| JgBA Jg BANS NSAL AL CLASSES CLAS SES I p T a r g e f CLASS: XII
Q. 1 Q.2
MATHEMA MATHEMATICS TICS Daily Practice Problems
SI T J E E 2 0 0 7
d T ( ABC d T
DATE: 11-12/12/2006
TIME: 50 Min.
DPP. NO.-53
Revision Dpp on Permutation & combination Select the correct alternative. (Only one is correct) Number Num ber of natural number s between betwee n 100 and 1000 such that at least one of their digits is 7, is (A) 225 (B) 243 (C) 252 (D) non e The number of ways in whic h 100 persons may be seated at 2 round tables T, and T 2 , 5 0 persons being seated at each is : ( A ) f
m
M !
m
l !
Q. 3
There are six peri ods in each work ing day of a school. Num ber o f way s in whi ch 5 subject s can be arranged if each subject is allotted at least one period and no period remains vacant is (A)210 (B)1800 (C)360 (D)12 0
Q. 4
The number of ways in whic h 4 boys & 4 girls can stand in a circle so that each boy and each girl is one after the other i s: (A) 4 ! . 4 ! (B) 8 ! (C) 7 ! (D) 3 !. 4 !
Q.5
If letters ofthe word "PARKAR" are written down in all possible manner as they are in in a dictionary dictionary,, then the rank of the word "PARKAR" is : (A) 98 (B) 99 (C) 100 (D) 101
Q. 6
The number of different wor ds of three letters which can be formed fr om the word "PROPOSAL", "PROPOSAL ", if a different words vowel is always always in the middle a re: (A) 53 (B) 52 (C) 63 (D) 32
Q.7
Consider 8 gula r octagon and its centre. centre. If T denotes the number of triangles and and S denotes 8 vertices of are gular the number of straight lines that can ca n be formed with these 9 points then T - S has the value equal to (A) 44 (B )4 8 (C) 52 (D )5 6
Q. 8
A polygon has 170 diagonals. How many sides it will have ? (A) 12 (B) 17 (C) 20
(D) 25
Q. 9
The number of ways in whi ch a mixe d double tennis game can be arranged from from amongst 9 married couple if no husband & wife plays in the same game is ; (A) 756 (B) 1512 (C) 3024 (D) 4536
Q. 10
4 normal distinguishable distinguishable dice are rolled once. The number ofpossible outcomes in which atleast atleast one die shows up 2 i s: (A) 216 (B) 648 (C) 625 (D) 671
Q- l l
OQ Il-l X nr x . p
f
is equal to :
( B ) f ^
( Q ^
Q. 12
There are counters counte rs available in x different colours, The counters are all alike except for the colour. colour. The total number of arrangements arrang ements consisting c onsisting of y counters, assuming sufficient number of counters of each colour, if no arrangement arrange ment consists consist s of all all counters of the same colour i s: y (A) x y - x (B) x - y (C) y x - x (D)yx-y
Q. 13 13
In a plane plan e a set of 8 parallel lines, that goes in another direction, forming 8 parallel lines intersects a set of n parallel a total of 1260 parallelograms parall elograms.. The value of n i s: (A) 6 (B) 8 (C) 10 (D) 12
Q. 14
A team of 8 students goes on an excursion, in two cars, of which one can seat 5 and the other only 4. If internal arrangement inside the car does not matter then the numb er of ways in which they can travel, is (A) 91 (B) 126 (C) 182 (D) 39 20
Q. 15
In a confe rence 10 speakers are pre sen t, If S5 wants to speak before S 2 & S 2 wants to speak aft er S3 , then the number o f way s all all the 10 speakers can give their speech es with the abov e restriction if the remaining seven speakers hav e no obj ection to speak at any number is (A)
10
C3
(B) (B )
10
Pg
(C) (C )
I0
P3
( D) i i l
Q. 16 There are 8 different consonants and 6 different vowels. Number of different words o f 7 7 letters which can be formed, ifthey are to contain 4 consonants and 3 vowels ifthe three vowels are to occupy even places is (A) 8 P 4 . 6 P 3 (B) (B ) 8 P 4 . 6 C 3 (C) (C ) s P 4 . 7 P 3 (D) (D ) 6 P 3 . 7 C 3 . 8 P 4
Q.17
Number o f ways in which 5 different books can be tied up in three bundles is (A) 5 (B) 10 (C) 25 (D) 50
Q. 18 18
How many word s can be made with the letters letters of the words "GENIUS" if each word neither begins with G nor ends en ds in S is : (A) 24 (B) 240 (C) 480 (D) 504
Q. 19
Number of of numbers greater greater than 1000 1000 which can be formed using only the digits 1, 1, 2, 3 ,4 ,0 taken four at a time is (A) 332 (B) 159 (C) 123 (D) 112 Select the correct alternative. (.More than one are correct)
Q.20 Identify the correct statement(s). (A) Numb er of naughts standin g at the end of 112 1125 is 30. (B) Atelegraph has 10 arms and each a im is capable of 9 distinct positions positio ns excluding exclud ing the position of rest. 10 The number o f signals that can be transmitted is 10 - 1 . (C) In a table tennis tourn ament, every player plays with every other player. player. If the numbe r of games played play ed is i s 5050 50 50 the n the nu mb er of players playe rs in the tour to urnam nament ent is 100. (D) Numb er of numb ers greater than 4 lacs which can be formed by using only the digits 0 ,2 ,2 ,4 , 4 and 5 is 90. Q.21
n+
'-Cg + «C 4 >
n+2
C 5 - n C 5 for all ' n ' greater than :
(A) 8 Q.22
(B) 9
(C) 10
(D) 11
The number of ways in which 200 different things can be divided into groups of 100 pairs is : ( 1 0 f l (102 (102^1 ^1 (103 (103^1 ^1
I t J
(A) (A) 2 ( 1 . 3 . s..199) s..199) ( C) -,100200! / •ln n\ i
-
r r J
(20 0^
I T
_ 200! (D) ->100
00
Q.23
Q.24
2' (100)! The continued continued product, 2 . 6 . 1 0 . 1 4 (A) 2n P n (C) ( n + 1)( n + 2) (n + 3) (n + n)
to n factors is equal to : (B) (B ) 2»Cn (D )2 n • (1 - 3 - 5
2n-l)
The Numb er of way s in which five diffe rent books to be distributed among 3 persons so that each person per son gets at least o ne boo b ook, k, is i s equal equa l to the th e num ber of ways in whic wh ich h (A) 5 persons are allotted 3 different residential residential fl ats so that and each pers on is alloted at most one flat and no two person s are alloted the same flat. (B) number numb er of parallelogr ams (some (s ome of which may be overlapping) f ormed orm ed by one set of 6 parallel lines and other set of 5 parallel lines that goes in other d irection. (C) 5 different toys are to be distributed a mong 3 childr en, so that each child gets at least one toy. (D) 3 mathematics pr ofessors are assigned five different lecturers to be delivered, so that each professor prof essor gets at least one lecturer.
4
J BANSAL CLASS CLA SSES ES {Target BIT JEE 2007
CLASS: XII (ABCD) This is the test paper ofClass-XI
Q. l (a) (b)
MATHEMATI MATHEMATICS CS Daily Practice Problems
DATE: 22-23/11/2006
TIME: 75 Min.
DPR
NO.-S2
(PQRS & J) held on 19-11-2006. Take exactly 75 minutes.
Consider the quadratic polyno mial f (x) = x 2 - 4a x + 5 a 2 - 6a. Find the smallest positive positi ve integral value of 'a' 'a ' for which f (x) is positive positiv e for every real x. Find the largest distance between the roots of the equation f (x) = 0. [2.5 + 2.5]
Q.2(a) Q.2(a ) Find the greatest value of c such that system of equation s x 2 + y 2 = 25 x+y=c has a real solution. (b) The equations to a pair of opposite sides of a parallelogram are x 2 - 7x + 6 = 0 and y 2 -14y + 40 = 0 find the equations to its diagonals. Q. 3
Find the equation of the straight line with gradient 2 if it intercepts intercep ts a chord of length 4^/5 on the circle x 2 + y 2 - 6x - 1 Oy + 9 = 0.
[5]
cos^ 2x + 3 cos 2x 7 7 wherever defined is independent of x. Without allotting cos x-sin x a particular value of x, find the value of this constant. [5]
Q.4
The value ofthe ofthe expression,
Q. 5
Find the general solution of the equation si n 3 x(l + cot x) + cos 3 x( l + tan x) = cos 2x.
Q. 6
[2.5+2.5]
[5]
If the third and fourth terms of an arithmetic sequence are increased by 3 and 8 respectively, then the first four terms form a geometric sequence. Find (i) the sum of the first four fo ur terms term s ofA.P. (ii) (ii) second term of the th e G.P. [2.5+2.5]
Q.7(a) Let x = — or or x = - 15 satisfies the equation, log 8 (& x 2 + wx + / ) = 2 . If I f k, w and/are relatively prime pos itive iti ve inte i nteger gerss then the n find the th e value val ue of k+w (b)
Q. 8
Q. 9
+f.
The quadratic equation x 2 + mx + n - 0 has roots which are twic e those of x 2 + px + m = 0 and n an d p* 0. Find the value of ~ . [2.5+2.5] m, n and x y Lme Lm e — + + — = = 1 intersects the x and y axes at M and N respectively. I f the coord inates of the point P 6 8 lying inside the triangle OMN (where 'O' is origin) are (a, b) such th at the areas of the triangle P OM, PON and PMN are equal. Find (a) the coordinates of the point P and (b) the radius of the circle escribed opposite to the angle N. [2.5+2.5] Starting Starting at the origin, a beam oflig of lig ht hits a mirror (in the fo mi of a line) at the point A(4 ,8) and is reflected at the point B(8 ,12 ). Comput e the slope of the mirror. [5] [5]
lo g x +3 (x 2 - x) < 1.
Q. 10
Find the solution set of inequality,
[5]
Q . ll
If the first 3 conse cutiv e term s of a geometric al progress ion are the roots of the equation 2x 3 - 1 9 x 2 + 57x - 5 4 = 0 find the sum to infinite number of terms of G.P. [5]
Q. 12
Find the equation to the straight lines joining 1 lie o- "m to the points of intersection of the straight line 2L + L = i and the circle 5(x 2 +y 2 + bx+ ay) = 9ab. Also find the linear relation between a and b so that a b these straight lines may be at right angle. [3+2]
Q. 13
Q.14
L et /( x) = | x - 2 | + | x - 4 | — | 2 x - 6 j . Find the sum of the largest and smallest values of f (x) if x e [2, 8], [5] If
x+1 x+ 2 x+ 3
x+ 2 x+ 3 x+4
x+a x + b = 0 then all lines represented by ax + by + c = 0 pass through a fixed point. x+c
Find the coordinates of that fixed point. Q. 15
If Sj, S7 , S3 ,. .. S ,.... are the sums of infinite geometric series whose first terms are 1, 2, 3, .. . n,. .. and 1 1 1 1 whose comm on ratios are —, - , —,.... , ,. .. respectively, then find the value of O *T* 1 2* J nr
Q. 16
[5]
A 5 B 20 In any triangle if tan — = 7 and tan — = — then find the valu e of tan C. 2 6 2 3/
2(1-1
r=l
-
. [5]
[5]
Q.17
The radii r p r 2 , r 3 of escribed circles of a triangle ABC are in harmonic progression. If its area is 24 sq. cm and its perimete r is 24 cm, find the lengths of its sides. [5]
Q. 18
Find the equation of a circle passing through the origin if the line pair, xy - 3x + 2y - 6 = 0 is orthogonal to it. If this circle is orthog onal to the circle x 2 + y 2 - kx + 2ky - 8 = 0 then find the value of k. [5]
Q. 19
Find the locus of the centres of the circles which bisects the circumference of the circles x 2 + y 2 - 4 and x 2 + y2 — 2x + 6y + 1 = 0. [5]
Q.20
Find the equation of the circle whose radius is 3 and which touche s the circle x2 + y 2 - 4x — 6y - 12=0 internally at the point ( - 1 , - 1 ) . [5]
Q.21
Find the equation of the line such that its distance fiom the lines 3x - 2y - 6 = 0 and 6x - 4 y - 3 = 0 is equal. [5]
Q. 22
Find the range of the variable x satisfying the quadratic equation, x 2 + (2 cos (j))x - sin2c|> = 0 V
Q.23
( n y^ (n sin x(3 + sin 2 x) If tan ~ + ~ ! = t a r r ~ + ~ then prove that s i n y = 5 . 2.) \ 4 J,) l + ^s in ^x
[5]
[5]
1
i BANSAL CLASSES
MATHEMATICS
Target IIT JEE 2007
CLASS : XII (ABCD)
Daily Practice Problems
DATE: 10-11/11/2006
TIME: 60 Min.
DPP. NO.-51
Select the correct alternative. (Only one is correct) There is NEGATIVE marking and 1 mark will be deducted for each wrong answer.
Q. l
1 1 1 1 1 Find the sum of the infinite series 7T + 7T: + T r + 7 7 + 7r + 9 18 30 45 63
(A) } Q. 2
(B) i
(C) |
Number of degrees in the smallest positive angle x such that 8 sin x cos 5 x - 8 sin 5 x cos x = 1, is (A) 5° (B) 7.5° (C )1 0°
(D) f
(D) 15°
Q. 3
There exist positive integers A, B and C with no common factors greater than 1, such that Alog 20 0 5 + B log 200 2 = C. The sumA + B + C equals (A) 5 ~ (B) 6 (C) 7 (D) 8
Q. 4
A triangle with sides 5,12 and 13 has both inscribed and circumscribed circles. The distance between the centres of these circles is (A) 2
(B)|
(C) V65
(D)^f
Q. 5
The graph of a certain cubic polynomial is as shown. Ifthe polynomial can be written in the form / ( x ) = x 3 + ax 2 + bx + c, then (A) c = 0 (B) c < 0 (C) c > 0 (D) c = - 1
Q. 6
The sides of a triangle are 6 and 8 and the angle 0 between these sides varies such that 0° < 0 < 90°. The length of 3rd side x is (A) 2 < x < 14 (B) 0 < x < 10 (C) 2 < x < 10 (D)0
Q.7
The sequence a t , a^ a 3 ,... . satisfies a { = 19. first n - 1 terms. Then a2 is equal to (A) 179 (B) 99
Q.8
= 99, and for all n > 3, a n is the arithmetic mean of the (C) 79
(D )59
If b is the arithmetic mean between a and x; b is the geometric mean bet ween 'a' and y; 'b' is the harmonic mean between a and z, (a, b, x,y,z> 0) then the value of xyz is (A) a 3
Q.9
y
(B,b3
( C ) ' t a 2b-a
2a-b
Given A(0 ,0) , ABC D is a rhombus of side 5 units where the slope of AB is 2 and the slope of AD is 112. The sum of abscissa and ordinate of the point C is ( A ) 4 V5
(B) 5V5
(C )6 V5
(D) 8V5
Q. 10
A circle of finite radius with points (- 2, - 2) , (1,4) and (k, 2006) can exist for (A) no value of k (B) exactly one value of k (C) exactly two values of k (D) infinite values of k
Q. 11
If a A ABC is formed by 3 staright lines u = 2x + y - 3 = 0; v = x - y = 0 and w = x - 2 = 0 then for k = - 1 the line u + kv = 0 passes through its (A) incentre (B) centroid (C) orthocentre (D) circumcentre
Q. 12
x 2 +1 0x -3 6 a b — = — — ++ ——~ ++ If a, b and c are number s for which the equation - — x(x - 3 ) then a + b + c equals (A) 2
Q. 13
Q. 15
(C )1 0
1 1 1 If a, b, c are in G.P. then ~ , —, b - a 2 b b - c (A) A. P.
Q. 14
(B) 3
(B) G.P.
)
(C) H.P.
B
(
(x - 3)
are in
The sum of the first 14 term s of the sequen ce
A
x -3
is an identity,
(D )8
(D) none
How many terms are there in the G.P. 5,20 , 80, 20480. (A) 6 (B )5 (C) 7
(
x
c
)
(D )8
1
1 1 j= + h t= + 1-X 1 + Vx 1 —v x 7
is
^ f >
14 (C)
(l + V x ) ( l - x ) ( l - V x )
(D) non e
10 Q. 16
If x, y > 0, log yx + log xy = — and xy = 144, then arithmetic mean of x and y is (A) 24
(B) 36
(C )1 2V 2
(D) 13V 3
Q. 17
A circle of radius R is circumscribed about a right triangle ABC. If r is the radius of incircle inscribed in triangle then the area of the triangle is (A)r( 2r + R) (B)r (r + 2R) (C) R(r + 2R) (D) R(2 r + R)
Q. 18
The simplest form of 1 +
£
— is 1 — 1-a
(A) a for a * 1 (C) - a for a * 0 and a * 1
(B) a for a * 0 and a * 1 (D)lfora*l
Select the correct alternatives. (More than one are correct)
Q. 19
If the quadratic equation ax 2 + bx + c = 0 (a > 0) has sec 2 9 and cosec 2 0 as its roots then which of the following must hold good? (A) b + c = 0 (B) b 2 - 4ac > 0 (C) c > 4a (D) 4a + b > 0
Q.20
Which of the following equations can have sec 29 and cosec 29 as its roots (9 e R)? (A) x 2 - 3x + 3 = 0 (B) x 2 - 6x + 6 = 0 (C) x 2 - 9x + 9 = 0 (D) x 2 - 2x + 2 = 0
Q.21
The equation | x - 2 | 10x2_1 = | x - 2 | 3x has (A) 3 integral solutions (C) 1 prime solution
Q. 22
(B) 4 real solutions
(D) no irrational solution
Which of the following statements hold good? (A) If Mi s the max imum and m is the minimum value of y = 3 sin 2x + 3 sin x • cos x + 7 cos2x then the mean of M and m is 5, 71 .71 (B) The value of cosec— sec — is a rational which is not integral. 18
^
18
(C) If x lies in the third quadrant, then the expression 1/4 s i n 4 x + sin 2 2x
+
4 cos 2
4
2
is
independent ofx. (D) There are exactly 2 val ues of 9 in [0, 2tt] whic h satisfy 4 cos29 - 2 -J l cos 9 - 1 = 0 .
MATCH THE COLUMN
Q. l
INSTRUCTIONS: Column-I and column-II contains four entries each. Entries of column-I are to be matched with some entries of column-El. One or more than one entries of column-I may have the matching with the same entries of column-H and one entry of column-I may have one or more than one matching with entries of column-II. Column-I Column-II (A) Area of the triangle formed by the straight lines (P) 1 x + 2y - 5 = 0, 2x + y - 7 = 0 and x - y + 1 = 0 in square units is equal to (Q) 3/4
(B) (C)
(D)
Abscissa of the orthocentre of the triangle whose vertices are the point s (- 2, -1) ; (6, - 1) and (2, 5) Variable line 3x(A. + 1) + 4y(A. - 1) - 3 ( 1 - 1) = 0 for different values of A, are concurrent at the point (a, b). The sum (a + b) is The equation ax 2 + 3xy - 2y 2 - 5x + 5y + c = 0 represents two straight lines perpendicular to each other, then | a + c | equals
(R) (S)
2 3/2
Q.2
(A)
(B)
Column-I
Column-II
In a triangle ABC, AB = 2 ^ 3 , BC = 2-J6 , AC > 6,
(P)
60°
and area of the triangle ABC is 3 V<5 . Z B equals
(Q)
90o
In a triangle AB C is b = S , c = 1 a n d A= 30°
(R)
120o
(S)
75°
then angle B equals
Q.3
(C)
In a A ABC if (a + b + c)(b + c - a) = 3bc then Z A equals
(D)
Area of a triangle ABC is 6 sq. units. If the radii of its excircles are 2, 3 and 6 then largest angle of the triangle is
(A) (B)'
Column-I The sequence a, b, 10, c, d is an arithmetic progressio n. The value o f a + b + c + d
Column-II (P) 10
The sides of right triangle form a three term geometric sequence. The shortest side has length 2. The length
(Q)
20
of the hypotenuse is of the for m a + Vb where a e N
(R)
26
(S)
40
and 7 b is a surd, then a2 + b 2 equals (C)
The sum of first three consecutive numbers of an infinite G .P. is 70, if the two extremes be multipled each by 4, and the mean by 5, the products are in A.P. The first term of the GP. is
(D)
The diagonals of a parallelogram have a measure of 4 and 6 metres. They cut o ff formin g an angle of 60°. If the perimeter of the parallelogram is 2[-Ja + V b) where a, b e N then (a + b) equals
J g BANSAL CLASSES
MATHEMATICS
I B Target I1T JEE 2007 CLASS: XII (ABCD)
Q. 1
Pa/7/ Practice Problems
DATE: 04-07/10/2006
TIME: 40 Min.for
each
DPP. NO.-49, 50
-49 8 clay targets have been arranged in vertical column, 3 being in the first column, 2 in the second, and 3 in the third. In ho w many way s can they be shot (one at a time) if no target below it has been shot. [4]
Q.2
Evaluate: /x (s in 2 (sinx) + cos 2 (cosx))dx o
[4]
Q.3
Evaluate: jx( sin (c os 2 x)cos(sin 2 x))dx
[4]
J - . x dx 111x YJ-fAQY * V . xYQ sin + cos x / 0
Q.4
Q.5
VI Prove that 2 ^
1 3 n + 1
[6]
71
_ J _ 1= 3n + 2 j
^
[9]
- S O Q. 1
If cos A, cos B and cos C are the roots of the cubic x3 + ax 2 + bx + c = 0 whe re A, B, C are the angles of a triangle then find the value of a 2 - 2b - 2c. [4]
Q.2
Find all fu nc ti on s, /: R- >R satisfying ( x / ( x ) - 2 F ( x ) ) ( F ( x ) - X 2 ) = 0 V x e R where f (x) = F'(x). [4]
0 33 Q'
Q.4
¥
J
2
j f ^ f * l3-xJ
00 J For a > 0, b > 0 verify that J—^ dx reduc es to zero by a substitut ion x = 1 /t. Using this or „o ax" +bx + a °f fax otherwise evaluate: i 2 0
tan - 1 x Q.5
HI
1 v
x
d
aAx
[7]
"\3
y
d x
[81
A
JABANSAL CLASS ES l ^ P T a r g e t HT J EE CLASS: XII (ABCD)
MATHEMATICS Daily Practice Problems
2007 DATE: 29-30/9/2006
DPP. NO.-47
This is the test paper-1 of Class-XIII (XYZ) held on 24-09-2006. Take exactly 60 minutes.
Q. 1
P A R X - A Select the correct alternative. (Only one is correct) [24 x 3 = 72] There is NEGATIVE marking. 1 mark will be deducted for each wrong answer. The area of the region of the plane bounded above by the graph of x 2 + y 2 + 6x + 8 = 0 and below by the graph of y = | x + 3 is (A) jc/4 (B) ti 2/4 (C) 7c/2 (D) it
Q.2 7'
Consider straight line ax + by = c where a , b , c e R + and a, b, c are distinct. This line meets the coordinate axes at P and Q respectively. If area of AOPQ, 'O' being origin does not depend upon a, b and c, then (A) a; b. c are in G.P. (B) a, c, b are in G.P. (C) a, b. c are in A.P. (D) a, c, b are in A.P.
Q. y
If x and y are real numbers and x 2 + y2 = 1, then the maximum value of (x + y) 2 is (A) 3
Q.4
(B) 2
(C) 3/2
(D) J 5
dx
The value of the definite integral j n (a > 0) is q (1 + x )(1 + x ) (A) ti/4
(B) nil
(C)
(D) some function of a.
tc
a b e cos — cos—cos Let a, b, c are non zero constant number then Lim —-— — equals r-»co
... a 2 + b 2 - c 2 (A) 2bc Q.6 ^
(B)
c2 + a 2 - b 2 2bc
.
C
^ x b 2 + c 2 - a 2 (C) 2bc
(B) - 15
(C) - f sinx
Q.8
.
. . . _ _ _ J (D) independent of a, ba nd c
A curve y =/(x) such that /"(x) = 4x at each point (x, y) on it and crosses the x-axis at (-2, 0) at an angle of 450. The value of / (1), is (A) - 5
Q.7/ v
b
sin—sin r r
(D) -
y
cosx
tanx cotx = The minimum value of the functi on/(x) = 1 + / + 7 + ~T as 2 9 Vl- cos x vl -s i n x vsec x - 1 Vcosec x - 1 x varies over all numbers in the largest possible domain of / ( x ) is (A) 4 (B) - 2 (C) 0 (D) 2 A non zero polynomial with real coefficients has the property that f (x) = / ' (x) • f"(x). The leading coefficient of / (x) is (A) 1/6 ' (B) 1/9 (C) 1/12 (D) 1/18 l_
r tan -1 (nx) ^ Q-9
Let Cn = J s i n - V )
X
2 2
then Lim n -C f l
equai s
"n+l
(A) 1 / Q. 10
(B) 0
(C) - 1
(D) 1/2
Let Zj, z2, z3 be complex numbers suchthat z x + z2 + z3 = 0 and | z x \ - \
2 2 2 | = | z 31 = 1 then z, + z 2 + z 3 ,
is (A) greater than zero
(D) equal to 1
(B) equal to 3
(C) equal to zero
Q.ll
Number of rectangles with sides parallel to the coordinate axes whose vertices are all of the form (a, b) with a and b integers such that 0 < a, b < n, is (n e N)
(A)
n 2 (n + l) 2
2
(B)
(n -l ) n
2
(C)
(n + 1)2
(D) n2
1 Q.12
Number of roots of the function/(x) ~ (A) 0
^.13
+
^ 3 - 3x + sin x is
(B) 1
(C )2
(D) more than 2
If p (x) = ax 2 + bx + c leaves a remainder of 4 when divided by x, a remainder of 3 when divided by x + 1, and a remainder of 1 when divided by x - 1 then p(2) is (A) 3 (B) 6 (C) - 3 (D) - 6 Let/(x) be a function that has a continuous derivative on [a, b],/(a) and/(b) have opposite signs, and / ' (x) * 0 for all numbers x between a and b, (a < x < b). Number of solutions does the equation /(x) = 0 have (a < x < b). (A) 1
V^l5
Q. 16
vXl7
\J2[- 18
(B) 0
A circle with center A and radius 7 is tangent to the sides of an angle of 60°. A larger circle with center B is tangent to the sides of the angle and to the first circle. The radius of the larger circle is (A) 30V3 (B) 21 (C) 20V3
(D) 30
The value of the scalar (p x q)-(r x s) can be expressed in the determinant form as qs p-s
p-r p-s (B) q-s q-r
a/x jf Lim x • In 0 x->00 1
1 1/x 0
y p 1/x
p-r
(C) q-r
q-s p-s
p-r p-s q-r q-s
5, where a, p, y are finite real numbers then
(B) a =2, p=2, y = 5
(C) a e R, p= l, yeR
(D) a e R, p = 1, y = 5
If / (x. y) = sin _1( | x [ + | y |), then the area of the domain of / is (A) 2
Q.21
^
-3tt/2 j Sin(3x + 7t)dx
Let set A consists of 5 elements and set B consists of 3 elements. Number of functions that can be defined from A to B which are neither injective nor surjective, is (A) 99 (B) 93 (C) 123 (D) none
(A) a = 2, p= l, yeR Q.20
(D) cannot be determined
Which of the following definite integral has a positive value? 2it/3 0 0 Jsin(3x + 7i)dx (g) Jsin(3x + 7t)dx ^ q Jsin(3x + Jt)dx 2tc/3 -3it/2
q-r (A) p-r
Q.19
(C) 2
(B) 2 / 2
(C) 4
(D) 1
A, B and C are distinct positive integers, less than or equal to 10. The arithmetic mean of A and B is 9. The geometric mean of A and C is 5 / 2 • The harmonic mean of B and C is „ 9 (A) 9— v ' 19
(B)
(C) v_/ 2 ~ 19
9
(D) 2-^r 17
Q.22
If x is real and 4y2 + 4xy + x + 6 = 0, then the complete set of values of x for which y is real, is (A) x < 2 or x > 3 ( B ) x < - 2 or x > 3 ( C ) - 3 < x < 2 ( D ) x < - 3 or x > 2
Q.23
I alternatively toss a fair coin and throw a fair die until I, either toss a head or throw a 2. If I toss the coin first, the probability that I throw a 2 before I toss a head, is (A) 1/7 " (B) 7/12 (C) 5/12 (D) 5/7
Q.24
Let A, B. C, D be (not necessarily square) real matrices such that AT = BCD; BT = CDA; CT = DAB and DT = ABC for the matrix S = ABCD, consider the two statements. I
3
S = S
(A) II is true but not I
II
(B) I is true but not II
s 2 = s 4
(C) both I and II are true
(D) both I and II are false.
JsBANSAL CLASS ES
MATHEMATICS Daily Practice Problems
V S Target Target NT JEE 20 07 CLASS: XII (ABCD)
DATE: 02-03/10/2006
DPP. NO.-48
This is the test paper-2 of Class-XIII (XYZ) held on 24-09-2006. Take exactly 60 minutes. Select the correct alternative. (More than one is/are correct) There is NEGATIVE marking. 1 mark will will be deducted f or each wrong answer.
Q. 1
[ 3 x 6 = 18]
The fun ct io n/ (x ) is defined for x > 0 and and has its inverse g (x) which is differentia differentiable. ble. I f / (x) satisf satisfies ies g(x) J f (t) dt = X2 and g (0) = 0 then (A)/(x) is an odd linear polynomial (C) /(2 ) = 1
Q. 2
(B )/ (x ) is some some quadratic polyno polynomial mial (D)g(2) = 4
Consider a triangle ABC in xy plane with D, E and F as the middle points of the sides BC, CA and AB respecti respectively vely.. If the coordinates of the points D, E and F are are (3/2, 3/2); (7/2,0 (7/ 2,0 ) and (0, (0, - 1 / 2 ) then which of the following are correct? (A) circumcentre circumc entre of the triangle tri angle ABC does not lie inside the triangle. tr iangle. (B) orthocentre, orth ocentre, centroid, circumcentre and incentre incentre of triangle DEF are collinear collinear but of triangle ABC are non collinear. (C) Equation o f a line passes through thro ugh the orthocentre orthocen tre of triangle ABC and perpendicular to its plane is is r = 2(i - j) + A.k 5V2 (D) distance between centroid and and orthocentre or thocentre of the triangle ABC is —— . X
Q. 3
X
If a continuous function/ function / ( x ) satisfies the rela relati tion on,, j t / ( x - t ) dt = j / ( t ) dt dt + s j n X-+ cos cos x - x - 1 „ for 0 0 . all all real real numbers x, then which of the following does not hold good? good ? it (A)/(0) = 1 ( B) / ' (0) = 0 (0) = 2 (D) (D) J / ( x ) d x = e * (C)f" (0) 0 MATCH THE COLUMN [ 3 x 8 = 24] There is is NEGATI VE marking. marking. 0.5 mark will will be deducted deducted for each wron g match within a question. Column I
Q.l
,.. (A)
Column II
In x r dt Lim — IS J Y—VtYl X V J3 In / n t t X-*co
. T
(P)
0
(Q)
:
(R)
1
(S)
non existent existent
e
(B) (C) (C)
z2 ' „vx /~T7 +1 - „x +l e e Lim
is
Lim Lim (-1)" s i n f W n 2 + 0.5n + l l sin sin J tan
(D)
The value of the integral j 0
tan" 1
f
-1
/
VX
is where n e N
4n „
\
+ ly
l + 2x -2 x
9A A
dx is
Q.2
andB :
Consider the matrices A =
a 0
b let P be any orthogo nal matrix and Q = PAP T 1 and let
and R = P T Q K P also S = PBP T and T = P T S K P Column I (A) If we vary K fro m 1 to n then the first row first column column elements at Rwill fo rm (B) If we vary K fr om 1 to n then the 2 nd row 2 nd column elements elements at Rwill for m (C) If we vary K fro m 1 to n then th e first row first column elements of T will form (D) If we vary K fr om 3 to n then the first row 2 nd column elements of T will represent the sum of Q.3
Column II
(P) G.P. G.P. with common comm on ratio a (Q) A. P. with common difference 2 (R) GP. with common ratio b (S) A. P. with common commo n difference - 2.
Column I
(A)
(B) (B) (C) (D)
Column II
Given tw o vec tor s a and b such that | a | = | b | = |a + b| = 1 The angle between the vectors vect ors 2a + b and a is In a scalene scalene triangle triangle ABC, if a c o s A = b c o s B then Z C equals In a triangle ABC, BC = 1 and AC = 2. The maximum possible possib le value which the Z A can have is In a A ABC Z B = 75° and BC = 2AD wher e AD is the altitude from fro m A, A, then Z C equals
(P)
30°
(Q)
45°
(R)
60°
(S)
90°
[5x10 = 50]
SUBJECTIVE: tc/2
Q.l
96V • 2 1 Sup pos eV= J x sin x — dx, find the value of 71 2
Q. 2
One of the roots of the equation 2000x 6 + 100x5 + 1 Ox3 + x - 2 = 0 is of the form for m
m +
r
" , where m
is non zero in teger and n and r are are relatively relatively prime natural numbers. Find the th e value of m + n + r. Q.3
circle C is tangent to the th e x and y axis in the first quadrant at the poi nts P and Q respectively. respectively. BC and A circle AD are parallel parallel tangents t o the circle with slope - 1 . If the points A and B are on the y-axis while while C and and D are on the x-axis and the area of the figureABCD is 900 V2 sq. units then the n find the radius of the circle. circle.
Q. 4
Le t/ (x ) = ax2 - 4 ax + b (a > 0) be defined defined in 1 < x < 5. Supp ose the av erage of the maximum value and and the minimum minimum value of the function 14, and the difference between t he maximum value and minimum funct ion is 14, 2 2 value is 18. 18. Find the value of a + b .
Q.5
If the Lim x-*0 x- *0
1 x
1 + ax
Vl + x
1 + bx
1 2 3 exists and has the value equal to I, then find the value of — - y + — .
JGBANSAL CLASS>ES
MATHEMATICS Daily Practice Problems
Target IIT JEE 2007 DATE: 27-28/9/2006
CLASS: XII (ABCD)
DPR NO.-46
This is the test paper of Class-XI (J-Batch) held on 24-09-2006. Take exactly 75 minutes. Q. l
If tan a . tan P are the roots of x 2 - px + q = 0 and cot a, co t p are the roots of x 2 - rx + s = 0 then find the value of rs [4] rs in terms of p and q.
Q. 2
Let P(x) = ax 2 + bx + 8 is is a quadratic polynomial. If the mini mum value va lue of P(x) is 6 when x = 2 , find the values of a a and b. b . 14] ( \_\ .n-i
Q.3
L e tP tP = f j 1 0 2 " n=l
Q. 4
Prove the identity
Q.5
Find the general solution set of the equation loglan x(2 + 4 cos hi) - 2.
Q.6
Find the value of
then find log 00 1 (P).
sec 8A - 1
tan 8 A
sec 4A - 1
tan 2 A
[4]
[4]
sin a + sin 3a + sin 5a + — cos a + cos 3a + cos 5a +
[4]
+ sinl7a n — - when a = — . + c o sl 7 a 24
[4]
Q.7(a) Sum the following series to infinity 1 1-4-7
1 +
1 +
4-7-10 7-10-13
+
(b) Sum the following series upton-t erms. 1 -2 -3 -4 + 2 - 3 - 4 - 5 + 3 - 4 - 5 - 6 +
[3 + 3]
Q.8
The equation cos 2 x - sin x + a = 0 has h as roots when x e (0, rc/2 rc/2)) find fi nd 'a'.
Q. 9
A, B and C are distinct positive integers, less than or equal to 10. The arithmeti c mea n ofA of A and B is 9. The geometric mean ofA and C is 5 / 2 • Fi F i n d the harmonic mean of B and C.
Q. 10
Express cos 5x in terms of cos x and hence find general solution o f th e equation cos 5x = 16 cos 5 x.
Q. 11 11
If x is real and 4y 2 + 4xy + x + 6 = 0, then f ind the comple te set of valu es of x fo r whi ch y is real.
Q. 12
Find the sum of all the integral solutions ofthe inequality 2 1 o g 3 x - 4 1 o g x 27<5.
—, show sh ow that tha t 2
(i-f)HI) l + tan — [ i 2 j
+
f 1 tan—1 1 — tan—1 I 2 J ( y^ l + tan — t»|] I 2>
sin a + sin P + sin y - 1 cos a + cos cos p +c os y [7]
j y i 4(a ) In any an y A ABC prove that C C c 2 = (a - b) 2 cos co s 2 — + + (a + b) 2 sin si n 2 — . (b) In any A ABC prove that a 3 cos( B - C) + b 3 cos(C - A) + c 3 cos( A - B) = 3 abc.
[4 + 4]
d
l BANSAL CLASSES
MATHEMATICS
5Targe* liT JEE 2007
Daily Practice Problems
CLASS: XII (ABCD)
DATE: 20-21/9/2006
DPP. NO.-44
This is the test paper of Class- XI (PQRS) held on 17-09-2006. Take exactly 75 minutes. n n
Q. 1
r O i f r ^ S r
s
Evaluate £ 8 n -. 2 • 5 where 5 rs = r=l s=l Will the sum hold if n - > oo?
1
if r = s [4]
Q.2
x x Find the general solution of the equation, 2 + tan x • cot — + cot x • tan — = 0.
Q.3
Given that 3 sin x + 4 cos x = 5 where x e (0, n/2 ). Find the value of 2 sin x + cos x + 4 tan x.
|4J
14]
log 0 3: (1 x - 1 ) ' <— ==• < 0. 2 V2x~- x +8
Q.4
Find the integral solution of the inequality
Q.5
In A ABC, suppos e AB = 5 cm, AC = 7 cm, Z AB C :
(a)
Find the length of the side BC.
(b)
Find the area of A ABC.
Q. 6 Q. 7
The sides of a triangle are n- \,n and n + 1 and the area is n-Jn • Determine n. With usual notions, prove that in a triangle ABC, r + r { + r 2 - r 3 = 4R cos C.
{4]
K
[4] [4]
[5]
Q.8
Find the general solution of the equation, sin %x + cos nx = 0. Also find the sum of all solutions in [0,1 00], [5]
Q. 9
Find all negative values of 'a' which makes the quadratic inequality sin 2x + a cos x + a 2 > 1 + cos x true for every x e R
[5]
Q.10
Solveforx, s i°g 2 * 2 ^ l o g J x V s ) = ^ l o g ^ x 2 _ 5 i o g 2 * 1 ,
[5]
Q. 11
„2 ™ 2 2 In a triangle ABC if a + b = 101 c then find the value of &
Q.12
Solve the equation for x, 5 2
+52
+!0g5(smx)
= 1 52
cot C cot A + cot B
+l08l5(C0Sx)
.
1[5] 1
[5]
00
Z~n . n=l
6
[5]
Q. 14
Suppose that P(x) is a quadratic polynomial such that P(0) = cos 3 40°, P( 1) = (cos 40°)(sm240°) and P(2) t 0 . Find the value of P(3). [8]
Q . 15
If /, m, n are 3 number s in G.P. prov e that the first term of an A.P. who se 7th, mth , n th terms are in H.P. is to the common difference as (m + 1) to 1. [8]
BAN SAL CLA SSES
MATHEMATICS
y g Target I IT JEE 2007 CLASS : XII (ABCD)
Q. 1
Daily Practice Problems
DATE: 22-23/9/2006
DPP. NO.-45
Let a, b, c, d, e, f e R such that ad + be + cf = ^ ( a 2 + b 2 + c 2 ) ( d 2 + e 2 + f 2 ) use vectors or otherwise to pro ve that,
Q.2
TIME: 55 to 60 Min.
a+b+c Va2+b2+c2
d+e+f V d 2 + e 2 + f 2 '
Let the equation x 3 - 4x 2 + 5x - 1.9 = 0 has real roots r, s, t. Find the area of the triangle with sides r, s, and t. 50
Q. 3
2
Suppose x J + ax 2 + bx + c satisfies f (- 2) = - 1 0 and takes the extreme value — where x = — . Find the value of a, b and c.
Q- 4
Le tI
f i-y ^/nxx+xy-l
H v d X
and
r / n x x + xy _ I 1 — y
dy
x
d dy wher e ~ = x y . Show that I • J = (x + d)(y + c) where c, d e R. Hen ce show that — (I J) = I + J — y dx dx
Q.5
Let a;, i = 1, 2, 3, 4, be real numbers such that aj + % + % + a 4 = 0. Show that for arbitrary real numbers bi5 i = 1,2, 3 the equ ation a, + bjx + 3a 2 x 2 + b 2 x 3 + Sa^x4 + b 3 x 5 + 7a 4 x 6 = 0 has at least one real root which lies on the interval - 1 < x < 1.
V3
Q.6
Q. 7
xx 2— - l l Evaluate: — t = x J I x +x +3x" + X r
dx + 1
Let x, y e R in the interval (0, 1) and x + y = 1. Find the minimum value of the expr ession x x + yy
r | (1 - sin x) (2 - sin x) ^ y (1 + sin x)( 2 + sin x)
^
i l l SBANSAL
C L A S SE S
M A T H E M A T I C S
l U l a r g e t NT JEE 2 0 0 7
Daily Practice Problems
CLASS: XII (ABCD)
DATE: 08-12/9/2006
DPP. NO.-42, 43
DATE : 08-09/09/2006 O P P - 4 2 This is the test paper of Class-XIII (XYZ) held on 27-08-2006. Take exactly 60 minutes.
TIME : 60 Min.
S^' S-y V Select the correct alternative, (Only one is correct) There is NEGATIVE marking. 1 mark will be deducted for each wrong answer. Q. I
[16 x 3 = 48]
sin 2 (x 3 + x2 + x - 3 ) rn ~~ ~ ~~ has the value equal M to x->i 1 — cos(x — 4x + 3)
Li
(A) 18
(B) 9/2
(C) 9
(D) none
dt Q.2 / Let/(x)= r . . If g'(x) is the inverse of / ( x ) then g'(0) has the value equal to 4 2 * 3-v t +3 t +13 (A) 1/11 (B) 11 (C) Vl3 (D) l/ Vn Q.3
The function/(x) has the property that for each real number x in its domain, 1/x is also in its domain and / ( x) + / ( l / x ) = x. The largest set of real numbers that can be in the domain of /( x), is (A){x|x*0)
Q.4 j 6/
Let w = (A) 2, -
Z2
( B ) { x | x > 0) 37 + 6
z +1
n/4
,
(C) { x | x * - l a n d x * 0 a n d x * 1)
(D) {-1, 1}
and z = 1 + i. then | w | and amp w respectively are (B)
, - 71/4
(C) 2, 3TC/4
(D) ^ , 3n/4
k cos a 1 - cos a - tan 2 (a/2) Q.5 A If . j/ " ~= where k, w and pF have no common factor other than 1, then the ./! sin (a/2) w + pcosa value of k 2 + w 2 + p2 is equal to (A) 3 (B)4 (C)5 (D)64 Q.6
In a birthday .party, each man shook hands with eveiyone except his spouse, and no handshakes took place between women. If 13 married couples attended, how many handshakes were there among these 26 people? (A) 185 (B)234 (C)312 (D)325
Q.7
If x and y are real numbers such that x 2 + y2 = 8, the maximum possible value of x - y, is (A) 2 (B) (C) V2/2 (D) 4
Q.8/
Let w(x) and v(x) are differentiable functions such that
u(x)
= 7. If
U^x)
~ P a n d
p+q
' u(x) v(x)
= q, then
M to p - q has the value equal
(A) 1 Q.9
(B)0
(D)-7
The coefficient of x9 when (x + (2/Vx j)30 is expanded and simplified is (A) 30 C| 4 • 29
Q. 10
(C)7
(B) 30 C]6 • 214
(C ) 30 C 9 -2 21
(D) 10C9
Let C be the circle described by (x - a)2 + y2 = r 2 where 0 < r < a. Let m be the slope of the line through the origin that is tangent to C at a point in the first quadrant. Then r
Q. 11
Q.l 2
V a 2 - r 2 r (A) m = r ^ 7 (B) m = — (C) m = (D) m = Va - r r a What can one say about the local extrema of the func tion/(x) = x + (1/x)? (A) The local maximum o f f (x) is greater than the local minimum o f/(x). (B) The local minimum off (x) greater than the local maximum off (x). (C) The functi on/( x) does not have any local extrema. (D)/ (x) / r _ 2 ^ + arctan(5) equals tan arc tan I 3 v (A) - / 3 (B)-l
(C)l
(D)V3
a r
has one asymptote.
/ y/Q- ip
^gf. 14
A line passes through (2, 2) arid cuts a triangle of area 9 square units from the first quadrant. The sum of all possible values for the slope of such a line, is (A) - 2.5 (B) - 2 (C) - 1.5 (D) - 1 Which of the following statement is/are true concerning the general cubic /(x) = ax3 + bx2 + cx + d (a * 0 & a, b, c, d e R) I The cubic always has at least one real root II The cubic always has exactly one point of inflection (A) Only I (B) Only II (C) Both I and II are true
(D) Neither 1 nor II is true
Q. 15
If S = 12 + 3 2 + 5 2 + (A) S + 2550
Q. 16
Through the focus of the parabola y 2 = 2px (p > 0) a line is drawn which intersects the curve at A(x,, y,) y\y 2 and B(x,, v.). The ratio x x equals l 2 (A) 2 (B) - 1 (C) - 4 (D) some function of p
'! 7
18 Q.l 9
Q.20
Select the correct alternative. (Only one is correct) There is NEGATIVE marking. 1 mark will be deducted for each wrong answer, i • n-3 n ^ i I f D+1 n = 6 N) ^n(x-9)»+n-3 -3 3 ^ ^ ^ ° f X iS (A) [2,5) ' ' (B) (1,5) (C) (- 1, 5) (D)(-co ,oo)
[ 9 x 4 = 36 j
The area of the region(s) enclosed by the curves v = x 2 and y = ^ | x | is (A) 1/3 (B) 2/3 (C) 1/6 (D) 1 Suppose that the domain of the fun ct io n/ (x ) is set D and the range is the set R, where D and R are the subsets of real numbers. Consider the fun cti ons :/( 2x), /(x + 2), 2/ (x ), / (x / 2) , / ( x ) / 2 - 2 . If m is the number of functions listed above that must have the same domain a s/ an d n is the number of functions that must have the same range as f (x), then the ordered pair (m, n) is (A) (1, 5) (B) (2, 3) (C )( 3, 2) (D) (3, 3) r x 2 + 2mx - 1 for x < 0 / : R -» R is defined as / ( x ) = - mx - 3 for x > 0 If / ( x ) is one-one then m must lies in the interval (A) (— oo, 0)
Q.2 1
+ (99)2 then the value of the sum 22 + 4 2 + 6 2 + + (100)2 is (B)2S (C) 4S (D) S + 5050
Let
(B) (— oo, 0]
(C )( 0, oo )
2
A = { x | x + (M - l )x - 2(m + 1 ) = 0 , X G R } ;
(D) [0, co) B = { x | (m - 1)X2 + mx + 1 = 0, X e R }
. Number of values of m such that A u B has exactly 3 distinct elements, is (A) 4 (B) 5 (C) 6 (D) 7 ^Q.22
If the function/(x) = 4x2 - 4x - tarr a has the minimum value equal to - 4 then the most general values of 'a' are given by (A) 2n7t + ti/3 (B) 2n n - rc/3 (C) im ± n/3 (D) 2nn/3 where n e I Direction for Q.23 to Q.25.
^/ Q. 23
sinx-xcosx x 0 anc Consider the function defined on [0, i] -> R, / ( x ) = 5 ® f (0) = 0 * The function/(x) (A) has a removable discontinuity at x = 0 (B) has a non removable finite discontinuity at x=0 (C) has a non removable infinite discontinuity at x = 0 (D) is continuous at x = 0 1
^jQ.24
J/( x )dx equals (A) 1 - sin (1)
(B) sin (1) - 1
(C) sin (1)
( D )- si n( l )
t
^ . 2 5
1 L i m j / ( x ) d x equals t->o t1z7 0 (A) 1/3 (B) 1/6
(C) 1/12
(D) 1/24
i>B>S>-43
DATE : 11-12/09/2006
TIME : 60 Min.
[ 7 x 4 = 28]
Select the correct alternative. (More than one are correct)
There is NO NEGATIVE marking. Marks will be awarded only if all the correct alternatives are selected. xe x Q.26
Let / (x) =
L
x<0 2
then the correct statement is
J
x+x - x x>0 ( A ) / is continuous and di ffe ren ti ate for all x. ( C) / ' is continuous and di ffe ren ti ate for all x. Q.27
( B ) / is continuous but not di ffe ren ti at e at x = 0. ( D ) / ' is continuous but not di ffe ren tia te at x = 0.
x2-l Sup po se/ is defined from R —> [—1, 1] as / ( x ) = —z where R is the set of real number. Then the x" + 1 statement which does not hold is ( A ) / is many one onto ( B ) / increases for x > 0 and decrease for x < 0 (C) minimum value is not attained even though f is bounded (D) the area included by the curve y = f (x) and the line y = 1 is n sq. units. 2
Q.28
The value of the definite integral
(A) n ] l n ( Jdx J V3 — cosx J
0-29
(B)
r , (3 + cosx V J x ' n i 3 _ c o s x J > is v 0
]dx J ^3- cos x J
'
(D)
V* 0 V3 + cos x;
r x 3 ( l - x ) s i n ( l / x 2 J if 0 R is defined as / ( x ) = j 0 if x = 0 ( A )/ is continuous but not derivable in [0, 1 ] (C)/ is bounded in [0, 1 ]
Q.30
( CV) z e r o
(B )/ is dif feren tiat e in [0, 1 ] (D ) / ' is bounded in [0, 1]
Let 2 sin x + 3 cos v = 3 and 3 sin y + 2 cos x = 4 then (A) x + y = (4n + 1)TE/2, n e l (B) x + y = (2n + l)rc/2, n E I (C) x and y can be the two non right angles of a 3-4-5 triangle with x > y. (D) x and v can be the two non right angles of a 3-4-5 triangle with y > x.
Q.31
Q.32
The equation cosec x + sec x = 2V2 has (A) no solution in (0, n/ 4)
(B) a solution in [tc/4 , n/2)
(C)no solution in (n/2, 3n/4)
(D) a solution in [37r/4, tc)
For the quadratic polynomial / ( x ) = 4x2 - 8kx + k, the statements which hold good are (A) there is only one integral k for whi ch /( x) is non negative V x e R (B) for k < 0 the number zero lies between the zeros of the polynomial. (C )/ (x ) = 0 has two distinct solutions in (0, 1) for k e (1/4, 4/7) (D) Minimum value of y V k e R is k(l + 12k) I^ A. l^ TI -S ^ [ 3 x 8 = 24]
MATCH THE COLUMN
Q. i
Column-I contain four functions and column-II contain their properties. Match every entry of column-1 with one or more entries of column-II. Column -I Column -II ] 1 (A) / ( x ) = sin" (§in x) + cos"" (cos x) (P) range is [0,71] (B) (Q) is increasing V x e (0, 1) g (x) = sin-'j-x | + 2 ta ir 'j x | (C)
( 2x 1 h (x) = 2sirr>! — j j , x 6 [0, 1]
(R)
period is 2%
(D)
k (x) = co t( co r' x)
(S)
is decreasing V x e (0, 1)
Q.2
Column-I
Column-II
(A)
(P) Centre of the parallelopipeci whose 3 coterminous edges OA, OB and OC have position vectors a, b and c respectively where O is the origin, is
(B)
OABC is a tetrahedron where O is the origin. Positions vectors of its angular points A, B and C are a, b and c respectively. Segments joining each vertex with the centroid of the opposite face are concurrent at a point P whose p. v.'s are
(Q)
(C)
Let ABC be a triangle the positio n vectors of its angular points are a, b and c respectively. If\a-b\ = \b-c\=\c-a\then the p.v.of the orthocentre of the triangle is
(R)
(D)
Let a, b,c be 3 mutually perpendicular vectors of the same magnitude. If an unbiown vector x satisfies the equation a x[fx -b)xaj+b x[(x-c)xbj+c x({x -a)xc) = G. Then x is given by
(S)
Column-I
Column-II
Q.3 (A)
If
1 a a~
1 b
(a - b)(b - c)(c - a)(a + b + c) then the solution
1 (x-a)2 of the equation (x-b )(x- c) L X ™
1 (x-b)2 (x-c )(x-a )
1 (x-c)2 =0, is (x-a )(x -b)
(B)
The value of the limit,
(C)
Lim x->0
(D)
Let a, b, c are distinct reals satisfying a 3 + b 3 + c3 = 3abc. If the quadratic equation (a + b - c)x2 + (b + c - a)x + (c + a - b) = 0 has equal roots then a root of the quadratic equation is
f
, X a X +b +c X
(P)
a + b + c 3
a + b + c 2
a + b + c
(Q) a+b+c
(^/(x + a)( x + b)(x + c) - x), iis
(R)
equals
(S)
SUBJECTIVE: Q.l
+c
[ 4 X 6 = 24]
Let / ( x ) = (x + l)(x + 2)(x + 3)(x + 4) + 5 where x e [- 6, 6], If the range of the function is [a, b] where a, b e N then find the value of (a + b). tu/4
Q.2
Let I
j (TCX - 4x2) /n(l + tan x)dx. If the value of 1 o
Q.3
7i "7n 2
where k e N, find k.
k
Suppose/and g are two functions such that f g : R -> R, 2 /(x) ^/n ^l + V l ^ ]
then find the value of x egW
and (
g(x ) = /n! x + \ / l T x 2
fiW
+ g'(x) at x = 1. /
Q.4
If the value of limit
-1 l + 7( k- l )k (k + lXk + 2) L,m Z cos k(k + l) k=2
120ti is equal to —-—, find the value of k. K
JHBANSAL CLASSIES
MATHEMATICS Daily Practice Problems
^ T a r g e t 1ST J EE 2 0 0 7 DATE: 04-07/9/2006
CLASS: XII (ABCD)
DPR N0.-40,
41
DATE: 04-05/09/2006 TIME: 50 Min. 3 Q. 1 Let/(x ) = 1 - x - x . Find all real values of x sati sfying the inequality, 1 - / ( x ) - / 3 ( x ) > / ( 1 - 5x)
g2x _ gX
j
Q.2
Integrate: j — dx 3 x x ( e sin x + cos x) (e cos x - sin x)
Q.3 (i) (ii)
The circle C : x 2 + y 2 + kx + (1 + k)y - (k + 1) = 0 pass es thro ugh th e same two p oint s for every real number k. Find the coordinates of these two points. the minimum value of the radius of a circle C.
Q. 4
i Commen t upo n the nature of roots of the quadratic equation x + 2 x = k + J| t + k | dt depending on the 2
0
value of k e R.
3n
Q.5
Given Lim n->oo
1/n
C„ 2n f\ \ n y
a = — where a and b are relatively prime, fi nd the value of (a + b). b DFP-41
DATE: 06-07/09/2006
Q. 1
Q.2
Q.3
TIME: 50 Min.
Let a, b, c be three sides of a triangle. Suppo se a and b are the roots of the equ atio n x 2 - (c + 4)x + 4(c + 2) = 0 and the largest angle of the triangle is 9 degre es. Find 0. 71 Find the value of the definite integral j| V2 si nx + 2 co sx jd x. o 1 Let tan a • tan (3 = 7 ^ 5 . Find the value of (1003 - 1002 cos 2a) (10 03 - 1002 cos 2(3) 1+V5
0 * 4
2
r
/ Q.5
X2 + l — /. j( .l n + X — X +1 V
x
— n dx X J
Two vecto rs Sj and e 2 with | e ( | = 2 and \ e 2 | = 1 and angle betwee n
and e 2 is 60°. The angle
betwe en 2t e, + 7 e 2 and ej +1 e 2 belon gs to the interval (90°, 180°). Find the range of t. Q.6
Afimction fix) continuous on Ra nd periodic with period 2% satisfies f (x) + sin x -/(x + n) = sin 2 x. Find /(x ) and evaluate f / ( x ) dx .
4
| BAN SAL CLASSES
MATHEMATICS^
glTarget SIT JEE 2007
CLASS: XII (ABCD)
DATE; 30-31/8/2006
Daily Practice Problems TIME: 60 Min.
DPP. NO.-39
This is the test paper of Class ~XI (J-Batch) held on 27-08-2006. Take exactly 60 minutes. Q. 1
Find the set of values of' a' for which the quadratic polynomial (a + 4)x 2 - 2ax + 2a - 6 < 0 V x e R . x+1 x+5
[3] I 31
Q. 2
Solve the inequality by using method of interval, ——- ^
Q.3 Q.4
Find the minimum vertical distance between the graphs of y = 2 + sin x and y = cos x. d (3 ^ co s x - c o s J x Solve: dx 4 wh en x = 18°.
Q.5
•
[3]
If p, q are the roots o f the quadratic equation x 2 + 2bx + c = 0, pro ve that 2 l o g [ j y - p + y f y - q } = lo g2 + lo g( y + b +
Q. 6
[3]
j,
[4]
x 2 +14x + 9 Find the maximu m and minim um value of y = —, VxeR. x +2x + 3
[4]
Q.7
Suppo se that a an d b are positive real numbers such that log 2 7 a + log 9 b = 7/ 2 an d log 27 b + l og 9 a=2 /3 . Find the value of the ab. [4]
Q. 8
Given si n 2 y=sin x • sin z where x, y, z are in an A.P. Find all poss ible values o f the commo n difference of the A.R and evaluate the sum of all the comm on differences which lie in the interval (0,315) . [4] tan 86 Prov e that = (1 + sec29 ) (1 + sec40) (1 + sec86) . [4]
Q.9 Q.10
371 571 •jl In Find the exact value of tan 2 —: + tan 2 — + tan 2 —~ + tan 2 — . 16 16 16 16 89
[4]
i
Q. l l
Evaluate Y ^ l + ( t a nn ° ) 2
151
Q. 12
Find the value of k fo r whic h one root of the equati on of x 2 - (k + 1 )x + k 2 + k- 8= 0 exceed 2 and other is smaller than 2. [5]
Q. 13
Let a n be the 0 th term of an arithmetic progression. Let S n be the sum of th e first n terms of the arithmetic pro gressi on with aj = 1 and a 3 = 3a g . Find the largest possible valu e of S n . [5]
( C^ C A B Q. 14(a) If A+B +C = n & sin A + — = k sin —, then find the value of tan — -tan — in terms of k. V
Z. J
\ ( X +x (b) Solve the inequality, log.'0.5 lo g 6 - <0 . x+4
[2 + 4]
Q. 15
Given the prod uct p of sines of the angles of a triangle & prod uct q of their cosines, find the cubic equation, whose coefficien ts are functions o f p & q & whose roots are the tangents of the angles of the triangle.
Q. 16
If each pair of the equations x 2 +pjX + q j = 0 x2 +p 2 x + q2 = 0 x 2 + p 3 x - i - q3 = 0 has exactly one root in commo n then sh ow that 2 (p, + p 2 + p 3 ) = 4(pjp 2 + p 2 p3 + p3pj - q, - q 2 - q 3).
[6]
4
| BANSAL CLASSES
MATHEMATICS Daily Practice Problems
j Target III JEE 20 0 7
CLASS: XII (ABCD)
Q. 1
DATE: 23-24/8/2006
TIME: 60 Min.
DPP. NO.-38
r 2 1/2 Find the value of a and b whe re a < b, for whi ch the integral j (24 - 2x - x ) dx } i a s the largest a
value.
sin x - cos x
Q.2
Solve the differential eqaution: y' +
Q.3
Integrate: J. -dx (x cos x - sin x)( x sin x + cos x)
Q.4
5
Q-
Ve
-cosx
y = —x
e
-cosx
In a A ABC, given sin A: sin B : sin C = 4 : 5 : 6 and cos A : cos B : cos C = x : y: z. Find the ordered pair that (x, y) that satisfies this extended proportion.
V sin
1
V x
FCN
dx
X
X
Q.6
Find the general solution of the equ ati on, 2 + tan x • cot — + cot x • tan — = 0
Q.7
Let a, (3 be the distinct positive roots of the equation tan x = 2x then evalua te J(sinax-sin[3x) dx , o
independent of a and {3.
J | BANSAL CLASSES
MATHEMATICS Daily Practice Problems
Ig gT ar g ef HT JEE 20 07 CLASS: XII (ABCD)
"
DATE: 18-19/8/2006
TIME: 75 Min.
DPR NO.^37
This is the test paper of Cl ass -XI (PQRS) held on 13-07-2006. Take exactly 75 minutes. Q. 1
The sum of the first five terms of a geometric series is 189, the sum of the first six terms is 381, and the sum of the first seven term s is 765. What is the com mon ratio in this series. [4]
Q.2
Form a quadratic equation with rational coefficients if one of its root is co t 2 l 8°.
Q.3
Let a and (3 be the roots ofth e quadratic equation ( x - 2 ) ( x - 3)+(x- 3)( x +l )+ (x + l)( x-2) =0. Find 1 the value of
1
( a + 1)(p +1 }
+
(a
_ 2 ) ( p _ 2 ) +
[4]
1 (a
_ m
_ 3 ) •
W
Q.4
If a sin2 x +M ie s in the interval [- 2, 8] forev eryx <= R then find the value of ( a - b ) .
Q.5
For x > 0, what is the smallest possible value of the expression log(x 3 - 4x 2 + x + 26) - log(x + 2)? [4]
Q. 6
The coeffic ients of the equation ax 2 + bx + c = 0 where a * 0, satis fy the inequality (a + b + c)(4a - 2b + c) < 0. Prove that this equation ha s 2 distinct real solutions.
[4]
[4]
Q.7
In an arithm etic progr essio n, the third term is 15 and the eleve nth term is 55. An infinite geometric progression can be f ormed beginning with the eighth term of this A.P. and f oll owed by the fourth and second term. Find the sum o f this geometric progression upto n terms. Also compute Srjo if it exists. [5]
Q.8
Find the solution set of this equatio n log )sin X|(x2 - 8x + 23) > l o g ( s i n x j ( 8 ) in x e [0 ,2 n) .
[5]
Q.9
Find the positive integers p, q, r, s satisfying tan — = ( j p - Jq) (yfr - s)-
[5]
Q. 10
Find the sum to n terms of the series. 1 -
2 +
—
3 +
-
4 +
—
5 +
—
+
2 4 8 16 32 Also find the sum if it exist if n - > oo.
[5]
Q. 11
If sin x, sin 2 2x and cos x • sin 4x form an increasing geometric seque nce, find the numerial value of cos 2x. Also find the common ratio of geometric sequence. [5]
Q. 12
Find all possible para mete rs 'a' for which , f (x) = (a2 + a - 2)x 2 - (a + 5)x - 2 is non positive for every x e [0, 1 ].
Q.13 (a) (b) (c) Q. 14
The 1st, 2nd and 3 rd terms of an arithmetic series are a, ban d a 2 where 'a' is negative. The 1st , 2 nd and 3rd terms of a geometric series are a, a2 and b find the val ue of a and b sum of infinite geometric series if it exists. If no then find the sum to n terms of the G P sum of th e 40 term of th e arithmetic series. [5] j) The nth term, a n of a sequence of numbers is given by the formul a a n = a n _ } + 2n for n > 2 and aj = 1. Find an equation expressing a n as a polynomial in n. Also find the sum to n terms of the sequence. 00
Q. 15
[5 j
2x x Le t/ (x ) denote the sum of the infinite trigonometric series, / ( x ) = ^ sin — sin — . 3 n=J 3 Find/ ( x ) (independent of n) also evaluate the sum of th e solutions o ft he equation f (x) = 0 lying in the interval (0, 629 ). [8]
MATHEMATICS
.kBANSAL CLASSES
Daily Practice Problems
I B T a r g et I I T J E£ 2 0 0 7
DPP. NO.-35, 36
CLASS: XII (ABCD) I > I* I " - 3
5
DATE: 16-17/08/2006
TIME: 45 Min.
x 2
d3
Q.l
If y = Jx V^ nt dt, find l
at x = e .
Q.2
Find the equation of the norma l to the curve y = (l +x ) y + sin -1 (sin 2 x) at x = 0. x
Q.3
f f(t)dt
Find the real num ber 'a' such that 6 + J
-j— = 2 v x •
a
Q.4
7 The tangent to y = ax + bx + - at (1 ,2 ) is parallel to the normal at the poin t (- 2, 2) on the curve 2
y = x 2 + 6x + 10. Find th e value of a and b. Q.5
Let f be a real valued funct ion satisfying f(x) + f( x+ 4) = f(x + 2) + f(x + 6) then prove that the function x+8
g(x) =
| f(t) dt is a constant functi on. X
Q. 6
A tangent drawn to the curve C l = y = x 2 + 4 x + 8 at its point P touches the curve C 2 = y = x 2 + 8x + 4 at its point Q. Find the coord inates of the point P and Q, on the c urves C j and C 2 . 3 « S
DATE: 16-17/08/2006 TIME: 45 Min. 2 3 4 Q. 1 Given real numbers a and r, consider the following 20 numb ers : ar, ar , ar , ar , , ar 20 . If the sum of the 20 numbers is 2006 and the sum of the reciprocal of the 20 numb er is 1003, find the product of the 20 numbers.
Q.2
Let f(x) and g(x) are differentiable functions satisfyingthe conditions; (i)f (0) = 2 ; g ( 0 ) = l (ii) f'(x ) = g(x) & Find the functions f(x ) and g(x). 3
Q.3
Let f( x) =
( b 3 - b 2 +b-l) — Y 2 (b + 3b + 2 j
L 2x-3
(iii)g '(x) = f(x).
_ ,0
Find all possi ble real valu es of b such that f( x) has the smalle st value at x = 1. Q. 4
There is a function f defined and contin uous for all real x, which satisfies an equation of the form Xf
V 2
X1 6
X18
J f (t) dt = j t f( t) dt + _ _ + _ + c , where C is a constant. Find an explicit formula for f(x) and o x 8 9 also the value of the constant. r
Q.5
Given Jf(tx) dt = nf ( x ) then find f(x) where x > 0. o
Q. 6
Tangent at a point P j [other than (0, 0) ] on the curve y = x 3 meets the curve again at P 2 . The tangent at P 2 meets the curve at P 3 & so on. Show that the abscissae of P, , P 2 , P 3, P n , form a GP. Also find the ratio
^( P^P,) area (P 2 P 3 P 4 )
ft
4
| BANSAL CLASSES |Target 8iT JEE 2007
CLASS 7 XII (ABCD)
Q.l
MATHEMATICS Daily Practice Problems
DATE: 11-12/8/2006
TIME: 60 Min.
DPP. NO.-34
Let F (x) = j V4 +1 2 dt and G (x) = JV4 +1 2 d t then compu te the value of (FG)' (0) where dash -1
X
denotes the derivative.
Q.2
10 identical balls are to be distributed in 5 different boxes kept in a ro w and labelled A, B, C, D and E. Find the num ber of ways in which the balls can be distributed i n the boxes if no two adjacen t boxes remain empty.
Q. 3
If f (x) = 4x 2 + ax + (a - 3) is negative for atleast one negative x, find all possibl e values of a.
Q.4 (a) (b) (c)
Let /(x ) = sin6x + cos 6 x + k(si n 4 x + cos 4 x) for some real numb er k. De termi ne all real numbers k for wh ic h/ (x ) is constant for all values of x. all real numbers k for which there exists a real number 'c' such that f (c) = 0. I f k = - 0 . 7 , determine all solutions to the equatio n/(x) = 0.
7T
Q.5
,
Letx 0 = 2cos— andx n = ^2 + x ^ , n = 1, 2,3 , n-*>o find Lim 2< n+1) -V2^T n ~.
Q.6
Q.7
Q.8
f Le t/ (x ) = — — - the n fin d the valu e of the sum y j 20C>6 / + ^
V j ^ d * 8 + sin x
x
3
^2006
^ +f f [2006J
.
Va For a > 0, fm dt he minimum value of the integral J( a 3 + 4 x - a 5 x 2 ) e a x dx. 0
(2005^ 2006
I BANSAL CLASSES
MATHEMATICS Daily Practice Problems
Target liT JEE 2007
DPP. NO.-33
DATE: 31/7/2006 to 5/08/2006
CLASS: XII (ABCD)
O P P
1
O
F
X
H E
W
E
E
K
This is the test paper of Class-XIII (XYZ) held on 30-07-2006. Take exactly 2 Hours. N O T E : Leave Star ( *) marked problems.
" P A R T ' - A . Select the correct alternative. (Only one is correct) Q.l
[26 x 3 = 78]
Number of zeros of the cubic f (x) = x3 + 2x + k V k e R, is (A) 0 (B) 1 (C) 2
(D)3
/x
Q.2
t
The value of Lim dr, is x->°° dx yL(r + l ) ( r - l ) (A) 0
Q.3
(B) 1
Q.5
-2 x 4
There are two numbers x making the value of the determinant these two numbers, is (A )- 4
Q.4
(D) non existent
(C) 1/2
(B)5
5 - 1 equal to 86. The sum of 2x
(C )- 3
(D)9
A function / (x) takes a domain D onto a range R if for each y e R , there is some x e D for which / (x) = y. Number of function that can be defined from the domain D = {1,2,3} onto the range R = {4, 5} is (A) 5 (B) 6 (C)7 (D) 8
f/(x),„
Suppose/,/' and/" are continuous on [0, e] and that/' (e) =/(e) =/(l) = 1 and j e
the value of 5 Q.6
1
1
f / " ( x ) / n x d x equals I 3 1 (B) j -
1
1
(D) 1 -
(C)
2
X
1
= Z, then 1
1
A circle with centre C (1, 1) passes through the origin and intersect the x-axis at A and y-axis at B. The area of the part of the circle that lies in the first quadrant is (A) n + 2 (B) 2n - 1 (C) 2n - 2 (D) n + 1 The planes 2x - 3y + z = 4 and x + 2y - 5z = 11 intersect in a line L. Then a vector parallel to L, is (A) 13i + l l j + 7 k
(B) 1 3 i + l l j - 7 k
(C) 1 3 i - l l j + 7 k
(D) i + 2 j - 5 k
&Q.8 A fair dice is thrown 3 times. The probability that the product of the three outcomes is a prime number, is (A) 1/24 (B) 1/36 (C) 1/32 (D) 1/8 Q.9
Period of the function, / ( x ) = [x] + [2x] + [3xj +
+ [nx] -
where n e N and [ J denotes the greatest integer function, is (A) 1 (B) n (C) 1/n Q. 10
Q. 11
n(n +1) n
J.
(D) non periodic
2i - i 1 Let Z be a complex number given by, Z = 3 i - 1 the statement which does not hold good, is 1 (A) Z is purely real 10 1 (B) Z is purely imaginary (C) Z is not imaginary (D) Z is complex with sum of its real and imaginary part equals to 10 Let /(x , y) = xy2 if x and y satisfy x2 + y2 = 9 then the minimum value off (x, y) is (A) 0
(B) - 3-^3
(Q-6V3
( D) - 3 V6
Q. 12
Vl + 3 x - l - x Ei m — — ^ has the value equal to x^o (1 + x ) -l -1 0 1 x 3 (A)-
Q. 13
(B)-
5050
(C)
5050
(D)
5051
4950
Number of positive solution which satisfy the equation lo g 2 x • lo g 4 x • lo g 6 x = log 7 x • lo g 4 x + log 2 x • lo g 6 x + log 4 x • loggX?
(A) 0
(B) 1
(C) 2
(D) infinite
Q.14
Number of real solution of equation 16 sin"'x tan _1x cosec"'x = n 3 is/are (A) 0 (B) 1 (C) 2 (D) infinite
Q. 15
Length of the perpendicular from the centre of the ellipse 27x 2 + 9y2 = 243 on a tangent drawn to it which makes equal intercepts on the coordinates axes is (A) 3/2
(B) 3/ V2 f,
Q.l 6
1
Le t/ (x ) = cos" (A) 0
2n
1 — x
1 + x 2
+ tan -
(C) 3V2
(D) 6
2x 1-x2
(B) ti/4
where x e (- 1, 0) then /sim plif ies to (C) n/2
(D) 7t
Q. 17A person throws four standard six sided distinguishable dice. Number of ways in which he can throw if the product of the four number shown on the upper faces is 144, is (A) 24 (B) 36 (C) 42 (D )4 8
Q.18
a Let A = p x
b q y
(A) det(B) = - 2 Q. 19
Q.20
4x c r and suppose that det.(A) = 2 then the det.(B) equals, where B = 4y 4z z (B) det(B) = - 8
(C) det(B) = - 16
The digit at the unit place of the number (2003)2003 is (A) 1 (B) 3 (C) 7
2a 2b 2c
-p -q -r
(D) det(B) = 8 (D)9
AB AF Let ABCDEFGHIJKL be a regular dodecagon, then the value of — + — is Ar AB (A) 4
(B)2- s/3
(C) 2V2
(D )2
&Q. 21 Urn A contains 9 red balls and 11 white balls. Urn B contains 12 red balls and 3 white balls. One is to roll a single fair die. If the result is a one or a two, then one is to randomly select a ball from urn A. Otherwise one is to randomly select a ball form urn B. The probability of obtaining a red bail, is (A) 41/60 (B) 19/60 (C) 21/35 (D)35/ 60 Q.22
L e t / be a real valued function of real and positive argument such that / ( x ) + 3x / (l /x ) = 2(x + 1) for all real x > 0. The value of /( 10 09 9) is (A) 550
(B) 505
(C)5 050
(D) 10010 a
\2
/
„
P + a + 1
Q.23
If a and P be the roots of the equation x2 + 3x + 1 = 0 then the value of
Q.24
(A) 15 (B) 18 (C) 21 (D) none The equation (x - l)(x - 2)(x - 3) = 24 has the real root equal to 'a' and the complex roots b and c. Then the value of b c / a , is (A) 1/5
Q.25
(B) - 1/5
(C) 6/5
1 + P
is equal to
(D) - 6/5
If m and n are positive integers satisfying 1 + cos 20 + cos 40 + cos 60 + cOs 80 + cos 100 = (A) 9
(B) 10
(C) 11
cos m0 • sin n0 — then m + n is equal to sin0 (D) 12
Q.26
A circle of radius 320 units is tangent to the inside of a circle of radius 1000. The smaller circle is tangent to a diameter of the larger circle at the point P. Least distance of the point P from the circumference of the laiger circle is (A)300 (B)360 (C)400 (D) 420
[8 x4 = 32]
Select the correct alternative. (More than one are correct)
Q.27
In which of the following cases limit exists at the indicated points.
(A) /(x)
[x+|x|]
at x = 0
(B )/ (x ) =
x e 1/x
at x = 0 1/x x l+e where [x] denotes the greatest integer functions. -1 tan 1 x | 1/5 (C )/ (x ) = (x - 3) Sgn(x - 3) at x = 3, (D )/ (x ) = at x = 0. x where Sgn stands for Signum function. &Q.28 Let A and B are two independent events. If P(A) = 0.3 and P(B) = 0.6, then (A) P(A and B) = 0.18 (B) P(A) is equal to P(A/B) (C) P(A or B) = 0 (D) P(A or B) = 0.72 Q.29
Let T be the triangle with vertices (0, 0), (0, c 2) and (c, c2) and let R be the region between y = cx and y = x2 where c > 0 then (A) Area (R)=-
c3 (B) Area of R= — 3
Area (T) Area(T) _ 3 (C) Lim+ — =3 (D) Lim c-»o Area (R) c-»o+ Area(R) 2 ( x+3
Q.30
Q.31
In Consider the graph of the function f (x) = e U+i . Then which of the following is correct. (B) / (x) has no zeroes. (A) range of the function is (1, oo) (D) domain of f is ( - oo, - 3) u (- 1, oo) (C) graph lies completely above the x-axis.
1
Let /,(x ) = x, / 2 (x) = 1 - x; / 3 (x) = - ,/ 4 (x) = X
Suppose that (A) m = 5
1 I
X
; / 5 (x ) =
x x-1
; /6(x) =
x-1
/ 6 ( / m (x)) =/ 4 (x) and / n ( / 4 ( x ) ) = / 3 (x) then (B) n = 5
(C) m = 6
(D) n = 6
Q.32
The graph of the parabolas y = - (x - 2)2 - 1 and y = (x - 2)2 - 1 are shown. Use these graphs to decide which of the statements below are true. (A) Both function have the same domain. (B) Both functions have the same range. (C) Both graphs have the same vertex. (D) Both graphs have the same y-intercepts.
Q.33
Consider the function / ( x ) =
f a x + l"\ vbx + 2y
where a2 + b2 * 0 then Lim / ( x )
(A) exists for all values of a and b
X-»CO
(B) is zero for a < b
(D) is e~ (5/a) or e~ (l/b) if a = b (C) is non existent for a > b Q.34 Which of the following fiinction(s) would represent a non singular mapping. (A) / : R -» R f (x) = | x | Sgn x (B) g : R -> R g(x) = v 3/5 where Sgn denotes Signum function 3x 2 - 7 x + 6 4 2 : (C) h : R R h (x) = x + 3x + 1 (D) k : R R k (x) x -x2 - 2 MATCH THE COLUMN ^^^E^TT-S [ 4x4 = 16] INSTR UCTIONS: Column-I and column-II contains four entries each. Entries of column-I are to be matched with some entries of column-II. One or more than one entries of column -I may have the matching with the same entries of colurnn-II and one entry of column-I may have one or more than one matching with entries of column-II. Q.l (A)
Column I Constant fu nc tion /( x) = c, c e R
Column II (P) Bound
(B)
The function g (x) = P — (x > 0), is
(Q)
periodic
(C)
The function h (x) = arc tan x is The function k (x) = arc cot x is
(R)
Monotonic neither odd nor even
(D)
Ji t
(S)
Column I
Q.2
1
Column II 0
(A)
co r (tan(-37 ))
(P)
143°
(B)
cos"1 (cos(-233°)) A 1 T sin -cos v9,
(Q)
127°
(C)
(D)
cos
(R)
- arc cos
(S)
4 2 3 Column II
Column I
Q.3
3
x-1
2
(P) 4 x-3 (B) The quadratic equations 2006 x2 + 2007 x + 1 = 0 and x 2 + 2007x + 2006 = 0 have a root in common. Then the product of the uncommon roots is (Q) 3 3 (C) Suppose sin 9 - cos 9 = 1 then the value of sin 9 - cos 9 is (9 e R) (R) (A) Number of integral values of x satisfying the inequality
(D)
Q.4
The value of th e limit,
L l
sin2x-2tanx ™ — ~ ; 3;—is /n(i + x )
(S)
A quadratic polynomial / ( x ) = x 2 + ax + b is formed with one of its zeros being
1 - 2 - 1 0
4 + 3^3
where a and b 2 + V3 are integers. Also g (x) = x 4 + 2x 3 - 10x2 + 4x - 10 is a biqua drati c polynomial such that 8
4 + 3y 3
=
+
d where c and d are also integers.
2 + V3 Column II
Column I
(A) (B) (C) (D)
(P) (Q) (R) (S)
a is equal to b is equal to c is equal to d is equal to
SUBJECTIVE:
4 2 -1 -11 13 x 8 = 24]
Q.l
Let y = sin"'(sin 8) - tan _1(tan 10) + cos _i (cos 12) - sec"'(sec 9) + co r '(cot 6) - cosec "'(cosec 7). If y simplifies to an + b then find (a - b).
Q.2
Suppose a cubic polynomial / (x) = x3 + px2 + qx + 72 is divisible by both x 2 + ax + b and x2 + bx + a (where a, b, p, q are constants and a ^ b). Find the sum of the squares of the roots o ft he cubic polynomial.
Q.3
The set of real values of 'x ' satisfying the equality
~3~ —
V
r 4 4-
—
X
= 5 (where [ ] denotes the greatest integer
( b function) belongs to the interval a , - where a, b, c e N and ~ is in its lowest form. Find the value of c I c. a + b + c + abc.
4
| BANSAL CLASSES
MATHEMATICS
| Target IIT JEE 2007
CLASS: XII (ABCD)
Daily Practice Problems
DATE: 26-27//07/2006
TIME: 45 Min.
DPP. NO.-32
This is the test paper of Class -XI (J-Batch) held on 23-07-2007. Take exactly 45 minutes. Q. 1
If (sin x + cos x) 2 + k sin x cos x = 1 holds V x e R then find the value of k.
Q.2
If the expression r
371
r>.371 +x + sin
+ sin (327t + x) - 18 cos(19rt - x) + v2 , V 2y is expressed in the fo rm of a sin x + b cos x find the value of a + b. cos
Q.3
X
c o s ( 5 6 t c +
[3]
x) - 9 sin(x + 17 tc ) [3]
3 statements are given below each of which is either True or False. State whether True or Fals e with appropriate reasoning. Marks will be allotted only if appropriate reasoning is given. I (log3 169)(log13 243) = 10 cos( cos 7t) = cos (cos 0°) II III
cos x +
1
3 =T cosx 2
„4
3
S3]
1
1
Q.4
Prov e the identity cos t = ~ + - cos 2t + r cos 4t. o 2 o
Q. 5
Suppose that for some angles x and y the equations
[3]
• i 3a 0 sin^x + cos^y = — and
Q. 6
a2 cos x + sin y = — J 2 2
2
hold simultaneously. Determine the possible values of a.
[3]
Find the sum of all the solutions of the equation (log 27 x 3) 2 = log 27 x 6 .
[3]
7i % 10y-10~y If - — < x < — and y = log 10 (tan x + sec x). Then the expression E = — simplifies to one £ ** JL the six trigonometric functions,findthe trigonometric function. 13] Q .8
If log 2 (log 2 (log 2 x) )= 2 then find the number of digits in x. You may use l og ?0 2 = 0,3010. [3]
Q. 9
Assuming that x and y are both + ve satisfying the equation log ( x +y ) =l o g x +l og y find y in terms of x. Base of the logarithm is 10 everywhere. [3]
Q.10
If x = 7.5° then find the value of
Q. 11
Find the solutions of the equation, log ^
cosx ~~ cos 3x : . sin 3x - sin x sm x (1
+ cos x) = 2 in the interval x e [0 ,2 n] .
[3]
[4]
Q. 12
Given that log a2 (a 2 +1) = 16 find the value of log a32 (a + - )
Q, 13
If cos e = - find t he values of
[4]
a
(i)
[4]
(ii)tam
cos 36
[5]
Q. 14
If log 12 27 = a find the value of log 6 16 in term of a.
Q . 15
sin x - c o s x + 1 1 + sinx Prove the identity, — r = =t an —+— 4 2 , wherever it is defined. Starting with left sin x + c o s x - 1 cosx hand side only.
[5]
Q. 16
Find the exact value of cos 24° - cos 12° + cos 48° - cos 84°.
[5]
Q. 17
S olve the system of equations 5 (log xy + log y x) =2 6 and xy = 64.
[6]
r=4
Q.18
Prov e that
£ sin r=l V
(2r - l)7c'
8
r=4
-2 r=l
cos
(2r-l)7t
-\
4
8
Also find their exact numerical value.
0, 19
2
Solve for x: log (4 - x ) + log (4 •- x) . log f x + - 1 - 2 log V 2J
2
r
i a 1 x + — = 0. 2,
4
| BANSAL CLASSES
MATHEMATICS
STarget iiT JEE 2007
CLASS : XII (ABCD)
The value of Lim
X-»00
(A)
1
Daily Practice Problems
DATE: 05-06/06/2006
TIME: 50 Min.
DPR
NO.-28
/ n x - / n Vx +1 +x
(B)/n
(C) does not exist
(D) 0
(C) 3/4
(D)l
vw
7t/4
Evaluate J(tanx-sec 4 x ) d x . (A) 1/4
(B) 1/2
The product of two positiv e numbers is 12. The smallest possible value of the sum of their squares is (A) 25 Q4
(B) 24
(C) 18 V2
(D) 18
Given that log (2) = 0.3010 numb er of digits in the numbe r 200 02000 is (A) 6601 (B) 6602 (C) 6603 (D) 660 4 , , 1 1 1 Given that a, b and c are the roots of the equation x" - 2x 2 - 1 1 x + 12 = 0, then the value of — + — + ~ (A)
(B)
n 12
(C)
13 12
(D)
7
If Jta n x dx = 2, then b is equal to (A) arc cos(2e) t. Q/7
(B) arc sec(2)
The sum of all values of x so that 16 ( " 2+3x (A) 0
(B)3
(C) arc sec 2 (e)
(D)none
= 8 ( x 2 + 3 x + 2 ) , is (C)-3
(D)-5
A certain fu nc ti on /( x) satisfies f (x) + 2 / ( 6 - x) = x for all real numbers x. The value of / ( l ) , is (A) 3 (B )2 (C)l (D) not possible to determine
{
Q.9
Number of ways in which the letters A, B, C and D be arranged in a sequence so that A is not in position 3, B is not in posi tion 1, C is not in position 2 and D is not in posit ion 4, is (A) 8 (B) 15 (C) 9 (D) 6
Q.10
Using only the letter from the word WILDCATS with no repetitions allow ed in a codeword, numb er of 4 letter codeword s are possibl e that both start and end with a consonant, are (A)3 60 (B) 900 (C) 1680 (D) 220 4
Q: ll
Find j(x /nx )dx
(A)-
(B)-
(C)-l
(D)l
Q.12
If P( x) is a polynomial with rational coefficients and roots at 0,1, -J l
anci
1 - \/3 , then the degree of
P(x) is at least (A) 4
(B) 5
(C )6
7 Sum of the infinite series, 4 - ^ + — (B)
(A) ^ Q.14
00
>
cc ua t0
l
l
(D)
49
12! (D) 56
(B)
3!-6!
e3x - 1 . if x * 0 x then /'(0 ), is
L et /( x) =
3
if x = 0
(A)-9
(D) nonexistent
(C) 9/2
(B)9
I f f "(x) = 10 and f ' (1) = 6 and f ( l ) = 4 then f (- 1) is equals (A)-4 (B) 2 ' (C) 8 The coefficient of x 3 in the expansion of
x" v
4
2 +
(D)12
\12
xy
,is (D)100
(C) 99
(B)98
(A) 97 Q.18
(C)
9!
( ) 2!-6!
Q.17
+
24
8!
n6
V +
A florist has in stock several dozens of each of the following: roses, carnations, and lilies. How many different bouquets of half dozen flowers can be made? A
^ 1 5
(D )8
In how many ways can six boys and five girls stand in a row if all th e girls are to stand together but the boys cannot all stand together? (A) 172,800 (B) 432,000 (C) 86,400 (D)no ne The composite of two funct ions f and g is denoted by fog and defined by (fog)(x) = f (g(x) ). When f(x)
6x
5x and g (x) = — which one of the followin g is equal to (fog)( x)? x — 1 x-2 3 Ox
4-x
(C)
x-2 k iA
The equation In
>
( k + 1)i/(k +D
= F(k)
In 1 -
x-2 4x + 2 1
k + 1
+—Ink k
(D)
15x 2x + l
is true fo r all k wherev er def ined.
F(100) has the value equal to (A) 100
Q.21
Compu te
(B)
f ,_ ^Vx+K/x
1
101
(C)5050
(D)
1
100
ill
BANSAL CLASSES H Target I I I JEE 2007
CLASS: XII (ABCD)
MATHEMATICS Daily Practice Problems
DATE: 28-29/06/2006
TIME: 50 Min each DPR
DPR
NO.-25
I > P P - 2 5
DATE: 28-29/06/2006
Q. l
TIME: 50 Min
tan 9 =
1 2—+ -
where 9 e (0, 2n) , find the possible value of 6.
{2]
~
2 +
'--co
Q. 2
Find the sum of the solutions of the equation 2e2x
Q.3
5e
x
+
4
=
o_
[2 ]
Suppose that x and y are positi ve numbe rs for which log 9 x = log 12 y = log 16 (x + y). If the value of --2
Q. 4
_
cos 9, where 9 e (o, rc/2) find 9.
[3]
Using L Hospitals rule or otherwise, evaluate the following limit: Limit
Limit
X - > 0 +
n->eo
[l 2 (sinx)" j + 22 (sinx) x
+
+
n2 (sinx) x " where [. ] denotes the
n3
greatest integer function. Q .5
1 Consider f ( x ) = - ^ =
, |1 + I
~
sin2
x
— . I V a + htan 'x , f o r b > a > 9 & the functions g(x)&h(x)
sinx
are defin ed, such that g(x) = [f(x>] - j - ^ J & h(x) = sgn (f(x)) for x e domain of »f, otherwise g(x) = 9 = h(x) for x £ domain o f ' f , where [x] is the greatest integer function of x & {x} is the fractional 7t part of x. Then discuss the continuity o f' g' & *h' at x =— and x = 9 respectively. ~ ^
Q. 7
f x 2 tan
_1
x
,
Using substitution only, evaluate: jc os ec 3 x dx.
DATE: 30-01/06-07/2006
Q. l
[5j
JIME: 50 Min.
12 A If sin A = — . Find the value of tan — , x
Q.2
[5]
[2]
v
The straight line - •+ ^ = 1 cuts the x-axi s & the y-axis in A& B respectively & a straight line perpendicular to AB cuts them in P & Q respectively. Find the locus of the point of intersection of AQ & BP. [2]
Q.J
tan 9 1 cot 9 If - - — - — — = —, find the value of - — . tan 9 - tan 39 3 co t9 -c ot 39
HI
Q.4
If a A ABC is form ed by the lines 2x + y - 3 = 0; x - y + 5 = 0 and 3x - y + 1 = 0, then obtain a cubic equation whose roots are the tangent of the interior angles of the triangle. [4] dx
Q.5
Integrate
f
15]
(a>b)
J a 2 - tan2 x)Vb2 - tan2 x
xsmxcosx I ((a „ 2 cos 2 x„ +, bT,2sin „;„2 x)\2 dx
Q.6
[5]
d dy Let ~— (x 2y) = x - 1 where x ^ 0 a nd y = 0 whe n x = 1. Find the set of valu es of x for whic h — dx [5] is positive.
Q.7
DATE:
TIME;
03-04/07/2006
50Min.
Q. 1
Let x = (0.15) 20 . Find the characteri stic and mantis sa in the logarithm of x, to the base 10. Assume log10 2 = 0.301 and log 10 3 = 0.477. [2]
Q. 2
Two circles of radii R & r are externally tangent. Find the radius oft he third circle which is between them and touches those circle s 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 5 \ 5 5 S
then compute the value ofthe integral j( det A) dx . P]
Q. 4
Using algebraic geometry prove that in an isosceles triangle the sum ofthe distances from any point of the base to the lateral sides is constant. (You may assume origin to be the middle point of the base of the isosceles triangle) [4]
Q.5
Evaluate:
Q.6
dx
Jf1 ++-xx
Vx +
X2
If the three distinc t point s,
fa v
Q.7
[5]
+x3 3
a 2 -3]
a - l ' a -1
;
fb ?
3
[ b - r
b 2 -3^1 b-ij
r c3 ?
[c-l '
c 2 -3^1 c- l j
are collinear then
show that abc + 3 (a + b + c) = ab + be + ca.
[5]
Integrate: j^/ tan x dx
[5]
ill BANSAL CLASSES
MATHEMATICS
IglTarget HT JEE 2007 CLASS: XII (ALL)
Daily Practice Problems
DATE: 23-24/06/2006
TIME: 50 Min.
DPP. NO.-24
[16 [16 x 3 = 48]
Select the correct alternative: (Only one is correct)
Q. 1
A circle of radius 2 has center at (2, 0). Acircl Aci rclee of radius 1 has center at (5,0) . Aline is tangent tangent to to the two circles at points in the first quadrant. Which of the following foll owing is the y-intercept ofthe line? (A) 3
Q.2
8
V2
(B)
(D) 2a/2
(Q3
In a triangle ABC, the length ofAB ofAB is 6, the length of BC is 5, and the length of C Ai s 4. If K lies on BC BK 3 such that the ratio of length r — is —, then th e length of AK is KC
(A) 2V3 Q. 3
Q. 4
2
(B)4
(D) 2,
(C) 3V2
Which Whic h one of the foll owing owin g quadrants quadr ants has the most mos t solutions to the inequality, x - y < 2? (A) I quadrant (B) II quadrant (C) ID quadrant (D) I and III quadr ant have same The range of the function /( x) = sin sin _1 x + tan~'x + cos _1 x, is 7t
(A) (0,71)
(B)
371
(D)R
(C) [0,71]
4'T
Q.5
The area of the reg ion of the pla ne consist ing of all poin ts whose coordinates (x, y) satisfy the conditions 4 < x 2 + y 2 < 36 and y < | x is (A) (A ) 24 n (B) (B ) 27TI 27TI (C) (C ) 20TT (D) (D ) 32tc
Q. 6
A straight straight wire 60 cm long is is bent into the shape of an L. The shortest possible possibl e distance between the two ends of the bent wire, is (A) 30 cm
(B) 3 0 V 2 c m
(C)1 0V26
'7t •X holds, is Sum of values of x, in (0, n/2) for whic h tan — + X = 9 tan 4 ' 4 v _1 (A) 0 (B) 71 - tan (2) (C) cor'(O) (D) tan -1 (2) N
Q.7
Q. 8
(B) (B) sinx + (e+ l)x
(C )( e+ l)x + cosx
(D) (D) (e+ l ) x - c o s x + 2
Evaluate Evaluate the integral : j x e c o s x 2 sin x 2 dx (A) | e c o s x 2 + C
Q.10
71
Giv en/ "(x ) = cos x, / ' ^ y J = e an d/ (0 ) = 1, 1, the n/ (x ) equals. (A )s i nx - (e +l )x
Q.9
(D) 20^ 5
The value of Lim x->n x->n (A)0
1
(B)- -es mx
e~n - e " x sin si n x
+C
(C)
1 _sin x 2
+ C
(D)- i e c o s x 2 + C
is
(B)-e-
(D)e-
(D)nonexitent
Q.LL
kx ^" 2 x -k
Let/( x) =
+ 1
all possible values of k for wh ic h/ is continuous for every every . The interval(s) of all
x e R, is (A) (-«,,-2] Q.12
(B)[ -2,0 )
(D )( -2 ,2 )
Suppose F (x) = / (g( x)) and g(3) = 5, g'(3) = 3, /' (3 ) - 1 ,/ '( 5 ) = 4. Then the value of F'(3), is (A) 15
Q, 13 13
(C) (C) R - ( - 2,2)
(B) 12
(C) 9
( D) 7
Fro m a poi nt P outsi de of a circle with centre at O, tang ent seg ment s PA PA and PB arc drawn. If
1 ( A O )
1 2
"
~
+
1
Ye ' t
(A) 6
b e n l e n t b
chord AB is (C) 8
(B)4 a
Let
a
a
l2
l3
a21
a22
a23
a31
a32
a33
b
i,
b 12 b 2 2
b 2 3
b 3 ]
b 3 2
b 3 3
C12
C13
C2 1
C22
C 23
C31
C32
C 33
n
, Aj * 0
b 1 3
b 2 1
c
and
n
(D) 9
wher e b- is cofactor of a^ a^ V i, j = 1, 2, 3
where c^ is cofac tor of
V i, j = 1, 2, 3.
then which one of the fo llowin g is always correct. (A) Aj, A2, A 3 are in A.P. (B) Aj, A2 , A3 are in G.P. A
2
(C) A Q. 15
3
(D) A,
The first first three thre e ter ms of an arithmeti c sequence, in order, are 2x + 4, 5 x - 4 and 3x + 4. The sum of the first 10 terms of this sequence, is (A) 176 (B) 202. 4 (C) 352 (D) 396 r
Q. 16 16
The value of
7L 7t
4
. .
71 \ /wI 7t
cos—+ zsin — 8 8
V3 i (A) — + w 2 2
71 7i
K J
Subjective: Evaluate:
dx Q.l
J"
q . 2
ff s * s* j 5 x5
xV a x - : 5
rsin j , sm
1
x 5
dx
V x - co s 1 yfx r, r* r— dx + COS
2
\5
7T n. cos — +1 sin — is equal to 15 15,
/ n
Q.3 Q. 3
A0
2
. .
(C)
S
i
(D)- ^--i w 2 2
BANSAL CLASSES
MATHEMATICS
8Targe* IIT JEE 2007
Daily
CLASS: XII(ALL)
Practice Problems
DATE: 16-22/06/2006
2
DPR NO.-21, 22, 23
1
DATE: 16-17/06/2006 Q. 1 For x > 0 and ^ 1 and n e N, evaluate, 1
Lim
n-»co V
log 2 . log=>x 4' °X
+
TIME: 45 Min.
1
log 4 . log 8 • °x • •
+
+
1 n_1 n log .Iog_ ~o x 2 • ~ o x -2
y
Show that (a + b + c), (a 2 + b 2 + c 2 ) are the facto rs of the determinant
Q. 2
a2
(b + c) 2
be
b 2
(c + a) 2
ca . Also find the remaining factors.
c2
(a + b) 2
ab
Q. 3
Prove that a non singular singular idempotent matrix is always an involutaiy involut aiy matrix.
Q. 4
Find an upper uppe r triangular matrix A such that A 3 =
Q.5
Q. 6
8 0
-57 27
^2
d2
y „ dy I f ' y' is a twice differentiable funct ion of x, transf orm the equation, (1 - x ) -—7 - x —- + y = 0 by dx dx means of the transf ormation, ormati on, x = sin t, in terms of the independent independent var iable' t '. Atangent Atang ent line is drawn to a circle circle of radius unity at the point A and a segment seg ment AB is laid offwhose length is equal equal to that of the arc AC. A straight line BC is drawn to intersect the extension of the diameter AO at the point poin t P. Prove that: 9 (1 - cos 0) ( i i ) L ^ t p A =3. (i) PA = e - sin e 1 Use of series series expansion or L L Hospital's rule prohibited.
DATE:
TIME:
19-20/06/2006
45Min.
\ l-x\
Q. 1
Without using any series expansion or L' Hospital' s rule, Evaluate: Lim x la e| 1 + x/ VT3+V3
2V5
/
V5
Q. 2
Find the value of the determinant V15+V26 3 + V65
5 V10 VPS VPS 5
Q. 3
/ ( x ) is a diffrentiable function satisfy the relationship relationship f 2 (x) + f 2(y) + 2 (xy - 1 ) = f 2(x + y) V x, y e R. Also f (x) > 0 V x e R , and f (V2 Determin e f (x). (V2 )= 2 . Determine
Q.4
Let, y = t a n - | j — 5
+ tan"
x z
2.3 + x j
dy Find —- expressing your answer in two terms, dx
+ tan -1
j 3.4 3. 4 + x^
+
up to n terms te rms .
Q. 5
0 Without expanding the determinant show that the equatio n x + a x+b
x-a 0 x+c
x-b x-c 0
:
0 has zero as a
root. Q.6
Let a j , a 2 & p j, (3, be the ro ots of a x 2 +bx + c = 0 & px 2 + qx + r = 0, respectively. If the system b ac of equations a, y + a 2 z = 0 & p t y + p 2 z = 0 has a non-tr ivia l solution, then prov e that — = — .
D r » P - 2 DATE:
Q. 1
TIME: 45 Min.
21-22/06/2006
Compute x in terms x 0 , x, , and n. Also evaluate Lim x n =
a Q.2
3
A— 2 v b
5 c 8 2
d is Symmetric and B = b - a -2
3 e 6
X0 + ^X1 ~
a - 2b - c -f
Asi
~ Zs>-J
j-2..
is Skew Symmetric, then find AB.
Is AB a symmetric, Skew Symmetric or neither of them. Justify your answer. x +1
Q.3
Let f(x)=e x , x<0 = 0 ,7 x=0 2 =x , i x> 0 Discuss continuity and differentiability of f (x) at x = 0 .
Q. 4
Show that the matrix A =
evaluate the matrix
1 2
0 1
1 2
0 can be decomposed as a sum of a unit and a nilpotent marix. Henc e 1
2007
Q. 5
dv Find — , if (tan"1 xV + y cotx = 1. dx
Q.6
•f w -)_ bY^) f If / is di ff ere nti ate and Lim ^ h
^ = n
' th en fm d
the value
L'Hospital's rule. 1 + e"
Q.7
Consider the function / ( x ) =
x + 2 , 0 < x <3 3
6—
x (a) (b) (c)
x <0
x >3
Find all points where f (x) is discontinu ous. Find all points when f (x) is not differentiable. Draw the graph, showing clearly the points of discontinuity or non derivability.
Without using
| h B A N S A L C L A S S IES
MATHEMATICS Daily Practice Problems
V S T a r g e t I I T J EE 2 0 0 7 TIME: 60 Min.
CLASS: XII (EXCEPT A-2)
Q. 1
DPR
NQ.-20
3 10 The set of all x for which — > — 2 —7 consists of the un ion o f a finite and an infinit e interval. The length x x +1 ofthe finite interval is (A) 3
(B)2
1
CO I
(D)2T
Q.2
Five perso ns put their hats in a pile. When they pick up hats later, each one gets some one else's hat. Nu mb er of way s this can happen, is (A) 40 (B )4 4 (C) 96 (D) 120
Q.3
Suppo se the origin and the poi nt (0,5 ) are on a circle who se diam eter is along the y-axis and (a, b) lies on the circle. Let L be the line that pas ses throu gh the origin an d (a, b). If a2 + b 2 = 16 and a > 0 then the equation of the line L is ( A) 3 x - 4 y = 0 ( B) 2 0 x - 3y = 0 (C)2x-y = 0 ( D) 4 x -3 y = 0
Q.4
If 1 lies between the roots o f the equation y2 - my + 1 = 0 then the value of
4[x]
IxI +16 has the valu e equal to (Here [x] denotes gratest integer funct ion ) (A) 0 (B) 1 (C) 2
(D) non e
Q.5
The sum of the squares of the three solutions to the equatio n x3 + x 2 + x + 1 = 0, is (A)~ 1 (B)0 (C )l (D)2
Q.6
Let / ( x ) = 1 + x 3 . If g (x) =/ _ 1 (x), i.e. if g is the inverse /, then g'( 9) equal to (A) 1/12 (B) 1/243 (C) 1/8 (D) 1/24
Q.7
.Lim j V x - V x - Vx + Vx x-»oo v (A) equal to 0
Q. 8
is
(B) equal to 1
(C) equal t o - 1
(D) equal t o - 1/2
Suppose f is a differentiable function such that / ( x + y ) = /( x) + /( y) + 5xy for all x, y and f'(0) = 3. The minimum value of f (x) is (A) - 1 (B) —9/10 (C) - 9/25 (D) non e x-1
Q-9„ f i i n Jfg . x + l = 3x th en the value of g (3), is v y (A)Q. 10
Q. 11
15
V2 (B)-
(D)
(C)9
V3
9 sin ( A + B) For acute angl es A and B if (tan A)(c ot B) = - the n the val ue of — — equal to 5 sin(A — d ) (A) 7/4 (B) 2/7 (C) 4/7 (D) 7/2 The value of this produ ct of 98 numb ers 1 - -
! (A) Q. 12
Vx eR
3y 1
1-2
1 - -
98
5y
1-
98 (B) 100
10
99
1 - -
100
,is
(C) 5050
(D)
1 4950
2
If T = 3 /n( x + £x ) with £ > 0 and x > 0, the n 2x + £ is equal to
(A) V-f'2 + 4eT/3
(B) V^2 + 4e-T/3
(C)
2
T 3
(D) V^ -4e /
Q.13
Q.14
-2 - , ^ U - 1 2 X + 35 (A)-1. 25 (B) -1. 5
Evaluate:
(D)-2
(C)- 1.75
Le t/ be a polynomial function such that for all real x f( x 2 + 1) = x 4 + 5x 2 + 3 then the premitive o f / ( x ) w.r.t. x, is
3 2 x3 3x 2 x3 3x 2 x 3 3x 2 x + C v(D ) —+ ^ + x + C (KAJ )— + — — x + C w(B) —•- — + x + C (C) „ 3 2 3 2 3 2 3 Q.15 Number of regular polygons that have integral interior angle measure, is (A) 20 (B) 21 (C) 22 (D )2 3
Q.16
Suppose/ is a differentiable function such that for every real number x, / ( x ) + 2 / ( - x ) = sin x, then f'(n/ 4) has the value equal to
(A)l/V 2
( B ) - l/ V 2
(B) -1 /2 V2
(D)V2
Q.17
The number of permutation of the letters A A A A B B B C i n which the A's appear together in a block of four letters or the B's appear in a block of 3 letters, is (A) 44 (B) 50 (C) 60 (D)n one
Q.18
If {x} denotes the fractional part function then the number x = TT^a
{sf-iyif
(A) 1/2 Q.19
(B )0
(C) -1/2
+C
(C) Jxsinxdx = s in x- xc os x + C
Let/(x) =
tankx x 3x + 2k
(A) none Q.21
Find y->2 L™
1
Q.24
(D)none
. I f / ( x ) is continuous at x = 0 then the number of values of k is for
x>0
1
(C) 2
(D) more than 2
(C )- l/ x 2
(D) does not exist
1
y-2^x +y-2
x/
( B ) /n x
(B)
21
16
(C)
147
16 441
The sum (in radians) of all values of x with 0
(A) 1 Q.25
+C
Let p(x) be the cubic polynomia l 7x 3 - 4x 2 + K. Suppose the three roots of p(x) form an ar ithmetic progression. Then the value of K, is (A)
Q.2 3
(B) JtanOsec2OdO =
for x < 0
(B) 1
(A)0 Q.22
(D)none
Which one of the following is wrong? 2
(A) JtanOsec2 0dO =
Q.20
f/— )2 simplifies to
n=0v 2 (B) 2
128 1323
(D) 671
is
n
T
(D)
y
(C) 4
(D) none
If sin(x + 2y) = 2x cos y, the the value of dy /d x at the point (0,71) must be (A)-l (B) - 3/2 (C )0 (D)2
| | | BANSAL CLASSI E S
M A T HE M A TI C S Daily Practice Problems
Target IIT JEE 2007 DATE: 02-03/06/2006
CLASS: XII (ALL)
DPP. NO.-l 7, 18,19
Take approx. 40 to 45 min. for each Dpp.
-X Q.l
If y =
sin x 1 + cotx
cos x
, dy n . then — at x = — is dx 4 1 + ta nx
+
(A)0
(B)-l f
Q. 2
The value of cot
\
(C)l
(D)2
(C) (3 + VI0)
(D)( l0 + V3)
W
f
3k - tan
( A ) ( i o ( B ) (
Mv 3 J)
v i o
3 +
sm ^ yfH-ylcos Q.3
~7
l
equals
y
x equals
Lim x->-l+
x
(A)0
(C)
(B)l
ijn
f
Q.4
1_x I f/ (x )= 3 + l + 7
then
v Lim f (x) = 4 ' x->r
Q.5
(B)
(C) L ^ f ( x ) = 5
(D )/ ha s irremovable discontinuity at x = 1
I f/ (x ) = 3x 1 0 - 7 x 8 + 5 x 6 - 2 1 x 3 + 3 x 2 -7thenthevalueof
(A)Q.6
Q. 7
Q.8
Lim f (x) = 3
(A) v
53
22
(B)-
(C)
53
Lim x^r
f( 1
h) 3
h +3h
'is
22 (D) —
If the triangle formed by the lines x 2 - y 2 = 0 and the line Ix + 2y = 1 is isosceles then /= (A) 1 (B)2 (C) 3 (D)0 . (e 2x - l - 2 x z ) ( c o s x - l ) Using L'Hospital's rule or otherwise evaluate the limit, Lim x->o (sin 3x - /n(l + 3x )) x 4
Evaluate the limit Lim x-»o
e x -/ n( x + e) e
x
-l
. Use of L'Hospital's rule or surd expansion not allowed.
Q.9
Find all real number s t satisfying the equation ( 3 t - 9 ) 3 + ( 9 t - 3 ) 3 = (9 t + 3 t - 12)3.
Q. 10
Find g'(3) if g (x) = x • 2h where h (3) = •- 2 and h'(3) = 5.
P Q. l Q.2
Q.3
F
P
-
2 Find the value of the expression lo g 4 (2000)
1
8 3 \6
log 5 (2000)
'
Let f (x) = a cos(x + 1) + b cos(x + 2) + c eos(x + 3), where a, b, c are real. Given that f (x) has at least two zeros in the interval (0, n), find all its real zeroes. 1 . V63 Calculate, sin — arc sin
V
Q. 4
In an infinite pattern, a square is placed, inside a square, as shown, such that each square is at a constant angle 0 to its predecessor. The largest, outermost square is of side unity. Find the sum of the areas of all the square in the infinte pattern as a function of 0.
Q.5
If 0 is eliminated fr om the equations, a cos 0 + b sin 0 = c & a cos 2 0 + b sin 2 0 = c, show that the eliminant is, (a - b) 2 (a - c) (b - c) + 4 a2 b2 = 0.
Q.6
A triangle has side lengths 18, 24 and 30. Find the area of the triangle whose vertices are the incentre, circumcentre and centroid of the triangle.
Q. 1
Find the real solutions to the system of equations log10 (2000xy) - log10 x • log 10 y = 4 log 10 (2yz) - log10 y • log 10 z = 1 and log 10 (zx) - log1Qz • log 10 x =0
Q.2
Prove that, cos
Q.3
1 _i 1 24 Compute the value of cos - t a n — 4 7
1
1-cosx 12c os x + 13
= 71 - 2 cot
1
1
1
9
X
- t a n — where x e (0, n). 5 2
Q.4
If g (x) = x 3 + px 2 + qx + r wher e p, q and r are integers. If g (0) and g (- 1) are both odd, then prove that the equation g (x) = 0 cannot have three integral r oots.
Q.5
Sum the series, c o r ' ( 2 a + a) + cot" 1 (2a"1 + 3a) + cor ! ( 2 a - 1 + 6a) + cor 1 (2a _ 1 + 10a) + Also find the sum of infinite terms, (a> 0) . 44
^Tcosn Q.6
Let x = — 44 ^sinn n=l
find the greatest integer that does not exceed 1 OOx. c
+ to ' n ' terms.
l | | B AN SA L C LA S SE S
M AT H EM A TI CS
v B Target il T JEE 2007 CLASS: XII (ALL)
Daily Practice Problems
DATE: 12-13/05/2006
TIME: 60 Min.
DPP. NO.-14
This DPP will be discussed on Friday & Saturday.
2 cos x - sin 2x Q.l
f(x) =
-1 e g(*) = 8x - 47t
z
(7t-2x) ' f (x) fo r x < 7t/2 h(x) = g (x) for x > 7t/2 then which of the following holds? (A) h is continuous at x = n!2 ;
(B) h has an irremovable discontinuity a t x = 7t/2
(C) h has a removable discontinuity at x = tc /2 ( D ) /
(% 2
2 V J Two balls are drawn fro m a b ag containing 3 white, 4 black and 5 red balls then the number of ways in which the two balls of different colours are drawn is \
Q. 2
(A) 94 Q.3
(B) 47
Q.5
/
(C) 38
(D) 19
If A ABC if cosA, cosB, cosC ar einA. P. then which of th e following is also an A. P.? A
Q. 4
=g
( _ "N 71
B
C
A
B
C
(A) tan Y , tan—, tan—
(B) c o t y , c o t y , cot—
(C )( s- a) (s -b ), (s -c )
(D) none
1 1 If tan" t a nx4 (3x) (A) x =(x) ± 1+ tan" (2x) +(B) = 0 = n, then
The most general solutions of the equation x ( A) x = n7t + ( - l ) " - | ( B ) x = y - C - l ) " ^
(C) x = 1 3 sin2 x+ 2
= ^
(D )x G( j) is
( C )x = 0
(D) x = nrc - ( - l ) n
71 12
where n e I Q. 6
The sum of the squa re of the length of the chor ds inter cepte d by the line x + y = n, n e N on the circle x 2 + y 2 = 4 is (A) 11 (B) 22 (C) 33 (D) none
Q. 7
Which one of the following statements about the function y = f (x), graphed have is true?
Q.8
(A) Li mf (x ) = 0 x-»l
(B) Lim f (x) - 1 x-»0
(C) Lim f (x) exists at eveiy po int x 0 is ( - 1, 1) "
(D) Lim f (x) = 1 X->1
(B)
a-b a+b
(C)
a+b a~-b
,
.
o\
Lim is x->0 3-3cosx (A)
4
(a).
3 C
< >4
CD)
1y 1. .
+Y \ I " i /
(D)nonexistant
2x 2
2
y=f(x)
I
a sinbx-b sinax , Lim — — (a ^ b) is x-+o ta n bx - ta n ax (A) 1
Q.9
y
Q. 10
x2 - 9
Let / = LimX
3
2
^ V X +7 -4
"
- 9
then
+ 7-4
(D) / = m
(C) / = - m
If (2 - x2 ) < g (x) < 2 cosx for all x, then Li m g (x) is equal to x-»0
(A)l Q 1 2
x - > - 3 y jx 2
(B) I-2m
(A)Um
Q. 11
and m = Lim
x2
Le ,
+
+
x-1
(A)-5
(D)0
(C) 1/2
(B)2
5 then Lim f (x) is x-»l
(B)2
(C)-6
(D) non existent
(B)e e
(C) e-
(D)e
(C) e 3 / 2
(D) e 2
(C)/n2
(D) non existent
i Q. 13
Lim (/n x)
x_e
x—»e+
is
£ (A) e e Q.14
Lim (e x + x ) x x->0
Q. 15
1/2
(B)e
(A) e1 /n(x + l) Lim - 7 — is
X->QO l og 2 x
O)0
(A) log2e
30 + 71
Q. 16
Let / = Lim
e->--
—j
^ A
then [ I ] is, where [ ] indicates greatest integer function
7 1
sin 0 + I 3y (B) equal to 2 (A) equal to 3 Q.17
Lim
e
(D) none existent
(C) e 3 + 1
(D) e 3 - 1
- sin x - e x
x->0
(B)e- 3 + l
(A) e + 1 Q. l 8
(C) equal to 1
Lim cos(tan _1 (sin (ta n -1 x)]) is equal to
1 (A) - 1 Q. 19
(B)V2
Let f (x) =
aX
X
(A) 1 Q.20
If Lim x—>3
(A) 3
+
X - 3
S
, if Lim f (x) = 1 and Lim f (x) = 1 then f (2) + f (- 2 ) is equal to
+1
x- »0
(B) 2 x n —3n
(C)
x-»oo
(C )0
(D) 4
108 (n e N ) then the valu e of n is (B)4
(D)6
(C)5
SUBJECTIVE
_ Q. 21
, tan2x-2sinx Find the limiting value of ~ — — as x tends to zero.
Q. 22
Without using series expansion or L' Hospital's rule evaluate, ^ ™
Q. 23
Show that the sum of infinite series,
PI Li m l n Q + *
4 4 4 4 ta n - 1 — + tan" 1 — + tan -1 — + tan - 1 — +
x
2
+ x
(e -l)>
71 00 = — +cot _ 1 3.
4
)
[3]
[3]
Jig BANSAL CLASSES
MATHEMATICS Daily Practice Problems
vSTarget IIT JEE 2007 CLASS: XII (ALL)
Q. 1
DATE: 22-23/05/2006
Draw the grap h of the func tion / (x) =
TIME: 55 to 60 Min.
3X
, -1
4-x
, l
DPP. NO.-15
& discus s its continuity & defferentiability
at x = 1.
Q.2
Given f : [0, a] —» S, such that f (x) = 3 cos—. Find the largest value of'a', for which fhas an inverse function f _ 1 . Find f _ 1 . State the domain and the range of f & f - 1 . Find the gradient of the curve y = f _ 1 (x) at the point where the curve cross es the y - axis.
Q. 3
Give n/( x) = [x] tan (71 x) where [x] denotes greatest integer function, find the LHD and RHD at x = k , where k e I.
Q. 4
Examine the continuity at x= 0 of the sum function of the infinite series: .00
x + 1 (x +1) (2x +1) (2x+l)(3x + l)
Q.5
l + /nt If x = — 2 — t
Q.6
Let f (x) = tan — secx + tan
v/
and y =
Limit x-»0
t
X
X
2
TT
and g (x) = f (x) + tan
(a)
dv fdvA . Show that y — = 2x — +1. dx Vdx j
3 + 2/nt
'g(x)^ V X y
~
2
x
X
sec — + tan
2
where x e
^
71 7CN
2 2
(b) Limit x->0
2
v
A
y
sec
X t t
2
+
+t an ~
2
X
sec
X t tz t
2
and n eN. Evaluate the following limits.
(c) Limit x-^0 V
X
y
x + / n | V x 2 +1 - x Q. 7
Without using L' Hospitals rule or series expansion evalua te: Lim x-»0
ill BANSAL CLASSES
MATHEMATICS Daily Practice Problems
v S T a r g e t BIT JEE 2007 CLASS: XII (ALL) Q.l
DATE: 24-25/05/2006
TIME: 55 to 60 Min.
DPP. NO.-16
Suppose f( x) = tan (sin -1 (2x)) (a) (b)
Find the domain and range of / Express f (x) as an algebaric function of x.
(c)
Find f'
I
v4y
X (3 e
Q. 2
Discuss the limit, continuity & differentiability of the function f (x) =
1/ x
2-e
+4)
,XTK)
1/ x
atx=0.
,x=0
Q.3
Evaluate: Limit
r* 7t tan —hax In 4 sin bx
x->0
a
(b*0).
Use of series expansion and L'Hospital's rule is not allowed.
Q.4
The function/is defined by y= f( x) . Wh er ex =2 t- | t|, y= t 2 + t 11|, t g R. Draw the graph of / for the interval - 1 < x < 1. Also discuss its continuity & differentiability at x = 0.
Un
'
}
1
1
A
Q.5
Evaluate
Q. 6
If g is an inverse function of / & /' (x ) =
Q. 7
Given a real valued functions f(x) as follows
/nx
x-1
x 2 + 2 c o s x - 2 x" f(x)
Use of series expansion or L' Hospital's rule is not allowed.
n
— 7 , prove that g'(x) = 1 + [g(x)] . 1 ~h X
for x<0
p forx=0 x sinx - ^n(e cosx) for x>0 6x 2
Determine the value o f p if possible, so that the function is continuous at x = 0 . Use o f power series or L'Hospital's rule is not allowed.
ill BANSAL CLASSES
MATHEMATICS Daily Practice Problems
V g Target IIT JEE 2007 CLASS: XII (ALL)
DATE: 03-04/05/2006 ax-sinax
If 'a' is a fixed constant then Lim x-»0
a2
DPP. NO. -11
is equal to 2 _ 2
a3 (D)j
a ( O -
2
I f / (x) = x + bx + c and/ ( 2 +1) = / ( 2 - 1 ) for all real numbers t, then which of the following is true? (A)/(l)< /(2)( 4) (B)/(2) (l)(4 ) (C)/(2)(4) (l) (D )/ (4 ) ( 2) (l )
2
t^
a3 < B) £
x
TIME: 60 Min.
Let P > 0 and suppose AABC is an isosceles right triangle with area P sq. units. The radius of the circle that passes through the points A, B and C, is (A) VP V
Q4
(B) V2P
Q.6
(D) =
Number of real solutions of the equation cos x + cos {^[2 x ) 2, is (A) 0
X5
(C) 2-s/P
(B) 1
(C) 2
(D) infinite Y
2
A quadratic polynomial y = a x +bx+c has its vertex at (4, - 5) and has two x-intercept, one positive and one negative as shown. Which one o f the following must be negative? (A) only a (B)onl yb (C) only c (D) only b and c
\O B\
4
/A
"
-5'
If in a AABC, the altitudes from the vertices A, B, C on oppos ite sides are in H.P, then sinA, sinB, sinC are in (A) GP. (B)A.P. (C) A.G.P. (D) H.P. Suppose that two circles C[ and C 2 in a plane have no points in common. Then (A) there is no line tangent to both Cj and C2. (B) there are exactly four lines tangent to both CL and C2. (C) there are no lines tangent to both Ct and C2 or there are exactly two lines tangent to both C { and C2. (D) there are no lines tangent to both C1 and C2 or there are exactly four lines tangent to both Cj and C2.
Q8
Number of three term arithmetic progressions which exist in the set {1 ,2 , 3, difference d ^ 0, is (A)190 (B)200 (C)380 (D)400 _i x _t 1 The smallest positive integer x so that tan tan — + tan 10 x + 1 (A) 8 (B )9 (C)7
Q. 10
2 3 7
When (XV4 _ x / )
J&
rX
2
1 3
%
tan— is 4 (D)0 3
is multiplied out and simplified, one o f its terms has the form kx where 'k' is a
constant which is equal to (A) 7 (B)-7 Q.ll
:
,40} and common
(C) 35
(D)-35
„I f = - 1 + /V3 and, y = - I - / V 3 , x where i 2 - -1, then which of the following is not correct? 5 5 7 7 9 9 (A) x + y = - 1 (B) x + y — - 1 (C)x + y = l
(D) x
Number of solutions of the trigonometric equation 3 cos x - 3 cos x sin^ = cos 3x where x e (0,1), is (A) 0 (B) 1 (C) 2
(D)infhinte
In which one o f the following cases, limit does not tend to e? i / / x + 4 NX+3 i1 r x (A) Lim x ~' (B) Lim (C) Lim X— X— V x+2 X—>1 Xy (D) Lim (l +/ (x )) ?o o when Lim / ( x ) X->CC
0
11
+y
n
=- 1
Q. 14 *
y
The lines L and K are symmetric to each other with respect t o the line y = x. If the equation of the line L is y = a x+ b where a and b are non zero, th en the equat ion of K is x b x „ x b x b (A)y=--(B)y=---b (C)y= - - + (D )y =- + a a a a a a a 3X-4X
Q. 15
Domain of definition of the function f (x) =
is -3x-4 ( A) ( - o o , 0 ] ( B) [ 0, oo) ( C ) ( - oo, - l ) u [ 0 , 4 ) (D)(-oo,l)u(l,4) 2 The root s of x + bx + c = 0 are bot h real and greate r than 1. If s = b + c + 1, then' s' (A) may be less than zero (B) may be equal to zero (C) must be greater than zero (D) must be less than zero Vx
Q. 16
Q. 17
Which one of the following does not reduce to sin x for every x where th e expressions are defined? -2 s •m x ^s ^ sin xs ec x „ . . (A) — 9 5— w(B) csc x - cot x cos x (C) (D) all redu ce Ato sin x v ' sec x - ta n x tanx
Q' 1 8
Le t/ ( x) be a fiinction with two properties (a) for any tw o real number x and y, f( x + y) = x + / ( y ) and (b) f ( 0) = 2. The value of /( 10 0) , is (A) 2 (B) 98 (C) 102
Q.1 9
Q.20
(D) 100
Read the following statements carefully: I If a, b and c are posit ive numbers not equal to 1 and a < b, the n log a c < log bc. II The equatio n x 2 - b = 0 has a real solution for x for any real numbe r b. HI The sequence a n defined by a n = 3 (0.2)" n is a geometr ic sequence. IV cos(cos(x)) < 1/2, V x e R ' No w indicate the correct alternative. (A) exactly one is always true (B) exactly tw o are always true (C) exactly three are always tru e (D) exactly four are always true. If x = a + b/ is a complex number such that x 2 = 3 + 4i and x 3 = 2 + 11/ where i = J I \ , then (a + b) equal to (A) 2
(B) 3
(C )4
(D )5
MORE THAN ONE ARE CORR ECT Q 21 If x satisfies log 2 x + log x 2 = 4, then log 2 x can be equal to
(A) t a n ~ Q.22
(B )c ot y
(C )t an |
(D)cot^
In a triangle ABC, altitude from its verte x meet the opposite sides in D , E and F. Thenthe perimeter of the triangle DEF, is abc (A)-F
2A (B )T
R( a + b + c) ( C i - ^ — l
_ 2rs ( D) T
where A is the ar ea of the triangle ABC and all other symbols have their usua l meaning. , Q 23
In a triangle AB C if Z B = 3 0°, b = 3 V2 - a/6 and c = 6 (A) the triangle ABC is an obtus e triangle
t hen
(B) angle Z A can be 15°
(C) there can be only one value for the side BC
Q.24
(D) the value of tanA tanC will be unique. n Let z =( 0, l ) e C. Where C is the set ofcom plex numbers, then the sum ^ z for n e N can be equal to k=0
(A) 1 + i Q.25
( B) i
(C)0
(D)-l
Value of the expression log 1/ 2(sin6° • sin42° • sin45° • sin 66° • sin 78°) (A) lies bet wee n 4 and 5 (B) is rational which is not integral (C) is irrational which is a simple surd (D) is irrational whic h is a mixed surd.
i l l BANSAL CLASSES
MA T HE MA TI CS
\8Target IIT JEE 2007 CLASS:XII(ALL)
DATE: 05-06/05/2006 f
,
If a>0an d Lim x-»0 +
' 1l + ax
TIME: 50Min.
DPP. NO.-12
\l/x v
has the value equal to unity then'a'is equal to
l J+ 2x ^
1
( A ) l Q2
Daily Practice Problems
(B)2
(C)3
(D)4
Thefirstthree terms of a geometric sequence are x, y, z and these have the sum equal to 42. If the middle 5y term y is multiplied by 5/4, the number x, — , z now form an arithmetic sequence. The largest possible value (A) 6 of x, is
, y^f.3
^JQA
(B) 12
(C) 24
The value of the expression 2 2 2 sin 1° + sin 2° + sin 3° + (A) 0 (B) 45
(D) 30
2
+ sin 90°, is (C) 45.5
(D )9 0
In a triangleABC with altitude AD, ZB A C = 45°, DB = 3 and CD = 2. The area of the triangle ABC is (A) 6 (B) 15 (C) 15/4 (D) 12 3
2
When the polynomial 5x + Mx + N is divided by x + x + 1 the remainder is 0. The value of (M + N) is equal to (A)-3
(B) 5
(C) - 5
(D) 15
Number of real values of x for which the area ofthe triangle formed by 3 points A( -2 ,1 ) ;B( 1,3) and C(3x, 2 x - 3) is 8 sq. units is (A) 0 (B) 1 (C) 2 (D) infinitely many
p. 7
Assume that p is a real number. In order for ^/x + 3p + l - ^/x = 1 to have real solutions, it is necessary that (A) p > 1/4
(B) p > — 1/4
(C) p > 1/3
(D) p > — 1/3
SUBJECTIVE
, Q9
Find the equation of the circle which has its diameter the chord cut off on the line px +q y - 1 = 0 by the 2 2 2 circle x + y = a . [4] Obtain a relation in a and b, if possible, so that the function n
/ ( X )
QT10
n
n
j . x (a + sin(x ))+ (b - sin(x ) ) = n n n n-»oo ^ (l + x )sec(tan (x + x" ) j - ^ i n u o u s a t x = 1.
[6]
The interior angle bisector of angle A for the triangle ABC who se coordinates of the vertices are A (- 8, 5) ; B( - l 5 , - 19 ) and C(1, - 7) has the equation ax + 2y + c = 0. Find 'a' and V. [6]
i
MATHEMATICS
Target IIT JEE 2007
CLASS: XII (ALL)
v/
Q.l
v/42
DATE:
Daily Practice Problems 08-09/05/2006
TIME:
50Min.
In AABC (a + b)(a - b) = c(b + c), the measure of angle A, is (A) 30° (B) 60° (C) 90°
DPP. NO.-13
(D)120°
The point A (sin 9, cos 9 ) is 3 units away from the point B (2 cos 75°, 2 sin 75°). If 0° < 9 < 369°. Then 9 is (A) 15° (B) 165° (C) 195° (D )2 55 ° The radius of the circle inscribed in a triangle with sides 1 2,3 5 and 37, is (A) 4 (B) 5 (C) 6 (D )7
v ^
4
2 Consider the equation 19z - 3/z - k = 9, where z is a complex variable and i 2 - - 1. Which of th e
following statements is Tme? (A) For all real positive numbers k, both roots are pure imaginary. (B) For real negative real numbers k, both roots are pure imaginary. (C) For all pure imaginary numbers k, both roots are real and irrational. (D) For all complex numbers k, neither root is real.
J*'
.5
The set of values o f x for which the function defined as 1-x /(x)=
x
(1 - x)( 2 - x)
1 < x < 2
3-x
x >2
fails to be continuous or differentiable, is (A) (1)
Q.6
(B) {2}
(C ){ 1, 2)
(D)(1)
The digram shows several numbers in the complex plane. The circle is the unit circle centered at the origin. One ofthese numbers is the reciprocal of F, which is (A) A (B) B (C)C (D)D A triangle has side a -
(b + c) = 5. The ratio of the longest to t he shortest side of the triangle, is 3
(A) 1
„#
, the oppo site angle a = 69°, and the sum of the two other sides is
(B)V2
K)-,
SUBJECTIVE Q.8
Evaluate : Lim n—»oo
-
n+
Va) ( a > o ,
neN)
Use of series expansion and L'Hospital's rule is not allowed. Q.9
[4]
Show that the centroid of the triangle of which the three altitudes to its sides lie on the line y = nijx; y = m 2 x& y = m 3 x l i e o n t h e l i n e , y ( m 1 m 2 + m 2 m3 + m 3 m 1 + 3 ) = (mj +m2 + m3 + 3 m 1 m 2 m 3 )x. [6]
^Q Tl O
Find the equations of the circles which touch the co-ordinate axes and the line, 3x + 4 y = 12.
J s BANSAL CLASSES
MAT H EM AT IC S Daily Practice Problems
V g Target i lT JEE 2007 CLASS: XII (ALL)
DATE: 26-27/04/2006
TIME: 40 to 45 Min each Dpp.
DPR
NO,-8,9,10
-8
Q. 1
A variable straight line whose length is C moves in such a way that one o f its end lies on the x-axis and the other on the y-axis. Show that the locus of the feet of the perpendicular from origin on the variable line has the equation, (x 2 + y 2 ) 3 = C 2 x2y2. |5] cosx /n(x-a)
Q.2
Evaluate:
Q.3
Let t,, t 2 and t3 be the lengths of the tangents drawn from a point (x,, yj) to the circles x + y = a , 2 2 2 2 x + y = 2ax and x + y = 2ay respectively. The lengths satisfy the equation
x->a
[51
ln(e" - e ) 2
ty = t 2 t 2 + a 4 . Show that locus of (xj, y ^ consists of x + y = 0 and x2 + y 2 = a(x + y)
Q.4
T; 9 . 2c os 0 + l Lim Let an = 2 c o s9— l then show that (a,a,a,....a ,a ) = ,' 0 e R . n n v 7 " n~>oo 1 i i ii -l nl
Q. 5
Consider a function f: x —>
x+a x —1
2
2
[5]
; x e R - {1 } where a is a real constant. If / is not a constant
function, findthe following (ii) f~ x , is it exist
(i) the range of /
/
- /
(iii)/
V
/ V
1 W J J
-9 Q. 1
Given Lim x-»0
ffx)
(i) Lim [f( x)l x->0
Q. 2
- 2 then evaluate the following limits, giving explicit reasoning.
x
;(ii)Lim
f(x)
where [x] denotes greatest integer function.
X
x-»0
Find the sum to n terms of the series
S n = c o t - ^ 2 2 + £ )
+
c o r i
f 2
3
+
^ + cor 1 2* +'
23y
+,
upto n terms
Also deduce that Limit S n = c o t _ 1 2 .
[5]
n—
Q. 3
[5]
The vertices of a triangle are A(x t , x }tan 6j), B(x2, x^an 0 2 ) & CCxg, x3tan03). If the circumcentre O of '
. .
^
X
COS0, +CO S0~ +COS 0o
the tnangle ABC is at the ongin & H (x ,y ) beits orthocentre, then show that —= —=-. y sinOj+sinOj+sinOj
f5] Q.4
If (1 + sin t)(l + cos t) = - . Find the value of (1 - sin t)(l - cos t).
Q.5
10 identical balls are distributed in 5 different boxes kept in a row and labled A, B, C, D and E. Find the number ofways in which the balls can be distributed in the boxes ifno two adjacent boxes remain empty.
JF»3F»-;l_ Q. 1
Q
Tangents are drawn fr om any point on the circle x2 + y 2 = R 2 t o the circle x 2 + y2 = r 2 Show that if the linejoining the points of intersection ofthese tangents with th e first circle also touches the second, then R = 2r. [5] - / n ( 2 - c o s 2 x )
f o r x < Q
/n (l + sin 3x) Q.2
Let a fun cti onf (x) be defined as f( x) = sin 2 x _ i
for X > 0
/n(l + ta n9 x)
Find whether it is possible to define f (0) so that ' f ' may be continu ous at x = 0.
Q. 3
Find all possible values of a and b so that f (x) is continuo us for all x e R if
/(x)
Q.4
| ax+ 3 |
,
if x < - 1
13x + a ]
,
if - 1 < x < 0
*si a2*- 2b,
if 0 < x <7t
cos2x-3 ,
if x> 7t
[5]
Prove that in a AABC, the median throu gh A divides the angle Ain to tw o parts whos e cotangents are, 2 cot A + cot C and 2 cot A + cot B and it makes an angle with the side BC whose cotangent is |
(cot B - cot C).
[8]
/
Q.5
[5]
Find the value of y = sin co t-1 cos tan - 1 x wh ere x - cosec cos V
-i
2
J v 3
cos
V6+1 2A/3
>y
1
a BANSAL CLASSES
M A T H E MA T I C S
Target 11T JEE 2007
CLASS: XII (ALL)
Daily Practice Problems
DATE: 24-25/04/2006
TIME: 55 to 60Min.
DPP. NO.-7
Q.l
Let T = {1, 2, 3 ,4 , 5 }. A fu nc ti on /: T- »T is said to be one-to-one if tj * t 2 implies that /(t j) ^/ (t 2 ). Obtain a one-to- one function such that t + / ( t ) is a perfect square for every t in T. [4]
Q.2
-1 If a > b > c > 0 then find the value o f : co t
Q.3
Find the equation to the locu s of the centre of all circles which touc h the line x = 2a and cut the circle 2 2 2 x + y = a orthogonally. [4]
Q.4
Let/(x) = (* -4 )( X 2 ~ 4 X -5) (x
-2x-3)(4-x
(a) the domain o f f (x) (c) all x such th at /( x) >0
C a b+l ^i /bc+n / c a + l"] 1 1 r + co r t — + cot"" — ~ . vc—ay \a -b y vb-cy
[4]
,Fi nd ) (b) the root s o f f (x) (d) all x such th at /( x) < 0
[6]
Q.5
The points ( - 6, 1) , (6 ,1 0) , ( 9, 6) and (- 3, - 3 ) are the vertices of a rectangle. What is the area of the portion of this rectangle that lies above the x axis? [6]
Q.6
Let /(x )= V 2 ax + bx • Find the set o f real values of'a' for which there is at least one positive real value
of 'b' for which the domain of /a n d the range of / are the same set.
Q. 7
[6]
Two circles of different radii R and r touch each other externally. The three comm on tangents form a 2
2(RrW triangle. Show that the area of the triangle is — — - — . R-r
f8|
MATHEMATICS Daily Practice Problems
Target IIT JEE 2007 ZX4r£:
CLASS: XII (ALL)
TIME:
21-22/04/2006
sin 120° The value of
(A)
16 cos 15° • cos 3 0° • cos 120° • cos 240°
2-V3
(B)
8
V3-1
(C)
60Min.
DPP NO.-6
is
2-V3
(D)2-V3
Let S denote the set of all numbers m such that the line y = mx does not intersect the parabola y = x2 +1. S is a bounded interval. The length of S is (A) 3 (B) 3.5 (C) 4 (D) 4,5 . xQ 3
A line lx has a slope of (-2 ) and passes through the point (r, - 3). A second line l2, is perpendicular to lx, intersects at (a, b), and passes through the point (6, r). The value of' a' is equal to 2r
(A)r
V Q.4
(B)
(C) 2 r - 3
(D)
7 11 + 1511 when divided by 22 leaves the remainder (A) 0 (B) 1 (C) 7
5r
(D)10
The coefficient of x in the expansion of (1 + x)(l + 2x) (l + 3x) (A) 4950 (B) 5000 (C) 5050
(1 + lOOx), is (D) 51 00
Suppose AABC is an equilateral triangl e and P is a point interior to AABC. If the distance fro m P to sides AB, BC and AC is 6, 7 and 8 units respectively, then the area of the AABC, is (A) 147V3
^M 7
Q.9
, N 21V3 (C)
(D)
441
m (B) 105
(C) 210
(D) 504 0
If the graphs of y = cos x and y=t an x intersect at some value say 9 in the first quadrant. Then the value of sin 9 is (A)
V
147V3
The tune Twinkle Twinkle Little Star' has 7 notes in its first line, CCG GAA G All notes are held for the same length of time. If the notes are rearranged at random, number of different melodies that can be composed, is
(A) 72 /Q.8
(B)
-1 +
V2
(B)
-l + fi
(C)
- 1 + V5
If S = 1 + - + - + — + 4 9 16
, then 1 + - + — H—— + 9 25 49
S (A)-
3S ( B ) t
(C) > 4
(D)
-1±V5
equals
CD)S-i
Q 10
How many solutions are there for the equation cos 2x - sin 22x = 0 on [0, 2n]7 (A) 6 (B)4 (C )2 (D )l
Q. 11
Numbe r of ways in which 7 people can be divided into two teams, each team having at least one member, is (A) 72
Q. 12
(B) 32
(B) 2y = 2 - x
(C )y = x - 2
(D )y = 2 x - 1
The positive value o f x that satisfies VlO = e x + e~ x, is (A)|/n(4-Vl5)
( B ) ^ / n ( 4 + Vl5)
( Q ~/n(4 + Vl ?) f
^/Q.14
(D )6 3
Let P be a point on the complex plane denoting the complex number z. If (z - 2) (z + /') is a real number then the locus of P is (A) y = 2x + 1
13
(C) 144
Le t/ : R - {0} - > R be any function such that/ (x) + 2 /
n
—
vx;
(D)
^ - J r i )
- 3x. The sum of the values of x for which
/(x)=l,is (A)l ^ QTl 5
(B)2
(A) 5/2
19
(B)2
(D) 1/4
(C)3
1 If sin lf , a = —, then the value of 5
1
H
cos a (B )6
1
2 — h —I 1 + sin a 1 + sin a (C) 8
(D)4 4
, is 1 + sin a (D) 10
A variable circle touch es the x-axis and also touch es the circle with centr e at (0 ,3 ) and radius 2. The locus of the centre of the vaiable circle is (A) an ellipse (B) a circle (C) a hyperbola (D) a parabola
The set of real number s x satisfying ( A ) x >2
\J$.20
(C) 17/4
Sum of th e x an d^ i nterc epts of th e circle described on the line segment joining ( -2 , 1) and (1, 2) as diameter, is
(A) 4 . Q. i 8
the roots, is equal to
(B) 1/2
(A)l Q. 17
(D )- 2
Let r j, r 2 , r 3 , r 4 be the roots of the equation, x 4 - 9x 3 + ax 2 + bx + 16 = 0 r, + r, + r 3 + r. where a, b are constants. Then the difference between the arithmetic mean - — and the 4 geometric mean i ] ^ 2 h r 4
16
(C ) - l
(B )x< - 1
1
X
1 > -3—— is 1 X 2*
(C)x<2 an dx< - 1
(D)-l
A sequence of three real numbers forms an arithmetic progression wit h a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting number form a geometric progression. The smallest possible value for the third term of the geometr ic progression, is (A) 1 (B)3 ( C) 4 (D) 6
•.^M.l 1
The curve such that each point P on the curve has equal distances from the point (2, - 2) and from the line y = x has the equation (A)(x + y)2 = 4 (B) (x - y) 2 = 4 (C) (x + y)2 + 8(y - x) = - 16 (D) (x + y)2 + 2(x + y) = 16 For any straight line, let m and b represent its slope and y-intercept, respectively. Consider all lines having the property that 2m + b = 3. These lines all have the specific point (x l3 y } ) in common. The ordered pair, (Xj, y, ) is equal to (A) (2, 3) (B) (3, 2) (C) (1, 2) (D )( 2, l )
V.Q'.'23
Ifthe sum of the solutions of the equation sin 2x - sin x = cos 2 x on the interval-[0,2ri] is expressed as a7i/b, where a and b are positive integers, a/b in lowest terms, then (a + b) is (A) 8
Q. 24
Q2 5
(A) x 2 - 14x + 48 = 0
(B) x2 - 14x + 50 = 0
(C) x 2 + 14 x- 4 8 = 0
(D) x 2 + 14x + 49 = 0
(D) 11
Which of the following is equal to sec(t) + tan(t) ? n (A) cot H + 4
t
r
(B) cot -2 + — 4
( C ) - c o t -t+— v 2
71
(D)-cot
If / ( x ) = x + 4 and g(f (x>) = 2x + 1, then the function g (x) is (A) 2x - 7
Q.27
(C )1 0
Quadratic equation with real coefficients whose one root is (2 + /')(3 - z) where /' = ^T J , is
[
\ ^2 6
(B) 9
(B) 8x + 3
(C)2x + 9
(D)2x 2 + 5x + 4
In a triangle ABC, Z A = 7 2 ° , b = 2 a n d c = ^5 +1 t henthe triangle ABC is (A) obtuse isosceles (C) right isosceles
(B) acute isosceles (D) not isosceles
MATH EMATICS
Js BANSAL CLASSES
Daily Practice Problems
V8Target I1T JEE 2007 CLASS: XII (ALL)
DATE: 19-20/04/2006
TIME: 50 to 60 Min.
DPR
Only one is correct. There is NEGAT IVE mar king fo r each wro ng answer 1 mark will be deducted.
1
NO.-5
[6 X3 = 18]
Given an isosceles triangle, whose one angle is 120° and radius of its incircle is ^[j,. Then the area of triangle in sq. units is (A)7+I2V3
J%2
(C) 12 + 7^ 3
(E>) 4n
If 0 < 9 < 2n, then the intervals of values of 9 for which 2si n2 9 - 5si n9 + 2 > 0, is (A)
0,
(C)
Q.3
(B) 1 2 - 7 ^ 3
f i5n „ ^ u — , 2 n J v 6 u
n 5tx (B)
71 571
(D)
6'T
If w = a + ip wher e p
8'T 41TT 48
-,7t
w — wz and z ^ 1, safisfies the condition that — — — is purely real, then the set of 1 — z
values o fz is (A) {z: |z| = l } x_ Q.4
(B) {z: z = z )
(C) { z : z * l }
(D) {z : | z | = 1, z ^ 1}
Let a, b, c be the sides of triang le. No two of them are equal and X, e R. If the roots of the equ ation x 2 + 2(a + b + c)x + 3X(ab + be + ca) = 0 are real, then (A)
(B)^>3
(CH
'1 v3'3y
(D) X €
v3 '3 y
^ QC5
If r, s, t are prime num bers and p, q are positi ve integers su ch that the LC M of p, q is r 2 t 4 s 2 , then the numbers of ordered pair of (p, q) is (A) 252 (B) 254 (C) 225 (D) 224
A$.6
LetOe
V 4y
and t, = (tan0)tan0 , t 2 = (tan9) cote, ^ = (cot9) tane , t 4 = (cot0) cote , then
( A ) t j < t2 < t 3 < t 4
(B) t 4 > t 3 > t j > t 2
( C ) t 3 < t j < t 2 < t 4
(D)t2
One or more than one is/are correct. There is NEGA TIVE ma rkin g for each wro ng answer 1 mark will be deducted .
\JQ.1
[ 1 X 5 = 5]
Internal bisector of Z A of a triangle ABC meets side BC at D. A line drawn through D perpendicular to AD int ersects the sid e AC at E an d the side AB at F. If a, b, c repre sent sides o f AABC then 2bc
A cos— b+c 2
(A) AE is harmonic m ean o f b and c
(B) v 7 AD=
4bc . A (C)EF=—sin b + c 2
(D) the triangle AE F is isosceles
Only one is correct. [ 3 x 5 = 15] There is is NEGAT IVE m arkin g for each wrong answ er 2 marks will be deducted. Comprehension Let AB CD be a squa re of side length 2 units. C 2 is the circle thr ough ver tices A, B, C, D and C, is the circle touchi ng all the sides of the square ABCD. L is a line throug h A
, JQ. 8
PA2+PB2+PC2+PD2 I f P is a point on C,1 and Q in anothe r point on C 7 , then 7- 72 , , t^A + v^ ' V*-' V*-' v^D + (A) 3/4
- Q. 9
Q. 10
(B) 3/2
Q.12
( D) 9
A circle circle touches the line L and and the circle C, externally such that both the circles are on the same same side of the line, then the locus of cen tre of the circle is (A) ellipse (B) hyper bola (C) parabo la (D) parts of straight line Ali ne 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 loc us of S cuts M at T 2 and T 3 and AC at Tj, then area of AT j T 2 T 3 is i s (A) 1 /2 sq. units (B) 2/3 sq. units (C) 1 sq. unit uni t (D) 2 sq. units unit s
SUBJECTIVE: There is is NO NEGAT IVE marking. "4
11
(C) 1
ise qua lto
[ 2 x 6 = 12] 12]
If roots of the equa tion x 2 - 1 Ocx Ocx - 1 Id = 0 are a, b and thos e of x 2 - 1 Oax Oax — 1 l b — 0, 0, then find the value of a + b + c + d. (a, b, c and d are distinct numb ers ) v3 If A = — - — + — +• + V( - / l ) n ' — and Bn = 1 - An , then find the minim um natural n A AI A 1 4 J number n 0 such that B n > An . V n > nQ.
4
3BANSAL CLASSES
MATHEMATICS
8T ar ge t IIT JE1 20 07
CLASS : XI (P, Q, R, S)
Q. 1
Which I II III IV (A) I
DaiSy DaiSy Practice Practice Pr ob le ms
DATE: 26/01/2006
TIME: 45 Min.
of the followi ng sets does NOT represent represent a function ? {(x, y) | y = 2x + 1} {(x, y) j x 2 + y2 = 10, y > 0} {(3, 1) (4, 1),(5, 2), (6, 2), (7, 3)} {(x, y) | y = 2X + 1} (B) II (C) III
DPP. NO.-53
(D) IV
(E) none
Q.2
(x) is a fun cti on fro m R R, we say say that f (x) has property I f f (x) I if f (f (x) ) = x for all real num ber x, and we say that f (x) has prop erty II if f (- f( x) ) = - x for all real number x. How many linear functions, have both property I and II? (A) exactly one (B) exactly two (C) exactly three (D) infin ite
Q.3
The fun cti on f (x) is defi ned by f (x) = cos4x + K cos 2 2x + sin 4 x, where K is a constant. If the function f (x) is a constant function, the value of k is (A) - 1
Q.4
(B) - 1/2
(C) 0
(D) 1/2
(E) 1
Defin e the functi on f (n) where n is is a non negative integer satis fying f (0) = 1 and f (n) is define d n-l
. Let 2 m be the highest power of 2 that divides f (20). The value of
for n > 0 as f (n) = n • i=0
m is (A) 18
(B) 19
(C) 20
(D) 21
(E )2 2
Direction for Q.5 and Q.6 The graph of a relation is (i) Symmetri c wit h respect to the x-axis provide d that whene ver (a, (a, b) is a point on the so is (a, - b) (ii) Symmetric with respect to the y-axis provided that when ever (a, b) is a point on the so is ( - a, b) (iii) Symmetric with respect to the origin provided that whene ver (a, b) is a point on the so is is ( - a, - b) (iv) Symmetri c with respect to the line y = x, provided that when ever (a, b) is a point graph, so is (b, a)
Q.5
Q.6
The graph of the relati on x4 + y 3 = 1 is symmetric with respect to (A) the x-axis (B) the y-axis (C) the origin (E) both the x-axis and y-axis
graph, graph, graph, on the
(D) the line y = x
Suppose R is a relation relati on who se graph is symmetric to both the x-axis x-axi s and y-axis, y-axis , and that the point (1, 2) is on the graph of R. Which one of the following points is NOT necessarily necessarily on the graph graph of R? (A) (A) (-1 ,2) " (B) (B) (1 , - 2 ) (C ) (- l , -2 ) (D) (D) (2, (2, 1) (E) all of these points are on the graph of R.
Q.7
Q.8
Q.9
Suppose that f (n) is a real valued func tion whos e domain is the set of positi ve integers and that f (n) satisfies the foll owing two properties f (1) = 23 and f (n + 1) = 8 + 3 • f (n), for n > 1 It follows that there are constants p, q and r such that f (n) = p • q n - r, for n = 1, 2, then the value of p + q + r is (A) 16 (B) 17 (C) 20 (D) 26 (E )3 1 x r x Let f (x) = —— and let g (x) = . Let S be th e set of all real num ber s r such that 1+x 1-x f (g(x)) = g (f (x)) for infinitely many real number x. The number of elements in set S is (A) 1 (B) 2 (C) 3 (D)5
Let f be a linear fun cti on with the properties that f (1) < f (2), f (3) > f (4), and f (5) (5) = 5. Which of the following statements is true? (A) (A) f (0) (0) < 0 (B) (B) f (0) (0) = 0 (C)f(l)
Q. 10
1-x2 If g (x) = 1 - x and and f (g(x (g(x)) ) = — w h e n x * 0. 0. then then f (1/2 (1/2)) equals x (A) 3/4
Q. 11 11
*
2
(B) 1
(C )3
(D )V 2
A function f from from integers to integers is defined as follows f(n) =
- n + 3
if n is odd
L
if n is even
n/2
Suppose k is odd and and f ( f ( f (k))) = 27. The sum of the digit of k is (A) 3 Q. l2
(B) 6
(C) 9
Let f be the function defined by f (x) = ax2 - ^ 2
fo for
(D) 12 some
positive a. If f|f (V 2) )= - ^ 2
equals (A) V2
(B) |
(C) ^
(D)
^
then a
ill BAN SAL CLASSES I S
Ta rg et
MA THE MA TI CS Daily Practice Problems
II T J EE 2 0 0 7
CLASS: XII (ALL)
DATE: 10-11/04/2006
TIME: 35 to 45 Min.
DPP. NO.-l
DPP 1 to 4 complete revision of class XI. This DPP will be discussed on Monday (10-04-2006). Q. V
ABC is triangle. Circle s C p C 2 and C 3 are drawn with sides AB, BC and C Aa s their diameters. The radical axis betwee n any two circles w.r.t the AABC is one of its (A) angle bisector (B) altitude (C) median (D) perpendicular bisector of the sides.
Q. 2"
The function/ ( x ) defined on the real numbers has the property that / ( / (x) ) • (l + / (x)) = - / ( x ) for all x in the domain off. If the number 3 is in the domain and range of f, then the value of / ( 3 ) equals ( A) -3 /2
Q .3
( B) -3 / 4
(C) 1/4
(D) 1/2
If m and b are real numbers and mb> 0, then the line whose equation is y=mx + b cannot contain the point (A) (0,2006) (B) (2006,0) (C) (0 ,- 20 06 ) (D )( 19 ,- 97 )
Which is the inverse of the funct ion/( x) = - / n f x + V x 2 + l j ?
Q.4
(A) 3(e3x + e" 3x )
(B) | (e 3x + e- 3x )
sin (a + P)
p
sin( a - p) p + q (A) v ' p-q
q
if
Q-?
C6 /
(C) ^ (e~3x - e 3x )
' c o t P has the value equal to p-q p+q (C) v(B) ^ ' p+q q
(D) - (e 3x - e~3x)
ta n a
p-q (D)
4
Number of seven digit whole numbe rs in which only 2 and 3 are present as digits if no two 2's are consecutive in any number, is (A) 26 (B) 33 (C) 32 (D )5 3
t
If /(x ) = x 4 + ax3 +bx 2 + cx + d be polynomial with real coefficient and real roots. If | f (i ) | = 1, where i = ^/CT > then a + b + c + d i s equal to
(A) - 1 Q-8-
(B) 1
(C )0
(D) can not be determined
Let ABCD E is a regular pentagon with all sides equal to 4. Whic h one of the followin g is a correct solution for the length A C? (i) 2 csc(18°) (ii) 2 sec (72°) (iii)
V32-32cos(108°)
(A) only (i) and (ii) are correct (C) only (iii) and (i) are correct
(B) only (ii) and (iii) are correct (D) all are correct
9
A line x = k intersects the graph of y= lo g 5 x and the graph of y=l og 5 (x +4 ). The distance between the points of intersection is 0.5. Give n k = a + Vb , where a and b are integers, the value of (a + b) is (A) 5 (B )6 (C) 7 (D) 10
,
Q.10
^yQ. 11
Which ofthe foliowing sets ofrestrictions is true for the function /( x) = ax2 + bx + c represe nted by the graph as show n (A) a > 0 , b < 0 , c > 0 (B) a > 0 , b < 0 , c < 0 (C) a > 0, b > 0, c < 0 (D) none of these The radius of the circle passing through the vertices ofthe triangle ABC , is (A)
8y/l5
(B)
5
(C) 3^5 Q. 12
/ 0
3y/l5
CD) 3V2
The area of the region consisting of all points (x, y) so that x 2 + y 2 < 1 < | x | +1 y |, is (A) n (B)n-l (C )t c- 2 ( D) TC -3 1 3
j Q 1 3
A^.iJ
13
21
7 23
15
5 9 25
17
11 27
19
29
Consecutive odd number s are arranged in rows as shown. If the rows are conti nued in the same pattern, then the middle number of row 51, is (A) 2601 (B) 2500 (C) 270 4 (D) 2401 Q.14
The expression [x + (x 3 -l) 1 / 2 ] 5 + [x-( x 3 - l ) 1 / 2 ] 5 is a polynomial of degree (A) 5 (B )6 (C) 7 (D) 8
Q. 15
Let a, b, c, d, e, f, g, h be distinct elements in the set {-7, - 5 , - 3 , - 2 , 2 , 4 , 6 , 1 3 }. The minimum possible value of (a + b + c + d) 2 + (e + f + g + h) 2 is (A) 30 (B) 32 (C) 34 (D )4 0
J | BANSAL CLASSES
MA TH EM AT IC S
^BTarget IIT JEE 2007 CLASS: XII (ALL)
_ Q.l
Daily Practice Problems
DATE: 12-13/04/2006
TIME: 35 to 45 Min.
If the circles x 2 + y 2 + 2ax + 2by + c = 0 and x 2 + y 2 + 2bx + 2ay + c = 0 where c > 0, have exactly one point in comm on then the value of ^ (A)l
( / Q. 2
DPR NO.-2
(D) 1/2
(C) 2
(B)V2
is
2c
Sup pose / is a real function satisfyi ng/(x +/ (x )) - 4/ (x ) and /( I) = 4. Then the value of /( 21 ) is (A) 16 (B )2 1 (C) 64 (D) 105 100
Q.3
The value of
equals (where i=
)
n=0
(A) - 1 >jQ. 4
n
(B)
y
(D) 97+ i
2n
71
(C)
71
(D) 8
18
The value of b > 0 for which the region bounded by both the x-axis and y = - 1 2 x | + b has an area of 72, is (A) 12
, 0( 6
(C) 96 + i
Given AABC is inscribed in the semicircle with diameter AB. The area of AABC equals 2/9 of the area of the semicircle. If the measure of the smallest angle in AABC is x then sin 2x is equal to (A)
,
(B)i
(B) 36
(C)6V2
(D) 144
If 500 ! = 2 m • N, where N is an odd positive integer, then m is equal to (A) 452 (B) 494 (C )4 98 (D) none of these Le t / be a linear function for whi ch/ (6) - / ( 2 ) = 12. The value o f / ( 1 2 ) - / ( 2 ) is equal to (A) 12 (B) 18 (C) 24 (D )3 0
v^ r .8
a = tan
-1
V2-1
(A) is equal to 1
-tan
1rv2]
I2 J
and P = tan~'(3) - sin
(B) is equal to 0
-1
U s )
I5 J
t pB-) —§tn=i ^
(C) is equal to J 2 - 1
- —
W
, then cot(a - p)
(D) is non existent
v/Q.9
There are three teachers and six students. Numbe r of ways in which they can be seated in a line so that between any two teachers there are exactly 2 students, is (A) 3- 3! - 6! (B) 2 • 6! (C) 2 • 3! • 6! (D )3 -6 !
^QCIO
The average of the numbers n sin n° f or n = 2, 4 , 6, (A) 1
(B)cot 1c
180 (C)tan 1°
(D)
A circle with center O is tangent to the coordinate axes and to the hypotenuse of th e 30°-60°-90° triangle ABC as shown, where AB = 1. To the nearest hundredth, the radius of the circle, is (A) 2.37 (B) 2.24 (C) 2.18 (D)2 .41
^ 1 2
If s = l +
+
+
, t he n 1 - 1 - + I - - 1 - +
(B )^ -s
^ j Q . 13
(B) 10
Q. 14
s — 1
l/(x+l)
(C) 100
5050 (D) 1000
The set of points (x, y) whose distance from the line y = 2x + 2 is the same as the distance from (2, 0) is a parabola. This parabola is congruent to the parabola in standard fo rm y = Kx 2 f or some K which is equal to (A)
. 15
(D)
B
x1/x ^
' X
(A)l
1
equals
(C )" -2
Find the value of x that satisfies the equation log
C
V
12
(B)
V5
4
12
T
The number 2006 is made up of exactly two zeros and two other digits whose sum is 8. The number of 4 digit numbers with these properties (including 2006) is (A) 7 (B) 18 (C) 21 (D) 24
i l l BANSAL CLASSES
MA TH EM AT IC S
Target IIT JEE 2007 CLASS: XII (ALL)
Q. l
Daily Practice Problems
DATE; 14-15/04/2006
Let / be the function defined by /( x, y, z ) : y and z. The smallest possible value of / , is (A) 9 (B )8
(J3-2
^3.3
(x + y + z)(xy + xz + yz) xyz (C)6
DPP. NO.-3
for all positive real numbers x,
(D)3
The right-angled triangle has two circles touching its sides as shown. If the angle at R is 60° and the radius of the smaller circle is 1, then the radius of the larger circle is (A )2V3
(B)2
(C) 2V2
(D )3
Let 7 be a function defined from R +
R + . If [/(xy)] 2 =x (/ ( y ))2 for all positive numbers x and y and
/( 2) = 6 the n/( 50) is equal to (A) 10 (B )30 Q.4
TIME: 35 to 45 Min.
(C) 40
(D )5 0
An equilateral triangle, with sides of 10 inches, is inscribed in a square AB CD in such a way that one vertex is at A, another vertex on BC and one on CD. The area of the square is 100
(A) 25 (2 -V 3)
Q.5
(B) 25(2 + V3)
(C) 25
The coefficient of x 3 in the expans ion of (1 + x + x 2 ) 12 , is (A)352 (B)350 (C)342
(D) 332
The radius ofth e inscribed circle and the radii of the three escribed circles of a triangle are consecutive terms of a geometric progression then triangle (A) is acute angled (B) is obtuse angled (C) is right angled (D) is not possible ^A-l
A function/is defined for all positive integers and satisfies /(I) = 2005 and / l ) + / 2 ) + ... +y (n ) = n2y(n) for all n > 1. The value of /( 200 4) is 1 (A)
^ . 8
1002
1 (B)
2004
2004 (C)
2005
(D)2004
The line (k + 1 ) 2 x + ky - 2k 2 - 2 = 0 passes through a point regardl ess of the value k. Which of the following is the line with slope 2 passing through the point? (A) y = 2x - 8 (B) y = 2x - 5 (C )y = 2 x - 4 (D )y = 2x + 8 Ifthe solution set fo r/ (x ) < 3 is (0,00) and the solution set for /( x) > - 2 is (- 00,5), then the true solution set for (/ (x )) 2 >f(x) + 6,is (A) (-o o,+ 00)
(B)(-00, 0]
(C) [0,5]
(D )( -oo ,0] u[5 ,oo )
Q. 10
,
ABCD is a quadrilateral with an area of 1 and ZB CD - 100°, ZA DB = 20°, AD = BD and BC = DC shown in figure. The product (AC) x (BD) is equal to D^ V3 CA)V
(B)
(C) V3
(D)
2V3 — 4V3 3
Q. 11
Locus of the feet of the perpendicular from the origin on a variable line passing through a fixed point (a, b) (where a * 0, b ^ 0) is a circle with x-in terce ptp and y-intercept q, then (A) p = 0 and q = 0 (B) p = 0 and q * 0 (C) p * 0 and q = 0 (D) p * 0 and q * 0
Q. 12
Two rods AB and CD of length 2a and 2b respectively (a > b) slides on the x and y axes respectively such that the points A, B, C and D are concyclic. The locus of the centre of the circle th rough A, B, C and D is a conic whose length of the latus rectum is .2 (A) — (B) 2aVia 2 - b 2 a (C) 2abVa 2 - b 2
(D) 2 V a 2 - b 2
1 if x is rational v_X).13
Let / ( x ) = 0 if x is irrational
^ / Q . 14
15
A function g (x) which satisfies x f (x) < g (x) for all x is (A) g(x) = sin x (B )g (x ) = x (C )g (x ) = x2
(D) g(x ) = |x
How many of the 900 three digit number have at least one even digit? (A) 775 (B) 875 (C) 450
(D) 750
cot 10° + tan 50 equal to (A) sec 10° (B) sec 5°
(D )c os ec l0 °
(C) co se c5 °
ill BAN SAL CLASSES
MAT HEMAT IC S Daily Practice Problems
Target IIT JEE 2007 CLASS: XII (ALL)
DATE: 17-18/04/2006
TIME: 35 to 45 Min.
DPP. NO.-4
(^A A sequence ofequilateral triangles is drawn. The altitude of each is J 3 times the altitude ofthe preceding triangle, the difference between the area of the first triangle and the sixth triangle is 968 The perimeter oft he first triangle is (A) 10 (B) 12
(C) 16
square unit.
(D) 18
Two circles with centres at A and B, touc h at T. BD is the tangent at D and TC is a comm on tangent. AT has length 3 and BT has length 2. The length CD is (A) 4/3 (B) 3/2 (C) 5/3 (D) 7/4 Q.3
The value of cos 5° + cos 77° + cos 149° + cos 221° + cos 293 ° is equal to (A)0 (B) 1 (C)-l (D) 1/2 Let C be the circle of radius unity centred at the origin. If two positiv e numb ers x, and x 2 are such that the line passing through (x,, - 1 ) and (x 2 ,1) is tangent to C then (D)4x,x2= 1 (A) x,x 2 = 1 (B) X j X 2 = — 1 ( C ) x j + x 2 = L
Q.5
Suppose that (o and z are compl ex numbe rs such that both (1 + 20® and (1 + 2/)z are different real numbers. Th e slope of the line connecting © and z in the complex pla ne is (A)-2 ( B ) - 1 /2 (C) 2 (D) can not be determined
xse c0 + yt an 0 = 2cos 0 J *
6
If
then y equals xt an 0 + yse c0 = cot0 cos 20
(A)
sin0
(B) sin 0
(C) cos 0
(D) sin 20
The locus of the point of intersection of th e tangent to the circle x 2 + y 2 = a2 , which include an angle of 45° is the curve (x 2 + y 2 ) 2 = la 2 (x 2 + y 2 - a 2 ). The value of X is (A) 2 (B )4 (C) 8 (D) 16 Consider the circle x 2 + y 2 - 14x - 4y + 49 = 0. Let 1, and 1 2 be lines through the origin 'O' that are tangent to the circle at points 'A' and 'B'. If the measure of angle AO B is ta n -1 (X) then X equal to 2 (A)~
21 «
28
The value of the expression, 14 tan tan ' - + tan ' - + tan which is equal to (A) 2
(B)5
(C)7
(D)none
13
+ tan
21
+ tan
(D)10
1
31
is an integer
10
If a, b are positive real numb ers such that a - b = 2, the n the smallest value of the constant L for whic h V x 2 +ax - Vx 2 + b x < L for all x > 0, is (A) 1/2
^Q.ll
(B) 1/V2
(C)l
(D )2
If every solution of the equatio n 3 cos 2 x - cos x - 1 = 0 is a sol uti on of the equ at ion a cos 2 2x + bcos2x - 1 = 0 . Then the value of (a + b) is equal to (A) 5 (B) 9 (C) 13 (D) 14
12
What is the y-intercept of the line that is parallel to y= 3 x, and which bisects the area of a rectangle with corners at (0,0) , (4,0) , (4,2 ) and (0, 2)? (A) (0 ,- 7) (B )( 0, -6 ) (C) (0, - 5) (D )( 0, -4 )
13
Let / ( x ) = x 2 + kx ; k is a real number. The set of values of k for whic h the equ ation f (x) = 0 and / ( / ( x ) ) = 0 have exactly the same real solution set is (A) (0,4)
^ 1 4
Q^ l 5
(B) [0,4)
If X'°83(4) = 27, then the value of x 0og 3 4)2 (A) 4 (B) 16
(C) (0, 4]
(D) [0, 4]
(C) 64
(D) 81
If Q is the point on the circle x 2 + y 2 - 1 Ox+ 6y +2 9 = 0 which is farthest from the point P(- l, - 6) , then the distance from P to Q is (A) 2V5
(B) 2V7
(C )4V 5
(P)4y/7
ill BAN SAL CL ASSES
MATHEMATICS
V 8 T a r g e t IIT JEE 2007 CLASS: XI (P, Q, R, S)
Daily Practice Proble ms
DATE: 14/11/2005
TIME: 120 Min.
DPP. NO.- 50
DPP OF THE WEEK This is the test paper of Class-XI (J -Ba tch) held on 13-11-2005. Take exactly 120 minutes. To be discuss on Friday (18-11-2005) PART-A
Only one alternative is correct. There is NEGATIVE marking. For each wrong answer 0.5 mark will be deducted. ZERO for not attempted. Q. 1
If the solutions of the equation sin20 = k ( 0 < k < l ) i n ( 0 , 2 n ) are in A. P. then the value of k is (A )|
Q. 2
(B)^
(C)^
(D) i
Number of real x satisfying the equation | x - l | = | x - 2 | + | x - 3 | i s (A) 1
Q.3
[20 x 1.5 = 30]
(B )2
(C) 3
(D) more than 3
A rectangle has its sides of length sin x and cos x for some x. Largest possible area which it can have, is (A) --1 T
(B) 1
(C) ~
(D) can not be determined
Z
Q.4
Consider an A. P .t jj ^t g, If 5th , 9th and 16th terms of this A.P. form three consecutive terms of a GP. with non zero common ratio q, then the value of q is (A) 4/7 (B) 2/7 (C) 7/4 (D)none
Q.5
The new coordinates of a point (4,5 ) when the origin is shifted to the point (1, -2 ) are (A) (5,3) (B) (3, 7) (C) (3, 5) (D)none
Q.6
A particle is moving along a straight line so that its velocity at time t > 0 is v (t) = 3t2. At what time t during the interval from t = 0 to t = 9 is its velocity the same as the average velocity over the entire interval? (A) 3 (B) 4.5 (C) 3(3)1/2 (D)9
Q.7
Acute angle made by a line of slope - 3/4 with a vertical line is (A) cot _1[ A)
Q. 8
^
(B) tan" 1 - I
_
- i f 3^
(C) tan -1 ! 2
.-l S 3\ (D) cot v2y
If logAB + log B A 2 = 4 and B < A then the value of iog A 8 equals ( A) V2 - 1
(B) 2 V 2 - 2
(C) 2 - V3
(D) 2 - V 2
Q. 9
The sum of 3 real numbers is zero. If the sum of their cubes is 7ccthen their product is (A) a rational greater than 1 (B) a rational less than 1 (C) an irrational greater than 1 (D) an irrational less than 1
Q. 10
Three circle each of area 4n, are all externally tangent (i.e. externally touch each other). Their centres form a triangle. The area of the triangle is (A) 8V3
(B) 6v'3
(C) 3^ 3
(D) 4 ^ 3
Q.ll
I f th e li ne s L, : 2 x + y - 3 = 0 , L 2 : 5 x + k y - 3 = O a nd L 3 : 3 x - y - 2 = 0, are concurrent, then the value of k is (A )- 2 (B)5 ( C )- 3 (D)3
Q. 12
Suppose x, y, z is a geometric series with a common ratio o f'r' such that x ^ y . If x, 3y, 5z is an arithmetic sequence then th e value of'r 1 equals (A) 1/3 (B) 1/5 (C) 3/5 (D) 2/3
Q. 13
The radius of the incircle of a right triangle with legs of length 7 and 24, is (A) 3 (B) 6 (C) 8.5 (D) 12.5
Q. 14
Number ofintegers which simultaneously satisfies the inequalities | x | + 5 < 7 and | x - 3 | > 2, is (A) exactly 1 (B) exactly 2 (C) more than 2 but finite (D) infinitely many
Q. 15
The value of (V Tj) 2959 is (A)l
Q. 16
'
Q. 18
17^ 2,
If x 2 +
x
= 7 then the value of
(D)-V=l
(B )2 1
X
(D)none
( D ) - 8 + 3/
equals (x>0) (C) 24
(D) 27
S et of all real x satisfying the inequality ! 4i—1 - lo g 2 x | > 5 is, where i = ^ p l . (A) [4, oo)
Q.20
(C)(l,-8)
(B) (13,25)
If F (x) = 3x 3 - 2x 2 + x - 3, then F(1 + /') has the value e qual to (A) 8 + 3/ (B) 8 - 3 / (C) — 8 — 3/
(A) 18 Q. 19
(C)V^T
The points Q = (9 ,1 4) and R = (a, b) are symmetric w.r.t. the point (5, 3). The coordinates of the point R are
(A) v Q. 17
(B)-l
(B)
r i
r A i i (C) I 0, — 16.
(D)
( I
ll 16.
u [4 , °o)
Let Xj and X2 are two realnumb ers such that x 2 + x 2 = 7 and xj* + x 2 = 10. Find the largest possible value of Xj + X2 is (A) 8
(B) 6
(C) 4
(D )2
PART-B
Q. l
For what values of m will the expression y2 + 2xy + 2x + my - 3 be capable of resol ution into t wo rational factors?
[3]
Q.2
If one root of the quadratic equation x2 + mx - 24 = 0 is twi ce a roo t of the eq uation x 2 - (m + 1 )x + m = 0 then find the value of m. [3]
Q.3
Ifx is eliminated fro m the equation, si n( a+ x) = 2b and si n (a -x ) = 2c, then find the eliminant. [3] r
2(x - 2)
N
Q.4
Solve the logarithmic inequality, lo g j
Q.5
Find allx such that ^T k - x k=l
Q.6
Find the area of the convex quadrilateral whos e vertices are (0 , 0) ; (4, 5) ; (9 ,2 1) and (- 3, 7) .
,(x + l)(x-5),
=2 0.
P]
[3]
P] Q.7
Find the direction in which a straight line must be drawn thro ugh the point (1 ,2 ) so that its point of intersection with the line x + y = 4 may be at a distance ^ y[6 from this point.
Q. 8
[4]
We inscribe a square in a circle of unit radius and shade the region between them. Then we inscribe another circle in the square and another square in the new circle and shade th e region between th e new circle and the square. Ifthe process is repeated infinitely many times, find the area of the shaded region. [4]
Q.9
In a AABC, if a, b, c are in A.P, then pro ve that cos(A - C) + 4cosB = 3
[4]
i l l BAN SAL CLASSES
MATHEMATICS
v B T a r g e t IIT JEE 2007 CLASS: XI(P, Q, R, S)
Daily Practice Problems
DATE: 10/10/2005
TIME: 50Min.
DPP. NO.- 49
Q. 1
Identify whether the statement is Tr ue or False. 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 th an 10 metres, but the area of the first triangle is larger than the area of second triangle.
Q.2
Number of positive integers x for which/ ( x ) = x3 - 8x 2 + 20x - 1 3 , is a prime number, is (A) 1 (B)2 (C) 3 (D) 4
Q. 3
The value of m for the zeros of the polynomial P(x) = 2x2 - mx - 8 differ by (m - 1 ) is 10 (A)4,-y
10 (B)-6 ,—
10 (C)6, —
10 (D)6,-y A
Q. 4
Each side of triangle ABC is divided into 3 equal parts. The ratio of t he area of hexagon UV WX YZ to the area of triangle ABC i s u, 5 (A) -
2 (B) j
1 (C) 2
3 (D) 4
Q.5
If cos A, cos B and cos C are the roots of the cubic x3 + ax 2 + bx + c = 0 where A, B, C are the angl es of a triangle then (A) a 2 - 2 b - 2c = 1 (B) a 2 - 2b + 2c - 1 (C) a 2 + 2 b - 2c = 1 (D) a 2 + 2b + 2c = 1
Q.6
What quadrilateral has the points (- 3, 6) , (-1 , -2 ), (7 , - 4 ) and (5, 4) taken in order in the xy-plane as its vertices? (A) Square (B) Rhombus (C) Parallelogram but not a rhombus (D) Rectangle but not a square
Q. 7
Which of these statements is false? (A) A rectangle is somet imes a rhombus. (B) A rhombus is always a parallelogram. (C) The digonals of a parallelogram always bisect the angles at the vertices. (D) The diagonals of a rectangle are always congurent.
Q. 8
Points P and Q are 3 units apart. A circle centre at P with a radius of 3 units intersects a circle centred at Q with a radius of ^ 3 units at point A and B. The area of the quadrilateral APB Q is
(A)V99
a/99 (B) - f -
[99 (C) ^ f
(D)
199 ^
Directions for Q.9 to Q.ll: A straight line 4x + 3y = 72 inte rsect the x and y axes at A and B respectively. Then Q.9
Distance between the incentre and the orthocentre of the triangle AOB is (A) 2V6
(B)3V 6
(C) 6V6
(D) 6V2
Q. 10
The area of the triangle whose vertices are the incentre, circumcentre and centroid of the triangle AOB in sq. units is (A) 2 (B) 3 (C) 4 (D) none
Q.ll
The radii of the excircles of the triangle AOB (in any order) fon n (A)anA.P. (B)aG.P. (C)an H.P.
(D) none
Directions for Q.12 to Q.15: Consider two different infinite geometric progressions with their sums S j and S7 as 00 S ] = a + ar + ar 2 + ar 3 + 2 3 S 2 = b + bR + bR + bR + 00 If Sj = S 2 = 1. ar = bR and ar 2 = — then answer the followi ng: Q.12
The sum of their commo n ratios is (A)
Q. 13
Q. 14
(B )
4
The sum of their first terms is (A )l (B) 2
(C)l
(D>2
(C)3
(D)none
Common ratio ofthe first G.P. is (A)
Q.15
1
1 (B)
l-x/5
(C)
V5-1 4
(D)
V5+1
Common ratio of the second G.P. is (A)
3 + V5
(B)
3-V5
(C)
(D)none
ill BAN SAL CLASS ES
MATHEMATICS
V 8 T a r g e t I1T JEE 200 7 CLASS: XI (P, Q, R, S)
Daily Practice Prob lems
DATE: 03/09/2005
TIME: 120 Min.
DPP
NO.-48
This is the test paper of Class- XI (J-Batch) held on 02-10-2005. Take exactly 2 hours. PART-A
Only one alternative is correct. There is NEGATIVE marking. For each wrong answer 0.5 mark will be deducted. Q. 1
[20 x 1 = 20]
If n arithmetic means are between two quantities 'a' and 'b' then the /7th arithmetic mean is b + na (A) v ' n + 1
v(B) 7
a + nb
w(C)
n
n ( b - a )2 p n + 1
a + nb (D) v ' n + 1
Q. 2
If log a b + log bc + logca vanishes where a, b and c are positive reals different than unity then the value of (loga b) 3 + (log bc) 3 + (logca) 3 is (A) an odd prime (B) an even prime (C) an odd comp o site (D) an irrational number
Q. 3
Sum to n terms of the sequence
+ ^21 + T>4l+ .
77(3" -1)1 (C) — ^
Q.4
(D) none of these
If th ea rc so ft he same length in two circles S t and S 2 subtend angles 75° and 120° respectively at the S, centre. The ratio — is equal to S
2
, 1 (A) J Q. 5
a2 b + 1
(A) 3
25 (D)-
is equal to (B) 2
2 3 2 3 2 3 — + — + — + —r + —r + 5 5 5 5 5s 5 15 (A)^
Q. 7
64 ( O -
If th e roots of the cubic, x3 + ax2 + bx + c = 0 are three consecutive positive integers. Then the value of
Q.6
81 CB)-
(C)l
+
00
(D) none of these
isequalto
13 (B)^
Number of princip al solution of the equation tan 3x - tan 2x - tan x = 0, is (A) 3 (B) 5
3 (C)?
4 (D)?
(C )7
(D) more than 7
Q. 8
If the mth , nth and pth terms of G P. form three consecutive terms of another G.P. then m, n and p are in (A)A.P. (B)GP. (C)H.P. (D)A.GP.
Q. 9
Each of the four statements given below are either True or False. I.
1 sin765° = - ^
II.
cosec(-1410 °) = 2
m.
1371 1 tan— = ^
IV.
cot
1571
4 .
= -1
Indicate the correct order of sequence, where 'T' stands for true and 'F' stands for false. (A) F T F T (B )F FT T ( C ) TF F F (D ) FT FF Q. 10
Q . ll
oo 2 k+2 The sum ^T —— equal to k=i 3 (A) 12 (B) 8
/n21 -/nl2
^
/nl 2 + /n5
/n5 + /nl 2
/nl2 -/n 21
/nl44 -/n21
+
Q.15
(B) 3lo g 65
Q. 17
(C )c os2 (a-p)
(D)sin2(a-p)
(C) 3log 56
(D)3
The quadratic equation X 2 - 9X + 3 = 0 has roots r and s. If X2 + bX + c = 0 has roots r 2 and s2, then (b, c) is (A) (75,9) (B) (-75 ,9) (C) (-8 7,4 ) (D) (-8 7,9 ) ^ . ta n 2 2 0 ° - s i n 2 20° . simplifies to The expression T ; ta n 2 20°-sin 2 20° (A) a rational which is not integral (C) a natural which is prime
Q.16
/nl2-5/ n21
Which of the following is the largest? (A)2 1 o 8 s 6
n i r
^
y 2
2xy JL co s( a - P) is equal to a bl ab (A) sec2 ( a - P) (B) cosec2 ( a - P)
Q. 14
/nl 2
If 0 is eliminated from the equations x = a cos(0 - a ) and y = b cos (0 - P) then x2
Q. 13
(D) 4
The value of p which satisfies the equation 122p_1 = 5(3 p -7 p) is / n 5 - / nl 2
Q. 12
(C) 6
(B) a surd (D) a natural which is not composite
202 4 571 971 If sin 2x= r r r r , where — < x < — , the value of the sin x - cos x is equal to H 2025 ' 4 4
If a, b, c are real numbers such that a2 + 2b = 7, b2 + 4c = - 7 and c2 + 6a = - 14 then the value of a 2 + b 2 + c2 is (A) 14 (B )2 1 (C) 28 (D) 35
Q. 18
Q.19
Q.20
The value of x that satisfies the relation 00 x = l - x + x2-x3 + x 4-x5 + (A) 2 cos36° (B) 2 cos 144° (C )2 si nl 8° (D)none 2 If sin 0 and cos 9 are the roots of the equation ax - bx + c = 0, then (A) a 2 - b2 ^2 ac (B)a 2 + b 2 = 2ac (C) a 2 + b2 + 2ac = 0 ( D ) b 2 - a 2 = 2ac The equation, | sin x | = sin x + 3 in [0, 2tc] has (A) no root (B) only one root (C) two roots
(D) more than two roots
More than one alternative are correct. There is NO negative marking.
[ 5 x 2 = 10]
Q.21
Theval ue(s) of 'p' for which the equation a x 2 -px + ab = 0 and x 2 - a x - b x + ab = 0 may have a common root, given a, b are non zero real numbers, is (A) a + b 2 (B) a 2 + b (C) a(l +b ) (D) b( l+ a)
Q.22
If ax 2 + b x + c = 0 , b * l be an equation with integral co-efficients and A > 0 be its discriminant, then the equation b 2 x 2 - Ax - 4 a c = 0 has : (A) two integral root s (B) two rational roots (C) two irrational root s (D) one integral roo t independent of a, b, c.
Q.23
Fo rt he AP. given by a t , a^, (A) aj + 2a2 + % = 0 (C) a, + 3a2 - 3a3 - a 4 = 0
Q. 24
, an,
, the equations satisfied are (B) ^ +%=0 (D) aj + - 4a4 + a 5 = 0
V3s in( a + P)
•
It is known that sin P = — and 0 < P < % then the value of 5
r - r T cos(a
cosItc 6) v sin a
+ P)
' '
is:
5
Q. 25
(A) independent of a for all p in (0,7t/2)
(B)
(7 + 24cota) (C) — - — — for tan P < 0
(D) none
for tan p > 0
The sum of the first three terms of the G.P. in which the difference between the second and the first term is 6 and the difference between t he fourth and the third term is 54, is (A) 39 (B) -1 0. 5 (C) 27 (D)-27 PART-B
Q. 1
If c os (a + p ) + s i n ( a - p ) = 0 a n d t a n p = ^ ^ . F i n d t a n a .
[3]
Q.2
If a , p are the root s of ax2 + bx + c = 0, find the value of (a a + b)~3 + (ap + b)~3.
[3]
Q. 3
Find the largest integral value ofx satisfying the inequality log
2
(3-2x)>l.
[3]
Q.4
If between any tw o positive quantities there be inserted two arithmetic means A p A^; two geometric means G t , G 2 and two harmonic means Hj , F^, then show that G j G 2 : H , H 2 = A 1 +A2 : Hj + U 2 .
P] Q. 5
Find all the values of the parameter'm' for which every solution o f the inequality 1 < x < 2 is a solution of the inequality x 2 - mx + 1 < 0. [3]
Q, 6
Find the general solution of the equation, si n4 2x+cos 4 2x = sin 2x cos 2x.
Q. 7
Find the sum of the series,
Q. 8
Show that the triangle ABC is right angles if and only if si nA+ sinB + sinC = co sA + cosB + cosC + 1.
^
1.2.3
H
^
I
2.3.4
+
I +— —— — . n( n + l)( n + 2)
[3]
[41
[4] Q. 9
Find the real solutions to the system of equations log10 (2000xy) - log 10 x • l og 10 y = 4 l o g 1 0 ( 2 y z ) - l o g 1 0 y l o g 1 0 z=l and log 1 0 (zx)-log 1 0 z-log 1 0 x = 0.
[4]
i l l BANSAL CLASSES
MATHEMATICS
Target NT JEE 20 07 CLASS: XI (P, Q, R, S)
Dai ly Practice Pr ob le ms
DATE: 26-27/09/2005
TIME: 60 Min.
OB JE CT IV E PR AC TI CE Select the correct alternative. Only one is correct. For each wrong answer 1 mark will be deducted.
DPR
NO.-47
TES T
[3 x 20 = 60]
Q. 1
In a triangle ABC , R(b + c) = a Vbc where R is the circumradius of the triangle. Then the triangle is (A) Isosceles but not right (B) right but not isosceles (C) right isosceles (D) equilateral
Q.2
Starting with a unit square, a sequence of square is generated. Each square in the sequence has half the side length of its predeces sor and two of its sides bisected by its predeces sor's sides as shown. This process is repeated indefinitely. The total area enclosed by all the squares in limitin g situation, is
Q.3
5 (A) - sq. units
79 (B) — sq. units
75 (C) — sq. units
1 (D) — sq. units
1 1 1 + Thesum — — — — — — H — : — — — — — . sm 45 sin46° sin4 7°si n48° sin4 9°sin 50° (A) sec (1)°
Q.4
Q.5
(B) cosec (1)°
1 + . . is equal to M sin 133°sin 134°
(C )c ot (l ) 0
(D)none
8 _ Number of real values of x e (0, n) for which — — — ^ 3 sin2 x < 5, is d sin x. sin J X (A) 0 (B) 1 (C) 2 (D) infinite If f (x) = x 2 + 6x + c, wher e 'c' is an integer, then f (0) + f (-1 ) is (A) an even integer (C) an odd integer not divisible by 3
(B) an odd integer always divisible by 3 (D) an odd integer may or not be divisible by 3
Q.6
If abed = 1 where a, b, c, d are positive reals then the mini mum val ue of a 2 + b 2 + c 2 + d 2 + ab + ac + ad + be + bd + cd is (A) 6 (B) 10 (C) 12 (D) 20
Q.7
Minimum vertical distance between the graphs of y = 2 + s in xa nd y = co sx is (A) 2
Q. 8
(B)l
(C)V2
(D)2-V2
A square and an equilateral triangle have the same perimeter. Let Ab e the area of the circle circumscribed A about the square and B be the area of the circle circumscribed about the triangle then the ratio ~ is B 9 (A) .j g
Q.9
3 (B) -
27 (C) -
Iflog 10 sinx + lo g 1 0 c o s x = - 1 and log 10 (sinx + co s x )= (A) 24
(B) 36
(C) 20
(D)
( l o g ] 0 n)-l
3V6 - f
then the value of 'n'is (D )1 2
Q. 10
Let f (x) = x 2 +x 4 + x 6 + x 8 + oo for all real x such that the sum conver ges. Numb er of real x for which the equation f (x) - x = 0 holds, is (A) 0 (B) 1 (C) 2 (D)3
Q.ll ^
Find the smallest natural 'n' such that tan( 107n)° = (A) n = 2
Q. 12
(B) n = 3
(D)n = 5
ABC is an acute angled triangle with circumcentre 'O' orthocentre H. If AO - A H then the measure of the angle A is 71
7t
(A)Q.13
cos 96° +si n 96° — . — . cos96 -s in 96 (C)n = 4
71
(C )j
571
(D)~
Let a, b, c be the three roots of the equatio n x 3 + x 2 - 333x - 1002 = 0 then the value of a3 + b 3 + c 3. (A)2006 (B)2005 (C)2003 (D)2002 44
cosn Q. 14
Let x = j 44 y
then the greatest integer that does not exceed 1 OOx is equal to
sin n° Z n=l
(A) 240
(B) 241
(C) 242
(D )2 43
Q. 15
The number s b, c, d are all integers. The parabola y = x 2 + bx + c and the line y = dx have exactly one point in com mon. With these assumption, which one ofthe following statement is necessarily True? (A) b = 0 (B) d - b is even (C) | a |2 = | b | 2 (D )c = 0
Q. 16
The numbe r of solutions to the system of equations y 2 - x y - | x | y + x | x | = 0 and x 2 + y 2 = 1 is (A )l (B) 2 (C) 3
(D) 4
Answer the following questions on the basis of the information given below: (Q.18 to Q.21) Triangle ABC has vertices A (0,0 ), B (9, 0) and C (0,6) . The points P and Q lie on the side AB such that AP = PQ = QB. Similarly the points R and S lie on the side AC so that AR = RS = SC. The vertex C is joined t o each of the poi nts P and Q in the same way, B is jo in ed t o R an d S. Also the line segment PC and RB intersect at X and the line segments QC and SB intersect at Y. Q. 17
Equation of the line AX is ( A ) y =| x
Q.18
Q. 19
Equation ofthe line XY is (A) 3 x - 4 y = 0 (B) y = x + 1
(C)y=|x
( D )y = |x
( C ) 4 x- 4y + 3= 0
(D)none
Radius ofthe circle inscribed in the triangle APS is (A) 4
Q. 20
(B) y = x
(B) 1
(C)j
(D) 2
Distance betwee n centroid and circumcentre of the triangle ABC is Jl 3
2J13
Ju
J u
MATHEMATICS
y j T a r g e t IIT JEE 2007
Daily Practice Problems
CLASS: XI (PQRS)
DATE: 12-15/09/2005
DPR NO.- 43, 44
Take approx. 50 min. for each Dpp.
DPP
-
43
Q.l
A B C In a triangle ABC, prove that, t a n y + t a n — + t a n — > ^3
Q. 2
Find the general solution of the equation, cos ( 1 0x + 1 2 ) + 4V2 sin(5x + 6) = 4.
Q.3
If p, q, r be the roots of x 3 - ax 2 + bx - c = 0, show that the area of the triangl e whose sides are p, q & 1 r is — [a(4ab - a 3 - 8c) ] m .
ta n( a + f3- y) _ tany tan ( a - P + y) ~ ta np '
Q- 4
I f
Q.5
In the triangle A' B' C, having sides B' C = a ' , A' C = b' and A' B' = c' , a circle is drawn touching two of its sides a' & b' and having its diameter on the side c'. If A' is the area of the triangle A' B' C , find the radius of the circle. Further, a line segment parallel to A' B' is drawn to meet the sides C A ' , CB' (produced) in points A & B respectively and to touch th e given circle fo rming a triangle ABC with sides BC = a, AC = b and AB = c. If A denotes the area of the triangle ABC, show t hat ; CO
s h o w that eithe r sin
a b c a' + b' + c' = = 77 ~ 17 77 7T~u~r a' b' c' a' + b'
md
,
PPP
^ ~^
„ a w ) (
-
= 0
A A'
' o r ' s i n 2 a
+ sin2
^ +
sin2y = 0
( a ' + b' + c ' ^ V a' + b
44
Q. 1
If log 10 (l 5) = a and log 20 (50) = b then find the value of log 9 (40)
Q. 2
Find the general and principal solution ofthe trigonometric equation sec x - 1 = ( ^ 2 _ i ) t a n x
Q.3
The ratios of the lengths of the sides BC & AC of a triangle ABC to the radius of a circumscribed circle are equal to 2 & 3/2 respectively. S how that the ratio of the lengths of the bisectors of the interior angles B & C is,
7(V7-l) 9V2
Q.4
If two vertices of a triangle are (7 ,2 ) and (1 ,6 ) and its centroid is (4 ,6 ) find the third vertex.
Q.5
If A , B , C are the angles of a triangle & sin3 6 = sin (A - 0 ) . sin (B - 9) . sin (C - 0), pr ove that cot 0 = cot A + cot B + cot C.
Ja BANSAL CLASSES
MATHEMATICS
Target II T JEE 200 7 CLASS: XI(PQRS)
Da il y Practice Pro bl ems DATE: 16-17/09/2005
DPR
NO.-45~46
Take approx. 50 min. for each Dpp.
DPP
-
45
Q. 1
If the sum of the pairs of radii of the escribed circle of a triangle taken in order round the triangle be denoted by, s ,l s 2 , s 3 and the corresponding diff erence s by dj , d 2 , d 3 , prove th at, dj d 2 d 3 + d, s 2 s 3 + d ? s 3 s, + d 3 s, s 2 = 0;
Q. 2
Find the general solution of the trigonometric equation cosec x - co sec 2x = cosec 4x
Q.3
Let the incircle of th e A ABC touches its sides BC , CA & A B at A j , Bj & Cj respectively. If pj , p2 & p 3 are the circum radii of the triang les, Bj I C j , Cj I A, and A, IB, respectively, then prove that, 2 p, p 7 p 3 = Rr 2 where R is the circumradius and r is the inradius ofthe A ABC.
Q.4
Ifthe area of th e triangle formed by the points (1 ,2 ) ; (2, 3) and (x, 4) is 40 square units, find x.
Q. 5
If a , p, y are angles, unequal and less than 2n, which satisfy the e quation a
b — + + c = 0, then prov e that si n( a + P) + sin(P + y) + sin (y + a ) = 0 cosB sinQ
DPP
-
46
Q. 1
If dp d 2, d 3 are diameter s of the excircles of AABC, touchin g the sides a, b, c respectively then prove
Q.2
Show that for any triangle 2r < R (where R is the inradius and R is the circ umradius)
Q.3
Find the least positive angle satisfying the equation cos 5a = cos 5 a.
Q. 4 ^
Find the equation of the straight line which passes through the point (1,2) and is such that the given point bisects the part intercepte d betwee n the axes.
Q.5
In a A ABC, if cosA + cosB = 4sin 2 -y, prove that tan y . t a n ^ = ^ . Hence deduce that the sides of the triangle are i n A.R
J j B A N S A L C L A S S E S
MATHEMATICS
v S T a r g e t ||T JEE 200 7 CLASS: XI (PQRS)
Q. 1
Dai ly Practice Pro ble ms
DATE: 09/09/2005
TIME: 60Min.
DPP. NO.- 42
If a, b, c are positive real number such that log a
lo gb
log c
b - c
c-a
a-b
then prove that a b
+c
+
b
c +a
+
c
a + b > 3
Q.2
Find all values of k for which the inequality, 2x2 - 4k 2 x — k 2 + 1 > 0 is valid for all real x which do not exceed unity in the absolute value.
Q.3
Find the values of' p' for which the inequality, (
2
- ( p £ r ) )
x2 + 2 x
(1
+ 1o
& ph)
-2(
! +
p fr )
>0
is valid for all real x.
Q.4
1 — — — 8 If positive square root of , a* . (2 a ) 2 a . ( 4 a ) 4 a . (8a) 8 a . .. . .... °° is — , find the value of 'a1.
Q.5
Provethat
Q.6
x Find the general solution of the equation (1 + c o s x ) i j t a n — - 2 + sinx = 2 cos x
Q.7
1
2 4 - + - = — + ^ 4— + x + 1 x + 1 x +l
2" 2 1 + —2 = ——r^ T"; (x +1) 1-x2
Ifp, q, r be the lengths of the bisectors of the angles of a triangle ABC from the angular po in ts A, Ban d C respectively, prove that 1
A 1 B 1 C 1 1 1 cos —i— cos cos = —l b— and — H— — w p 2 q 2 r 2 a b c Q.8
pqr abc(a + b + c) nil ^ ^ = K > 4A (a + b)(b + c)(c + a)
If x, y, z are perpen dicul ar distances of the vertices of a A ABC from the opposit e sides and A is the area of the triangle, the n prove that + —r + -r2 - = — v (cot A + cotB + cotC ) —7 x2 y2 z A
ill BAN SAL CL AS SES
MATHEMATICS
V S Targe* II T JEE 20 07 CLASS : XI (PQRS)
Dai ly Practice Pro ble ms
DATE: 05/09/2005
TIME: 60 Min.
DPP. NO.- 41
Q. 1
Solve the inequality, ^j \o gy 2 x + 41og2 Vx < V2 (4 - lo g^x 4 ).
Q. 2
Find the set of real values of 'a' for which there are distinct reals x, y satisfying x=a-y2 and y = a-x2.
Q.3
A polynomial in x of degree greater than 3 leaves the remainder 2, 1 and - 1 when divided by (x - 1) ; (x + 2) & (x + 1) respectively. Find the remainder, if the polynomial is divided by, (x 2 - 1) (x + 2).
Q.4
Find the general solution of the equation sin 6x + cos 6 x = — . 4
Q.5
If pj, p2 are the roo ts of the quadratic equation, ax2 + bx + c = 0 and q ]5 q 2 are the roots of the quadratic equation cx2 + bx + a = 0 such that Pj , qj , p2 , q 2 is an A.P. of distinct terms, then prove th at a + c = 0 whe re a, b, c e R. 88
Q.6
r
Q. 7
Q. 8
1 cos k T\ ——: 7 — = —x— ^ cosnk • cos(n + l)k si n^ k
Let k = 1 t h e n prove that
S olve the equation,
1
2
x
2
2
J
+ V2 4 cos
J 1 UJ
cosx 2
y
Let a l , a2, a 3, a4 and b be real numbers such that 4 b + X a K = 8
4 ; b + Z4 = 1 6 2
K=1
K=1
Find the maximum value of b.
?
J j S A N S A L C L A S S E S
MATHEMATICS
^ B T a r g e t l i t JEE 2006 CLASS: XI (PQRS)
Daily Practice Problems
DATE: 29/08/2005
P P P
OF
Max. Marks: 60
T H E
DPR
NO.-40
W E E K
This is the test paper of Cla ss -XI (J-Batch) held on 28-08-2005. Take exactly 120 minutes.
Q. 1
22 x If sec x + tan x = — , find the value of tan—. Use it to deduce the value of cosec x + cot x. [3]
Q.2
Simplify the expression
Q.3
1 1 1 Prove that . • + — — r ~ + ~ — T ~ + sin2 x sin 2 x sm 2 x
-r + -r . lo g 4 (2000) 6 log 5 (2000)
1 + . = cot x - cot 2n x for any natural number sin 2 x
n and for all real x with sin 2 r x ^ 0 where r = 1,2, Q.4
Q.5
[3]
n.
[3]
Let X = sin 2 72° - sin 2 60° and Y = co s 2 48°-sin 2 12° Find the value of XY.
[3]
If A + B + C = ^ then prove that £ s i n 2 A + 2 ] ~ [ s i n A = 1-
Q. 6
PI
The position vector of a point P in space is given by r = 3 cos t i + 5 sin t j + 4 cos t k
(a)
Show that its speed is constant.
(b)
Show that its velocity vector v , is perpendicular to r .
[3]
Q.7
Find the value o f k for which the graph of the quadratic polynomial P (x) = x2 + (2x + 3)k + 4(x + 2) + 3k - 5 intersec ts the axis of x at two distinct points.
[3]
Q.8
Let u = 1 0 x 3 - 13x2 + 7x and v = l l x 3 - 15x 2 -3. du Find the integral values of x satisfying the inequality, ™
>
dv ^ •
42.
Q. 9
V6
Let a and b are two real numbers such that, sin a+ si n b = - y and cos a +c os b = - - - . Find the value of (i )c os (a -b ) and (ii) sin(a + b).
Q.10
[3]
[3]
Let a and b be real numbe rs greater than 1 for which there exists a positive real number c, different from 1, such that 2(logac + log b c) = 91ogabc Find the largest possible value of log a b. [5]
Q, 11
Find the product oft he real roots of the equation
x2 + 18x + 30 = 2a/x 2 +18x + 45
[5]
ix
Q.1 2
If a. p be two angles satisfying 0 < a, P < — and whose sum is a constant k„ find the maximu m value of (i) cos a • cos p
and
(ii) cos a + cos p.
[5]
Q. 13 Find a quadratic equation whose sum and product of the roots are the values ofthe expressions (cosec 10° - 7 3 sec 10°) and (0.5 cosec 10° - 2 sin70°) respectively. Also exp ress the roots of this quadratic in terms of tangent of an angle lying in
(n
~
A
.
Q. 14
x +2x-3 If y = —5 then find the interval in which y can lie for every x e R wherever defined. x + 2x — 8
Q.1 5
Prove the inequality, 1 1 sinx + - sin2x+ - sin 3x> 0
for0
[6]
[6]
[6]
4
gBANSAL CLASSES
MATHEMATICS
B Target NT JEE 20 07
Hol id ay Ass ign men t
CLASS: XI(P, Q,R,S)
DPP. NO.-37, 38, 39
RAKSHA.BANDHAN
HOLIDAY
ASSIGNMENT
These DPP will be discussed on the very first day after vacation. Take approx. 50 to 55 min. for each Dpp.
DPP For 9 = 1°, prove that 2 sin20 + 4 sin40 + 6 sin60 + Q. 2
-
37
+ 180 sinl 800 = 90 cot 0
Find all the solutions of the equation
- x ) = Vc os x
which satisfy the condition x € [0, 2n]
Q.3
Solve the equation, l + l o g x
=[log !0 (log, 0 p ) - l ] l o g x 10V 10 y
How many roots does the equation have for a given value of p? Q. 4
Find the set of values of 'a ' for which the equation, 2 ( 2 \2 X x 3 a —i H 4 a = 0 have real roots . (1+a) x2 + 1 VX 2 + 1,
Q. 5
Find four numbers, such that the first three form a G.P and the last three an A.P., while the sum ofthe first and last terms is 14 and the sum of the inner terms is 12.
DPP Q. 1
-
38
59 Each angular of a regular r-gon is — times larger than each angle of a regular s-gon. Find the largest 58 possible value of s.
Q.2
i
Solve for '0' satisfying c os(0 ) • cos (7t0) = 1. 31x1-2
> 2
Q. 3
Find the solution set ofthe inequality
Q. 4
The sum of an infinite GP is 2 & the sum of the GP made from the cubes of the terms of this infinite series is 24. Find the series.
Q. 5
A circle is inscribed in an equilateral triangle ABC ; an equilateral triangle in the circle, a circle again in the latter triangl eand so o n; in this way (n + 1) circles are describ ed; if r, Xj, x 2 , , xn be the radii of the circles, show that, r = xt + x 2 + x 3 + + x n _ } + 2 xn .
|x|-l
DPP Q. 1
39
If the equation sin4x + cos 4 x = a has real solutions then find the range of values of' a'. Find the general solution of the equation when a :
Q.2
-
1 2'
Find the complete set of real values of 'a ' for which both root s of th e quadratic equation ( a 2 - 6a + 5) x 2 - y a 2 + 2a x + (6a - a 2 - 8) = 0 lie on either side of the origin.
Q.3
Show that In (4 x 12 x 36 * 108
n(n -1) , _ up to n terms) = In In 2 + — - — 3
Q. 4
Find the value of x satisfying the equation
x -x +4 -2
Q.5
Show that
<2 + x - 12.
f 7 r s•i n x \ ^TtCOSX ^ 7Z TC 7t t an + tan > 1 for 0 < x < - and - < y < - . 2 6 i v 4 sin y y v4cosyj
ill
BA NSA L CL AS SES
MATHEMATICS
Target I8T JEE 20 07 CLASS:XI(P,
Da il y Practice Proble ms
Q, R, S)
DATE: 10-13/08/2005
DPP. NO.-35
Take approx. 50 min. for each Dpp.
Q. 1
DPP - 35 If sec( a - 2P), se ca and sec ( a + 2p ) are in arithmetical progression , show that co s 2 a = 2 cos 2|3 (P & nrc, n e l )
Q. 2
Show that the sum to n terms of the series : . . sin2(n + l)a. sin 2na n . + sm(2n~ l) a. co s( 2n + l ) a = 2sin ^a ~ ~2 s
sma cos 3 a + sin 3 a cos 5a + Q. 3
a
Find the set of real values of p for which the equation, VP c o s x - 2 sinx = V2 +V 2 - P possess solutions.
Q.4
Solve the equation
cos0 1 + sinB — :—— + l + sm 0 cos0
Q. 5
Prove that if (ac)' 08a
b
2
—. cos0
= c 2 , then the numbers loga N, log b N and log c N are three successive terms of an
arithmetic progres sion fo r any positive value of N ^ 1. Q.6
x2 - a x -2 Find all values o f ' a ' fo r which —5 lies betwe en -3 and 2 for all real values of x. x + x +1
Q.7
Solve the inequality for every a e R
x
2(a - 1 ) a
DPP Q. 1
Q.2
36
Let Aj, ^ , A 3 A n are th e vertices of a regular n sided poly gon inscribed in a circle of radius R. 2 If (Aj A 2 ) + (Aj A3)2 + + (Aj A n ) 2 = 14 R 2 , find the number of sides in the polygon. 3 + c os x Showthat —— - — V x e R can not have any value between - 2 V 2 and 2V2 . What inference can you draw about the values of
Q.3
-
2 < — (x + 1). 3a
sinx 3 + c os x
?
Find the solution set of the equatio n, log _ x 2 _ 6x (sin 3 x + sinx) = log x2_ 6x (sin 2x). 10
10
Q.4
Find the set of values o f x satisfying the equation sin |x| tan5 x = cosx
Q.5
Find the general solution of the equation, tan 2 (x + y) + cot 2 (x + y) - 1 - 2x - x 2.
Q.6
If p, q are the roo ts of the quadratic equation x2 + 2bx + c = 0, prove that 2 log (i/y-~i) + \/y~~q)
Q. 7
=
log 2 + log f y + b + ^j y 2 + 2by + c
Find all real values of x for which the expression Jlogl/2 [
^ | is a real number.
J a B A N S A L CLASSES JEE 2007 V S Target II T JE CLASS:XI(P,
Q,R,S)
DATE:
03-04/08/2005
TIME:
70Min.
DPP. NO.-3 4
Q. 1
If (x j, yj ) is the solution of the equation, log22 5(x) + log 64 (y) - 4 and (x 2 , y 2 ) as the solution of log x(225) - logy (64) = 1 then show that the value of log30 (x 1 y 1 x 2 y 2 ) =12.
Q.2
Let P (x) = x 2 + bx + c, where b and c are integer. If P (x) is a factor of both x 4 + 6x 2 + 25 and 3x 4 + 4x 2 + 28x + 5, find the value o fP (l ).
Q.3
Given a, b, c are +ve integer formin g an increasing geometr ic sequence, b - a is a perfect square, and log 6 a + log6 b + l og 6 c = 6. Find the value of a + b + c.
Q. 4
S olve the inequality, 2 log,/2 (x - 1 ) < ^ -
Q. 5
Let there be a quotient of two natural numbers in which the denominato r is one less than the square of the numerator. If we add 2 to both numerator & denomenator, the quoti ent will exceed 1/3 & if we subtract 3 fro m numer ator & denomenator, t he quotient will lie betw een 0 & 1/10. Determine the quotient.
Q. 6
The number of terms of an A. P. is even; the sum of the odd terms is 31 0; t he sum of the even terms is 34 0; the last term exceeds the first by 57. Find the number of terms and the series.
Q. 7
Find the two smallest po sitive values ofx for which sin x° = sin (x c)
J | BANSAL CLASSES
MATHEMATICS
y S T a r g e t NT JEE 2007 CLASS: XI (P, Q, R, S)
Daily Practice Probl ems
DATE: 01-02/08/2005
TIME: 60 Min.
OB JE CT IV E PR AC TI CE Select the correct alternative. Only one is correct. For each wrong answer 1 mark will be deducted. Q. 1
DPP. NO.-33
TES T [3 x 25 = 75]
Solution set of the inequality, 2 - log, (x 2 + 3x) > 0 is : (A) [ - 4 , - 3 ) u (0,1] ' " (B) [- 4, 1] ( C) ( - 0 0 , - . 3 ) U ( 1 , 0 0 )
( D ) ( - 0 0 , - 4 ) U [1 ,00)
Q.2
If A + B + C = 7r & cosA = cosB . cosC then tanC . tanB has the value equal to : (A) 1 (B) 1/2 (C) 2 (D) 3
Q.3
If a, b, c be in A.P., b, c, d in G.P. & c, d, e in H.P., then a, c, e will be in: (A)A.P. (B)G.P. (C)H.P, (D) none of these
Q.4
If the roots of the equation x 3 - px 2 - r = 0 are tan a, tan (3 and tan y then the value of se c 2 a • sec2|3 • sec2y is (A) p2 + r 2 - 2rp + 1 (B) p2 + r 2 + 2rp + 1 (C) p 2 - r 2 - 2rp + 1 (D)N one
^ „ Q. 5
The sum to n terms of the series, (A) 2 n - n - 1
1
3
7
(B) 1 - 2 " n
15
1S
equal to:
(C) 2~ n + n - 1
(D)2n-1
Q. 6
sinx - cos 2x - 1 assumes the least value for the set of values of x given b y: (A) x - tm + (~l) n +1 (n/6) (B) x = nn + ( - l ) n (n/6) (C) x = n% + (-l ) n (ti/3) (D) x = nw - (-l) n (tt/3) where n e l
Q.7
If the equation a (x - l) 2 + b(x 2 - 3x + 2) + x - a2 = 0 is satisfied for all x e R then the number of ordered pairs of (a, b) can be (A) 0 (B) 1 (C) 2 (D) infinite *
Q. 8
The base angles of a triangle are 22.5° and 112.5°. The ratio of the base to the height of the triangle i s: (A)V2
Q.9
If
(B )2 V2 -1
2x+i) ^ '
(C)2V2
1
1) ^
1 are in Harmonical Progression then
(A) x is a positive real (C) x is rational which is not integral Q. 10
(D)2
(B) x is an integer (D) x is a negative real
The absolute term in tile quadratic expression ' t l (A) zero
X
x-
3k+ 1A (B) 1
1
^
3k-2y
as n —» oo is 2 (C) -
(D)
1
Q. 11
Given four positive number in A.P. If 5 , 6 , 9 and 15 are added respectively to these numb ers, we get a G.P., then which of the following holds? (A) the common ratio of G.P. is 3/2 (B) common ratio of G.P. is 2/3 (C) common difference of the A.P. is 3/2 (D) common difference of the A.P. is 2/3
Q.12 x
The equati on, sin2 9 - — 3r r — ' = 1 H . sin 0 - 1 (A) no root
Q.13
(B) one root
The equation (x e R)
+ 1—, 4
(A) has no root
Q.14 "
sin3 0 - 1
has : (D) infinite roots
(C) two roots
=x : x?
V ' - i
(B) exactly one root
(D) four roots
(C) two root s
If xs in 9 = y c o s 9 then — L — 4 — is equal to sec29 cosec29 (A) x
(C )x 2
( B) y
(D)y2
Q. 15
An H.M. is inserted between th e number 1 /3 and an unknow n number. If we diminish the reciprocal of the inserted number by 6, it is the G.M. of the reciprocal of 1/3 and that of the unknown number. If all the terms of the respective H.P. are distinct then (A) the unknown number is 27 (B) the unknown number is 1/27 (C) the H.M. is 15 (D) the G.M. is 21
Q. 16
The number of integers 'ri such that the equation nx2 + (n + l) x + (n + 2) = 0 has rational roots only, is (A )l (B)2 (C) 3 (D) 4
Q. 17
The roots of the equation , cot x - cos x = 1 - cot x . cos x are : (A)mi+j (C) mt +
(B) or 2 nn±n
(D) ( 4 n + l ) ^
or
(2n+l)n
where n e I Q. 18
Q. 19
If x2 + Px + 1 is a factor of the expression ax 3 + bx + c then (A) a 2 + c 2 = - ab (B) a 2 - c 2 = - ab (C) a 2 - c 2 = ab
(D) none of these
The expression (ta n49 + tan29) ( 1 - tan2 39 tan 29 ) is identical to (A) 2 cot 39 . sec 2 9 (B) 2 sec 39 . tan 2 9 (C)2t an39 . sin2 9
(D) 2 tan 39. sec29
x2~3x + c Q.20
Ifthe maximum and minimum values of y = c is equal to (A) 3
(B) 4
X
"i" i X H~ C
(C )5
i are 7 and — respectively then the value of /
(D) 6
Q. 21
The general value of x satisfying the equation 2cot2x + 2 V3 cotx + 4 cosecx + 8 = 0 is (A) nn -
n
(B) nn +
71
n
(C) 2nTX -
(D) 2mc +
7t
6
Q. 22
Ifthe sum of n terms of a G.P. (with common ratio r) beginning with the p * term is k times the sum of an equal number of terms of the same series beginning with the q111 term, then the value of k i s: (A) r p/q (B) r^P (C)rP^ (D)rP + i
Q.23
The sum of th e roots oft he equation (x + 1) = 2 log2(2 x + 3) - 2 log 4 (l 980 - 2"x) is (A) 3954 (B) lo g 2 ll (C)log 2 395 4 (D) indeterminate
Q.2 4
If the expression, 2 ( ^ 2 _ i) sin x - 2 cos 2x + 2 -
is negative then the set of values of x lying in
(0,2n) is: 'it 57r"
(A)
W 6 , /
(C)
Q.25
71 7T
1
u
r57i l b O U
' 6 J
(5n
3T^ 1 4 2J —
v 6 ' 2 J
(B)
f 5n llTl")
(D)
—
U '
6 J u
V
6/
571 5tc
T ' T
u
In
,2n
Solution set of the inequality log 3 x - log? x < : log, .4 is %/2j2) (A) [3,9]
(B)
(H
u[9,oo)
( C ) f - o o , i u[9, oo) (D)
u (1,9]
J j B A N S A L C L A S S E S
MATHEMATICS
v B Target I IT JEE 200 7 CLASS: XI (P, Q, R, S)
Dai ly Practice Pro bl ems
DATE: 28/07/2005
TIME: 60 Min.
OB JE CT IV E PR AC TI CE Select the correct alternative. Only one is correct. For each wrong answer 1 mark will be deducted. Q. 1
TES T [3 x 25 = 75]
A regular hexagon & a regular dodecagon are inscribed in the same circle. If the side of the dodecagon is
- i j , then the side of the hexagon is :
(A) 1 Q.2
DPP. NO.-32
If
(B) 2
(C) V2
+ upto 00 = 8, then the value of d is :
3 + ^ (3 + d) + ^ y (3 + 2d) +
(A) 9
(D) 2V2
(B) 5
(C)l
(D) none of these
Q.3
If in a A ABC, sin3 A + sin 3 B + sin3 C = 3 sinA • sinB • sinC then (A) A ABC may be a scalene triangle (B) A ABC is a right triangle (C) A ABC is an obtuse angled triangle (D) A ABC is an equilateral triangle
Q.4
The value of (0.2 )loSvI ^ + » + ^ (A) 4 (B) 6
Q.5
+
^ is equal to (C )8
The set of angles btwe en 0 & 2n satisfyi ng the equation 4 cos2 0- 2- ^2 cos 0 — 1 — 0 is (A)
23n J j L ^L ] I12 ' 12 ' ~12 ' T T j
(B)
f ^ L l^ZL 1 9 7 t ] (C) I 1 2 ' 12 ' 12 J Q.6
Let x=
1 1-4
+
1 4. 7
(A)y = 3x
Q.7
Q.9
1 +
7.10
2 3n
! JL l2L ]2JL 1 12 ' 12 ' 12 ' 12
f 7C 1% 1771 2371
+
00 and
(B)y = 2x
1 1 1 y = ~ + 7TT + TT 4 " ' 1-2 2-3 3- 4
°° the n
(C)x + >>=1
(D)x+y=^
If cos a = "" c o s P—- then tan ^ 1 cot ^- has the value equal to, wher e(0 < a < n and 0 < B < 71) 2 - cosp 2 2 (A) 2
Q.8
(D )2
( B )V 2
1/ 8 1/16 1/32 1/64 2m . 4 . 8 .16 . 32 (A) 2 (B) 1
(D) Vs
(C) 1/2
(D) 1/4
00 is equal to
If xs in0 = ys in |e + y j = z sin^e + (A) x + y + z = 0
(C) 3
(B)xy + yz + zx = 0
then: (C)xyz + x + y + z = 1 (D) none
Q. 10
If x AM's are inserted between xr and 1 then the value of the x th arithmetic mean is (A) I- x (C)x 2 — x + 1 (D ) x (B)l+x
Q. 11
If a cos 3 a + 3a cos a sin 2 a = m and a sin 3 a + 3a cos 2 a sin a = n . Then (m + n)2/ 3 + (m - n) 2/ 3 is equal to : (A) 2 a2 (B) 2 a 1/ 3 (C) 2a 2 / 3 (D) 2 a 3
Q. 12
Consider
the A.P. a t , s^ ,
,a n ; the G.P. b>, b2 , 9 such that a 1 = bj = 1 ; a9 = b 9 and ]!Ta r = 369 then r=l (A) b 6 = 27 (B) b ? = 27 (C) b g = 81
, bn
(D )b 9 = 1 8
Q. 13
If tan A & tan B are the root s of the quadratic equat ion x 2 - ax + b = 0, the n the val ue of sin 2 (A + B) is : a2 (A) 2, « a + (1-b) 2
Q. 14
Q.15 v
(B
V
a2
_ a2 ( Q - ^ - T2 (b + a)
a 2 + b2
(C) Vl + sinA
Q. 17
(B) - V 1 + sinA - Vl -s in A
- -\/l - sin A
^ D ) - Vl + sinA + V l - s i n A
Conside r a decr easi ng G.P.: g 1 ,g 2 ,g 3 ,
gn
such that g 1 + g 2 + g 3 = 13 and gj + g 2 + g 3 =91
then which of the follow ing does not hold? (A) The greatest term of t he G.P. is 9.
(B) 3g4 = g 3
(C) g, = l
(D) g 2 = 3
*\/3 + 1 "J3 Numbe r of roots of the equation cos 2 x + — - — s in x - — - 1 = 0 which lie in the interval [-71, tt] is
(A) 2 Q.18
( B) 4
(C) 6
(D) 8
The sum of th e first three terms of an increasing G.P. is 21 and the sum oft hei r squares is 189.The n the sum of its first n terms is
r
n
(A) 3 (2 - 1) Q.19
Q.20 V
b 2 (1 - a) 2
If a, b, c are distinct positive reals in G. P., the n; log a n, log b n, log c n (n > 0, n * 1) are in: (A) A. P. (B) G. P. ( C) H .P . (D) none A IfA= 34 0° then 2 sin — is identical to 2 (A) Vl + si nA + Vl - sin A
Q.16
(D)
a2
1 1 \ ( B) 12 1 - ^ r V
2
1 1-^r)
(C)6
/
Ifsin ( 6 + a ) = a & sin(G + p) = b (0 < a , p9 0 < tc/2) then cos2 ( a - (3) - 4 ab co s( a - P) = (A) 1 - a 2 - b 2 (B) 1 - 2a 2 - 2b 2 (C.) 2 + a 2 + b 2 J .f S= 4- + - r — r + n
l3
(A) 1/2
1 +2
* r \ '
+
1 +2 +3 + (B) 1
(D )6 (2 «- l)
\
- , n = 1, 2, 3,
(D)2-a2-b2 The n S„ is not greater than n
+n (C) 2
S
(D )4
Q.21
The exact valu e of cos 2(B) 73° 1/2 + cos 2 47° + (cos73°. cos47°) is (A) 1/4 (C)3/4
Q.22
Let Sj , S 2 , S 3 be the sums of the first n , 2n and 3n terms of an A.P. respectively. If S 3 = C (S 2 - S,) then, 'C' is equal to (A) 4 " (B )3 (C )2 (D )l
(D) 1
\
Q.23
Maxim um value of the express ion cos6 • sin ® v 1 (A) j
Q.24
1 (C) 4
V 9 e R, is
(D)l
The value of the expre ssion (sinx + cosec x) 2 + (cosx + secx) 2 - (tanx + cotx) 2 wherever defined is equal to (A) 0
Q.25
V3 ( B ) ^
6y
(B )5
( C) 7
(D) 9
The roots of the equat ion 2 + co tx = cosec x always lie in the quadra nt numbe r (A) I only (B) I and II (C) II and IV (D) II only
i k BANSAL CLASSES
MATHEMATICS
^ S T a r g e t IIT JEE 2007 CLASS: XI (P, Q, R, S)
Dally Practice Problems
DATE: 24/07/2005
OB JE CT IV E
TIME: 60 Min.
PR AC TI CE
DPP. NO.-31
TES T
Select the correct alternative. Only one is correct. For each wrong answer 1 mark will be deducted.
Q. l
[3 x 25 = 75]
The set of values of x satisfying simultaneously the inequalities 2 X - 3 - 31 > 0 is : (A) a unit set (C) an infinite set
J(x-8) (2-x) t— -y > 0 and iogo.3 ("T (log2 5 - 1))
(B) an empty set (D) a set consisting of exactly two elements.
Q.2
The root s of the equatio n (x—l) 2 — 4|x—1 | + 3 = 0, (A) form an A.P. (B) form a GP. (C) form an H. P (D) do not form any progr essio n.
Q. 3
The perimeter of a certain sector of a circle is equal to the length of the arc of a semicircle having the same radius. The angle of the secto r in radians i s: (A) 2 (B) 7i - 1 (C) 7i - 2 (D) none
Q.4
If the roots of the equation, x3 + Px2 + Qx - 19 = 0 are each one more than the roots of the equaton, x3 - Ax2 + Bx - C = 0 where A, B, C, P & Q are constants then the value of A+B+C = (A) 18 / (B) 19 (C) 20 (D) none
Q.5
Number of ordered pair(s) of (x, v) satisfying the system of simultaneous equations I x 2 - 2x j + y = 1 and x2 + | y f = 1 is (x, y e R) : (A) 1 (B) 2 (C) 3 (D) infinitely many
Q. 6
Given log 2x • log,xyz =10 log2y-log 2xyz = 40 log2z • log 2xyz = 50 w he re x > 0 ; y > 0 ; z > 0 then which of the following inequalities may be true? (A )x y>z & z< x< y (C)x>y>z&x z > y & z < x < y
Q 7
The quadratic equation who se roots are the A.M. and H.M. betwe en the roo ts of the equation, 2x2 - 3x + 5 = 0 is : (A) 4x2 - 25x + 10 = 0 (B) 12x2 - 49x + 30 = 0 (C) 14x2 - 12x + 35 = 0 (D) 2x2 + 3x + 5 = 0
Q.8
If the sum of the first n natural numbers is 1/5 times the sum oftheir squares, then the value of n is : (A) 5 (B) 6 (C) 7 (D) 8
Q. 9
A particle begins at the origin and moves successively in the following manner as shown, 1 unit to the right, l/2uni tup, l/4u nitt othe righ t, 1/8 unit down, 1/16 unit to the right etc. The length of each move is half the length of the previous mo ve and movement continues in the 'zigzag'manner indefinitely. The co-ordinates of the point to which the 'zigzag' converges is : (A) (4/3, 2/3) (B) (4/3, 2/5) (C) (3/2, 2/3) (D) (2, 2/5)
1/4 CJ 1 0
"
s U 1/16 v X
Q.10
A quadratic equation defined over rational coefficient whose one root is sin237i/10 is: (A) 16x2 + 12x - 1 = 0 (B) 4x2 + 2x - 1 = 0 (C) x2 - 3x + 1 = 0 (D) 16x2 - 12 x+ 1 = 0 1 00
Q. 11
Let an be the n
th
ter m of a G.P. of positive numbers . Let X
1 00
a2n = a & X
n = 1
n
=
P such that
n = 1
a * p. Then the common rati o of the G.P. is : W f p
C B ) a£
( C ) j\j fp
( D )Vj ai
Q. 12
Given a sequence a p a2, a3, an, in which the sum of the first m terms is 2 Sm = m - 5m then which of the followi ng is not true? (A) a5 = 0 (B) a5 = 4 (C )a 6 = 6 (D) iti sanA P.
Q.13
l° g 2 x + l°S 4 y + l ° g 4 z = 2 log 3y + log 9z + log9x = 2 log 4z + log16 x + log16y = 2 then which oft he following is true? (A) y > z (B) x > y Given
(C)x>y>z
(D)x
Q. 14
The number of integral values of m, for which the root s of x 2 - 2 mx+m 2 - 1 = o will lie between - 2 and 4 is (A) 2 (B) 0 (C )3 ( D )l
Q.15
Given a regular triangle with side 'a', a new regular triangle is forme d by the length ofits altitudes. This pro cess is repeated. This proc edure being rep eated n times. The limit of the sum of areas of all t he > oo is triangle as n — (A) 3a2
(B)V3a2
r Q.16
Let y =
v2cosx+sin2x
(A) [l,oo) Q. 17
• • 2sinx + sin2x
1- co sx
(C) 2a2
( D ) V ^ a2
\ 2/3
1-sinx^
(B )(-a,, co)
The interval in which y can lie for V x e R . ( C )[0,1]
(D)[0,oo)
A horse is teethered to a stake by a rop e 9 m long. If th e horse moves along the circumfer ence of a circle always keeping the rop e tight then the distance traversed by the horse wh en the rope has raced an angle of 70°, is (Assume n = 22/ 7) f (A) 7 m
(B) 9m
(C)llm
(D) 22m
Q.18
If x g R, the numbe rs ( 51+x + 5 1"*), a/2, (25 x + 25' x ) form an A.P. then 'a' must lie in the interval (A) [6,o) ) (B)[ 12,o o) (C) [24, oo) (D) [24, oo)
Q. 19
If 7 times the 7th term of an A.P. is equal to 11 times its eleventh ter m then the 18th term of the A.P. is (A) 0 (B) 7 (C) 11 (D) 18
Q.20
ABC is a triangle such th at , sin ( 2 A + B ) = sin (C - A ) = - sin (B + 2C) = ^ • If A, B, C are in A.P, A B & C are respectively. (A) 30°, 60°, 90° (B) 45°, 60°, 75°
(C) 15°, 60°, 105°
(D) non e of these
3
Q.21
Set ofintegral solution of the equation x (A) 1
(B) 2
5
2
+iog2X
4 _ ^
(C) 3
js (D) 0
y
Q. 22
If sin (x - y) , sin x and sin (x+y ) are in H. P., then sin x . sec
has the value equal to (x, y, z are +ve
acute angles) (A) 2
Q.23
If
(B) V?
.
imi
(A )p + 9q
Q.25
(D) none
1025 10
Q.24
(C)l
-
P an c *
^
=
Q then th e value of log 10 4100 in ter ms of p and q is equal to
(B) p + lOq
(C )1 2p + q
2
( D ) p + 12q
Number ofv alu es of x satisfying the equality 2 cos2x = 3.2C0S x - 4 and the inequality x 2 < 30 is (A) 0 (B) 1 (C) 2 (D) 3 3 2 Let a, P, y be the roots of the equation x + 3 ax + 3 bx + c = 0 . If a, p, y are in H.P. then P is equal to: (A) a (B) c/b (C) - a (D) - c / b
J j B A N S A L C L A S S E S
MATHEMATICS
v B Target II T JEE 2007 CLASS: XI (P, Q, R, S)
Daily Practice Probl ems
DATE: 18/07/2005
TIME: 60Min.
OB JE CT IV E P RA C T IC E Select the correct alternative. Only one is correct. For each wrong answer 1 mark will be deducted. Q. 1
(A) independent of b (C) independent of both a & b
^ Q.4
Q. 7
(B) independent of a (D) dependent on bot h a & b .
(B) a + 1, P + 1
(C) ~
, ^
(D) ^
,
^
(D) infinitely many
Given a, b, c are non negative real number s and if a2 + b 2 + c 2 = 1, then the value of a + b + c is (B) > 2
The value of (0.2 ) ! ° S ^ & + * (A) 2
Q. 6
is :
The number of the integral solutions for the equation x + 2 y ~2xy is (A) 2 (B) 1 (C) 4 '
(A) > 3 Q. 5
[3 x 25 = 75]
If a , P are the root s of the equation ax2 + bx + c = 0 , the n the roots of the equation ax 2 + bx(x+ l) + c (x +1) 2 = 0 are (A) a - 1 , 0 - 1
VQ 3
TE ST
If a > b > 0 are two real numbers, the value of , ib + (a - b) -,/ab + (a - b) ^/ab + (a - b) -v/ab +
Q. 2
DPP. NQ.-30
+
(C)
+
i s
(D )< V2
equal t o
(B) 4
(C) 5
(D) none
The maximum valu e of the sum of the A.P. 50, 48, 46 ,4 4, (A)325 (B)648 (C)650
is (D)652
Solution set of the inequation, glo§2x - 2x 2 > x - 2 is (A) (0, 1)
(B) (2, oo)
(C) (0, 2)
(D) (0, 1) u (2, oo)
Q. 8
The quadratic equation (3 + sin9)x2 + (2 cos9)x + 2 - sinQ = 0 has (A) equal roots for all 0 (B) real and distinct roo ts for all 8 (C) complex roo ts for all 0 (D) real or complex roo ts depending upon 0
Q. 9
If the roots of the quadrati c equation ax2 + bx + c = 0 are imaginary then fo r all values of a, b, c and x e R , the expression a2 x 2 + abx + ac is
Q. 10
(A) positive
(B) non - negative
(C) negative
(D) may be positi ve, zero or negative
If \ 2 + -V2 + 4 "2 + l 2 3 (A)~
Q. l l
u
P t0
(B)
00 =
then
6
TT2 + \ I 3 (C)~
+
~7 + 5
equal to
(D) none
In a potato race, 8 potato es are placed 6 metres apart on a straight line, the first being 6 metres from the basket which is also placed in the same line. A contestant starts from the basket and puts one potato at a time into the basket. Find the total distance he must run in order to finish the race. (A) 420 (B) 384 (C) 43 2 (D)non e
Q.12
The sum of th e first 100 terms common to the series 17 ,2 1, 25 , (A) 101100 (B) 111000 (C) 110010
and 16 ,2 1, 26 , (D) 100101
is
Q.13
If x, y, z e N then the number of ordered triplets of (x,y, z) satisfying the equation x + y + z =1 0 2i s (A) 4950 (B) 5050 (C) 5150 (D )N one
Q. 14
Consider an A.P. with first term 'a' and the common difference d. Let S k denote the sum of the first K SjQJ
terms. Let "77" is independent of x, then (A) a = d/2 Q. 15
Q. 16
(B) a = d
(C) a = 2d
(D)none
If p, q, r in harmonic progression and p & r be different having same sign then the roots oft he equation, px 2 + q x + r = 0 are : (A) real and equal (B) real and distinct (C) irrational (D) imaginary. The numbers
1
log 32
, —-— & — - — consitute log 62 j log 12 2
(A) an A. P.
(B)aG.P .
(C )a H. P.
(D) None
Q. 17 1
If log2, lo g ( 2 x - 1) and log(2 X + 3 ) are in A.P,, then xi s equal to : (A) 5/2 (B) log 3 2 (C) log 2 5 (D)none
Q. 18
Complete set of the values of x satisfying the inequality, log 7 x - log x (l /7 ) < - 2 is (A) (0 ,1 ) (B) (1, 2) (C) (1, co) (D) (2, co)
Q.19
If the equation sm4 x - (k + 2) sin 2 x - (k + 3) = 0 has a solution then k must lie in the i nter val: (A) ( - 4 , - 2 ) (B) [-3, 2) ' (C ) (- 4, - 3) (D) [ - 3 , - 2 ]
Q. 20
The first term of an infinitely decreasing GP. is unity and its sum is S . The sum of the squares ofthe terms of the progression is : (A) — ( 2S-1
Q.21
)V
S2
V ;
B
The expression
sin (a + 0) - sin( a - 0) is c o s ( p _ e ) _ c o s ( p + e )
;
2S-1
(A) independent of a (C) independent of 0 Q. 22
(C)
2- S
(D) S 2 '
K
(B) independent of |3 (D) independent of a and p
Number of values of 0 e [ 0,2 n ] satisfying the equation cotx - c o s x = 1 - cotx. cosx (A) 1
(B) 2
(C )3
(D )4
Q. 23
The solution set of th e inequality log }/3 x + 2Iog I 9(x - 1) < log 1/3 6 i s (A) ( - « , - 2] U [3, *>) (B) [- 2, 3] ' ( C ) R - [- 2, 3] (D) [3, oo)
Q.24
4 sin5° sin55° sin65° has the values equal to V3 + 1
Q.25
V3-1
•
S - l
_
3(V3 - l)
The values of x smaller than 3 in absolutevalue which satisfy the inequality log (2 a _ x 2 ( x - 2 a x ) > 1 for alla>5 is (A) - 2 < x < 3
( B) - 3 < x < 3
(C)-3
(D)-3
MATHEMATICS
fi BAN SAL CLAS SES
Daily Practice Problems
Target IIT JEE 2007 DATE:
CLASS : XI (P, Q, R, S)
TIME:
16/07/2005
DPR
60Min.
OB JE CT IV E PR AC TI CE Select the correct alternative. Only one is correct.
NO.-29
TE ST [3 x 25 = 75]
For eacii wrong answer 1 mark will be deducted. Q. 1
a2 p2 If a, p are the roots of th e equation ax + 3x + 2 = 0 (a < 0) the n — + — is 2
(A) > 0
(B) > 1
(C) < 1
(D)<0
Q.2
The valu e of f (x) = x 2 + (p - q)x + p 2 + pq + q 2 for real values of p, q and x (A) is always negative (B) is always positive (C) is some times zero for non zero value of x (D) none of these
Q.3
For an increas ing A.P. a, , a 2 , a 3 ,a n,.... if a! + a 3 + a 5 = - 12 ; a,a 3 a 5 = 80 then which ofthe following does not hold? (D) a, = 2 1 (A) a,= - 10 (B )a 2 (C) a, = - 4
Q. 4
The solution set of the inequality log<
1
1
o 2 2x - x
u
(A)
3
> 1 is
(B)
f 3
\
(C)
2' 4
u
(D)
)
U
-co.
J
u
T'
00
^
J
Q.5
If a , p are the roots of the quadra tic equati on (p 2 + p + l)x 2 + (p - 1 )x + p 2 = 0 such that unity lies bet ween the roo ts then the set of values of p is (A)* (B)peO,-l)U(0,oo) (C) p e (- 1,0 ) (D) (-1 ,1)
Q.6
If cos9 + cos(|) = a and sinB + sincj) = b, then the valu e of cos9-cos(|) has the valu e equal to 2
I -4a (A) Q.7
41
hb 2 )
2
|( a
2
+ b
2
2
)! -4b
2(a2+b2)I
2 2 2 ( a + b ) 1 1
2
^
-
4a 2
2(a2+b2)
|( a 2 + b 2 ) |2 - 4 b 2 (U) "
2 2 41( a + b ) 1
If both roots of the equat ion (3X + l) x 2 - (21 + 3 p)x + 3 = 0 are inifin te then (A) A, = p
1
3
1 2 (C)X = - - ; p = -
SPACE FOR ROUGH WORK
1 (D)?i = --;n
= - -
Q.8
If p & q are distinct reals, then 2 { ( x - p ) ( x - q ) + ( p - x ) ( p - q ) + ( q - x ) ( q - p ) } = (p - q)2 + (x - p)2 + (x - q)2 is satisfied by: (A) no value of x (B) exactly one value of x (C) exactly two values of x (D) infinite values of x.
Q.9
The expression cot 9° + cot 27° + cot 63° + cot 81° is equal to (A )V l6
(B) V64
(C )V 80
(D) none of these
If the quadratic equa tion ax2 + bx + 6 = 0 does not have two distinct real roots, then the least value of 2a + b is (A) 2
(B )- 3
(C )- 6
(D )l
Q. ll
The set of values of' p' for which the expression x 2 - 2 px + 3 p + 4 is negative for atleast one real x is: (A)<1) (B) ( - 1 , 4 ) (C) (-00, -1)1^ (4,00 ) (D) {- 1, 4}
Q.12
The equation
5log"
X+ 1
+ 5l o g o
(A) no integral solution (C) one irrational solution Q. 13
25 X
"
1
= y
has
(B) only one rational solution (D) two real solutions
If a, b, c are positive reals and b2 < 4ac, then the difference between the maximum and minimum values of the function, f (0) = a sin20 + b sin0-cos0 + c cos 20 V 9 e R, is (A) 0
(B) a + c
( C ) V ^ V
(D )a c
Q. 14
Let a > 0, b > 0 & c > 0. Then both the roots of the equation ax2 + bx + c = 0. (A) are real & negative (B) have negative real parts (C) are rational numbers (D) none
Q.15
Greatest integer less than or equal to the number log 2 15 . log |/f) 2. log 3 1/6 is: (A) 4 (B)3 (C) 2 " ' (D) 1
Q. 16
Integral value of x satisfying the equation (x2 + x + 1) + (x2 + 2x + 3) + (x2 + 3x + 5) +....+ (x2 + 20 x + 39) = 4500 is (A) .10 (B) - 1 0 (C) 20.5 (D) Non e
Q.17
Given a2 + 2a + cosec 2 ~ ( a + x) = 0 then, which of the following holds good? V.2 y ( A)a = l ; | e l
( B)a = - l ; |
(C) a e R ; x e ^
(D) a , x are finite but not possible to find SPACE FOR ROUGH WORK
el
Q.18
If a, P are the roots of the equation, x2 + (sin
Q. 19
Q.20
(B) 3
(C) 9/4
If In 2 x + 3 In x - 4 is non negative then x must lie in the interv al: (A) [e, co) (B) (-oo, e~ 4)u[e, °o) (C ) (l / e, e)
(D) 2
(D) none
If the quadratic polyn omial, y = (cot a)x 2 + 2 (V sin a ) x + ^ tan a, a e [0, 2 TT| can take negative values for all x e R , then the value of a must in the interval: (A) a e
Q. 21
, it)
(B) a
If a , p are roots of the equation x2 - 2mx + m2 - 1 = 0 then the number of integral values of m for which a, p € (-2, 4) is (A) 0
(B )l
(C) 2
(D)3
P.T.O. X
X
Q.22
The value of the expre ssion, log 4 4
v (A) - 6
y
(B) - 5
- 2 log 4 (4 x4) whe n x = - 2 is :
(C) - 4
If x, and x 2 are the roots of the equation x 2 + px -
Q.23
xj1 + x 2
(D) meaningless
1
= 0 , (p e R ) then the mini mum value of
is equal to
(A)V2
(B)V2(2-V2)
(D) 2 + 2^/2
(C)2+V^ { ^ \6x+10-x2
Q. 24
Number of integral values of x satisfying the inequality v4y (A) 6
(B) 7
(C) 8
(D) infinite
If the roots of the equat ion ax 2 + bx + c = 0 are real and of the fo rm
Q.25
(a + b + c) 2 is (A) b 2 - 4ac
27 . < — is 64
(B) b 2 - 2ac
a
r and a-1
(C) b 2 + 4ac
a +1 a
then the value of
(D) b 2 + 2ac
X
- X
ANSWER KEY Class - XI
Date: 16-07-2005
Max. Marks: 75
Max. Time: 1 Hr.
Roll No.
[25x3 = 75]
Only alternative is correct. There is NEGATIVE Marking. For each wrong answer 1 mark will be deducted.
A
B
O
o
O
o
o
4
o o
o
5
o
o
1
2 3
6o 7
o
8o o 10 o 1 1o
9
o 13 o 12
o
o o o o
o o o
c o o o o o o o o o
o o o o
o o o
14 o
15 o 16 o
o
17
o
19
o o o
o o o o o
B
A
D
o
18 o O
20 O 21 O 22 O 23
O
24
o
25
o
o o o o o o o
o
o
o o o
c o o o
o o o o o o
o o o
D o o o
•
o
o o o
o o o o o
• X
J j B A N S A L C L A S S E S y S T a r g e t
II T JEE 200 7
CLASS: XI (P, Q, R, S)
Q. 1
Q.2
Daily Practice Pro ble ms
DATE: 08-09/07/2005
TIME: 40 Min.
DPP. NO.-27
Select the correct alternative : (Only one is correct) Which of the statement is false. (A) If 0 < p < n the n the quadrati c equation, (cos p - 1) x 2 + cos px + si np = 0 has real roots. (B) If 2a + b + c = 0 (c * 0) then thequadrat ic equation, ax2 + bx + c = 0 has no root in (0 ,2 ). (C) The necessary & suffi cient condition for the quadratiic func tion f(x) = ax 2 + bx + c to take both positive & negative values is, b 2 > 4ac. (D) The sum of the roots of the equation cos 2 x = 1 which liei in the interval [0,314] is 49 5071.
For every x e R, the polynom ial x 8 - x 5 + x 2 - x + 1 is : (A) positive
Q.3
MATHEMATICS
(B) never positive
(C) positve as well as negati ve
(D) negative
If x ] 5 x 2 & x 3 are the three real solutions of the equation; 2 3 xlog 0x+log10x +3
2
=
w h g r e
X ] > X 2 > X 3 ;
t h £ n
^x+T-i ,/x+i+i (B) x , . x 3 = x 2 2
(A) Xj + x 3 = 2 x 2 Q.4
Q.5
*2
(D) x f 1 + x f 1 = x f 1
X, + x 2
(A) (-co , 0) u (6,oo)
( B) ( - o o, 0 ] u( 6 , oo )
(C) (-00 , 0] vj [6,oo)
CD ) (0 , 6)
Three roots of the equation, x 4 - px 3 + q x 2 - rx + s = 0 are tan A, ta nB & tan C where A, B, C are the angles of a triangle. The fo urth root of the biquadratic is : ( B ) - ^ 1+ q- s
. . The value of the expression
(A )V 3- 2 Q. 7
2Xl
If exactly one root of the quadratic equation x 2 - (a +1 )x + 2a = 0 lies in the interval (0,3) then the set of values 'a' is given by
(A)-P^_ 1- q+ s „ , Q.6
(C) x 2 =
( C ) - ^ ' 1- q+ s
(D) —PJlI— y 1+ q- s
sin 8x cos x - sin 6x cos 3x
n . when x = — is sin 3x sin 4x - cos x cos 2x 24
(B)^
(C)V2 -1
( D ) V2 + 1
If the roots of th e quadratic equation ( 4 p - p 2 - 5 ) x 2 - (2p - 1 )x + 3p = 0 lie on either side of unity then the number of integral values of p is (A) 0 ' (B) 1 (C ) 2 " (D) infinite
Q.8 « The inequalities y ( - 1) > - 4, y( 1) < 0 & y(3) > 5 are kno wn to hold for y = ax 2 +b x + c then the least value of 'a' is : (A) - 1 / 4 Q.9
(B) - 1 / 3
(C) 1/4
Number of ordered pair(s) satisfying simultaneously, the system of equations, 2 r x + f y -256 & log 10 A/xy -log 1 0 1.5 = 1, is: (A) zero (B) exactly one (C) exactly two
Q. 10
(D) 1/8
(D) mor e than two
Find the value s of 'a' for which one of the roots of the quadratic equation,x 2 + (2 a + 1) x + (a2 + 2)=0 is twice the other ro ot . Find also the roots of this equation for these values of 'a'.
i l l BANSAL CLASSES
MATHEMATICS
v 8 T a r g e t IIT JEE 2007 CLASS: XI (P, Q, R, S)
Daily Practice Problem s
DATE:
Select the correct alternative:
11-12/07/2005
TIME: 40 Min.
DPP. NO.-28
(Only one is correct)
Q. l
If a, b, p, q are non- zer o real number s, the two equat ions, 2 a 2 x 2 - 2 ab x + b 2 = 0 and p 2 x 2 + 2 pq x + q 2 = 0 have : (A) no com mon root (B) one co mmo n root if 2 a 2 + b 2 = p 2 + q 2 (C) two co mmo n root s if 3 pq = 2 ab (D) two co mmo n root s if 3 qb = 2 ap
Q. 2
The equati ons x 3 + 5 x 2 +px + q = 0 and x 3 + 7 x 2 + p x + r = 0 have two roots in common. If the third root of each equation is repres ented by Xja nd x 2 respectively, then the ordered pair (x 15 x 2 ) is: (A) ( - 5 , - 7 ) (B) (1, -1) (C )( -1, 1) ( D ) ( 5, 7 )
Q.3
If the roots of the quadra tic equati on x 2 + 6x + b = 0 are real and distinct and they differ by atmost 4 then the range of valu es o f b is : (A) [ - 3 , 5 ] (B) [5,9 ) (C) [6,10 ] (D) none
Q.4
The expre ssion (A) positive
sec 4 x - 4 ta n 3 x + 4 tan x is alw ays : (B) negative (C) non-positive
(D) non-negative
Q.5
The value of the biqua drati c expre ssion, x 4 - 8 x 3 + 1 8 x 2 - 8 x + 2 w hen x = 2 + V3 is (A) 1 (B) 2 (C) 0 (D) non e
Q.6
If one root of th e quadratic equation px 2 + qx + r = 0 (p ^ 0) is a surd
—7 Va + Y a — b
where p, q, r ; a, b are all rationals then the other root is Va
Q. 7
(B) - cos 1
(C) cos 1
(D) - 1
(B ) 2 7i a 2 (l + cot 2 1 ) ( Q % a2 (l + cot 2 | J
(D) 4 71 a 2 (l + cot 2 1)
The equati on a sinx + cos 2x = 2a - 7 has a soluti on, if (A) a > 2
( B) a < 2
Subjective:
Q. 10
Va - V a - b
The area of the circle in whic h a chord of length 2a make s an angle 9 at its centr e is (A) 7i a 2 c o t 2 |
Q.9
a + ,/ a( a- b)
The min imu m valu e of co s (cos x) for every x e R is : (A) 0
Q. 8
Ja(a-b)
Solve the inequality, log 2x (x 2 - 5x + 6) < 1.
(C)2
(D)a<2ora<6
Jj BANSAL CLASSES
MATHEMATICS
V B Target IIT JEE 2007 CLASS: XI (P, Q, R, S)
Dail y Practice Prob lems
DATE: 06-07/07/2005
TIME: 40 Min.
DPP. NO.-26
Select the correct alternative : (Only one is correct)
Q. 1
x2 + 2x + c If x is real, then —can take all real values if : x + 4x + 3c (A) 0 < c < 2
Q.2
(B) 0 < c < 1
(D) none
27c
1 8TC 3TT 671 27 n 9n c os ec — + co s— co se c— + c o s — cosec—— isequalto 28 28 28 28 28 28 (B) 1/2 (C)l (D) 0
Th e exac t val ue of c os —
( A) - 1/ 2 Q.3
(C) - 1 < c < 1
If a, b, c are real numbers satisfying the condition a + b + c = 0t he nt he roots oft he quadratic equation 3ax 2 + 5bx + 7c = 0 are : (A) positive (B) negative (C) real & distinct (D) imaginary 3
Q.4
8l'° g 59 +3 Let, N = — 409 (A) 0
^ , Q.5
„ It
7a 2 +B2
2n 4TT 8TC and B = cos — + c o s — + c o s —
I l to (B) V2
(C )2
ZX ~ I
V3
X 4" 4
(B>(x>{l
( C )
("
4 < X <
1)
The equation | s i n x | = si nx + 3 has in [0,2 71] : (A) no root (B) only one root (C) two roots
(D)
(x<-4)u(x>|
(D) more than two roo ts.
The number of solution of the equation, lo g( - 2x) = 2 log (x + 1) is : (B) 1
(C) 2
(D) none
IfA and B are complimentary angles, then : (A) [l + ta n| -] [l + t a n | j = 2
(B) [l + c o t ^ j (l + cot|j = 2
(C)
(D) fl - tan~j fl - t a n | j = 2
+ se cy j (l + co se c | j = 2
Subjective:
Q.10
(D)
x- 3 x- 2 If — — - < then the most general values are :
(A) zero Q. 9
then lo g 2 N has the value = y (C) - 1 (D) none
ec ua
(A) ( x < - 4 )
Q. 8
f l r-\ 2 , . (V7 H 7 -125 log25 6 (B) 1
(A )l
Q.7
_3
^
. 271 . 47t . 8rc A A = sin — + s i n — + s m —
then
Q.6
log
n
I f un = sin"0 + cos 0, prove that ^
u3-u5 u,
U5-U7 =~ u3
i l l BANSAL CLASSES
MATHEMATICS
™ T a r g e t l i t JEE 2007 CLASS: XI (P, Q, R, S)
Daily Practice Problems
DATE: 04-05/07/2005
TIME: 40 Min.
DPR
NO.-25
Q. 1
Fill in the blank : If (x + 1 ) 2 is greater then 5x - 1 and less than 7x - 3 then the integral value of x is equal to
Q.2
If x 2 - 4x + 5 - sin y = 0, y e (0, 2n) the n x =
Q.3
If the vectors, p =( lo g 2 x) i — 6 j — k and q =(log 2 x) i + 2j +(log 2 x) k are perpendicular to each other, then the valu e of x is
& y=
.
.
.
Select the correct alternative : (Only one is correct)
Q.4
The equat ion, 7tx = - 2 x 2 + 6 x - 9 has : (A) no solution (B) one solution
(C) two solutions
(D) infinite solutions
Q.5
cos a is a root of the equa tion 2 5x 2 + 5x - 12 = 0, - 1 < x < 0, th en the value of sin 2 a is : (A) 12/25 (B) -1 2/ 2 5 (C) - 24 /2 5 (D) 20/25
Q. 6
Number of ordered pair(s) (a, b) for each of which the equality, a (cos x - 1) + b2 = cos (ax + b 2 ) - 1 ho lds true for all x e R are : (A) 1 (B) 2 (C) 3
Q.7
Q. 8
Let y = cos x (cos x - cos 3 x) . The n y is : (A) > 0 only wh en x > 0 (C) > 0 for all real x
(B) < 0 fo r all real x (D) < 0 only when x < 0
For V x e R , the diff eren ce betwee n the greatest and the least value of y =
(A)l
Q.9
(D) 4
( B )2
(C)3
x 2
^ is
(D)|
In a triangle ABC , angle A = 36°, A B = AC = 1 & BC = x. If x =
t h e n t h e o rd e r ed
pair
(p, q) is : (A ) (l ,- 5)
(B) ( 1 , 5 )
Subjective:
Q.1 0
(C )( -l ,5 )
(D) ( - 1 , - 5 )
n
Find the value(s) ofthe positive integer n for which the quadratic equation, ^ ( x + k- l ) ( x + k) = 10n k=l
has solutions a and a + 1 for some a .
J j B AN SA L C LA SS ES
MATHEMATICS
V S Target I IT JEE 2007 CLASS:XI(P,
Q,R, S)
Dail y Practice Prob lems
DATE:20-21/06/2005
TIME:40Min.
DPR
NO.-22
Take approx. 40 min. for each Dpp.
PPP Q. l
Q.2
-
22
If 0 is eliminatedfromthe equations as ec 0- xt an 0= y and bs ec 0+ yt an 9= x thenfindthe relation between x and y, where a, b are constants.
.6% 2TT 4TI 7r .37 i.57 t Provethat: si n— + si n— - s i n - y = 4 si n y sin — sin —
Q.3
IfA, B, C denote the an gles of a triangle ABC then prove that the triangle is right angled if and only if sin4A + sin4B + sin4C = 0.
Q.4
Solve the inequality:
Q.5
Let p & q be the two roots of th e equation, mx2 + x (2 - m) + 3 = 0. Let mt , m 2 be the two values o f m p q 2 nil m2 satisfying— + — =-.Determine the numerical value of m— + m— j . q p 3 2 i 17 Jgj Find the value oft he continued product ] ~ [ s i n — 18 $ $ $$ * # * * * *k=i ******* ##** ***** *
Q.6
1
1 1 - — ^ —
XTI
X
X
1 I
Z
PPP 4
4
-
23
6
Q. l
If 15 sin a + 10cos a = 6, evaluate 8co sec a + 27sec 6 a
Q.2
Prove that the funct ion y = (x + x + l)/(x + 1 ) cannot have values greater than 3/2 and values smaller than 1/2 for V x eR.
Q. 3
If a, |3 are the roots of the equation
2
2
(tan 2135°)x 2 - (co secl 0° - V3 secl 0°)x + tan2240° = 0 2 then prove that the quadratic equation who se roots are (2 a + (3) and (a + 2P) is x - 12x + 35=0. Q.4
John has 'x' children by his first wife. Mary has x + 1 children by herfirsthusband. They many and have children of their own. The whole familyhas 24 children. Assuming that the children ofthe same parents do notfight,find the maximum possible number of fightsthat can take place.
Q. 5V
Solve the following equation for x, 3x3 = [x2 + Vl8 x + a/32] [x2 - Vl8 x - V32] - 4x 2 , where x e R.
Q.6**
If cosA = tanB, cosB = tanC and eo sC =t an A, then prove that sinA = sinB = si nC =2 sin 18°. * * * * ** * * * t *** *** **** *
PPP
-
2JU
Q. 1
Find the minimum value of the expression 2 log10 x - log x 0.01 ; where x > 1.
Q.2
If x, y,z be all positive acute angle thenfindthe least value of tanx (cot y + cot z) +t an y (cot z + cot x) + tanz (cot x + cot y)
Q.3
r, ,, . ... sinx - 1 , 1 . Prove the mequality + - > smx ~ 2 2
Q.4
Prove that: 5 sin x = sin(x + 2y) =>2 tan(x + y) = 3 tan y.
Q.5
If cos 0 + cos
Q.6
. it . 2% 371 sin— + s i n — + s i n — + n n n
2 - sinx 3 - sinx
„ V x e R.
w
_ Deduce the value of n if this sum is equal to 2 + J3 . n y
«
MATHEMATICS
aba nsa l cl as se s
Daily Practice Problems
Target IIT JEE 2007 DATE:
CLASS: XI (P, Q, R, S) Q.l
17-18/06/2005
TIME:
DPP. NO.-21
40Min.
Identify whether the statement is True or False. tan 2 x sin 2 x — tan 2 x - sin 2 x o
o
o
a
sin 8 2 - .cos 3 7 - and sin 1 2 7 - .sin 9 7 - have the same value. 2 2 2 2 VI
& tanB =
VI
then tan (A - B) must be irrational.
(iii)
If tan A
Ov)
If tanA = 1, ta nB = 2 and tanC = 3 then A, B, C can not be the angles of a triangle.
(v)
If tanA=
(vi)
4-^3
1
c sB ° sinB
4+
S
,' then tan 2A= tan B.
There exists a value of 0 between 0 & 2n which satisfies the equation, sin 4 0 - sin2 0 - 1 = 0 . Select the correct alternative : (More than one are correct)
Q.2
Q.3
j then : If x = sec (|) - tan <|) & y = cosec (A) x = Z j t i ( B ) y - — (C) x = - — -1 w v N y-l ' " l1-- xx ' y +' l
If the sides of a right angled triangle are { cos2a + cos2p + 2cos (a + P)} and {si n2a + sin2p + 2sin (a + p)}. then the length of the hypotenuse is : ( B ) 2 [ l - c o s ( a + P)] ( C ) 4 c o s 2 -^^
( A ) 2 [ l + c o s( a - P ) ] Q.4
(B)
(Q _ si n 2x- cos2x
Q.6
(D)4sin 2 - a
+ f3
Which of the following functions have the maximum value unity ? (A) sin 2 x - c o s 2 x
Q.5
(D) xy + x - y + 1 = 0
sin2x - cos2x V2
(D) M 5
l
. I sinx + —p^cosx V3 '
For a positive integer n , let f n (0 ) = (2 co s0 + l ) (2 co s0 - l) (2 cos 2 0 - I) (2 cos 2 2 0 - l)
(2 cos 2 n~10 - 1). Then :
(A) f 2 (ti/6) = 0
(D) f 5 (ti/1 28) = V2
(B) f 3 (?t/8) = - 1
(C) f 4 (tt/32) = 1
Two parallel chords are drawn on the same side of the centre of a circle of radius R . It is found that they subtend an angle of 0 and 2 0 at the centre of the circle. The peipendicular distance between the chords is Q . . . _ _ , 30 . 0 (A) 2 R sm — sin — (B) 1 - cos 1 + 2 co s- 1 R 2 2 2j 30 0 (D) 2 R sin -— sin — 4 4
(C) (l + cos^J fl - 2 c os ^j R
Q.7
Subjective Determine the smallest positive value of x (in degrees) for which tan( x + 100°) = tan(x + 50°) tan x tan (x - 50°).
Q.8
Let
Q.9
If X = sinf© + Y ^ j + s i n
sin
cos ( 9 - a ) = -, & ( 9 -°0 — sin (0 - P) b cos (0 - P)
then prove that
X
Y
Y
X
0-
71
12
-2 tan20.
4 then r prove that cos ( a - p) = a ° ' b d ad + be d
+ sin
0 +
3tt 12
, Y=cos
0 +
7rc 12
+ cos 0 -
71
12
+ COS 0 +
3tt 12
t
f it BANSAL CLASSES
MATHEMATICS
Target I1T JEE 2007
CLASS: XI (P, Q, R, S)
Daily Practice Problems
DATE:
15-16/06/2005
TIME:
DPP. NQ.-20
40Min.
Q. 1
Select the correct alternative : (Only one is correct) c \ 71X : 2 The number of solutions of th e equation cos x + 2A/3X + 4 is 2V3 (A) more than 2 (C)l (D)0 (B)2
Q.2
r r il The value of cot 7 — + tan 67 — - cot 67— 2 2 2
r -tan7—
(A) a rational numbe r
(C) 2(3 + 2 v 3 )
Q.3
If
x2 - x
ta n a
X
- x + 1
(B) irrational number and tan p :
the value equal to : (A)l Q.4
(B)-l
7 (
10
X5fc
is:
10
(B) V5 - 1
(D)2(3-V3) 71
0, l) , w he r e 0< a , P < —, z then tan (a + P) has (D) 3/4
(C)2
The value of 4 cos — - 3 se c— - 2 tan — (A) 1
Q.5
1 -j 2 ZX ZX.
2
is equal to
10
(C) V5 + 1
(D) zero
( 7 cosO + 24 sinO ) x ( 7 sinG - 24 cos9 ) for every 0 e R . 625 625 (B) 625 (C)
The ma xim um valu e of (A) 25
Q.6
As shown in the figure AD is the altitude on BC and AD pr od uc ed me et s the ci rc um ci rc le of AABC at P wh er e DP = x. Similarly EQ = y and FR - z. If a, b, c respec tively a b c denotes the sides BC, CA and AB then — + -— + — 2x 2y 2z has the value equal to (A) tanA + tanB + tanC (B) cotA + cotB + cotC (C) cosA + cosB + cosC (D) cosecA + cose cB + cosecC
Q.7
The graphs of y = sin x, y = cos x, y = tan x & y = cosec x are draw n on the same axe s from 0 to u/2. A vertical line is drawn through the point where the graphs of y = cos x & y = tan x cross, intersecting the other two graphs at points A & B. The length oft he line segment AB is : (A) 1
(B) nsinAcosA
Q.8
If tanB (A)
Q.9
Q.1 0
Q. 11
1-ncos2 A sin A
( l - n ) c o s A
V5-1
(C) V2
(D)
V5 + 1
then tan(A + B) equals (n - 1 ) cos A (B) K±JJ
sinA
sin A v(C) w
(n-l)cosA
sin A (D) v ' (n + l) eo sA
In a triangle ABC , angle A is greater than angle B. If the meas ures of angles A & B satisfy the equation, 3 si nx - 4 sin 3 x - K = 0, 0 < K < 1 , then the measure of angle C is (A) n/3 (B) TI/2 (C) 2tc/3 (D) 5TC/6 si n 2 9
sin 9 +c os 9 for all permissible vlaues of 9 , si n9- co s9 ta n 9 - 1 (B) is greater than 1 (A) is less than - 1 (D) lies be twe en - J 2 a n d (C) lies between - 1 and 1 including both
The value of
The number of solution of th e equation log 3x (3/ x) + l og 2 x = l is (A) 3
(B)2
(C)l
(D)0
including both
MATHEMATICS
ft B AN SAL CL AS SE S 4
Paiiy Practice Problertis
Target IIT JEE 2007 DATE:
CLASS: XI (P, Q, R, S)
TIME:
13-14/06/2005
DPP. NO.-19
40Min.
Select the correct alternative : (Only one is correct) Q.l
5rc If — < x < 371, then the value of the expression x (A) -cot-;
Q.2
(A) 12 Q.3
Q.5
is
(D)-tan|
96 sin8 0 sin65 sin35
° ° ° is equal to sin20° + sin50° + sinllO 0 (B) 24 (C )- 12
(D) 48
The valu e of cot x + cot (60° + x) + cot (120° + x) is equal to (A) cot3x
Q.4
V l - s i n x - VI + sinx
(Q ta n|
(B)cot|
The exact value of
Vl - si n x + vl + sinx
(B) tan3x
(C) 3t an 3x
(D)
3 - 9tan x 3tanx - tan 3 x
3 + cot 76° cot 16°
The value of
is: cot 76 + cot 16 (D)cot46° (A) cot 44° (B) tan 44° (C) tan 2° a, p, y & 5 are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity k. The value of a p Y 8 4 sin — + 3 sin— + 2 sin— + sin— is equal to : 2 2 2 2 (A)
2 v n
(B) 2VTTk 2 > t
Q. 6
In A ABC, the mini mum value of
Q.7
Q. 8
Q-9
Q.1 0
2A +2B —, co t — 2 2 is 2A 2
n«*
(B)2
(A) 1 i
71 \
The value of l+cos— 9y
r
l + cos
(D) non existent
(C)3
3tt 9 ,
V
(D) 2k
(C) 2Vk
l + cos
571 9
l+cos
7tc
is
(B)l? (C)1 2 16 16 For each natural number k , let Ck denotes the circle with radius k centime ters and centre at the origin. On the circle C k , a particle moves k centimeters in the counter- clockwise direction. Afte r completing its motion on C k , the particle moves to C k+ 1 in the radial direction. The motion of the particle continues in this manner .The particle starts at (1,0 ).l f the particle crosses the positive direction of the x- axis for the first time on the circle C n then n equal to (A) 6 (B) 7 (C) 8 (D) 9 ( n) ^(x-f) The set of value s of x satisfy ing the equation, 2taa\ x 'V _ 2 (0.25 )" cos2x + 1= 0, is : (A) an empty set (B) a singleton (C) a set containing two values (D) an infinite set
If 0 = 3 a and sin 0 = 2
. The value of the exp res sio n, a cosec a - b sec a is
Va + b (A)
1 2
2
Va + b
(B) 2i/a 2 + b2
(C) a + b
(D) none
,|i BA NSA L CL ASS ES
MATHEMATICS
gTa rge t i l T JEE 2007 CLASS: XI (P, Q, R, S)
Daily Practice Prob lems
DATE: 08-09/06/2005
TIME: 50 Min.
DPP. NO.-17
Fill in the blanks :
Q.l
If log 2 14 = athen log 49 32 in terms of 'a 'i s equal to
Q.2
1 The value of 1 0 § cos cosiht 6
^
97
.
is equal to
. 27t
Q.3
The solution set of the system of equ ati ons, x + y = — , cos x + cos y = — , where x & y are re al , is
Q.4
3
If a < sinx. s i n
-
.
xj . sin
+ x j < b then the ordered pair (a, b) is
.
Select the correct alternative : (Only one is correct) Q. 5
Which of the following conditions imply that the real number x is rational? I x 1/2 is rational II x 2 and x 5 are rational III x2 and x 4 are rational (A) I and II only
Q.6
(B) I and III only
(C) II and III only
(D) I, II and III
If a 3 + b3 and a + b * 0 then f or all permissib le values of a, b; log (a + b) equ als 1 (A) - (log a + log b + log 3)
1 (B) - (loga + logb + log2) f
2
3
2
(D)log ' a + b 3ab i he number of all possib le triplets ( a p a^ a 3) such that a, + a2 cos2 x + a 3 sin 2 x = 0 f or all x is : (A) 0 (B) 1 ~ (C) 3 (D) infinite (E)n one (C) log(a - ab + b )
Q.7
Q-8
T -L ^ r + V3sin2 50° cos 290° (A) ~
(B) ^
(C) V3
(D) none
Q.9
The product cot 123°. cot 133°. cot 137°. cot 147°, when simpl ified is equal to : (A)-l (B) tan 37° (C) cot 33° (D) 1
Q. 10
V x e R the greatest and the least valu es of y = j cos 2x + sin x are respectively 3 1 (A )- ,~
Q. ll
3
3
1 3 (C)-,--
(D)
3 1,--
Given si nB = ~ sin (2 A+ B) then, ta n( A+ B) = kt an A, where k has the value equal to (A) 1
Q.1 2
(B)
(B) 2
(C) 2/3
(D) 3/2
( c\ C A B If A + B + C = 7i & sin A + — = k s i n 2- , t hen ta n— t a n - = 2 2 V 2 J (A)
k + 1
(B) i l l k- 1
(C) - A k + 1
(D)
k
^
, | i BANSAL CLASSES
MATHEMATICS
Target NT JEE 200 7 CLASS: XI (P, Q, R, S)
Q. l
Dail y Practice Probl ems
DATE: 10-11/06/2005
TIME: 50 Min.
If tan A & tan B are the roots of the quadratic equation, ax 2 + b x + c = 0 the n evaluate a sin 2 (A + B) + b sin (A + B). cos (A + B) + c cos 2 (A + B).
Q.2
3n 7i 571 7TT Find the exact value of ta n 2 — + t a n 2 — + t a n 2 — +ta n 2 — 16 16 16 16
Q.3
If A + B + C = 7i:,pr ovethat r
tanA
^
v tanB.tanC y
Q.4
DPR NO.-18
X (tan A ) - 2 X (cot A).
If a cos (x + y) = sin (y - x) then prove that, 1 l + asin2x
1 l- as in 2y
2 1-a2
Q. 5
In any triangle, if (sin A + sin B + sin C) (sin A + sin B - sin C) = 3 sin A sin B, find the angle C.
Q. 6
If cos9 + coscj) = a and sinG + sintj) = b then prove that, ( a 2 - b 2 ) ( a 2 + b 2 -2) cos29 + cos2(j)
Q.7
2 7t If a = — , prove that , sec a + sec2a + sec4a = - 4.
,|i BAN SAL CLA SS ES
MATHEMATICS
9T ar ge t 1IT JEE 2007 CLASS: XI (P, Q, R, S)
Dai ly Practice Probl ems
DATE: 06-07/06/2005
TIME: SO Min.
DPP. NO.-16
Fill in the blanks :
Q. l
A rail road curve is to be laid out on a circle. If the track is to change direction by 28° in a distance of 44 meters then the radius of the curve is
Q.2
. (use n = 22/ 7)
If 'm ' is the number of integers whose logarithms to the base 10 have the characteristic 5, and 'n' is the number of integers the logarithm s of whose reciprocals to the base 10 have the i H characteristic (-3) then Iogio ~~ has the value equal to Vn )
Q.3
ln (a b) -l n| b| simplifies to
Q.4
The least value of the express ion
Q. 5
The greatest value of the expression sin 2 f • -
4xj - sin 2
cot
^ x 1 + sin
• - 4x j
.
tan2x f o r Q < x < _ j s - 8xj ^
for 0 < x <
is
.
Select the correct alternative : (Only one is correct)
10
Q.6
9
If £ ka k = 22 and j ] pa p = 32 then a ] 0 =
k=l
p=l
(A )- 10 Q.7
(B)-l (B) 19
(D) none
(C)
( B ) V 6 + V2
7 Fl
(D) V6-V2
V37 T
In a triangle ABC, angle A is greater than angle B. Ift he measures of angles A and B satisfy the equation 2 tanx - k(l +tan 2 x) = 0,w he re k e (0,1), then the measure of the angle C is (A)
Q.10
(C) 39
The side of a regular dodecagon is 2 cm. The radius ofthe circumscribed circle in cms. is: (A )4 (V 6- V2 )
Q.9
(D) 1
Let x + y = 1 and x 3 + y 3 = 19 then the value of x 2 + y 2 is equal to (A) 9
Q. 8
(C)10
Let
71
571
(B)f
6 sin 30 cos20
= p whe re 0 e
(A) p > 0 and q > 0
Us
71 (D)~2
(C)12
2371^1 0 sin 3(3 & cos2p " ' 48
J
(B )p >0 an dq <0
wher e p e
(C )p <0 an dq <0
(I3n
1471
Then
t 48 ' 48
(D )p <0 an dq >0
Select the correct alternative : (More than one are correct)
Q.ll
Which of the following statement(s) does/do not hold good ? (A) log 10 ((1.4)2 - 1 ) is positive c o t
(C) log0.1 Q.12
T _
If
3% — is negative
8
^
(B) log 1 + log 2 + log 3 = log (1 7
(D) If m = 4>°S4 and n =
f i V 2' 0637
sin39 11 . 0 , , . ^ = — then tan — can have the value equal to : smO 25 2
(A) 2
(B) 1/2
(C) - 2
'(D)-1/2
3) then n = m 4 .
MATHEMATICS
8 Target I IT JEE 200 7 CLASS: XI (P, Q, R, S)
Q. l
Dai ly Practice Pro blems
DATE: 03-04/06/2005
TIME: 45 Min.
DPP. NO.-15
If secA - tanA = p, p # 9, fi nd the value of sinA. 2n-l
Q.2
Evaluate the product
| " | t a n ( r a ) where 4 n a = 7t. r=l
3 Q. 3
If cos ( y - z ) + c os ( z - x ) + cos ( x - y ) = - ~ , prove that
cos x + cosy + cosz = 0 = sinx + siny + sinz
sinx 0 V
'
4
Q.5
If
If 9 =
sin3x =
At
i 1
a3
sin5x =
a5
aj -2 a, +a 5 a 3 -3aj th en sh ow th at — —- — a3 a}
, prove that 2" cos9 cos29 cos2 29
cos2n"19 = 1. What the val ue of the product
71
whould be if 9 =
7.
co sec l9 0 + cosec50°-cosec70 0 .
Q.6
Find the exact value of
Q.7
Prove that from the equalities, * ^ x
yyX-yZzy=zx
xz
+ 2 ~ * > - y ( z + X " log x log y
y )
=
2
(*
+
y~2> log z
follows
4
g BAN SAL CL ASS ES
MATHEMATICS Dail y Practice Problems
8T ar ge t i l l JEE 200 7
CLASS: XI (P, Q, R, S)
DATE: 01-02/06/2005
Q .l
cos4 a sin4 a i f — t w ^ -2d cos p sin P
Q.2
' . (3n "l If [1 - sin (7t + a ) + cos (rc + a) ] + 1 - sin| — + a j +
=
TIME: 50 Min.
DPR NO. -14
cos 4 B sin4 3 1 men rind the value o f — • ? • cos a sin a
2
C0S
(3% |/7,—
2
a
= a + b sin 2a then find the
value of a and b. co s( A- B) cos(C + D) ^ ~ g7 + _ j) ) = 0 then prove that tanA • tanB • tanC • tan D = - 1
Q.3
If
Q.4
Prove that
Q.5
Express sin 2 a + sin 2 p - sin 2 y + 2 sin a sinP cosy as a product of two sines and two cosines.
Q.6
Find the solution set of th e equation Yln
5.
Q.7
tan 80
x
+
25
= ( 1 + sec20 ) (1 + sec40) (1 + sec80 )
4
5
COS2X
= (25)
sin2x 2
where x
e
[ 0,2 n]
Show that x = ^J 2 cos 36° is the only solution of the equation log x (x 2 + l) = A/log^(x 2 (l + x 2 ) ) + 4
,|i BANSAL CLASSES
MATHEMATICS
pTerget SIT i i i 2007 CLASS: XI (P, Q, R, S)
Dally Pra ctice Problems.
DATE: 30-31/05/2005
TIME: SO Min.
DPP. NO.-13
Fill in the blanks :
Q. 1
The exact valu e of cos 4 9 + cos 4 29 + cos 4 30 + cos 4 40 if 0 = 7t/3 is
Q.2
The expression
Q.3
sin24 0 cos6 0 -sin6°sin66 Q The exact valu e of s i n 2 i° C os39°-cos51 0 sin69°
Q. 4
. — 1 + logjL 3j cos — fj^log 3
Q.5
Exact value of tan2 00° (cot 10° - tan 10°) is
Q.6
^— . TC 96 V3 sm —
Q.7
If co sa =
sin 4 1 + cos 4 1 - 1 — when simplified reduces to sin 61 + cos 61 - 1
COS
71 48
COS
tan
7t 24
C0 S
1 and sin(3 =
cot
.
7t 7t , L2 C0S "6 n a S
va
*ue
=
•
wher e a e 4th quadrant and p e 2 nd quadrant then
f a - p V
2
Select the correct alternative:
Q. 8
is.
when simplified reduces to
1
a + p")
.
(More than one are correct)
Identify the sta tem ents ) which is/are incorrect (A) Vl + s i n a - V l - s i n a =2sin— 2 (%
\
(n
i f a e f e t >
)
(B) sin 2 a + c o s ! y ~ a j • cos [—+ a J is independe nt of a. u/ (C)
Q. 9
log, (cos 2 (8 + ) + cos2 (0 - (j>) - co s 20-cos2<))) is equal to 1.
3 (D) If ta na = — where a e 3 rd Q then co s3a is positive. Which of th e following when simplified reduces to unity? , . (A)
l- 2s in 2 a T T v 7 2cot —+ a cos — a U J U )
4 sm a co s' a Q. 10
„ (B)
4 tan a
sin(Ti-a ) * J — +c os (7 i- a) s m a - c o s a tan — 2
(s ma + co sa )
The equation log x+1 (x - 0.5) = log x _ 0s (x +1) has (A) no real solution (C) an irrational solution
(B) no prime solution (D) no composite solution
BANSAL CLASSES P Target SIT
MATHEMATICS D a l l y
J EE 2 0 0 7
CLASS: XI (P, Qj R, S)
DATE: 27-28/05/2005
Practice Problems.
TIME: 40 Min.
DPR
NO.-12
Fill in the blanks :
Q. l
Exact value of tan 12° tan 24° tan 3 6° tan 12 °+ tan 24° - tan 36°
Q.2
iS £q Ua l t0
'
The logarithm of 32.5 to the base 10 is 1.5118834. The number whose logarithms to the same base is 4.51188 34 is
.
Q.3
If cos0 = log9log log log 3273 then the set of values of 6 lying in [0, 2TC] is
Q.4
Let a and (3 be the solution ofthe equation log x2 • lo gx/16 2 = log x/642
where ( a > (3) then a =
.
& P=
Select the correct alternative ; (Only one is correct)
Q.5
Q.6
If 7T < 29 < —-, then v ; 2 + V2 + 2 cos 40 equals : (A) - 2 cos 0 (B) - 2 sin 0 (C) 2 cos 0
(D) 2 sin 0
In a right angled triangle the hypotenuse is 2 times the perpendicular drawn from the opposite vertex. Then the other acute angles ofthe triangle are 7T ( A
> I
7T &
i
71 ( B )
i
3ir &
i
7T ( c )
i
71 &
I
71
i
371
»
Q.7
If sin 0 + cosec 0 = 2 , then the value of sin8 0 + cosec8 0 is equal to : (A) 2 (B) 28 (C) 24 (D) none of these
Q. 8
If the expression 4s in5 ac os3 ac os 2a is expressed as the sum of three sines then two of them are sin4a and sinl 0a.Th e third one is {A) sin 8a (B) sin 6 a (C) sin 5 a (D) sin 12a
Q. 9
Given a system of simultaneous equations 2x.5y = 1 an d 5x+1.2y = 2. Then (A) x = log 1 0 5andy = log 10 2
(B )x = log 1 0 2andy = log 10 5
(C) x - log1Q
(D) x - log i0 5 and y = log 10 ( j
Q.10
Ifae
and y = log 10 2
J then the expression ( .
•\Asin 4 a + si n 2 2 a + 4cos2( ^ - ^ J equals (A) 2
(B) 2 - 4cosa
(C )2 - 4s in a
(D)none
MATHEMATICS
fit BANSAL CLASSES
4!Target I1T JEE 2007 CLASS: XI (P, Q, R, S)
Daily Practice Problems
DATE: 25-26/05/2005
DPP.
TIME: 45 Min.
NO.-ll
Select the correct alternative : (Only one is correct)
Q. 1
If cosQ = ^ ( a
+
t^11
(A)
Q.2
The expression
( C ) 4 ( a 3 + - i
cos (2a - 2%) . tan (a - 4r)
sin3 e - c o s 3 9 sinG - cosG
(D)none
a '3* cot— + cot sin 2 a 4 2 2
-
2 )_
when simplified
(D) sin 2 a
(C) sin 2 (a/2)
(B) 0
COS0 - 2 tan 0 cot 0 ~ — 1 if ^ i + cot2 9 ( B ) e e ( ^ ]
(0)0
6
^
(D) 9 e
Exact valu e of cos 20° + 2 sin2 55° - V2 sin 65° is : (A) 1
Q.5
°f ' a ' =
1 + sin 2a
( A )9 e |0 , |)
Q.4
te rm s
( B ) j ( ^
reduces to: (A) 1
Q.3
co s
(B) -j L
. ( C )
(D) zero
V2
If cos (0 + ) = m cos (0 - (j>), then tan 0 is equal to : f
(A)
\ + m^l , , tan<)> 1 - my
f l - m) tanij) (B) U + my
(C)
1- m 1 + my
cot
(D)
, m "N f i1 + 1 - my
COt
Subjective:
Q.6
Solve the equa tio n, 3 . 2,o g * (3 x ~ 2) + 2 . 3 lo g * (3 x " 2> = 5 . 6 '°8*2 (3x " 2)
Q.7
Prove the identity, c o s ^ y - + 4 a ] + sin (3tt - 8a ) - sin (47t - 12 a) = 4 cos 2 a cos 4a sin 6 a.
Q.8
Solve the following equation for x :
-i . aA - 3 B = 9 C
where A = log a x . l o g 1 0 a . l o g a 5 , B = log 10 (x/10 ) & C = lo g 1 0 0 x + log 4 2.
Q.9
cos5x + cos4x Prove that: 2c os 3x - 1 ~ ° ° S
Q. 10 ^
Prove the identity, sin 2 a (1 + tan 2 a . tan a) + * + s m a - = tan 2a + tan 2 ^ + ^ V4 2 1 - sma
X+
°° S
2x
'
JgBANSAL CLASSES
MATHEMATICS
Target l! T JEE 200 7 CLASS: XI (P, Q, R, S)
Q. 1
Q.2
Dai ly Practice Pro ble ms
DATE: 23-24/05/2005
TIME: 45 Min.
DPP. NO.-IO
Select the correct alternative : (Only one is correct) Exact value of cos 2 73° + cos 2 47° - sin 2 43° + si n 2 107° is equal to : (A) 1/2 (B) 3/4 (C) 1 (D) none
Theexpression sin22° cos8° + cosl58°cos98° sin23 cos7 + cos 157 cos97
whensimplified
(A) 1
(C) 2
(B) - 1
reduces to : (D) none
Q. 3
The tangents of two acute angles are 3 and 2. The sine of twice their difference is : (A) 7/24 (B) 7/48 (C)7/5Q (D) 7/2 5
Q.4
If
sin2a
~ s i n 3 a + s i n 4 a = tan k a is an identity then the value k is equal t o : cos 2a - cos 3a + cos 4a
(A) 2
Q.5
(B )3
(C)4
log 3 fl + ^ ) + l o g 3 f 1 + ~ ] + l o g 3 f l + ^ j + equal to (A) 1
(B)3
(D ) 6
+ log_ 5 1 + 2~42~
(C )4
w
^ e n simplified has the value
(D)5
Select the correct alternative : (More than one are correct)
5 Q.6
The sines of two angles of a triangle are equal to — & — . The cosine of the third angle is: 245 T3T3
Q.7
99
255 a n
735 T3T3
^
765 oil
17 5 If secA = — and cosecB = — then sec(A + B) can have the value equal to (A) |
(B ) -f
(C)-£
(D)|
Subjective: Q. 8
If log 6 (l 5) = a and log 12 (l 8) = p, then compute the value of log 25 (24) in terms of a & p.
Q.9
Solve the equation : 2>x + 1 1 - 2X = 12X - 11 + I.
Q.10
If si nx + co sx + ta nx + co tx + secx + cose cx = 7 then sin 2x = a - W 7 ordered pair (a, b).
where a, b e N. Find the
#
MATHEMATICS
f it BANSAL CLASSES Target I1T JEE 2007
CLASS : XI (P, Q, R, S)
Daily Practice Problems
DATE: 20-21/05/2005
TIME: 40 Min.
DPP. NO.-9
Fill in the blanks.
cosa I then the value of th e expression — s 3— is equal to r 2 J sin a + cos a
0. 1
( If ta na = 2 a n d a e ! v
Q.2
If log,4 7 = a and log 14 = b then the valu e of log175 56 is equal to
Q.3
logo.75 logj cos — - l o g j cosec i ) 75
Q.4
3«
3%
has the value equal to
4
The value of (cos 15° - cos75 0 ) 8 is. Select the correct alternative ; (Only one is correct)
Q. 5
Number of roots of the equation tan2 x + cot2x = 2 which lie in the interval (0 ,4 n) is (A) 4 (B) 6 (C) 8 (D) 10 /-5„ 371 V2
hypotenuse is (B)2
(A)l
Q.7
• 4 f 3n If f (x) = 3 sm I 2
\ -e
J
and cosO - cos| — - 0 . The length of its 2 / (D) some function of 0
(C) V2
)
X + sin (3TC + X) - 2 sin
J
6
f 7~C
^
— + X
U
J
+ sin (5 7t -x ) then, for all permissible
values of x, f (x) is (A)-l Q. 8
(C) l
(B) log .
2
(C) log (log 9)
Identify the correct statement (A) If f( x) - sinx - cosx then f( l) < 0 .371 sin
. 5n sm—
w—£—
t
sec
7T
5n tan—
^ 7C 71 COS H COS cosec—+ cot— 4 3 4 4 ( q 5 iog5 2+iog5 3 is equal to 6 (D) log:j5 + log s 3 is greater than 2
have the same value
Subjective:
Q.10
(D) not a constant function
Select the correct alternative; (More than one are correct) Which ofthe following numbers are positive?
(A) log log32l Q. 9
(B) 0
Solve the equation, | x - 1 | + |x + 2 | - | x - 3 | = 4.
(D) log (sinl25 °)
4
| BANSAL CLASSES
MATHEMATICS
pT or ge t SST JEE 20 07
CLASS : XI (P, Q, R, S)
Daily Practice Probl ems
DATE: 18-19/05/2005
TIME: 45 Min.
Q. l
Select the correct alternative : (Only one is correct) Numbe r of solutions of the equation. log C0 , J£ sinx = 1 when x e [ - 2%,2 %] is : (A) 4 (B) 3 * (C) 2 (D) 1
Q.2
If tanG = J— where a, b are positive reals and 0 e 1st quadrant then the value of
DPR NO.-8
sinG sec 70 + cos0 cosec 79 is (a (A)
+
b)3(a4+b4)
(a + b) 3 (a 4 - b 4 )
(ab)7' 2
(ab)7' 2
(a + b) 3 (b 4 - a 4 ) (C)
Q.3
(a + b) 3 (a 4 + b 4 )
(ab)7' 2
(D )
(ab)7'2
The solution set of the equation, 3 ^/log10 x + 2 log10 (A) {10, 102 }
(B) {10, 10 3 }
= 2 is :
(C) {10, 104}
(D) {10, 102, 104}
Select the correct alternative : (More than one are correct)
Q.4
If J -——— + - —- —, for all permissible values ofA, then A belongs to Vl + sinA cos A cos A (A) First Quadrant
Q.5
(B) Second Quadra nt
The solution set of the system of equations, log 12 x log 2 x. (log 3 (x + y)) = 3 log 3 x is : (A) x = 6 ; y = 2 (B)x = 4 ; y = 3
Q.6
( i logx2
(D) Fourth Quadrant
> + log2 y = log 2 x and /
(C)x = 2 ; y = 6
(D)x = 3 ; y = 4
The equation ^ l + logx V27 log 3 x +1 = 0 has : (A) no integral solution (C) two real solutions
Q. 7
(C) Third Quadrant
(B) one irrational solution (D) no prime solution
Which of the following are correct ? (A) log 3 19. log 1/ 7 3 . log 4 HI >2 (C) log10 cosec (160°) is positive
(B) log5 (1/23) lies betwe en - 2 & - 1 (D) lo g^ sinj — . log V5
- 5 simplifies to an irrational number
Subjective:
Q. 8
If an equilateral triangl e and a regular hexagon have the same perimeter then find the ratio of their areas. ta n 3 0
Q.9
Prove the identity,
Q.10
Solve the equation,
+
cot 3 0
_ 1 - 2 s i n 2 Qcos 2 0 = sin 0 cos 0 '
| x - 1 | - 2 j x - 2 | + 3 | x - 3 | = 4.
4
;BANSAL CLASSES
MATHEMATICS
B Target ilT JEE 2007
CLASS • XI (P, Q, R, S)
Dally Practice Pro ble ms
DATE: 16-17/05/2005
TIME: 45 Min.
DPR NO.-7
Fill in the blanks.
Q. l
If 3logl0 x = 5 4 - x l08l ° 3 , the n x has the value equa l to
Q.2
If log 7 2 = m, then log 49 28 in terms of m has the value equal to
Q.3
1 3 5108,5 + • simplifies to y V - l o g " (0.1)
Q.4
If x 2 - 5x + 6 = 0 and log 2 (x + y) = log 4 25, then the set of ordered pair(s) of (x, y) is
. .
.
Select the correct alternative : (Only one is correct)
Q.5
Let m = tan 3 and n = sec 6, then which one of the following statement holds good? (A) m & n both are posit ive (B) m & n both are negative (C) m is positive and n is negative (D) m is negative and n is positive.
Q.6
Solution set of the equation
x + 1 x — 1
= 1 is (B)
i (A) a singleton \ (C) a set consisting of two elements Q.7
Number of values o f ' x ' in (-2%,2%) satisfying the equation 2 s i n " x +4.2 C0S (A) 8
(B )6
(C) 4 2
Q.8
Solution set of th e equation 3 2 x -2.3 X (A) {- 3, 2} (B) {6, -1}
2
+ 3 2 ( x + 6 ) = 0 is J C ) {—2, 3}
x
= 6
is
(D )2
+x +6
(D){l,-6}
Subjective:
Q.9
Find the set of values o f ' x ' satisfying the equation ^6 4 - ^ 2 3 x + 3 + 12 = 0.
Q. 10
Find 'x' satisfying the equation 4 log io X+1 - 6 lo g io x - 2.3 log io x 2 + 2 - 0.
.
«
MATHEMATICS
&BANSAL CLASSES Target
I IT
JEE 20 07
Daily Practice Problems DATE:
CLASS: XI (P, Q, R, S)
DPR NO.-6
13-14/05/2005
Time: Take approx. 40 min. Fill in the blanks.
0.1
^ . tan(l 80° - a)c os (l 80° - a)ta n(9 0° - a ) , .. , . , . l4 The expression — ^ { } r-—-h ^ wherever it is defined, is equal to sin(90° + a)cot( 90° - a)tan( 90° + a )
Q.2
If 2 cos2 (7i + x) + 3 sin (tt +x ) vanishes then the values of x lying in the interval from 0 to 2tt are
Q.3
tan1 15 If tan 25° = a then the value of t a n 2 Q 5 in terms of 'a ' is tan245° + tan335°
Q.4
The product, (log 2 17) x (log1/5 2) * (log3 - ) lies between two successive integers which are _ and
Q.5
The value of the sum, —- — + —-— + -—-— + log2 N log3 N log4 N
+
, where N denotes the
- — i s l°g2oooN
continued product of first 2000 natural numbers.
Q.6
Q.7
Q.8
Q.9
Select the correct alternative : (Only one is correct) If Xj and x 2 are two solutio ns of the equation log3 j 2x — 7 j =1 where Xj < x 2 , then the number of integer(s) between Xj and x 2 is/are: (A) 2 (B) 3 (C) 4 (D) 5 f X - 2 l o g 4 ( 4 x 4 )w hen x = - 2 is: The value of the expression, log 4 V4. (C) - 4 (D) meaningless (A)-6 (B)-5
The number of real solution(s) of the equation, sin (2X) = n x + it x is: (A) 0 (B) 1 (C) 2 (D) none of these f TT^ (3n ^ . 3 (lit 1Z tan X — .COS + x - s i n I 2 ; 12 2) I The expression ) n) .tan fH COS f x — — + X J I 2 2j I
(A) ( 1 + cos 2 x) Q.10
(B) sinhi 1
Let y = 2 +
3 + -
1
J
simplifies to
(C) - (1 + cos 2 x)
(D) cos 2x
, then the value of y is
1 -
X
1
2 + -
1
3 + ..
(A)
Vl3+3
Vl3-3
.(C)
V15+3
(D)
Vl5~3
d
jBANSAL CLASSES
MATHEMATICS
8Target SIT JEE 200 7
Daily Practice Proble ms
CLASS: XI (P, Q, R, S)
DATE: 11-12/05/2005
DPR NO.-5
Time: Take approx. 40 min. True and False : Q. 1 State whether the following statements are True or False. (a) sec2 8 . cosec 2 9 = sec 2 0 + cosec 2 9.
(b)
There exist natural numbers, m & n such that m2 = n 2 + 2002. Fill in the
Q.2
Q.3 Q.4
Q.5
blanks.
If the eighteen digit number A 3 6 4 0 5 4 8 9 8 1 2 7 0 6 4 4 B i s divisible by 99 then the ordered pair of digits (A, B) is . . The pos iti vei nte ger sp, q&r are all primes. If p 2 - q 2 = r then the set of al l possible values of ri s
The solution set of th e equation x loga
x
= ( a K )log *x is
, (where a > 0 & a * 1)
Select the correct alternative : (Only one is correct) The number of real solution of th e equation log 10 (7x - 9)2 + log ]0 (3x - 4)2 = 2 is (A) 1 (B) 2 (C) 3 (D) 4 2 3 2 log 2 ,/4a_ 3 i og27 ( a + i) _ 2a
Q.6
The ratio
_ 1 — 4fo^a 7 4iog4 9 a_ a _|
(A) a 2 - a - 1 Q. 7
" simplifies to:
(B) a 2 + a - 1
( C ) a 2 -a+l
( D ) a 2 + a + l
Which one of the following denotes the greatest positive proper fraction? / | \ l ° S 2 6 (A)
/lY°g3 5 (B) V-V
7
i (C) 3
3
-log, 2
(D)8
Select the correct alternative : (More than one are correct)
Q. 8
Which of the following when simplified, vanishes ? 1 2 (A) t — r + log3 2 log9 4
3 log27 8
(B) log^fj +log4(j(C) - log8 log 4 log 2 16 (D) logjQ cot 1° + logjq cot 2° + logjQ cot 3° + Q. 9
Which of the following numbers are positive (A) log 9(2.7) -0 -3
(B) log 1/ 2 (l/3)
Subjective:
Q. 10
+ log 10 cot 89°
Compare the numbers
log3 4 and log 56.
(C) logvT5 VlT
(D) log1/2 ~
- 2
.
K BANSAL CLASSES
c
M A
I Hf ct Vl AS
Daily Practice Problems
Target SIT JEE 2007
DPP. NO.-4
DATE : 09-10/05/2005
CLASS: XI (P, Q, R, S) Time : Take approx. 40 min. Fill in the blanks.
Q. 1
The expression -Jlog 0 5 8 has the value equal to
Q.2(a) Solution set of the equation 1 - !ogi x
+
2 -
3 - log. x is
a b (b) If (a 2 + b2)3 = (a 3 + b 3 ) 2 and ab * 0 then the numerica l value of — + — is equal to b a Q.3
A mixt ure of wine and water is made in the ratio of wine : total = k ; m. Adding x units of water or removing x units of wine (x * 0), each produces the same new ratio of wine : total. The numerical value ofthe new ratio is
Q.4
A polynomial in x of degree three which vanishes when x = 1 & x = - 2 , and has the values 4&2 8wh en x " — 1 and x = 2 respe ctively is .
Q.5
The solution set of th e equati on 4 !o g 9 x - 6.x lo g 9 2 + 2 lo g 3 27 = 0 is
Q.6
The smallest natural number of the form 1 2 3 X 4 3 Y, whic h is exactly divisibl e by 6 where 0 < X, Y< 9, is .
.
Select the correct alternative : (Only one is correct)
Q.7
x+1 1 The equation, log2 (2x 2 ) + log 2 x . x. logx 0°S2 ) + -^ lo g 4 2 (x 4 )+ 2" 3 1 o g , ' 2(log2x)
(A) exactly one real solution (C) 3 real solutions
has
(B) two real solutions (D) no solution.
Select the correct alternative : (More than one are correct)
Q.8
The equation
(log 8 x)-
= 3 has :
(A) no integral solution (C) two real solutions
(B) one natural solution (D) one irrational solution
Subjective
Q.9
Which is smaller ? log,--
or
log, u s + v:
Q. 10
8 ax It is known that x = 9 is a root of th e equation lo g. (x 2 + 15 a2 ) - log^ ( a - 2 ) = log,, -— ; -. Find the other root(s) of this equation.
MATHEMATICS
K BANSAL CLASSES Target SIT JEE 2007
Daily Practice Problems
DATE:
CLASS: XI (P, Q, R, S)
DPR
02-03/05/2005
iva-i
Time: Take approx. 30 min. Fill in the blanks :
Q.l
The value of b satisfying log ^- b = 3 - is
Q.2
The number of integral pair(s) (x ,y ) whose sum is equal to their product is
Q.3
Q-4
Q.5
Select the correct alternative : (only one is correct) 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) infinite
In a triangle ABC, 3 coins of radii 1 cm each are kept so that they touch each other and also the sides ofthe triangle as shown. The side of the triangle is (A) 3 + V3
(B) 3V3
(C) 2(1 + 7 3 )
(D)3(3-V3)
The equation
= x _ 2
has
(B) one prime solution (D) one integral solution
(A) two natural solution (C) no composite solution Q.6
116 people participated in a knockout tennis tournament. The players are paired up in the first round, the winners of the first round are paired up in the second round, and so on till the final is played between two players. If after any round, there is odd number of players, one player is given a bye, i.e. he skips that round and plays the next round wi th the winners. The total number of matches played in the tournament is (A) 115 (B) 53 (C) 232 (D )1 16
Subjective:
Q.7 Q.8 (i) (Hi)
Prove that x 4 + 4 is prime only for one value of x e N. Establish tricotomy in each of this following pairs of n umbers (ii) log4 5 and log I/16 (1 / 25) 3 log273 and2 lo g 4 2 4 and log 3 10 + log
10
81
(iv) log 1/ 5 (l / 7) and !o g 1 /7 (l / 5) 4
Q.9 Q. l 0
Compu te the value of
^ ^log53
+
27
log 9 36
3
79
The length of a com mon interna l tangent to two circles is 7 and a com mo n external tangent is 11, Compute the product of the radii of the two circles.
1 st
bpp
OM
rue
mn-i
of
succee&
J^f
t
MATHEMATICS
f it BANSAL CLASSES Target
I1T JEE 2 0 0 7
Daily Practice Problems
CLASS: XI (P, Q, R, S)
DATE:
04-05/05/2005
DPR NO.-2
Time: Take approx. 40 min. Fill in the blanks :
Q.l
1 1 1 — — + — + — has the value equal to log _ abc log ^ca abc log ab abc J be v' V (Assume all logarithms to be defined)
^Q-2 Q.3
Solution set of the equation, log, 20 x + log.0 x2 = logj20 2 - 1 is log (o T) The expression (0.05) " v 2 0 ' ' ' is a perfect square of the natural number (where o.T denotes 0.111111
Q.4 OL
oo)
The line AB is 6 meters in length and is tange nt to the inner one of the t wo concentric circles at point C. It is known that the radii of the two circles are integers. The radius of the outer circle is _ _ _ _ _ _
Q.5
The expressi on, xlo»" ~ loez • y 1 ^ - log* • zlo ° x - w h e n simplified reduces to Select the correct alternatives
Q.6
: (More than one are correct)
If p, q G N satisf y the equati on
= jV xj then p & q are :
(A) relatively prime (C) coprimc
Q.7
(B) twin prime (D) if logq p is defined then log pi is not & vice versa
The expression, log log
^
where p > 2, p e N, when simplified is :
n radical sign v(A) indepe ndent of p, but dependent on n _(€) dependent on both p & n Q. 8
(B) independent of n, but dependent on p j(D) negativ e.
Which of th e following when simplified, reduces to unity ? 21og2 + log3
(A) log ]0 5 . log1Q20 + log20 2
log 48 - log 4
(C)-log5log3^
Q.9
The number N :
-1- logs 2 when simplified reduces to •
(i + iog 3 2) (A) a prime number (C) a real which is less than log,7t
(B) an irrational number (D) a real which is greater than log76
Subjective :
Q.10
Given, log7 12 = a & log 12 24 = b . Show that, log54 168 =
1 + ab a (8 - 5 b)
J| BAN SA L CLASSES
MATHEMATICS
H g T a r g e t 1ST JEE 200 7
Pa ll y Practice Pro blem s
CLASS: XI (P, Q,R,S)
DATE: 06-07/05/2005
DPR NO.-3
Time: Take approx. 40 min. Fill in the blanks
Q. 1
The solution set of the equation 4/| x - 3jx+1 =
- 3|x~2 is
Q.2
If x = ?/? + 5V2 -
Q.3
Iflog x Iog1B(V2 + Vs) = - . Then the value of 1000 x is equal to
•
, then the value of x3 + 3x - 14 is equal t o
1
.
Select the correct alternative : (Only one is correct)
Q. 4
Which one of the following when simplified does not reduce to an integer? *°g2 3 2
2log6
( 1 N " 2
log, 16- log , 4
Q. 5*
Let m denotes the number of digits in 2 64 and n denotes the number of zeroes between decimal point and the first significant digi t in 2 _6 4 5 then the ordered pair (m, n) is (y ou may use log i0 2 = 0.3) (A) (20 .21 ) (B) (2 0.2 0) (C) £19. 19) CD) (20 .19 )
Q. 6
PQRS is a square. SR is a tangent (at point S) to the circle with centre O and TR = OS. The n, the ratio of area of the circle to the area oft he square is 7i (A) j
11 (B) -
3 (C) -
(D)
7 -
Q.7 V Let u = (logjx) 2 - 6 log2x +12 where x is a real number. Then the equation x li - 256 has (A) no solution for x (B) exactly one solution for x (C) exactly two distinct solutions for x (D) exactly three distinct solutions for x Subjective:
Q. 8
If x, y, z are all different real numbers, then prove that 1 (x-y) 2
Q.9
Solve
Q.10
If
( 1 + 1 1 1 + r = 2 2 (y-z) (z-x) Vx-y y - z
1 z-x)
3 X + 1 - 13* - 11 = 2 log5 j 6 - x j.
log )g 36 = a & log24 72 = b, then find the value of
4 (a + b) -5 ab .
K
YJI
k..!^ 7
.
MATHEMATICS
I QBANSAL CLASSES
Daily Practice Problems
1 8 T a r g e t SIT JEE 2 0 0 7 CLASS : XII (ABCD)
TIME: 50 Min each DPR
DATE : 28-29/06/2006
DATE: 28-29/06/2006
Q. l
tan 9 =
DPR
NO.-25
TIME: 50 Min.
—j
where 0 e (0,2%), find the possibl e value of 0.
[2]
2 —+
2 + '--oo Q. 2 Q.3
Find the sum of the solutions of the equation 2 e 2 x ~ 5e x + 4 = 0.
[2]
Suppos e that x and y are posi tive numb ers for whic h log 9 x = log 12 y = log 15 (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
x - > 0 +
l 2 (sinx)* ] + [22 (sinx) x ] +
Limit n->«>
+ [n2 (sinx) x ] where [ . ] denotes the
n3
I
greatest integer function. Q.5
1 Consider f ( x ) = - j = Vb
[4] /b-a
,
VT~
M v
S1.
b - a
.
»2x .
\ 2
I Va + b t a n
x
, , f or b > a > 0 & th e functions g(x)&h(x)
1
-j— sinx I
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 of''f, where [x] is the greatest integer function of x & {x} is the fractional 7C part of x. Then dis cuss the continuity o f 'g' & 'h' at x=— and x = 0 respectively. Q.6 Q.7
[5]
J f ^ d x
[5]
Using substitution only, evaluate: j co se c 3 xdx.
DATE: 30-01/06-07/2006
[5] TIME : 50 Min.
Q. l
12 A If sin A = — . Find the value of tan ~ .
Q.2
The straight line - + - = 1 cuts the x-ax is & the y- ax is in A& B respectively & a straight line perpendicular
x
[2]
y
to AB cuts them in P & Q respectively. Find the locus of the point of intersection of AQ & BP.
Q.3
tanO 1 co t0 If -———- —— = —, find the value of . tan 0 - t a n 30 3 cot0 ~cot3 0
[3] 1 J