. 10.
(a)
Find the integral values of ‘ a ’ for which (a + 2) x2 + 2 (a + 1) x + a = 0 will have both roots integers.
(b)
Find the integral values of ‘m’ for mx 2 + (2m 1)x + m 2 = 0 are rational.
(c)
which
the
ro ots
of
the
equation
Find the values of a so that x2 – x – a = 0 has integral roots, where a N, and 6 a 100.
(d)
If a , b – 1 and c are odd prime numbers and ax2 + bx + c = 0 has rational roots then, prove that one root of the equation will be independent of a, b and c .
(e)
Show that the quadratic equation x2 + 7 x no integral roots .
(f)
Let x 2 px + q = 0 and x 2 qx + p = 0 both have unequal integral roots, where p , q N . Prove that the possible number of solutions of the ordered point (p , q) is 2. Find them.
14
(q 2 + 1) = 0 , where q is an integer, has
37
MATHS 11.
12.
(a)
Find the value of a for which inequality ax 2 + 4x + 10 0 has atleast one real solution and every solution of the inequality x2 x 2 < 0 is larger than any solution of the inequality ax 2 + 4x + 10 0 .
(b)
Find all values of the parameter ‘ k ’ for which the solution set of the inequation x 2 + 3 k 2 – 1 2 k (2 x – 1) is a subset of the solution set of the inequation x 2 – ( 2 x – 1) k + k 2 0 .
(c)
Find all values of k for which there is at least one common solution of the inequalities x 2 + 4 k x + 3 k 2 > 1 + 2k and x2 + 2 k x 3 k 2 – 8 k + 4 .
(d)
Find all values of ‘k’ for which any real x is a solution of at least one of the inequalities x 2 + 5 k 2 + 8 k > 2 (3 k x + 2) and x2 + 4 k 2 k (4 x + 1) .
(a)
Find all the values of the parameters c for which the inequality has at least one solution
1 + log2 2 x 2 (b)
13.
7 2 x log 2 (c x2 + c ) . 2
Find the value of ‘ b ’ for which the equation , 2 log1/25 (b x + 28) = – log5 (12 – 4 x – x 2) has (i) on ly on e s olu ti on (ii) two different solutions (iii) no solution
Solve the following for real values of x (depending upon the real parameter if any)
a x
(a)
x <
(c)
x 3 + 1 = 2
(e)
x+
3
x x2
1
2x
1
>
35 12
(b)
x+
(d)
x 2 –
a
x = a
a x = a .
38
MATHS
Exercise - 06
IIT NEW PATTERN QUESTIONS Section I
Fill in the blanks
1.
The set of values of p for which the roots of the equation 3x2 + 2x + p (p – 1) = 0 are of opposite signs, is ________ .
2.
If a, b, c are real numbers satisfying the condition a + b + c = 0 then the roots of the quadratic equation 3ax2 + 5bx + 7c = 0 are ________ .
3.
If a and b are the odd integers, then the roots of the equation, 2ax2 + (2a + b)x + b = 0 , a 0 , will be ________ .
4.
Equation x2 + x + a = 0 will have exactly one root in the interval (0, 1] . Then value of ‘a ‘ lies in ________ .
5.
Let (x –
, be the roots of the equation ) (x – ) + c = 0 are ________ .
Section II 1.
(x – a) (x – b) = c, c 0 . Then the roots of the equation
More than one correct :
A quadr atic equat ion with real roots is formed suc h that , its roots remain unch anged even aft er squaring them. The root can be (A) 0, 0 (B) 1, 0 (C) 1, 1 (D) –1, –1
2.
If roots of equation x 2 – (2n + 18) x – n – 11 = 0 (n I) are rational, then n is (A) 8 (B) –8 (C) 10 (D) –11
3.
From the following graphs it can be interpreted that (A) (C)
4.
5.
c>0 a > 0,
(B) (D)
c<0 abc < 0
X
If the difference of the roots of the equation x2 + kx + 7 = 0 is 6, then possible values of k are (A) 4 (B) – 4 (C) 8 The roots of ax2 + bx + c = 0. Where a a + c < b. Then (A) 4a + c > 2b (B) 4a + c < 2b
Section III
Y
(D) – 8
0 and coefficients are real, are nonreal complex and (C) a + 4c > 2b
(D) a + 4c < 2b
Condition and Result :
Each question has a conditional statement followed by a result statements. If condition result, then condition is sufficient and
If result condition, then condition is necessary If condition is necessary as well as sufficient for the result, mark (A) If condition is necessary but not sufficient for the result, mark (B) If condition is sufficient but not necessary for the result, mark (C) If neither necessary nor sufficient for the result, mark (D) Consider the following example : Condition : a > 0, b > 0 Result : a+b>0 Here, if a > 0 and b > 0, then it always implies that a + b is positive but if a + b is positive, then a and b both need not to be positive. So condition result but result does not always implies condition hence condition is sufficient but not necessary for the result to be hold. So answer is ‘C’. 39
MATHS 1.
Condition : Result :
Let f (x) = x 2 + bx + c , f (2) > 0 and b2 – 4 c > 0 Both roots of the quadratic equation x2 + bx + c = 0 are distinct and more than 2 .
2.
Condition : Result :
x 2 + b x + c = 0 has integral roots . For the quadratic equation x 2 + bx + c = 0, b2 – 4c is a perfect square of an integer and b , c Integer .
3.
Condition :
One of the root of the quadratic equation ax 2 + bx + c = 0 , (a , b , c R) is 2 +
3
. Result :
Section IV (A) (B) (C) (D)
Other root of the quadratic equation a x 2 + b x + c = 0 , (a , b , c
R)
is 2 –
Assertion/Reason
Statement 1 is True, Statement 2 is True, Statement 2 is a correct explanation for Statement 1 Statement 1 is True, Statement 2 is True ; Statement 2 is NOT a correct explanation for Statement 1 Statement 1 is True, Statement 2 is False Statement 1 is False, Statement 2 is True
Statement 1 (A) : Statement 2 (R) :
x R , x2 + x + 1 is positive. If D < 0 , a x 2 + b x + c , ‘ a ‘ have same sign
2.
Statement 1 (A) : Statement 2 (R) :
If x (2 , 3) then x2 – 5 x + 6 > 0 If < x < , a x 2 + b x + c = 0 and ‘ a ‘ have opposite sign ( < ) .
3.
Statement
The equation , a sin x + cos 2 x = 2 a – 7 , possesses a solution. If a [2 , 6]
1.
4.
3
1 (A) :
sin x 1
x
R.
R.
Statement 2 (R) :
– 1
Statement
The set of all real numbers ‘ a ‘ such that a2 + 2 a , 2 a + 3 and a2 + 3 a + 8 are the sides of a triangle is (5 , ). Since in a triangle sum of two sides is greater the third side and also sides are always positive.
1 (A) :
Statement 2 (R) :
Section V
x
Comprehensions
Write Up I Consider the quadratic equation ax2 + bx + c = 0 then condition for both the roots to be positive are (i) root must be real and (ii) sum of the roots > 0 and (iii) product of the roots > 0. Similarly we can find the condition for both the roots to be negative , both roots are of opposite sign and both roots of opposite sign but equal in magnitude. 1. The set of values of a for which quadratic equation x2 – (a – 3) x + a = 0 have both roots positive, is : (A) (0 , ) (B) (0 , 9) (C) (– , 1] [9, ) (D) [9, ) 2.
The least positive integral value of a for which quadratic equation x2 + 4x + a = 0 have both roots negative, is : (A) 1 (B) 4 (C) 3 (D) none of these
3.
The values of a and b for which quadratic equation (a – 3) x2 + 4bx + (a +3) = 0 has one root negative and other root positi ve, are : (A) a (– 3 , 3) , b > 0 (B) a (– 3 , 3) , b < 0 (C) a (– 3 , 3) , b R (D) None of these
40
MATHS Write Up II Let f(x) = ax2 + bx + c be a quadratic expression and y = f(x) has graph as shown in figure 1.
2.
3.
Which of the f ollowing is false ? (A) ab > 0 (C) ac < 0
(B) abc < 0 (D) bc < 0
Which of the following is true ? (A) a + b + c > 0 (C) a + 3b + 9c < 0
(B) a – b + c > 0 (D) a – 3b + 9c > 0
y
–1 – 1 0 2
1 2
1
x
If f(x) is an integer whenever x is an integer, then which of the following is always correct ? (A) a is an integer (B) 2 a is an integer (C) b is an integer
(D)
c is an integer 2
Write Up III If a , b are the roots of the quadratic equation , x 2 – 10 c x – 11 d = 0 and c , d are the roots of the quadratic equation , x2 – 10 a x – 11 b = 0 (where a b c d 0) 1.
2.
The value of a c is : (A) 22 The value of
(A) 3.
1 11
c b
(B) 1210
(C) 11
(D) 121
(C) 11
(D) 9
(C) 11
(D) 22
a is : d (B)
1 9
The value of a + b + c + d is : (A) 121 (B) 1210
Write Up IV Consider the quadratic equation , a x 2 + b x + c = 0 and d x2 + e x + f = 0. Then the required condit ion for the two equations to have a com mon root is (d c – a f )2 = (b f – c e) (a e – b d) and condition for both
c a b = = . d e f Condition for the quadratic equation , x 2 + 3x + 4 = 0 and a x 2 + bx + c = 0 (a , b , c R) has exactly one root common is : the roots to be common is ,
1.
(A)
(c – 4 a)2 = (b – 3 a) (3 c – 4 b)
(C) a , b , c are of the same sign 2.
If the quadratic equation, x2 + 3 x + 2 = 0 and x2 + (m – 3) x + m = 0 has one root common , then the value of ‘ m ‘ is : (A) 10
3.
b a c = = 3 1 4 (D) None of these
(B)
(B) 1
(C)
2 3
(D) 3
The equations , a x 2 + b x + a = 0 and x3 – 2 x 2 + 2 x – 1 = 0 have two roots in common . Then a + b must be equal to : (A) 1 (B) – 1 (C) 0 (D) None of these
41
MATHS
Section VI 1.
x2
If
Subjectives
5x 4 1 , then x2 4
find the least positive integer which does not come in the solution
set of ‘ x ‘ . 2.
If
,
are the roots of the equation, k (x 2 – x ) + x + 5 = 0 . If k1 and k 2 are the two values of ‘ k ‘
for which the roots
, are connected by the relation
+ =
4 k1 k2 . Find the value of + . 5 k2 k1
3.
If a x2 – b x + 5 = 0 does not have two distinct real roots, then find the minimum value of 5a + b .
4.
If the equation , x2 + 2 = 1 – 2 x and x2 + 2 = 1 – 2 then find the value of , – .
5.
If f (x) = 8 x 3 +
Section VII 1.
1 x
3
and
, are the roots of
2x +
2.
have one and only one root in common,
1 = 3 . Then find the value of f ( ) . x
Match the Column
Column I (Quadratic Equation) (A) x 2 – x – 6 = 0 (B) x 2 – 3 x + 1 = 0 (C) x 2 – 2 x cos + 1 = 0 ( n ) (D)
x
4 3 x 2 – 20 3 x + 9 3 = 0
Column II (Nature of roots) (p) roots are rational (q) roots are integer (r) roots are irrational (s)
Column I (A) (B) (C) (D)
7
7
7 .....
roots are non-real Column II
2
is equal to
If x2 + p x + q = 0 , has one root as 2 + i 3 the value of q is (p , q are real with q > 0) If x2 – h x – 21 = 0 and x2 – 3 h x + 35 = 0 (h > 0) have common root then h is equal to If the difference between the roots of x2 + a x + b = 0 and those of x2 + b x + a = 0 (a b) are equal then (a + b) is equal to
(p)
–4
(q)
7
(r)
49
(s)
4
3.
For the quadratic equation , x 2 – (k – 3) x + k = 0 , match the complete set of solution of the following. Column I Column II (A) Both roots are positive (p) (– , 1] (B) Both roots are negative (q) [9 , ) (C) Both roots are real (r) (– , 0) (D) Both roots are opposite in sign (s) (0 , 1]
4.
The roots of the quadratic equation , x2 – 5 x + 6 = 0 is Column I – (A) (B) 2 + 2 (C)
(D)
3 3 35
4 4 62 3 3
and . Then find the f ollowing. Column II (p) 1 (q) – 13 (r)
–1
(s)
13
42
MATHS
Exercise - 07
AIEEE FLASH BACK 1.
If
but 2 = 5 – 3 and 2 = 5 – 3 , then the equation having
(A) 3 x 2 – 19 x + 3 = 0 (C) 3 x 2 – 19 x – 3 = 0 2.
3.
4.
5.
6.
[ 2002 ]
Difference between the corresponding roots of x 2 + a x + b = 0 and x2 + b x + a = 0 is same and a b, then : (A) a + b + 4 = 0 (B) a + b – 4 = 0 (C) a – b – 4 = 0 (D) a – b + 4 = 0 [ 2002 ] Product of real roots of the equation, t2 x 2 + x + 9 = 0 (A) is always positive (B) is always negative (C) does not exist (D) None of these If p and q are the roots of the equation, x2 + p x + q = 0, then : (A) p = 1, q = – 2 (B) p = 0, q = 1 (C) p = – 2, q = 0 If 2 a + 3 b + 6 c = 0, (a , b , c
R)
[ 2002 ] (D) p = – 2, q = 1 [ 2002 ]
then the quadratic equation, a x 2 + b x + c = 0 has :
(A) atleast one root in [ 0 , 1 ]
(B) atleast one root in [2 , 3 ]
(C) atleast one root in [4 , 5]
(D) None of these
[ 2002 ]
If the sum of the roots of the quadratic equation, ax 2 + b x + c = 0 is equal to the sum of the squares a
,
b
and
c
c a b (A) Arithmetic - Geom et ric Progression (C) Geometric Progression
are in : (B) Ar ithm etic Progression (D) Harmonic Progression
[ 2003 ]
The value of ‘ a ‘ for which one root of the quadratic equation, (a2 – 5a + 3) x2 + (3 a – 1) x + 2 = 0 is twice as large as the other is : (A) –
8.
as its roots is :
(B) 3 x 2 + 19 x – 3 = 0 (D) x2 – 5 x + 3 = 0
of their reciprocals, then
7.
and
1 3
(B)
2 3
(C) –
2 3
The number of real solution of the equation, x2 – 3 x + 2 = 0 is : (A) 3 (B) 2 (C) 4
(D)
1 3
(D) 1
[ 2003 ]
[ 2003 ]
9.
Let two numbers have arithmetic mean 9 and geometric mean 4 . Then these numbers are the roots of the quadratic equation : (A) x 2 – 18 x – 16 = 0 (B) x2 – 18 x + 16 = 0 2 (C) x + 18 x – 16 = 0 (D) x2 + 18 x + 16 = 0 [ 2004 ]
10.
If (1 – p) is a root of quadratic equation , x2 + p x + (1 – p) = 0 , then its roots are : (A) – 1 , 2 (B) – 1 , 1 (C) 0 , – 1 (D) 0 , 1
11.
[ 2004 ]
If one root of the equation, x2 + px + 12 = 0 is 4, while the equation , x2 + p x + q = 0 has equal roots, then the value of ‘ q ‘ is : (A) 4
(B) 12
(C) 3
(D)
49 4
[ 2004 ]
43
MATHS 12.
If 2 a + 3 b + 6 c = 0 , then atleast one root of the equation, a x 2 + b x + c = 0 lies in the interval : (A) (1 , 3) (B) (1 , 2) (C) (2 , 3) (D) (0 , 1) [ 2004 ]
13.
In a triangle PQR , (A) a = b + c
14.
R =
2
. If tan
Q P and tan 2 2
(B) c = a + b
are the roots of a x 2 + bx + c = 0 , a 0 then
(C) b = c
(D) b = a + c
The value of ‘ a ‘ for which the sum of the squares of the roots of the equation, x 2 – (a – 2) x – a – 1 = 0 assume the least value is : (A) 1 (B) 0 (C) 3 (D) 2
[ 2005 ]
[ 2005 ]
15.
If the roots of the equation, x2 – b x + c = 0 be two consecutive integers, then b2 – 4 c equals : (A) – 2 (B) 3 (C) 2 (D) 1 [ 2005 ]
16.
If both the roots of the quadratic equation, x2 – 2 k x + k 2 + k – 5 = 0 are less then 5 , then ‘ k ‘ lies in the interval : (A) (5 , 6 ] (B) (6 , ) (C) ( – , 4) (D) [ 4 , 5] [ 2005 ]
17.
If the roots of the quadratic equation , x2 + px + q = 0 a re tan 30º and tan 15º respectively, then the value of 2 + q – p is : (A) 2 (B) 3 (C) 0 (D) 1 [ 2006 ]
18.
All the values of ‘ m ‘ for which both roots of the equation, x2 – 2 m x + m 2 – 1 = 0 are greater than – 2, but less than 4, lie in the interval : (A) – 2 < m < 0 (B) m > 3 (C) – 1 < m < 3 (D) 1 < m < 4 [ 2009 ]
19.
The quadratic equations, x2 – 6 x + a = 0 and x2 – c x + 6 = 0 have one root in common. The other roots of the first and the second equations are integers in the ratio 4 : 3. Then the comm on root is : (A) 4 (B) 3 (C) 2 (D) 1 [ 2011 ]
44
MATHS
Exercise - 08
IIT FLASH BACK (OBJECTIVE ) (A)
Fill in the blanks :
1.
If 2 + i 3 is a root of the equation, x2 + px + q = 0, where p and q are r eal, then (p, q) = ( ). [ IIT 82 ]
2.
If the product of the roots of the equation , x3 – 3 k x + 2 e2 nk – 1 = 0 is 7, then the roots are real for k = ________ . [ IIT 84 ]
3.
If the quadratic equations x2 + ax + b = 0 and x2 + bx + a = 0 (a b) have a common root, then the numerical value of a + b is ________ . [ IIT 86 ]
4.
The sum of all the real roots of the equation , x 2
2
+
x – 2 – 2 = 0 is ________ . [ IIT
97 ]
[ IIT
83 ]
(B)
True or False :
1.
The equation , 2 x 2 + 3 x + 1 = 0 has an irrational root .
2.
If a < b < c < d , then the roots of the equation (x – a) (x – c) + 2(x – b) (x – d) = 0 are real and distinct . [ IIT 84 ]
3.
If P (x) = ax 2 + bx + c and Q (x) = – ax 2 + dx + c , where ac two real roots .
(C)
Multiple choice questions with one or more than one correct answer :
1.
2 2 The equation , x (A) at least one real solution (C) exactly one irrational solution
3 / 4 (log x )2
log
x
5/4
=
0 , then P (x) Q (x) = 0 has at least [ IIT 85 ]
2 has : (B) exactly three solutions (D) complex roots
[ IIT
91 ]
(D)
Multiple choice questions with one correct answer :
1.
If , m , n are real, m , then the roots by the equation, ( – m) x2 – 5 ( + m) x – 2 ( – m) = 0 are : (A) real and equal (B) complex (C) real and unequal (D) None of these [ IIT 79 ]
2.
The equation, 2 cos 2
1 x sin 2 x = 2
x2 + x– 2 , 0 < x
(A) no real solution (C) more than one real solution 3.
The equation, x – (A) no root (C) two equal roots
4.
2 x
1
= 1 –
2 x
1
9
has :
(B) one real solution (D) None of these
[ IIT
80 ]
(B) one root (D) infinitely many roots
[ IIT
84 ]
has :
If and are the roots of x2 + px + q = 0 and 4 , 4 are the roots of x2 – r x + s = 0 , then the equation , x2 – 4 q x + 2 q2 – r = 0 has always : (A) two real roots (B) two positive roots (C) two negative roots (D) one positive and one negative roots [ IIT 89 ]
45
MATHS 5.
Let a , b , c be real number , a 0 . If is a root of a2 x 2 + b x + c = 0 , is the root of a2 x 2 – 2b x – 2 c = 0 and 0 < < , then the equation a 2x 2 + 2bx + 2c = 0 has a root that always satisfies [ IIT (A)
6.
=
(B)
2
= +
(C)
2
=
(D)
< <
Let f (x) be a quadratic expression which is positive for all real values of x . If g (x) = f (x) + f (x) + f (x) , then for any real x , (A) g (x) <0 (B) g (x) > 0 (C) g (x) =0 (D) g (x) 0 [ IIT
7.
8.
10.
(A)
(0 , 2 )
Let
, be the roots of the equation , (x – ) (x – ) + c = 0 are :
(B) ( –
, 0)
(C)
(B) b , c
The equation , x 1 – (A) no solution (C) two solutions
In a triangle PQR ,
R =
x 1 =
12.
(D) (0 , )
(C) a , b
[ IIT
4x
(D) a + c , b + c [ IIT
(B) one solution (D) more than two solutions
2
. If tan
P and 2
tan
Q 2
97 ]
[ IIT
99 ]
[ IIT
99 ]
are the roots of the equation
(C) a + c = b
(D) b = c
For the equation, 3 x 2 + p x + 3 = 0 , p > 0 if one of the roots is square of the other , then p is equal to : 1
(B) 1
3
, ( < ) , (A) 0 < < If
(C) 3
(D)
2 3
[ IIT
16.
2000 ]
are the roots of the equation , x2 + b x + c = 0 , where c < 0 < b , then : (B)
< 0 < <
(C)
< < 0
(D)
If b > a , then the equation, (x a) (x b) 1 = 0 has : (A) both roots in [ a , b ] (B) both roots in ( – (C) both root in
15.
92 ]
[ IIT
< 0 < < [ IIT
14.
91 ]
1 has :
If the roots of the equation x2 2 a x + a 2 + a 3 = 0 are real and less than 3 then : (A) a < 2 (B) 2 a 3 (C) 3 < a 4 (D) a > 4
(A) 13.
, 2 2
(x – a) (x – b) = c , c 0, then the roots of the equation
a x 2 + b x + c = 0 (a 0) then : (A) a + b = c (B) b + c = a 11.
90 ]
The equation , (cos p – 1) x2 + (cos p) x + sin p = 0 in the variable x, has real roots . Then p can tak e any value in the interval
(A) a , c
9.
89 ]
[ b , )
(D) one root in ( –
The set of all real numbers ‘ x ‘ for which , x2 –
x + 2 + x > 0 ,
(A) (– , – 2)
(2 , )
(B)
(C) (– , – 1)
(1 , )
(D)
2000 ]
, a )
, a ) and
other in (b , + ) [ IIT 2000 ]
is :
, 2 2 ,
2,
[ IIT
2002 ]
For all ‘ x ‘ x2 + 2 a x + (10 – 3 a) > 0 , then interval in which ‘a ‘ lies is : (A) a < – 5 (B) – 5 < a < 2 (C) a > 5 (D) 2 < a < 5 [ IIT
2004 ]
46
MATHS 17.
If one root is square of the other root of the equation , x2 + p x + q = 0 then relation between ‘ p ‘ and ‘ q ‘ is : (A) p 3 – (3 p – 1) q + q2 = 0 (B) p3 – q (3 p + 1) + q 2 = 0 (C) p3 + q (3 p – 1) + q 2 = 0 (D) p3 + q (3 p + 1) + q2 = 0 [ IIT 2004 ]
18.
Let f (x) = a x2 + b x + c , a 0 and (A) 0 (B) b = 0
= b 2 – 4 a c . If + , 2 + 2 and 3 + 3 are in G.P., then (C) c = 0 (D) b c 0 [ IIT
19.
a , b , c are the sides of a triangle ABC such that x2 – 2 (a + b + c) x + 3 (a b + b c + c a) = 0 has real roots :
<
(A)
4 3
(B)
>
5 3
(C)
3 , 3 4
5
(D)
3 , 3 1 5
[ IIT 20.
2005 ]
Let
the roots of the equation x2 – p x + r = 0 and
2
2006 ]
, 2 be the roots of the equation
x2 – q x + r = 0 . Then the value of ‘ r ‘ is : 2 (p – q) (2q – p) 9
(A)
22.
(q – 2p) (2q – p)
(D)
i
(A)
Let
and be the roots of x 2 – 6x – 2 = 0, with > .
a10
2a 8
2a 9 22.
2 (q – p) (2p – q) 9
2 (2p – q) (2q – p) [ IIT 2007 ] 9 9 A value of b f or wh ich the equations x2 + bx – 1 = 0, x 2 + x + b = 0, have one root in common is 2
(C)
21.
(B)
(B)
2
3
(C) i 5
(D) if an =
[ IIT
2
2011 ]
n – n for n 1, then the value of
is
[ IIT
2011 ]
(A) 1 (B) 2 (C) 3 (D) 4 Let (x0, y0) be the solution of t he following equations (2x)ln2 = (3y) ln3 , 3 lnx = 2 lny. Then x 0 is (A)
1 6
(B)
1 3
(C)
1 2
(D) 6
[ IIT
2011 ]
(E).
Assertion/Reason (A) Statement 1 is True , Statement 2 is True ; Statement 2 is a CORRECT explanation for Statement 1 (B) Statement 1 is True , Statement 2 is True ; Statement 2 is a NOT CORRECT explanation for Statement 1 (C) Statement 1 is True , Statement 2 is False (D) Statement 1 is False , Statement 2 is True
1.
Let a , b , c , p , q be real numbers . Suppose x 2 + 2 p x + q = 0 and , Statement - 1 (A) : Statement - 2 (R) :
1
,
are the roots of the equation ,
are the roots of the equation a x 2 + 2 b x + c = 0 , where
(p2 – q) (b2 – a c) 0 b p a or c q a
(F).
Integer Type
1.
The number of distinct real roots of x 4 – 4x 3 + 12x2 + x – 1 = 0 is
2 { – 1 , 0 , 1} [ IIT
2008 ]
[ IIT
2011 ]
47
MATHS
Exercise - 09
IIT FLASH BACK (SUBJECTIVE) 1.
If , are the roots of x2 + p x + q = 0 and r , are the roots of x2 + r x + s = 0, evaluate ( – ) ( – ) ( – ) ( – ) in terms of p , q , r and s . Deduce the condition that the equations have a common root . [ IIT 79 ]
2.
Show that the equation , esin x – e – sin x – 4 = 0 , has no real solution .
3.
If one root of the quadratic equation ax 2 + bx + c = 0 is equal to the nth power of the other, then
show that 4.
1
a c n n 1 +
a c n
[ IIT
1 n
1
+b=0.
Find all real values of ‘ x ‘ which satisfy x2 – 3 x + 2 > 0 and x2 – 2 x – 4 0 . x2
5 2 6
3
82 ]
+ 5
2
6
x2
3
= 10 .
[ IIT
83 ]
[ IIT
83 ]
[ IIT
85 ]
[ IIT
86 ]
5.
Solve for x :
6.
For a
7.
Let 1 , 2 1 , 2 be the roots of a x 2 + b x + c = 0 and p x 2 + q x + r = 0 respectively . If the system of equation 1y + 2z = 0 and 1y + 2z = 0 has a nontrivial solution , then prove that
0 , determine all real roots of the equation
b2 q2
=
x2 – 2 a x – a – 3 a2 = 0.
ac . p r
x2 + 4 x + 3 + 2 x + 5 = 0 .
[ IIT
87 ]
[ IIT
88 ]
8.
Solve
9.
If , are the roots of the equation x2 p x + q = 0 , then find the quadratic equation the roots of which are (2 – 2) (3 – 3) and 3 2 + 2 3 . [ REE 94 ]
10.
Let a , b , c be real , If a x 2 + b x + c = 0 has two real roots show that 1 +
11.
c + a
b a
and , where < – 1 and > 1 , then
< 0 .
Prove that the values of the function
[ IIT
95 ]
1 sin x cos 3 x do not lie between and 3 for any real x . 3 sin 3 x cos x [ IIT
96 ]
12.
If
,
are the roots of the equation x2 b x + c = 0 , then find the equation whose roots are , (2 + 2) (3 + 3) and 5 3 + 3 5 – 2 4 4 . [ REE 98 ]
13.
If
, are
14.
Let a , b , c be real numbers with a 0 and let , be the roots of the equation a x 2 + b x +c= 0. Express the roots of a3 x 2 + a b c x + c 3 = 0 in terms of , . [ IIT 2001 ]
15.
If x2 + (a b) x + (1 a b) = 0 where a , b unequal real roots for all values of ‘ b ’ .
16.
If
the roots of the equation, (x (x – ) (x – ) = c .
a)
(x
+ c = 0 , find the roots of the equation, [ REE 2000 ]
R . Then find the values of
‘ a ’ which equation has [ IIT 2003 ]
, are the roots of ax 2 + bx + c = 0 , (a 0) and + , + , are the roots of, A x 2 + B x + C = 0 , (A 0) for some constant
17.
b)
, then prove that ,
b2
4ac a2
=
B2
4 A C
. A 2 [ IIT 2004 ]
If x2 – 10 a x – 11 b = 0 have roots c and d and those of x2 – 10 c x – 11 d are ‘ a ‘ and ‘ b ‘ . Then find the value of a + b + c + d , where a b c d . [ IIT 2006 ]
48
MATHS
Exercise - 10 INTEGER T YPE 1.
If a, b, c > 0, a 2 = bc and a + b + c = abc, find the least possible value of a2
2.
If x1 + x 2 + x 3 = 1, x2x 3 + x 3x1 + x 1x 2 = 1 and x 1x 2x 3 = 1, find value of |x1| + |x2| + |x3|.
3.
Find the num ber of real roots of P(x) = x 100 – 2x 99 + 3x 98 – ... – 100x + 101 = 0.
4.
Fi nd the num be r o f p os it ive int egr al so lut ion s x 4 – y4 = 3879108
5.
Find the num ber of solutions of 2 cos 2
x 2 = (0.2) + (0.2) x
–x
6.
Let f(x) = ax2 + 2bx + c, a, b, c R If f(x) takes r eal values for values of x and imaginary value for imagi nary values of x, find the value of a.
7.
Fi nd t he r ea l val ue of – a for whi ch t he ro ot s x1, x 2, x 3 of x3 – 6x2 + ax – a = 0 satisfy (x 1 – 3 )3 + (x 2 – 3 )3 + (x 3 – 3 )3 = 0
8.
Sum of all the real values of x for which
9.
2x
4 2 x
3
2 x13
If x1 and x 2 are roots of the equation x 2 + x + c = 0, c 0, and
2 x2
x 32
2 x1
1 2
find value of –4c 10.
2
If x1, x 2, x 3 are such that x 1 + x2 + x3 = 2, x1
x 22 x32 6 ,
3
x1
x32 x 33 8
Find the value of –(x2 – x3) (x 3 – x1) (x 1 – x2).
49
MATHS
ANSWER SHEET CBSE FLASH BACK
Exercise - 01 1.
(i)
1
5
2
and
2.
3,–2
6.
48 km/hr. , 32 km/hr.
1
3.
5
(ii)
2 16
x=2, x=–1
4.
24
8.
–2
(iii) 5.
2 metres
OBJECTIVE
Exercise - 02 1.
D
2.
B
3.
8.
B
9.
B
10. B
11. A
12. C
13. B
16. C
17. B
18. D
19. C
20. A
15. C
2.
B
4.
A
5.
B
6.
A
7.
D
14. B
SUBJECTIVE
Exercise - 03 1.
1 1 , 3 4
–
(a)
x = 2 or 5
(b)
x = – 4 or – 1
(d)
x=1
(e)
(a)
x = 1 or – 2 log3 6
(b)
1 1 log 3
(d)
0
(f)
2 , 2
3.
(b)
x 2 – 3 x + 2 = 0
4.
(d)
A = 2 or
6.
a
9.
15
18 ,
x = – 1 or 1
(f)
x
( – 6 , – 5) ( – 3 , – 2)
(c)
( – 1 , )
x=±1
(c)
3 x 2 + 68 x – 18 = 0
(a)
(– , 1)
(25, )
(d)
(– , 0)
(h)
(– , 0]
[25, )
(k)
(– 7, 1]
(l)
(– 7, – 2)
(n)
(– 7, – 2)
(o)
(–
10.
(a)
6 , 27 4
(b)
a
12.
(a)
a
(b)
a = 1
13.
(a)
4 , 1 5
(b)
x 1 . 2 = 2 ± log3 2
15.
(a)
one , x =
4 3
–
17
2 – 1 or x = – 3
2 , x 4
x = ± 4 , ± 14
288
7.
1 0 , 3
x < –7, – 5 < x (h)
1 , 2
(e)
(e)
(g)
B = 32 or
7
x=
(where base of log is 2)
(c)
3
(b)
1
(c)
4
(d)
(b)
{1, 25}
(c)
(1, 25)
(f)
[25, )
(g)
(0, 1]
(i)
(– , –7)
(j)
[25, )
(m)
(p)
(– , – 7)
, – 7) [25, )
2 < a < 1
[25, )
6 (0 , 1) 1 , 5
(b)
2,5
a
10
4
a
where – 3 a < 0
2 , 0 3 50
MATHS
OBJECTIVE
Exercise - 04 1.
A
2.
A
3.
8.
B
9.
D
10. A
11. A
12. D
13. C
16. A
17. C
18. C
19. C
20. D
15. B
(a)
for a
(c)
a
a
7.
1 , 1 4
10.
(a)
B
5 (– 1 , 1) 4 3.
7 , 5 2 6
9.
{a, 1} for a
(c)
(a)
a
(c)
all
12.
(a)
(0 , 8 ]
13.
(a)
(b)
(d)
1 3 , , 2 2
(b)
a , 0 x if
1 2 x
for a = 1
(d)
for a = 1
7.
B
14. A
(– , 0) (0
5
if
,
5
1
2
x=
a
(f)
(5, 6), (6, 5)
[ – 1 , )
(d)
k
(– , 0] {1}
1 , 5 4
1 , 0 2
m = k (k + 1) , k
k
a
19 5 , 3
(b)
for a > 0 ,
I
4 , 14 3
(ii)
(iii)
[ – 14 , 4)
for a = 0
, 1) ;
2 a 1 2
3 if
4a
a
[1 , )
, 1
1 a , 4
(0 ,
(e)
(b)
14
a
a
( – , – 14) {4} , 3
(i)
for a < 0 , 0 , a
if a
(e)
D
( – , – 1) , {a , – 2 a2} for
2 0 , 5
x = {0} if a = 0 ;
(c)
6.
(b)
a
k
C
4.
{ – 4 , – 3 , – 1 , 0 } a {6 , 12 , 20 , 30 , 42 , 56 , 72 , 90 }
11.
5.
5 (– , – 1) , 4
5 , 2
2.
4.
SUBJECTIVE
Exercise - 05 1.
C
1) , x =
, x=
1
1 4a 2
4a 2
1
,
1
1
if
4a 2
a
1 , 0 4
3
if
a
[1 , )
1 , 5 5 , 4 3 51
MATHS
IIT NEW PATTERN
Exercise - 06 Section I 1.
(0, 1)
2.
real and distinct
4.
– 2 a < 0
5.
a, b
3.
rational
Section II 1.
BC
2.
BD
3. ACD
4.
CD
3.
D
5.
B
Section III 1.
B
2. A
Section IV 1. A
2.
D
3. A
4. A
Section V Write Up I 1. D Write Up III 1. D
2. A
2. A
3. C
Write Up II 1. B 2. D
3. B
3. B
Write Up IV 1. D 2. A
3. C
– 1
2
Section VI 1.
2
2.
254
3.
4.
5.
9
Section VII 1.
A - p q
B- r
C-s
D-p
2.
A - r
B- q
C-s
D-p
3.
A - q
B- s
C-p q
D - r
4.
A - p r
B- s
C-p
D-p
Exercise - 07 1.
A
2. A
8.
C
9.
15. D
AIEEE FLASH BACK 3.
C
4.
A
5.
A
B
10. C
11. D
12. D
16. C
17. B
18. C
19. C
6.
D
13. B
7.
B
14. A
52
MATHS
IIT-JEE FLASH BACK (OBJECTIVE)
Exercise - 08 (A) 1. (– 4 , 7)
2.
2
3. – 1
(B) 1.
4.
4
(C) F
2.
T
3.
T
1. ABC
1.
C
2.
A
3.
A
8.
C
9.
A
10. A
11. A
12. C
13. B
14. D
17. A
18. C
19. A
20. D
21.
(D)
15. B 22. C
16. B 23. C
4.
A
5.
D
6.
B
7.
D
B
(E) 1. A (F) 1. 2
IIT-JEE FLASH BACK (SUBJECTIVE)
Exercise - 09
1. (s – q) 2 + q (r – q) 2 – p (s – q) . (r – p) ,
2 , 1
4.
1
8.
4 , 1 3 x2
12.
5 ,1
5.
5
(q – s)2 = (r – p) . (ps – rq) ±2,±
6.
2
a a
2 , a
a
6
9. x 2 p (p4 5 p2q + 5q2) x + p 2q2 (p2 q2) (p2 q) = 0
(x1 + x 2) x +
13. (a, b)
x1 x 2 = 0 where 14.
= 2
x1 = (b2 2 c ) (b3 3 c b) and
= 2
or
; x2 c3 (b2 4 c )
= 2 and = 2
15.
a> 1
17. 1210
Exercise - 10 1.
3
8. 1
INTEGER T YPE
2.
3
3. 0
9.
5
10. 6
4. 0
5.
1
6.
0
7.
9
53
