| JgBANSAL CLASSES MATHEMATICS I p T a r g e f SIT J E E 2 0 0 7 Daily Practice Problems CLASS: XII (ABCdT DATE: 11-12/12/2006 TIME: 50 Min. DPP. NO.-53

Revision Dpp on Permutation & combination Select the correct alternative. (Only one is correct)

Q. 1 Number of natural numbers between 100 and 1000 such that at least one of their digits is 7, is (A) 225 (B) 243 (C) 252 (D)none

Q.2 The number of ways in which 100 persons may be seated at 2 round tables T, and T 2 ,50 persons being seated at each is :

( A ) f m M ! m l !

Q. 3 There are six periods in each working day of a school. Number of ways in which 5 subjects 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)120

Q. 4 The number of ways in which 4 boys & 4 girls can stand in a circle so that each boy and each girl is one after the other is: (A) 4 ! . 4 ! (B) 8 ! (C) 7 ! (D) 3 ! . 4 !

Q.5 If letters of the word "PARKAR" are written down in all possible manner as they are in a dictionary, then the rank of the word "PARKAR" is: (A) 98 (B) 99 (C) 100 (D) 101

Q. 6 The number of different words of three letters which can be formed from the word "PROPOSAL", if a vowel is always in the middle are: (A) 53 (B) 52 (C) 63 (D) 32

Q.7 Consider 8 vertices of aregular octagon and its centre. If T denotes the number of triangles and S denotes the number of straight lines that can be formed with these 9 points then T - S has the value equal to (A) 44 (B)48 (C) 52 (D)56

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 which a mixed double tennis game can be arranged 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 dice are rolled once. The number of possible outcomes in which atleast one die shows up 2 is: (A) 216 (B) 648 (C) 625 (D) 671 Il-l OQ

Q-l l X n r x . p is equal to :

f ( B ) f ^ ( Q ^ Q. 12 There are counters available in x different colours, The counters are all alike except for the colour. The

total number of arrangements consisting of y counters, assuming sufficient number of counters of each colour, if no arrangement consists of all counters of the same colour is: (A) x y - x (B) x y - y (C) y x - x ( D ) y x - y

Q. 13 In a plane a set of 8 parallel lines intersects a set of n parallel lines, that goes in another direction, forming a total of 1260 parallelograms. The value of n is: (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 number of ways in which they can travel, is (A) 91 (B) 126 (C) 182 (D)3920

Q. 15 In a conference 10 speakers are present, If S5 wants to speak before S2 & S2 wants to speak after S3, then the number of ways all the 10 speakers can give their speeches with the above restriction if the remaining seven speakers have no obj ection to speak at any number is

(A) 10C3 (B) 10Pg (C) I0P3 (D) i i l

Q. 16 There are 8 different consonants and 6 different vowels. Number of different words of 7 letters which can be formed, if they are to contain 4 consonants and 3 vowels if the three vowels are to occupy even places is (A) 8P 4 . 6P3 (B) 8 P 4 . 6C3 (C) sP4 . 7P3 (D) 6P3 . 7 C 3 . 8P4

Q.17 Number of ways in which 5 different books can be tied up in three bundles is (A) 5 (B) 10 (C) 25 (D) 50

Q. 18 How many words can be made with the letters of the words "GENIUS" if each word neither begins with G nor ends in S is : (A) 24 (B) 240 (C) 480 (D) 504

Q. 19 Number of numbers greater than 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) Number of naughts standing at the end of 1125 is 30. (B) Atelegraph has 10 arms and each aim is capable of 9 distinct positions excluding the position of rest.

The number of signals that can be transmitted is 1010 - 1 . (C) In a table tennis tournament, every player plays with every other player. If the number of games

played is 5050 then the number of players in the tournament is 100. (D) Number of numbers 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 + C4 > n + 2C5 - nC5 for all ' n ' greater than :

(A) 8 (B) 9 (C) 10 (D) 11

Q.22 The number of ways in which 200 different things can be divided into groups of 100 pairs is:

(10fl (102^1 (103^1 (200^ (A) 2 ( 1 . 3 . s..199) I t J r r J I T

200! - _ 200! 2'00 (100)!

(C) -,100 /lnn\ i (D) ->100

Q.23 The continued product, 2 . 6 . 1 0 . 1 4 to n factors is equal to : (A) 2nPn (B) 2Cn (C) (n+ 1)(n + 2)(n + 3) (n + n) (D)2 n (1 - 3 - 5 2 n - l )

Q.24 The Number of ways in which five different books to be distributed among 3 persons so that each person gets at least one book, is equal to the number of ways in which (A) 5 persons are allotted 3 different residential flats so that and each person is alloted at most one flat

and no two persons are alloted the same flat. (B) number of parallelograms (some of which may be overlapping) formed by one set of 6 parallel lines

and other set of 5 parallel lines that goes in other direction. (C) 5 different toys are to be distributed among 3 children, so that each child gets at least one toy. (D) 3 mathematics professors are assigned five different lecturers to be delivered, so that each professor

gets at least one lecturer.

4J BANSAL CLASSES MATHEMATICS {Target BIT JEE 2007 Daily Practice Problems CLASS: XII (ABCD) DATE: 22-23/11/2006 TIME: 75 Min. DPR NO.-S2

This is the test paper ofClass-XI (PQRS & J) held on 19-11-2006. Take exactly 75 minutes.

Q.l Consider the quadratic polynomial f (x) = x2 - 4ax + 5 a2 - 6a. (a) Find the smallest positive integral value of'a' for which f (x) is positive for every real x. (b) Find the largest distance between the roots of the equation f (x) = 0. [2.5 + 2.5]

Q.2(a) Find the greatest value of c such that system of equations x2 + y2 = 25 x + y = c

has a real solution.

(b) The equations to a pair of opposite sides of a parallelogram are x2 - 7x + 6 = 0 and y 2 - 1 4 y + 40 = 0

find the equations to its diagonals. [2.5+2.5]

Q. 3 Find the equation of the straight line with gradient 2 if it intercepts a chord of length 4^/5 on the circle x2 + y2 - 6x - 1 Oy + 9 = 0. [5]

cos^ 2x + 3 cos 2x Q.4 The value of the expression, 7 7 wherever defined is independent of x. Without allotting

cos x - s i n x a particular value of x, find the value of this constant. [5]

Q. 5 Find the general solution of the equation sin3x(l + cot x) + cos3x(l + tan x) = cos 2x. [5]

Q. 6 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 terms of A.P. (ii) second term of the G.P. [2.5+2.5]

Q.7(a) Let x = or x = - 15 satisfies the equation, log8(&x2+wx + / ) = 2 . If k, w and/are relatively prime

positive integers then find the value of k+w +f.

(b) The quadratic equation x2 + mx + n - 0 has roots which are twice those of x2 + px + m = 0 and n

m, n and p* 0. Find the value of ~ . [2.5+2.5]

x y Q. 8 Lme + = 1 intersects the x and y axes at M and N respectively. If the coordinates of the point P

6 8 lying inside the triangle OMN (where 'O' is origin) are (a, b) such that the areas of the triangle POM, 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]

Q. 9 Starting at the origin, a beam oflight hits a mirror (in the fomi of a line) at the point A(4,8) and is reflected at the point B(8,12). Compute the slope of the mirror. [5]

Q. 10 Find the solution set of inequality, logx+3 (x2 - x) < 1. [5]

Q. l l If the first 3 consecutive terms of a geometrical progression are the roots of the equation 2x3 - 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(x2+y2 + 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 Let / (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]

Q.14 If x + 1 x + 2 x + a x + 2 x + 3 x + b x + 3 x + 4 x + c

= 0 then all lines represented by ax + by + c = 0 pass through a fixed point.

Find the coordinates of that fixed point. [5]

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 2(1-1 whose common ratios are , - , ,...., ,... respectively, then find the value of . [5] 2* J nr O *T* 1 -

r=l

A 5 B 20 Q. 16 In any triangle if tan = 7 and tan = then find the value of tan C. [5] 2 6 2 3 /

Q.17 The radii r p r2, r3 of escribed circles of a triangle ABC are in harmonic progression. If its area is 24 sq. cm and its perimeter is 24 cm, 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 orthogonal to the circle x2 + y2 - 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 x2 + y2 2x + 6y + 1 = 0. [5]

Q.20 Find the equation of the circle whose radius is 3 and which touches the circle x2 + y2 - 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, x2 + (2 cos (j))x - sin2c|> = 0 V

1 i BANSAL CLASSES MATHEMATICS Target IIT JEE 2007 Daily Practice Problems CLASS : XII (ABCD) 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.

1 1 1 1 1 Q.l Find the sum of the infinite series 7T + 7T: + T r + 7 7 + 7 r +

9 18 30 45 63

(A) } (B) i (C) | (D) f

Q. 2 Number of degrees in the smallest positive angle x such that 8 sin x cos5x - 8 sin5x cos x = 1, is

(A) 5 (B) 7.5 (C)10 (D) 15

Q. 3 There exist positive integers A, B and C with no common factors greater than 1, such that Alog2005 + B log2002 = 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. If the polynomial can y be written in the form / ( x ) = x3 + ax2 + 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 < x < 1 4

Q.7 The sequence a t , a^ a3,.... satisfies a{ = 19. = 99, and for all n > 3, an is the arithmetic mean of the first n - 1 terms. Then a2 is equal to (A) 179 (B) 99 (C) 79 (D)59

Q.8 If b is the arithmetic mean between a and x; b is the geometric mean between '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) a3 (B,b3 ( C ) ' t a 2 b - a 2 a - b

Q.9 Given A(0,0), ABCD 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) 4V5 (B)5V5 (C)6V5 (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 If a, b and c are numbers for which the equation - = + ~ + is an identity, x2 + 1 0 x - 3 6 a b c

= + +

x(x - 3 ) x x - 3 ( x - 3 )

then a + b + c equals (A) 2 (B) 3 (C)10 (D)8

1 1 1 Q. 13 If a, b, c are in G.P. then ~ , , are in b - a 2 b b - c

(A) A. P. (B) G.P. (C)H.P. (D) none

Q. 14 How many terms are there in the G.P. 5,20, 80, 20480. (A) 6 (B)5 (C) 7 (D)8

1 1 1 Q. 15 The sum of the first 14 terms of the sequence j= + h t= + is

1 + Vx 1 - X 1 v x

( A ) ( B ) 7 ^ f >

14 ( C ) (l + V x ) ( l - x ) ( l - V x ) ( D ) n o n e

10

Q. 16 If x, y > 0, logyx + logxy = and xy = 144, then arithmetic mean of x and y is

(A) 24 (B) 36 (C)12V2 (D)13V3 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(2r + R)

Q. 18 The simplest form of 1 + is

1 1 - a

(A) a for a * 1 (B) a for a * 0 and a * 1 (C) - a for a * 0 and a * 1 ( D ) l f o r a * l

Select the correct alternatives. (More than one are correct)

Q. 19 If the quadratic equation ax2 + bx + c = 0 (a > 0) has sec29 and cosec20 as its roots then which of the following must hold good? (A) b + c = 0 (B) b2 - 4ac > 0 (C) c > 4a (D) 4a + b > 0

Q.20 Which of the following equations can have sec29 and cosec29 as its roots (9 e R)? (A) x2 - 3x + 3 = 0 (B) x2 - 6x + 6 = 0 (C) x2 - 9x + 9 = 0 (D) x2 - 2x + 2 = 0

Q.21 The equation | x - 2 |10x2_1 = | x - 2 |3x has (A) 3 integral solutions (B) 4 real solutions (C) 1 prime solution (D) no irrational solution

Q. 22 Which of the following statements hold good? (A) If Mis the maximum and m is the minimum value of y = 3 sin2x + 3 sin x cos x + 7 cos2x then the

mean of M and m is 5, 71 . 7 1

(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 + sin2 2x + 4 cos2

independent ofx.

(D) There are exactly 2 values of 9 in [0, 2tt] which satisfy 4 cos29 - 2 -Jl cos 9 - 1 = 0 .

4 2 is

MATCH THE COLUMN

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 Q.l (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) Abscissa of the orthocentre of the triangle whose vertices are the points (-2, -1); (6, - 1) and (2, 5) (R) 2

(C) Variable line 3x(A. + 1) + 4y(A. - 1) - 3 ( 1 - 1) = 0 for different values of A, are concurrent at the (S) 3/2 point (a, b). The sum (a + b) is

(D) The equation ax2 + 3xy - 2y2 - 5x + 5y + c = 0 represents two straight lines perpendicular to each other, then | a + c | equals

o

o

Column-I Column-II

Q.2 (A) In a triangle ABC, AB = 2^3 , BC = 2-J6 , AC > 6, (P) 60

and area of the triangle ABC is 3 V

J g BANSAL CLASSES MATHEMATICS I B Target I1T JEE 2007 Pa/7/ Practice Problems CLASS: XII (ABCD) DATE: 04-07/10/2006 TIME: 40 Min.for each DPP. NO.-49, 50

-49 Q. 1 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 how many ways can they be shot (one at a time) if no target below it has been shot. [4]

Q.2 Evaluate: /x(sin2(sinx) + cos2(cosx))dx o

Q.3 Evaluate: jx(sin(cos2 x)cos(sin 2 x))dx

[4]

[4]

Q.4 J - . x - dx [6] * V YQ111 YJ-fAQY 0 . x sin x + cos x /

R satisfying ( x / ( x ) - 2F(x) ) (F(x) -X 2 )=0 V x e R where f (x) = F'(x).

[4]

0 3 j f ^ f * Q ' 3 J 2 l 3 - x J HI

00 J Q.4 For a > 0, b > 0 verify that J^ dx reduces to zero by a substitution x = 1 /t. Using this or

ax" +bx + a o

f fax A otherwise evaluate: i 2 0 d a x [7]

Q.5 1 - 1 " \ 3 tan x

v x y d x [81

A

JABANSAL CLASS l ^ P T a r g e t HT J E E 2 0 0 7

ES MATHEMATICS Daily Practice Problems

CLASS: XII (ABCD) 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. 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.

Q. 1 The area of the region of the plane bounded above by the graph of x2 + y2 + 6x + 8 = 0 and below by the graph of y = | x + 3 is (A) jc/4 (B) ti2/4 (C) 7c/2 (D) it

Q.27' 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 x2 + y2 = 1, then the maximum value of (x + y)2 is

(A) 3 (B) 2 (C) 3/2 (D) J5

Q.4 The value of the definite integral j n (a > 0) is q (1 + x )(1 + x ) (A) ti/4 (B) nil (C) tc (D) some function of a.

a b e cos coscos -Let a, b, c are non zero constant number then Lim - equals

r-co . b . C sinsin -

r r

. . . a 2 + b 2 - c 2 c2 + a 2 - b 2 ^ x b 2 + c 2 - a 2 . . . J _ _ _ (A) (B) (C) (D) independent of a, band c 2bc 2bc 2bc

Q.6 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 (B) - 15 (C) - f (D) - y

sinx cosx tanx cotx Q.7/ The minimum value of the function/(x) = 1 + / 2 + 7 - + ~T 9 = as

v Vl-cos x v l - s in 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

Q.8 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) ^ 2 Q-9 Let Cn = J s i n - V ) X then Lim n2-C f l e q u a i s

"n+l (A) 1 (B) 0 (C) - 1 (D) 1/2

/ 2 2 2 Q. 10 Let Zj, z2, z3 be complex numbers suchthat zx + z2 + z3 = 0 and | zx \ - \ | = | z31 = 1 then z, + z 2 + z 3 , is (A) greater than zero (B) equal to 3 (C) equal to zero (D) equal to 1

dx

Q.ll

Q.22

Q.23

Q.24

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)

(n + 1)2 (A) n2(n + l)2

(B) (n -l) 2n 2 (C) (D) n2

Q.12

^ . 1 3

V ^ l 5

1 3 - 3x + sin x is

(C)2 (D) more than 2

Number of roots of the function/(x) ~ + ^

(A) 0 (B) 1 If p (x) = ax2 + 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 (B) 0 (C) 2 (D) cannot be determined Which of the following definite integral has a positive value?

2it/3 0 0 -3tt/2 Jsin(3x + 7i)dx (g) Jsin(3x + 7t)dx ^ q Jsin(3x + Jt)dx ^ j Sin(3x + 7t)dx

2tc/3 -3it/2

Q. 16

v X l 7

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 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

\J2[- 18 The value of the scalar (p x q)-(r x s) can be expressed in the determinant form as

(A) q-r q s p-r p-s (B)

p-r q-s

p-s q-r (C)

p-r q-s q-r p-s (D)

p-r q-r

p-s q-s

Q.19

Q.20

Q.21

jf Lim x In x->00 5, where a, p, y are finite real numbers then

a/x 1 y 0 1/x p 1 0 1/x

(A) a = 2, p=l, yeR (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 (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

(B) (C) 2 ~ (D) 2-^r 9

(A) 9 v ' 19 9 v _ / 19 17 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

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 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 S3 = S II s2 = s4 (A) II is true but not I (B) I is true but not II (C) both I and II are true (D) both I and II are false.

JsBANSAL CLASS V S Target NT JEE 2007

ES MATHEMATICS Daily Practice Problems

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) [ 3 x 6 = 18] There is NEGATIVE marking. 1 mark will be deducted for each wrong answer.

Q. 1 The function/(x) is defined for x > 0 and has its inverse g (x) which is differentiable. I f / (x) satisfies g(x) J f (t) dt = X2 and g (0) = 0 then

Q.l

(A)/(x) is an odd linear polynomial (C)/(2) = 1

(B)/(x) is some quadratic polynomial (D)g(2) = 4

Q. 2 Consider a triangle ABC in xy plane with D, E and F as the middle points of the sides BC, CA and AB respectively. If the coordinates of the points D, E and F are (3/2, 3/2); (7/2,0) and (0, -1/2) then which of the following are correct? (A) circumcentre of the triangle ABC does not lie inside the triangle. (B) orthocentre, centroid, circumcentre and incentre of triangle DEF are collinear but of triangle ABC

are non collinear. (C) Equation of a line passes through the orthocentre of triangle ABC and perpendicular to its plane is

r = 2(i - j) + A.k 5V2

(D) distance between centroid and orthocentre of the triangle ABC is . X X

Q. 3 If a continuous function / (x) satisfies the relation, j t / ( x - t ) dt = j / ( t ) dt + s j n X-+ cos x - x - 1 for 0 0 .

all real numbers x, then which of the following does not hold good? it

(A)/(0) = 1 ( B ) / ' (0) = 0 (C)f" (0) = 2 (D) J / ( x ) d x = e* 0

MATCH THE COLUMN [ 3 x 8 = 24] There is NEGATIVE marking. 0.5 mark will be deducted for each wrong match within a question.

Column I

,.. T. In x r dt (A) Lim YVtYl V J /n t X-*co X J3 In t

e

IS

Column II

(P) 0

' /~T7 2 vx +1 xz+l e - e is (B) Lim

(C) Lim (-1)" s i n f W n 2 + 0.5n + l l sin is where n e N J 4n

/ \

(Q) :

(R) 1

(D) The value of the integral j -tan -1

VX + ly f 9 A l + 2 x - 2 x A

dx is 0 tan"1

(S) non existent

Q.2 Consider the matrices A=

Q.3

andB : a b 0 1 and let P be any orthogonal matrix and Q = PAP

T

and R = PTQKP also S = PBPT and T = PTSKP Column I

(A) If we vary K from 1 to n then the first row first column elements at Rwill form

(B) If we vary K from 1 to n then the 2nd row 2nd

column elements at Rwill form (C) If we vary K from 1 to n then the first row first

column elements of T will form (D) If we vary K from 3 to n then the first row 2nd column (S) A. P. with common difference - 2.

elements of T will represent the sum of

Column II

(P) G.P. with common ratio a

(Q) A. P. with common difference 2

(R) GP. with common ratio b

Column I Column II

(A) Given two vectors a and b such that | a | = | b | = |a + b | = 1 (P)

The angle between the vectors 2a + b and a is (B) In a scalene triangle ABC, if a c o s A = b c o s B (Q)

then Z C equals (C) In a triangle ABC, BC = 1 and AC = 2. The maximum possible (R)

value which the Z A can have is (D) In a A ABC Z B = 75 and BC = 2AD where AD is the (S)

altitude from A, then Z C equals

30

45

60

90

SUBJECTIVE:

tc/2

Q.l SupposeV= J x 2 1 sin x 2

dx, find the value of 96V

71

[ 5 x 1 0 = 50]

Q. 2 One of the roots of the equation 2000x6 + 100x5 + 1 Ox3 + x - 2 = 0 is of the form m + " , where m r

is non zero integer and n and r are relatively prime natural numbers. Find the value of m + n + r.

Q.3 A circle C is tangent to the x and y axis in the first quadrant at the points P and Q respectively. BC and AD are parallel tangents to the circle with slope - 1 . If the points A and B are on the y-axis while C and

D are on the x-axis and the area of the figure ABCD is 900 V2 sq. units then find the radius of the circle.

Q. 4 Let/(x) = ax2 - 4ax+b (a > 0) be defined in 1 < x < 5. Suppose the average of the maximum value and the minimum value of the function is 14, and the difference between the maximum value and minimum value is 18. Find the value of a2 + b2.

Q.5 If the Lim 1

x-*0 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 Target I IT JEE 2007

>ES MATHEMATICS Daily Practice Problems

CLASS: XII (ABCD) DATE: 27-28/9/2006 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 x2 - px + q = 0 and cot a ,cot p are the roots of x2 - rx + s = 0 then find

the value of rs in terms of p and q. [4]

Q. 2 Let P(x) = ax2 + bx + 8 is a quadratic polynomial. If the minimum value of P(x) is 6 when x=2, find the values of a and b.

Q.3 LetP= f j n=l

( \_\ .n-i

10 2 " then find log001(P).

14]

[4]

Q. 4 Prove the identity

Q.5

sec 8A - 1 tan 8 A sec 4A - 1 tan 2 A

Find the general solution set of the equation loglan x(2 + 4 cos hi) - 2.

sin a + sin 3a + sin 5a + + s in l7a n Q.6 Find the value of - when a = .

cos a + cos 3a + cos 5a + + cos l7a 24

Q.7(a) Sum the following series to infinity

1 1 1

[4]

[4]

[4]

1-4-7 + 4-7-10 + 7-10-13 +

(b) Sum the following series upton-terms. 1 - 2 - 3 - 4 + 2 - 3 - 4 - 5 + 3 - 4 - 5 - 6 + [3 + 3]

[6] Q.8 The equation cos2x - sin x + a = 0 has roots when x e (0, rc/2) find 'a'.

Q. 9 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 Find the harmonic mean of B and C. {6]

Q. 10 Express cos 5x in terms of cos x and hence find general solution ofthe equation cos 5x = 16 cos5x. [6]

Q. 11 If x is real and 4y2 + 4xy + x + 6 = 0, then find the complete set of values of x for which y is real.

Q. 12 Find the sum of all the integral solutions of the inequality 21og 3 x-41og x 27 cos a + cos p +cosy

j y i 4(a) In any A ABC prove that

C C c2 = (a - b)2cos2 + (a + b)2sin2 .

(b) In any A ABC prove that a3cos(B - C) + b3cos(C - A) + c3cos(A - B) = 3 abc.

[7]

[4 + 4]

dl BANSAL CLASSES MATHEMATICS 5Targe* l iT 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 r O i f r ^ S Q. 1 Evaluate 8n-.2r 5s where 5 r s =

r=l s=l 1 if r = s Will the sum hold i f n - > oo? [4]

x x Q.2 Find the general solution of the equation, 2 + tan x cot + cot x tan = 0. |4J

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. 14]

log0 3 ( x - 1 ) ' Q.4 Find the integral solution of the inequality 1 + cos x true for every x e R [5]

Q.10 Solveforx, sig2*2 ^ l o g J x V s ) = ^ l o g ^ x 2 _5 io g 2 * 1 , [5]

cot C Q. 11 In a triangle ABC if a2 + b2 = 101 c2 then find the value of . [5] & cot A + cot B 1 1

Q.12 Solve the equation for x, 52 + 5 2 + ! 0 g 5 ( s m x ) = 152+l08 l5 (C0Sx) [5]

00

Zn ~ . [5] n=l 6

Q. 14 Suppose that P(x) is a quadratic polynomial such that P(0) = cos340, P( 1) = (cos 40)(sm240) and P(2) t 0 . Find the value of P(3). [8]

Q . 15 If /, m, n are 3 numbers in G.P. prove that the first term of an A.P. whose 7th, mth, nth terms are in H.P. is to the common difference as (m + 1) to 1. [8]

BAN SAL CLASSES MATHEMATICS y g Target I IT JEE 2007 Daily Practice Problems CLASS : XII (ABCD) DATE: 22-23/9/2006 TIME: 55 to 60 Min. DPP. NO.-45

Q. 1 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 )

a + b + c d + e + f use vectors or otherwise to prove that,

V a 2 + b 2 + c 2 V d 2 + e 2 + f 2 '

Q.2 Let the equation x3 - 4x2 + 5x - 1.9 = 0 has real roots r, s, t. Find the area of the triangle with sides r, s, and t.

50 2

Q. 3 Suppose xJ + ax2 + bx + c satisfies f (-2) = - 1 0 and takes the extreme value where x = . Find

the value of a, b and c.

f i - y H v r / n x x + xy_ I

Q-4 L e t I ^ / n x x + x y - l d X a n d 1 y d y

x d dy where ~ = xy. Show that I J = (x + d)(y + c) where c, d e R. Hence show that (I J) = I + J

y dx dx

Q.5 Let a;, i = 1, 2, 3, 4, be real numbers such that aj + % + % + a4 = 0. Show that for arbitrary real numbers bi5 i = 1,2, 3 the equation

a, + bjx + 3a2x2 + b2x3 + Sa^x4 + b3x5 + 7a4x6 = 0 has at least one real root which lies on the interval - 1 < x < 1.

V3 x 2 - l r x l Q.6 Evaluate: t = x dx J x + x +3x" + X + 1 I

Q. 7 Let x, y e R in the interval (0, 1) and x + y = 1. Find the minimum value of the expression xx + yy

r | (1 - sin x)(2 - sin x) ^ ^ y (1 + sin x)(2 + sin x)

i l l S B A N S A L C L A S S l U l a r g e t NT JEE 2 0 0 7

E S M A T H E M A T I C S 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 TIME : 60 Min.

This is the test paper of Class-XIII (XYZ) held on 27-08-2006. Take exactly 60 minutes. S ^ ' S - y V

Select the correct alternative, (Only one is correct) [16 x 3 = 48] There is NEGATIVE marking. 1 mark will be deducted for each wrong answer.

sin2(x3 + x2 + x - 3 ) Q. I Lirn ~~ ~ ~~ has the value equal to

x->i 1 cos(x 4x + 3) M

(A) 18 (B) 9/2 (C) 9 (D) none r dt Q.2 / Let/(x)= . . If g'(x) is the inverse of / (x) then g'(0) has the value equal to

* 3-v t4 +3t2 +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 ) (B) { x | x > 0) (C) { x | x * - l a n d x * 0 a n d x * 1) (D) {-1, 1}

Z2 37 + 6 Q.4 j Let w = - and z = 1 + i. then | w | and amp w respectively are 6/ z +1 ,

(A) 2, - n /4 (B) , - 71/4 (C) 2, 3TC/4 (D) ^ , 3n/4

1 - cos a - tan 2 (a/2) k cos a Q.5 A If . j / " ~ = where k, w and p have no common factor other than 1, then the

./! sin (a/2) w + pcosa F value of k2 + w2+ p2 is equal to (A) 3 (B)4 (C)5 (D)6 4

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 x2 + y2 = 8, the maximum possible value of x - y, is (A) 2 (B) (C) V2/2 (D) 4

u(x) U^x) ' Q .8 / Let w(x) and v(x) are differentiable functions such that = 7. If ~ P and

p+q has the value equal to p - q M

u(x) v(x)

= q, then

(A) 1 (B)0 (C)7 ( D ) - 7

Q.9 The coefficient of x9 when (x + (2/Vx j)30 is expanded and simplified is (A) 30C|4 29 (B) 30C]6 214 (C)30C9-221 (D) 10C9

Q. 10 Let C be the circle described by (x - a)2 + y2 = r2 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 V a 2 - r 2 r a (A) m = r ^ 7 (B) m = (C) m = - (D) m = -Va - r r a r

Q. 11 What can one say about the local extrema of the function/(x) = x + (1/x)? (A) The local maximum off (x) is greater than the local minimum of/(x). (B) The local minimum off (x) greater than the local maximum off (x). (C) The function/(x) does not have any local extrema. (D)/(x) has one asymptote.

Q.l 2 tan / r _2^ arc tan v I 3

+ arctan(5) equals

(A) - / 3 ( B ) - l (C)l (D)V3

/ y/Q- ip 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

^gf. 14 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 + 32 + 52 + + (99)2 then the value of the sum 22 + 42 + 62 + + (100)2 is (A) S + 2550 (B)2S (C) 4S (D) S + 5050

Q. 16 Through the focus of the parabola y2 = 2px (p > 0) a line is drawn which intersects the curve at A(x,, y,) y\y2

and B(x,, v.). The ratio equals x l x 2 (A) 2 (B) - 1 (C) - 4 (D) some function of p Select the correct alternative. (Only one is correct) [ 9 x 4 = 36 j There is NEGATIVE marking. 1 mark will-be deducted for each wrong answer,

i n-3n ^ i '! 7 I f ^ n ( x - 9 ) + n - 3 D + 1 - 3 n = 3 ^ ^ ^ f X iS 6 N )

(A) [2,5)' ' (B) (1,5) (C) (-1,5) (D)(-co,oo)

18 The area of the region(s) enclosed by the curves v = x2 and y = ^ | x | is (A) 1/3 (B) 2/3 (C) 1/6 (D) 1

Q.l 9 Suppose that the domain of the function/(x) is set D and the range is the set R, where D and R are the subsets of real numbers. Consider the functions:/(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 as/and 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)

2 r x + 2mx - 1 for x < 0 Q.20 / : 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) (B) ( oo, 0] (C)(0,oo) (D) [0, co)

Q.21 Let A = { x | x2 + (M - l ) x - 2(m + 1 ) = 0 , X G R } ; 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 . 2 2 If the function/(x) = 4x2 - 4x - tarra has the minimum value equal to - 4 then the most general values of 'a' are given by (A) 2n7t + ti/3 (B) 2nn - rc/3 (C) im n/3 (D) 2nn/3 where n e I Direction for Q.23 to Q.25. s inx-xcosx Consider the function defined on [0, i] -> R, / (x) = 5 x * 0 anc f (0) = 0

^/Q.23 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) - s in ( l ) t

1 ^ . 2 5 L i m 7 j / ( x ) d x

1 0 equals t->o tz

(A) 1/3 (B) 1/6 (C) 1/12 (D) 1/24

DATE : 11-12/09/2006 i > B > S > - 4 3 TIME : 60 Min.

Select the correct alternative. (More than one are correct) [ 7 x 4 = 28]

There is NO NEGATIVE marking. Marks will be awarded only if all the correct alternatives are selected.

Q.26 Let / (x) = xex x < 0

L x + x2 - xJ x > 0 then the correct statement is

(A)/ is continuous and differentiate for all x. (B) / is continuous but not differentiate at x = 0. (C)/ ' is continuous and differentiate for all x. (D) / ' is continuous but not differentiate at x = 0.

x 2 - l Q.27 Suppose/ 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.

2r , (3 + cosx V Q.28 The value of the definite integral J x ' n i 3 _ c o s x J > is

0 v

0-29 f : [0. 1] -> R is defined as / (x ) = j __ , then

(A) n ] l n ( Jdx (B ) ] d x ( C ) z e r o (D) V * J V3 cosx J J ^3-cosx J V ' 0 V3 + c o s x ;

r x3( l-x)sin(l /x2J if 0 < x < l

0 if x = 0

(A)/ is continuous but not derivable in [0, 1 ] (B) / is differentiate in [0, 1 ] (C) / is bounded in [0, 1 ] (D)/ ' is bounded in [0, 1]

Q.30 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 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) Q.32 For the quadratic polynomial / (x ) = 4x2 - 8kx + k, the statements which hold good are

(A) there is only one integral k for which/(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 ^ T I - S ^ MATCH THE COLUMN [ 3 x 8 = 24]

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 (A) / (x ) = sin"](in x) + cos""1 (cos x) (P) range is [0,71] (B) g (x) = sin-'j-x | + 2 tair'j x | (Q) is increasing V x e (0, 1)

( 2x 1 (C) h (x) = 2sirr>! j j , x 6 [0, 1] (R) period is 2%

(D) k (x) = cot(cor'x) (S) is decreasing V x e (0, 1)

Q.2 Column-I Column-II

Q.3

(A)

(B)

(C)

(D)

(B)

(C)

(D)

Centre of the parallelopipeci whose 3 coterminous edges OA, OB and (P) OC have position vectors a, b and c respectively where O is the origin, is

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

Let ABC be a triangle the position 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

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

(A) If

Column-I 1 a a~

1 b

of the equation 1

( x - a ) 2 1

(x -b ) 2 1

(x -c ) 2 ( x - b ) ( x - c ) ( x - c ) ( x - a ) ( x - a ) ( x - b )

The value of the limit, XL (^/(x + a)(x + b)(x + c) - x), i

=0, is

is

Lim x->0

f X , X X a +b +c equals

Let a, b, c are distinct reals satisfying a3 + b3 + 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

(Q)

+ c

a + b + c 3

(R)

(S) a + b + c

2

(a - b)(b - c)(c - a)(a + b + c) then the solution (P)

Column-II

a + b + c

(Q)

(R)

(S)

a + b + c

SUBJECTIVE: [ 4 X 6 = 2 4 ]

Q.l 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 7i "7n 2 o k

Q.3 Suppose/and g are two functions such that f g : R -> R,

where k e N, find k.

/ (x ) ^/n^l + V l ^ 2 ] and g(x) = /n! x + \ / lTx 2

then find the value of x egW ( fiW

/

Q.4 If the value of limit L ,m Z cos -1

k=2

+ g'(x) at x = 1.

l + 7(k- l )k(k + lXk + 2) k(k + l)

120ti is equal to -, find the value of k.

K

JHBANSAL CLASS ^ T a r g e t 1ST JEE 2007

IES MATHEMATICS Daily Practice Problems

CLASS: XII (ABCD) DATE: 04-07/9/2006 DPR N0.-40, 41

DATE: 04-05/09/2006 TIME: 50 Min. Q. 1 Let/(x) = 1 - x - x3. Find all real values of x satisfying the inequality, 1 - / ( x ) - / 3 ( x ) > / ( 1 - 5x)

g2x _ gX j Q.2 Integrate: j dx

3 (ex sin x + cos x)(ex cos x - sin x)

Q.3 The circle C : x2 + y2 + kx + (1 + k)y - (k + 1) = 0 passes through the same two points for every real number k. Find

(i) the coordinates of these two points. (ii) the minimum value of the radius of a circle C.

i Q. 4 Comment upon the nature of roots of the quadratic equation x 2 + 2 x = k + J| t + k | dt depending on the

value of k e R. 0

Q.5 Given Lim n->oo

3n C

2n f\ \ n y

1/n a

= where a and b are relatively prime, find the value of (a + b). b

DFP-41 DATE: 06-07/09/2006 TIME: 50 Min.

Q. 1 Let a, b, c be three sides of a triangle. Suppose a and b are the roots of the equation x2 - (c + 4)x + 4(c + 2) = 0 and the largest angle of the triangle is 9 degrees. Find 0.

71 Q.2 Find the value of the definite integral j |V2sinx + 2cosx jdx.

o

1 Q.3 Let tan a tan (3 = 7 ^ 5 . Find the value of (1003 - 1002 cos 2a)(1003 - 1002 cos 2(3)

1+V5

0 4 2r X2 + l . ( . n * / n j l + x / X X +1 V X J

dx

Q.5 Two vectors Sj and e2 with | e( | = 2 and \ e2 | = 1 and angle between and e2 is 60. The angle

between 2t e, + 7 e2 and ej +1 e2 belongs to the interval (90, 180). Find the range of t.

Q.6 Afimction fix) continuous on Rand periodic with period 2% satisfies f (x) + sin x - / (x + n) = sin2x.

Find/(x) and evaluate f / ( x ) dx.

4| BAN SAL CLASSES MATHEMATICS^ glTarget SIT JEE 2007 Daily Practice Problems CLASS: XII (ABCD) DATE; 30-31/8/2006 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)x2 - 2ax + 2a - 6 < 0 V x e R . [3]

x + 1 x + 5

Q. 2 Solve the inequality by using method of interval, - ^ I31

Q.3 Find the minimum vertical distance between the graphs of y = 2 + sin x and y = cos x. [3]

whenx = 18. [3]

d (3 ^ cos x - c o s J x 4

Q.4 Solve: dx

Q.5 If p, q are the roots of the quadratic equation x2 + 2bx + c = 0, prove that

2 l o g [ j y - p + y f y - q } = log2 + log(y + b + j , [4]

x 2 +14x + 9 Q. 6 Find the maximum and minimum value of y = , V x e R . [4]

x +2x + 3 Q.7 Suppose that a and b are positive real numbers such that log2 7a + log9b = 7/2 and

log27b + log9a=2/3. Find the value of the ab. [4]

Q. 8 Given sin2y=sin x sin z where x, y, z are in an A.P. Find all possible values of the common difference of the A.R and evaluate the sum of all the common differences which lie in the interval (0,315). [4]

tan 86 Q.9 Prove that = (1 + sec29) (1 + sec40) (1 + sec86). [4]

jl 371 571 In Q.10 Find the exact value of tan2: + tan2 + tan2~ + tan2 . [4]

16 16 16 16 89 i

Q.l l Evaluate Y 151 ^ l + (tann)2

Q. 12 Find the value of k for which one root of the equation of x2 - (k + 1 )x + k2 + k-8=0 exceed 2 and other is smaller than 2. [5]

Q. 13 Let an be the 0th term of an arithmetic progression. Let Sn be the sum of the first n terms of the arithmetic progression with aj = 1 and a3 = 3ag. Find the largest possible value of Sn. [5]

( C^ C A B Q. 14(a) IfA+B+C = n & sin A + = k sin , then find the value of tan -tan in terms of k.

V Z. J

(b) Solve the inequality, log. '0.5 ( \ X + x

log6 -x + 4

4| BANSAL CLASSES MATHEMATICS j Target III JEE 2007 Daily Practice Problems CLASS: XII (ABCD) DATE: 23-24/8/2006 TIME: 60 Min. DPP. NO.-38

r 2 1/2 Q. 1 Find the value of a and b where a < b, for which the integral j (24 - 2x - x ) dx } i a s the largest

a

value.

Q.2 Solve the differential eqaution: y' + sin x - cos x

V e - c o s x y = x e - c o s x

Q.3 Integrate: J. (x cos x - sin x)(x sin x + cos x)

-dx

Q.4 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 1V x Q-5 FCN dx

X X Q.6 Find the general solution of the equation, 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 evaluate J(sinax-sin[3x)dx , o

independent of a and {3.

J | BANSAL CLASSES MATHEMATICS IggTarge f HT JEE 2007 Daily Practice Problems CLASS: XII (ABCD) " DATE: 18-19/8/2006 TIME: 75 Min. DPR NO.^37

This is the test paper of Class-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 terms is 765. What is the common ratio in this series. [4]

Q.2 Form a quadratic equation with rational coefficients if one of its root is cot2l 8. [4]

Q.3 Let a and (3 be the roots ofthe quadratic equation ( x - 2 ) ( x - 3)+(x-3)(x + l )+ (x+ l)(x-2)=0.Find

1 1 1 the value of ( a + 1 ) ( p + 1 } + ( a _ 2 ) ( p _ 2 ) + ( a _ m _ 3 ) W

Q.4 If a sin2x +Mies in the interval [-2,8] foreveryx 0, what is the smallest possible value of the expression log(x3 - 4x2 + x + 26) - log(x + 2)? [4]

Q. 6 The coefficients of the equation ax2 + bx + c = 0 where a * 0, satisfy the inequality (a + b + c)(4a - 2b + c) < 0. Prove that this equation has 2 distinct real solutions. [4]

Q.7 In an arithmetic progression, the third term is 15 and the eleventh term is 55. An infinite geometric progression can be formed beginning with the eighth term of this A.P. and followed by the fourth and second term. Find the sum of this geometric progression upto n terms. Also compute Srjo if it exists.

[5]

Q.8 Find the solution set of this equation log)sin X|(x2 - 8x + 23) > log ( s i n x j (8) in x e [0,2n). [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, sin22x and cos x sin 4x form an increasing geometric sequence, find the numerial value of cos 2x. Also find the common ratio of geometric sequence. [5]

Q. 12 Find all possible parameters 'a' for which, f (x) = (a2 + a - 2)x2 - (a + 5)x - 2 is non positive for every x e [0,1 ]. [5 j

Q.13 The 1st, 2nd and 3rd terms of an arithmetic series are a, band a2 where 'a' is negative. The 1st, 2nd and 3rd

terms of a geometric series are a, a2 and b find the (a) val ue of a and b (b) sum of infinite geometric series if it exists. If no then find the sum to n terms of the G P (c) sum ofthe 40 term ofthe arithmetic series. [5]

j) Q. 14 The nth term, an of a sequence of numbers is given by the formula an = an _} + 2n for n > 2 and

aj = 1. Find an equation expressing an as a polynomial in n. Also find the sum to n terms of the sequence. [8]

00 2x x Q. 15 Let/(x) denote the sum of the infinite trigonometric series, / ( x ) = ^ sin sin .

n=J 3 3 Find / (x) (independent of n) also evaluate the sum ofthe solutions ofthe equation f (x) = 0 lying in the interval (0,629). [8]

. k B A N S A L CLASSES I B Target I I T JE 2007

MATHEMATICS Daily Practice Problems

CLASS: XII (ABCD) DPP. NO.-35, 36 I > I * I " - 3 5

DATE: 16-17/08/2006 TIME: 45 Min. x d3

Q.l If y = Jx2V^nt dt, find at x = e . l

Q.2 Find the equation of the normal to the curve y = (l +x)y + sin-1 (sin2 x) at x = 0. x

ff(t)dt Q.3 Find the real number 'a' such that 6 + J -j = 2 v x

a

7 Q.4 The tangent to y = ax2 + bx + - at (1,2) is parallel to the normal at the point (-2, 2) on the curve

y = x2 + 6x + 10. Find the value of a and b.

Q.5 Let f be a real valued function 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 function. X

Q. 6 A tangent drawn to the curve C l = y = x 2 + 4 x + 8 at its point P touches the curve C2 = y = x2 + 8x + 4 at its point Q. Find the coordinates of the point P and Q, on the curves C j and C2.

3 S DATE: 16-17/08/2006 TIME: 45 Min.

Q. 1 Given real numbers a and r, consider the following 20 numbers: ar, ar2, ar3, ar4, , ar20. If the sum of the 20 numbers is 2006 and the sum of the reciprocal of the 20 number 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 (i i)f ' (x) = g(x) & (iii)g'(x) = f(x). Find the functions f(x) and g(x).

Q.3 Let f(x) =

3 ( b 3 - b 2 + b - l ) _ Y , 0 < x < l (b2 + 3b + 2 j

L 2x-3 ,1 < x < 3

Find all possible real values of b such that f(x) has the smallest value at x = 1.

Q. 4 There is a function f defined and continuous for all real x, which satisfies an equation of the form

Xf V 2 X 1 6 X 1 8 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 = x3 meets the curve again at P2. The tangent at P2 meets the curve at P3 & so on. Show that the abscissae of P,, P2, P3, Pn, form a GP. Also find

the ratio ^ ( P ^ P , ) area (P2P3P4)

ft

4| BANSAL CLASSES MATHEMATICS |Target 8iT JEE 2007 Daily Practice Problems CLASS 7 XII (ABCD) DATE: 11-12/8/2006 TIME: 60 Min. DPP. NO.-34

Q.l Let F (x) = jV4 +12 dt and G (x) = JV4 +12 dt then compute 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 row and labelled A, B, C, D and E. Find the number of ways in which the balls can be distributed in the boxes if no two adjacent boxes remain empty.

Q. 3 Iff (x) = 4x2 + ax + (a - 3) is negative for atleast one negative x, find all possible values of a.

Q.4 Let/(x) = sin6x + cos6x + k(sin4x + cos4x) for some real number k. Determine (a) all real numbers k for which/(x) is constant for all values of x. (b) all real numbers k for which there exists a real number 'c' such that f (c) = 0. (c) I f k = - 0 . 7 , determine all solutions to the equation/(x) = 0.

7T ,

Q.5 Letx0 = 2cos a n d x n = ^ 2 + x ^ , n = 1 ,2 ,3 ,

find Lim 2o

Q.6 Let/(x)= - then find the value of the sum y j 20C>6 / + ^ f 1 1 f 2 1 f 3 ^ (2005^ U 0 0 6 j ^ ^2006 J +f [2006J 2006 J

Q.7 V j ^ d x . * 8 + sin x

Va Q.8 For a > 0, fmdthe minimum value ofthe integral J(a3 + 4 x - a 5 x 2 ) e a x dx.

0

I BANSAL CLASSES Target liT JEE 2007

MATHEMATICS Daily Practice Problems

CLASS: XII (ABCD) DATE: 31/7/2006 to 5/08/2006 DPP. NO.-33 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.

Q.l

Q.2

Q.3

Q.4

Q.5

" P A R T ' - A . Select the correct alternative. (Only one is correct) Number of zeros of the cubic f (x) = x3 + 2x + k V k e R, is (A) 0 (B) 1 (C) 2

[26 x 3 = 78]

(D)3

The value of Lim

(A) 0

/x

t x-> dx yL(r + l ) ( r - l ) (B) 1

dr, is

(C) 1/2 (D) non existent - 2 5 x - 1 4 2x

equal to 86. The sum of There are two numbers x making the value of the determinant

these two numbers, is ( A ) - 4 (B)5 ( 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 ) , 1 Suppose/,/ ' and/" are continuous on [0, e] and that/ ' (e) = / (e) = / ( l ) = 1 and j 2 = Z, then

e 1 X 1

the value of f / " ( x ) / n x d x equals I 3 1 5 1

(B) j -1 1

(C) (D) 1 -1

Q.6 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

n(n +1) Q.9 Period of the function, / ( x ) = [x] + [2x] + [3xj + + [nx] - n J.

(D) non periodic

the statement which does not hold good, is

where n e N and [ J denotes the greatest integer function, is (A) 1 (B) n (C) 1/n

2i - i 1 Q. 10 Let Z be a complex number given by, Z = 3 i - 1

(A) Z is purely real 10 1 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

Q. 11 Let/(x, y) = xy2 if x and y satisfy x2 + y2 = 9 then the minimum value o f f (x, y) is

(A) 0 (B) - 3-^3 ( Q - 6 V 3 ( D ) - 3 V 6

Vl + 3 x - l - x Q. 12 Eim - ^ has the value equal to x^o (1 + x) - l - 1 0 1 x

3 ( A ) - 5050 ( B ) - 5050 (C) 5051

(D) 4950

Q. 13 Number of positive solution which satisfy the equation log2x log4x log6x = log7x log4x + log2x log6x + log4x loggX?

(A) 0 (B) 1 (C) 2 (D) infinite

Q.14 Number of real solution of equation 16 sin"'x tan_1x cosec"'x = n3 is/are (A) 0 (B) 1 (C) 2 (D) infinite

Q. 15 Length of the perpendicular from the centre of the ellipse 27x2 + 9y2 = 243 on a tangent drawn to it which makes equal intercepts on the coordinates axes is

(A) 3/2 (B) 3/V2 (C) 3V2 (D) 6

Q.l 6 Let/(x) = cos"1 f , 2 n 1 x

1 + x 2 + tan-

2x

1 - x 2 where x e (-1, 0) then/simplifies to

(A) 0 (B) ti/4 (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)48

Q.18 Let A = a b c p q r x y z

(A) det(B) = - 2 (B) det(B) = - 8 (C) det(B) = - 16

Q. 19 The digit at the unit place ofthe number (2003)2003 is (A) 1 (B) 3 (C) 7

and suppose that det.(A) = 2 then the det.(B) equals, where B =

(D) det(B) = 8

(D)9

4x 2a - p 4y 2b - q 4z 2c - r

AB AF Q.20 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 Le 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 /(10099) is (A) 550 (B) 505 (C)5050 (D) 10010

\2 /

Q.23 If a and P be the roots of the equation x2 + 3x + 1 = 0 then the value of a

1 + P +

P a + 1

is equal to

(A) 15 (B) 18 (C) 21 (D) none Q.24 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 bc / a , is (A) 1/5 (B) - 1/5 (C) 6/5 (D) - 6/5

Q.25 If m and n are positive integers satisfying cos m0 sin n0

1 + cos 20 + cos 40 + cos 60 + cOs 80 + cos 100 = then m + n is equal to

(A) 9 (B) 10 (C) 11 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

Q.27

(A)300 (B)360 (C)400 Select the correct alternative. (More than one are correct) In which of the following cases limit exists at the indicated points.

(D) 420 [8x4 = 32]

(A) /(x) [ x + | x | ]

x at x = 0 (B)/(x) = xe

1/x

where [x] denotes the greatest integer functions. (C)/(x) = (x - 3)1/5 Sgn(x - 3) at x = 3, (D)/(x) =

where Sgn stands for Signum function.

l + e1/x

tan-11 x | x

at x = 0

at x = 0.

&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, c2) and (c, c2) and let R be the region between y = cx and y = x2 where c > 0 then

(A) Area (R)=-c3 Area (T)

(B) Area of R= (C) Lim 3 c-o+ Area (R)

=3 (D) Lim Area(T) _ 3

c-o+ Area(R) 2 In

Q.30

Q.31

Consider the graph of the function f (x) = e (A) range of the function is (1, oo) (C) graph lies completely above the x-axis.

1

( x+3 U+i . Then which of the following is correct.

(B) / (x) has no zeroes. (D) domain of f is ( - oo, - 3) u (-1, oo)

1 x x - 1 x - 1 ; /6(x) =

Q.32

Let /,(x) = x, /2(x) = 1 - x; /3(x) = - ,/4(x) = ; /5(x) = X I X

Suppose that / 6 ( / m ( x ) ) =/4(x) and / n ( / 4 ( x ) ) =/3(x) then (A) m = 5 (B) n = 5 (C) m = 6 (D) n = 6

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 f a x + l"\

Consider the function / (x ) = vbx + 2y

(A) exists for all values of a and b

(C) is non existent for a > b

where a2 + b2 * 0 then Lim / ( x ) X-CO

(B) is zero for a < b

(D) is e~(5/a) or e~(l /b) if 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 where Sgn denotes Signum function

(C) h : R R h (x) = x4 + 3x2 + 1 (D) k : R R

g(x)

k (x):

= v3/5

3x2 - 7 x + 6

x -x2 - 2 MATCH THE COLUMN ^ ^ ^ E ^ T T - 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 Column I Column II

(A) Constant function/(x) = c, c e R (P) Bound

(B) The function g (x) = P (x > 0), is Ji t The function h (x) = arc tan x is (R)

(Q) periodic

(C) (D) The function k (x) = arc cot x is (S)

Monotonic neither odd nor even

Q.2

Q.3

Q.l

Q.2

(A)

(B)

(C)

(D)

Column I cor1 (tan(-370))

cos"1 (cos(-233))

Column II

1 sin

cos

-cos T v9,

A

- arc cos

Column I

(A) Number of integral values of x satisfying the inequality x - 1 x - 3

(P) 143

(Q) 127

3 (R)

4

2 (S) 3

Column II 2 4

(P) 1

(B) The quadratic equations 2006 x2 + 2007 x + 1 = 0 and x2 + 2007x + 2006 = 0 have a root in common. Then the product of the uncommon roots is (Q) - 2

(C) Suppose sin 9 - cos 9 = 1 then the value of sin39 - cos39 is (9 e R) (R) - 1 s i n 2 x - 2 t a n x

(D) The value ofthe limit, L l ~ ; 3 ; i s (S) 0 /n(i + x )

Q.4 A quadratic polynomial / (x) = x2 + ax + b is formed with one of its zeros being 4 + 3^3 2 + V3

where a and b

are integers. Also g (x) = x4 + 2x3 - 10x2 + 4x - 10 is a biquadratic polynomial such that

8 4 + 3y3

2 + V3 = + d where c and d are also integers.

Column II (P) 4 (Q) 2 (R) - 1 (S) - 1 1

Column I (A) a is equal to (B) b is equal to (C) c is equal to (D) d is equal to

SUBJECTIVE: 13 x 8 = 24] Let y = sin"'(sin 8) - tan_1(tan 10) + cos_i(cos 12) - sec"'(sec 9) + cor '(cot 6) - cosec "'(cosec 7). If y simplifies to an + b then find (a - b).

Suppose a cubic polynomial / (x) = x3 + px2 + qx + 72 is divisible by both x2 + 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 ofthe cubic polynomial.

Q.3 The set of real values of'x' satisfying the equality ~3~ r 4 -

4 -

V X = 5 (where [ ] denotes the greatest integer

function) belongs to the interval

a + b + c + abc.

( b a , -

I c. where a, b, c e N and ~ is in its lowest form. Find the value of

c

4| BANSAL CLASSES MATHEMATICS | Target IIT JEE 2007 Daily Practice Problems CLASS: XII (ABCD) 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. [3]

Q.2 If the expression

371 r r>. cos

371 X

V 2 y + sin + x v2 , + sin (327t + x) - 18 cos(19rt - x) + c o s ( 5 6 t c + x) - 9 sin(x + 17tc)

is expressed in the form of a sin x + b cos x find the value of a + b. [3]

Q.3 3 statements are given below each of which is either True or False. State whether True or False with appropriate reasoning. Marks will be allotted only if appropriate reasoning is given. I (log3169)(log13243) = 10 II cos(cos 7t) = cos (cos 0)

1 3 III cos x + = T S3]

cosx 2

3 1 1 Q.4 Prove the identity cos4t = ~ + - cos 2t + r cos 4t. [3] o 2 o

Q. 5 Suppose that for some angles x and y the equations

i 0 3a sin^x + cos^y =

a 2 and cos2x + sin2y =

J 2

hold simultaneously. Determine the possible values of a. [3]

Q. 6 Find the sum of all the solutions of the equation (log27x3)2 = log27x6. [3]

7i % 10 y -10~ y If - < x < and y = log10(tan x + sec x). Then the expression E = simplifies to one ** JL the six trigonometric functions, find the trigonometric function. 13]

Q.8 If log2(log2(log2 x))= 2 then find the number of digits in x. You may use log?02 = 0,3010. [3]

Q. 9 Assuming that x and y are both + ve satisfying the equation log (x+y)=log x+log y find y in terms of x. Base of the logarithm is 10 everywhere. [3]

cosx ~~ cos 3x Q.10 If x = 7.5 then find the value of : . [3]

sin 3x - sin x

Q. 11 Find the solutions of the equation, log ^ sm x (1 + cos x) = 2 in the interval x e [0,2n]. [4]

Q. 12 Given that log 2 (a2 +1) = 16 find the value of log 32 (a + - ) a a a

Q, 13 If cos e = - find the values of

[4]

(i) cos 36 (ii)tam

Q. 14 If log1227 = a find the value of log616 in term of a.

[4]

[5]

sin x - c o s x + 1 1 + sinx Q . 15 Prove the identity, r = =tan sin x + c o s x - 1 cosx

+ 4 2 , wherever it is defined. Starting with left

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 (logxy + logyx) =26 and xy = 64. [6]

r=4 Q.18 Prove that sin

r=l V

(2r-l)7c' 8

r=4

- 2 r=l

cos ( 2 r - l ) 7 t

8

- \ 4

Also find their exact numerical value. [6]

0,19 Solve for x: log2 (4 - x ) + log (4 -x). log f x + - 1 - 2 log2 V 2J

r i a 1 x +

2 , = 0. [6]

4| BANSAL CLASSES MATHEMATICS STarget i iT JEE 2007 Daily Practice Problems CLASS : XII (ABCD) DATE: 05-06/06/2006 TIME: 50 Min. DPR NO.-28

The value of Lim X-00

/ n x - / n Vx +1 + x

(A) 1

(B)/n (C) does not exist (D) 0 v w

7t/4

Evaluate J(tanx-sec4 x ) d x .

(A) 1/4 (B) 1/2 (C) 3/4 (D) l

Q 4

The product of two positive numbers is 12. The smallest possible value of the sum of their squares is

(A) 25 (B) 24 (C) 18 V2 (D) 18

Given that log (2) = 0.3010 number of digits in the number 20002000 is (A) 6601 (B) 6602 (C) 6603 (D)6604

, , 1 1 1 Given that a, b and c are the roots of the equation x" - 2x2 - 1 1 x + 12 = 0, then the value of + + ~

(A) (B) n 12 (C)

13 12 (D)

7

If Jtan x dx = 2, then b is equal to

(A) arc cos(2e) (B) arc sec(2) (C) arc sec2(e)

t. Q/7 The sum of all values of x so that 16("2+3x = 8 ( x 2 + 3 x + 2 ) , is (A) 0 (B)3 ( C ) - 3

(D)none

( D ) - 5

Q.9

Q.10

A certain function/(x) satisfies f (x) + 2 / ( 6 - x) = x for all real numbers x. The value o f / ( l ) , is (A) 3 (B)2 ( C ) l (D) not possible to determine

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 position 1, C is not in position 2 and D is not in position 4, is (A) 8 (B) 15 (C) 9 (D) 6

Using only the letter from the word WILDCATS with no repetitions allowed in a codeword, number of 4 letter codewords are possible that both start and end with a consonant, are (A)360 (B)900 (C) 1680 (D)2204

{ Q : l l Find j (x/nx)dx

( A ) - ( B ) - ( C ) - l (D) l

Q.12 IfP(x) is a polynomial with rational coefficients and roots at 0,1, -Jl a n c i 1 - \/3 , then the degree of P(x) is at least

(A) 4 (B) 5 (C)6 (D)8

Sum of the infinite series, 4 - ^ + + + 00 > c c l u a l t 0 7 V

(A) ^ (B) 24 (C) 49 (D)

Q.14 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?

9! ( B ) 3!-6!

8! ( A ) 2!-6!

12! (D) 56

e3x - 1 .

^ 1 5 Let/(x) =

( A ) - 9

if x * 0 x then/'(0), is

3 if x = 0

(B)9 (C) 9/2

n 6

Q.17

Q.18

I f f "(x) = 10 and f ' (1) = 6 and f ( l ) = 4 then f (-1) is equals ( A ) - 4 (B) 2 ' (C)8

\12 x" 2

The coefficient of x3 in the expansion of

(D) nonexistent

(D)12

v 4 + x y

(A) 97 (B)98

,is

(C) 99 (D)100

In how many ways can six boys and five girls stand in a row if all the girls are to stand together but the boys cannot all stand together? (A) 172,800 (B) 432,000 (C) 86,400 (D)none

The composite of two functions f and g is denoted by fog and defined by (fog)(x) = f (g(x)). When

f(x) x 1

4 - x

6x 5x and g (x) = which one of the following is equal to (fog)(x)?

x - 2

The equation In

x - 2

3 Ox

k iA >

(C) x - 2

4x + 2 (D)

15x

( k + 1)i/(k+D

F(100) has the value equal to

1

= F(k) In 1 -1

k + 1 +Ink

k

2x + l

is true for all k wherever defined.

(A) 100 (B) 101 (C)5050 (D)

1 100

Q.21 Compute f ,_ ^Vx+K/x

ill BANSAL CLASSES MATHEMATICS H Target I I I JEE 2007 Daily Practice Problems

CLASS: XII (ABCD) DATE: 28-29/06/2006 TIME: 50 Min each DPR DPR NO.-25 I > P P - 2 5

DATE: 28-29/06/2006 TIME: 50 Min

Q.l tan 9 = 1 where 9 e (0,2n), find the possible value of 6. {2]

~ 2 + -2 + '--co

Q. 2 Find the sum of the solutions of the equation 2 e 2 x _ 5 e x + 4 = o_ [2]

Q.3 Suppose that x and y are positive numbers for which log9x = log12y = log16(x + y). If the value of

--2 cos 9, where 9 e (o, rc/2) find 9. [3]

Q. 4 Using L Hospitals rule or otherwise, evaluate the following limit:

Limit Limit X - > 0 + n - > e o

[l2 (sinx)" j + 22 (sinx)x + + n2 (sinx)x "

n3 where [. ] denotes the

greatest integer function. [4]

1 ~ s i n 2 x I . Q.5 Consider f ( x ) = - ^ = , Va + htan'x , f o r b > a > 9 & the functions g(x)&h(x)

(V5 |1 + I sinx

are defined, 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 of ' f , where [x] is the greatest integer function of x & {x} is the fractional 7t

part of x. Then discuss the continuity of'g' & *h' at x = and x = 9 respectively. [5]

~ ^ f x2 tan _ 1x ,

Q. 7 Using substitution only, evaluate: jcosec3 x dx. [5j

DATE: 30-01/06-07/2006 JIME: 50 Min.

12 A Q.l If sin A = . Find the value of tan , [2]

x v Q.2 The straight line - + ^ = 1 cuts the x-axis & 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]

tan 9 1 cot 9 Q.J If - -- = , find the value of - . HI

tan 9 - tan 39 3 c o t 9 - c o t 3 9

Q.4 If a A ABC is formed 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]

J dx

a2 - tan2 x)Vb2 - tan2 x Q.5 Integrate

f

I ( 2 2 , T,2 ;2 \2

(a>b)

xsmxcosx Q . 6

Q.7

(a cos x + b sin x) dx

15]

[5]

d dy Let ~ (x2y) = x - 1 where x ^ 0 and y = 0 when x = 1. Find the set of values of x for which dx is positive.

DATE: 03-04/07/2006

[5]

TIME; 50Min.

Q. 1 Let x = (0.15)20. Find the characteristic and mantissa in the logarithm of x, to the base 10. Assume log102 = 0.301 and log103 = 0.477. [2]

Q. 2 Two circles of radii R & r are externally tangent. Find the radius ofthe third circle which is between them and touches those circles and their external common tangent in terms of R & r. [2]

Q. 3 Let a matrix A be denoted as A= diag. 5 x , 5 5 \ 5 5 S then compute the value of the integral j( det A)dx.

P] Q. 4 Using algebraic geometry prove that in an isosceles triangle the sum of the 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]

f -J1 + x dx

+ x Vx + X 2 +x3 Q.5 Evaluate:

Q.6 If the three distinct points,

show that abc + 3 (a + b + c) = ab + be + ca.

Q.7 Integrate: j^/tanx dx

f a 3 a 2 - 3 ] f b 3 b2 -3^1 r c3 c2 -3^1 v a - l ' a - 1 ; ? [ b - r b - i j ? [ c - l ' c - l j

[5]

are collinear then

[5]

[5]

ill BANSAL CLASSES MATHEMATICS Ig lTarget HT JEE 2007 Daily Practice Problems CLASS: XII (ALL) DATE: 23-24/06/2006 TIME: 50 Min. DPP. NO.-24

Select the correct alternative: (Only one is correct) [16 x 3 = 48]

Q. 1 A circle of radius 2 has center at (2,0). Acircle of radius 1 has center at (5,0). Aline is tangent to the two circles at points in the first quadrant. Which of the following is the y-intercept ofthe line?

(A) 3 (B) V2 8

( Q 3 (D) 2a/2

Q.2 In a triangle ABC, the length of AB is 6, the length of BC is 5, and the length of CAis 4. If K lies on BC

BK 3 such that the ratio of length r is , then the length of AK is

K C 2

(A) 2V3 (B)4 (C) 3V2 (D) 2,

Q. 3 Which one of the following quadrants has the most solutions to the inequality, x - y < 2? (A) I quadrant (B) II quadrant (C) ID quadrant (D) I and III quadrant have same

Q. 4 The range of the function / ( x ) = sin_1x + tan~'x + cos_1x, is

(A) (0,71) (B) 7t 371

4 ' T (C) [0,71] (D)R

Q.5 The area of the region of the plane consisting of all points whose coordinates (x, y) satisfy the conditions 4 < x 2 +y 2 < 36 and y < | x is (A) 24n (B) 27TI (C) 20TT (D) 32tc

Q. 6 A straight wire 60 cm long is bent into the shape of an L. The shortest possible distance between the two ends of the bent wire, is

(A) 30 cm (B) 3 0 V 2 c m (C)10V26 (D) 20^5

Q.7 Sum of values of x, in (0, n/2) for which tan

(A) 0 (B) 71 - tan_1(2)

'7t N + X 4

= 9 tan

(C) cor'(O) v

71

4 ' X holds, is

(D) tan-1 (2)

Q. 8 Given/"(x) = cos x, / ' ^ y J = e and/(0) = 1, then/(x) equals.

( A ) s i n x - ( e + l ) x (B) sinx + (e+ l)x (C) (e+ l)x + cosx (D) (e+ l ) x - c o s x + 2

Q.9 Evaluate the integral: j x e c o s x 2 sin x 2 dx

1 (A) | e c o s x 2 + C ( B ) - - e s m x + C 1 _sin x

2

e~n -e"x Q.10 The value of Lim

x->n sin x is

(A)0 ( B ) - e -

(C) + C

(D)e-

( D ) - i e c o s x 2 + C

(D)nonexitent

Q.LL Let/(x) = ^" k x + 1 . The interval(s) of all possible values of k for which/is continuous for every x 2 - k

x e R, is (A) ( - , , - 2 ] (B)[-2 ,0) (C) R - ( - 2 ,2) ( D ) ( - 2 , 2 )

Q.12 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 (B) 12 (C) 9 (D)7

Q, 13 From a point P outside of a circle with centre at O, tangent segments PA and PB arc drawn. If

1 1 1 ( A O ) 2 " + ~ Ye ' t b e n l e n t b chord AB is

(C) 8 (D) 9

, Aj * 0

where b- is cofactor of a^ V i, j = 1,2, 3

where c^ is cofactor of V i, j = 1,2, 3.

(A) 6 (B)4

a n a l 2 a l 3 Let a 2 1 a 2 2 a 2 3

a 3 1 a 3 2 a 3 3

b i , b12 b 1 3 b 2 1 b 2 2 b 2 3 b3 ] b 3 2 b 3 3

c n C 12 C 13 and C21 C 22 C23

C 31 C 32 C33

then which one of the following is always correct. (A) Aj, A2, A3 are in A.P. (B) Aj, A2, A3 are in G.P.

A0 (C) A 2 A3 (D) A,

Q. 15 The first three terms of an arithmetic sequence, in order, are 2x + 4,5x - 4 and 3x + 4. The sum of the first 10 terms of this sequence, is (A) 176 (B) 202.4 (C) 352

\5 7L 71 I 71 7T. Q. 16 The value of

(D) 396

r \4/ 7t . . 7t w cos+ zsin

8 8 7i . . n cos +1 sin 15 15 ,

/ n V3 i (A) + -w 2 2

Subjective: Evaluate:

K J 2 2 (C)

is equal to

S i ( D ) - ^ - - i w 2 2

Q.l J" dx

xV a x - :

ff s5* s* q . 2 j 5 x 5 x 5 dx

rsin 1 V x - cos 1 yfx , Q.3 j , r- * rr dx

sm + COS

BANSAL CLASSES MATHEMATICS 8Targe* I I T JEE 2007 Daily Practice Problems

CLASS: XII(ALL) DATE: 16-22/06/2006 DPR NO.-21, 22, 23

2 1 DATE: 16-17/06/2006

Q. 1 For x > 0 and ^ 1 and n e N, evaluate, TIME: 45 Min.

L i m n-co

1 +

1 log 2 . log 4 log 4 . log 8

+ + 1

log 2 n _ 1 .Iog_ 2n V X =>x ' x ~ o x - ~ o x - y

Q. 2 Show that (a + b + c), (a2 + b2 + c2) are the factors of the determinant

a2 (b + c)2 be

b2 (c + a)2 ca . Also find the remaining factors.

c2 (a + b)2 ab

Q. 3 Prove that a non singular idempotent matrix is always an involutaiy matrix.

Q. 4 Find an upper triangular matrix A such that A3 = 8 - 5 7 0 27

^ d2y dy Q.5 I f ' y' is a twice differentiable function of x, transform the equation, (1 - x2) -7 - x - + y = 0 by dx dx

means of the transformation, x = sin t, in terms of the independent variable' t'.

Q. 6 Atangent line is drawn to a circle of radius unity at the point A and a segment AB is laid offwhose length is equal to that of the arc AC. A straight line BC is drawn to intersect the extension of the diameter AO at the point P. Prove that:

9 (1 - cos 0) (i) PA =

e - sin e ( i i ) L ^ t p A = 3 .

Use of series expansion or L1 Hospital's rule prohibited.

DATE: 19-20/06/2006

Q. 1 Without using any series expansion or L' Hospital's rule, Evaluate: Lim x la e| 1 + -x /

TIME: 45Min.

\ l -x\

/

Q. 2 Find the value of the determinant

VT3+V3 2V5 V5 V15+V26 5 V10

3 + V65 VPS 5

Q. 3 / ( x ) is a diffrentiable function satisfy the 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 )= 2. Determine f (x).

Q.4 L e t , y = t a n - | j 5

dy

+ tan" x

2.3 + x z j + tan -1

3.4 + x^ j + upto n terms.

Find - expressing your answer in two terms, dx

Q. 5 Without expanding the determinant show that the equation 0 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 roots of ax 2+bx + c = 0 & px2 + qx + r = 0, respectively. If the system b ac

of equations a , y + a 2 z = 0 & p ty + p2z = 0 has a non-trivial solution, then prove that = .

D r P - 2 3 DATE: 21-22/06/2006

X + ^ X Q. 1 Compute x in terms x0, x,, and n. Also evaluate Lim xn = 0 ~ 1 TIME: 45 Min.

Asi ~ Zs>-J j-2..

Q.2 A a

2 5 c b 8 2 v

is Symmetric and B = d 3 a

b - a e - 2b - c - 2 6 - f

is Skew Symmetric, then find AB.

Is AB a symmetric, Skew Symmetric or neither of them. Justify your answer.

Q . 3

x +1

Let f ( x ) = e x , x < 0 = 0,7 x = 0 = x2, i x > 0

Discuss continuity and differentiability of f (x) at x=0 .

Q. 4 Show that the matrix A=

evaluate the matrix 1 0 2 1

1 0 2 1

2 0 0 7

can be decomposed as a sum of a unit and a nilpotent marix. Hence

dv Q. 5 Find , if (tan"1 xV + ycotx = 1. dx

f w -)_ bY^) f ^ Q.6 If / is differentiate and Lim ^ h = n ' t h e n f m d t h e v a l u e Without using

L'Hospital's rule.

Q.7 Consider the function / ( x ) =

1 + e" x < 0

x + 2 , 0 < x < 3

3 6 x > 3 x

(a) Find all points where f (x) is discontinuous. (b) Find all points when f (x) is not differentiable. (c) Draw the graph, showing clearly the points of discontinuity or non derivability.

| h BANSALCLASSI V S T a r g e t I IT JEE 2007

ES MATHEMATICS Daily Practice Problems

CLASS: XII (EXCEPT A-2) TIME: 60 Min. DPR NQ.-20

3 10 Q. 1 The set of all x for which > 27 consists of the union of a finite and an infinite interval. The length

ofthe finite interval is

(A) 3 ( B ) 2

x x +1

1 CO I (D)2T Q.2 Five persons put their hats in a pile. When they pick up hats later, each one gets some one else's hat.

Number of ways this can happen, is (A) 40 (B)44 (C) 96 (D) 120

Q.3 Suppose the origin and the point (0,5) are on a circle whose diameter is along the y-axis and (a, b) lies on the circle. Let L be the line that passes through the origin and (a, b). If a2 + b2 = 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 ) 2 x - y = 0 ( D ) 4 x - 3 y = 0

Q.4 If 1 lies between the roots of the equation y2 - my + 1 = 0 then the value of 4[x] IxI +16

V x e R

has the value equal to (Here [x] denotes gratest integer function) (A) 0 (B) 1 (C) 2 (D) none

Q.5 The sum of the squares of the three solutions to the equation x3 + x2 + x + 1 = 0, is (A)~ 1 (B)0 ( C ) l (D)2

Q.6 Let / ( x ) = 1 + x3. 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 (B) equal to 1 (C) equal t o - 1 (D) equal t o - 1/2

Q. 8 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)none

is

Q-9 f i inJfg x - 1

(A)-

v . x + l y

15

= 3x then the value of g (3), is

V2 ( B ) - (C)9 (D)

V3

9 sin( A + B) Q. 10 For acute angles A and B if (tan A)(cot B) = - then the value of equal to

5 sin(A d) (A) 7/4 (B) 2/7

Q. 11 The value of this product of 98 numbers (C) 4/7 (D) 7/2

! 3y 1 - - 1 - 2

5y 1 - -

98 1-

99 1 - -

100 , is

(A) 1

(B) 98

(C) 10 100 5050 Q. 12 If T = 3 /n(x2 + x) with > 0 and x > 0, then 2x + is equal to

(A) V-f'2 + 4eT/3 (B) V^2 + 4e-T/3 (C)

(D) 1

4950

(D) V^2-4eT/3

Q.13

Q.14

- 2 Evaluate: - ,

^ U -12X + 35 (A)-1 .25 (B) -1 .5 (C) -1 .75

Let/be a polynomial function such that for all real x f(x2 + 1) = x4 + 5x2 + 3

then the premitive of / (x) w.r.t. x, is

( D ) - 2

x3 3x2 3 2 ( A ) + x + C (B)- + x + C (C) K J 3 2 w 3 2 3

x3 3x2 x3 3x2 x + C ( D ) + ^ + x + C v 3 2 Q.15

Q.16

Q.17

Q.18

Q.19

Number of regular polygons that have integral interior angle measure, is (A) 20 (B)21 (C) 22 (D)23

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/V2 (B)- l /V2 (B) -1/2V2 (D)V2 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)none

If {x} denotes the fractional part function then the number x = TT^a f/)2 simplifies to

(A) 1/2 (B)0

Which one of the following is wrong? 2

(A) JtanOsec2 0dO = +C (C) Jxsinxdx = s i n x - x c o s x + C

tankx

(C) - 1 / 2

{sf-iyif (D)none

(B) JtanOsec2OdO = +C

(D)none

Q.20

Q.21

Q.22

Let/(x) =

(A) none

Find L y->2

for x < 0 x . I f / (x) is continuous at x = 0 then the number of values of k is

3x + 2k for x > 0

1

(B) 1

1

(C) 2 (D) more than 2

1 y - 2 ^ x + y - 2 x /

(A)0 (B)/nx ( C ) - l / x 2 (D) does not exist

Let p(x) be the cubic polynomial 7x3 - 4x2 + K. Suppose the three roots of p(x) form an arithmetic progression. Then the value of K, is

(A) 21 (B) 16

147 (C) 16

441 (D) 128

1323 Q.23 The sum (in radians) of all values of x with 0

| | | BANSAL CLASS Target I IT JEE 2007

IES MATHEMATICS Daily Practice Problems

CLASS: XII (ALL) DATE: 02-03/06/2006 DPP. NO.-l 7, 18,19 Take approx. 40 to 45 min. for each Dpp.

Q.l I fy =

Q.6

-X ~7

sin x cos x , dy n . + then at x = is

1 + cotx 1 + tanx dx 4 (B)-l

f \ f 3k-tan M -v 3

W (A)0

Q. 2 The value of cot

( A ) ( i o ( B ) ( 3 + v i o y

(C) l (D)2

J) equals

(C) (3 + VI0) (D)(l0 + V3)

sm ^ yfH-ylcos l x Q.3 Lim

x->- l +

(A)0

Q.4 I f / ( x ) = 3 +

equals x

(B)l (C) ijn

f l + 71_x

v then

( A ) Lim f (x) = 4 v ' x->r

(C) L ^ f ( x ) = 5

( B ) Lim f (x) = 3

(D)/has irremovable discontinuity at x = 1

Q.5 I f / ( x ) = 3x 1 0 -7x 8 + 5x 6 -21x 3 + 3x 2 -7thenthevalueof Lim f ( 1 h ) '

( A ) -53

( B ) -22

(C) 53

x ^ r h3 +3h

22 (D)

is

If the triangle formed by the lines x2 - y2 = 0 and the line Ix + 2y = 1 is isosceles then /= (A) 1 (B)2 (C)3 (D)0

. (e2x - l - 2 x z ) ( c o s x - l ) Q. 7 Using L'Hospital's rule or otherwise evaluate the limit, Lim x->o (sin 3x - /n(l + 3x))x4

Q.8 Evaluate the limit Lim ex - / n ( x + e)

. Use of L'Hospital's rule or surd expansion not allowed. x-o e x - l

Q.9 Find all real numbers t satisfying the equation (3 t -9) 3 + (9 t -3) 3 = (9t + 3 t - 12)3.

Q. 10 Find g'(3) if g (x) = x 2h where h (3) = - 2 and h'(3) = 5.

Q.5

Q.6

P F P - 1 8 2 3

\6 ' Q.l Find the value of the expression -log4(2000) log5(2000)

Q.2 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.

Q.3 Calculate, sin 1 . V63 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.

If 0 is eliminated from the equations, a cos 0 + b sin 0 = c & a cos2 0 + b sin2 0 = c, show that the eliminant is, (a - b)2 (a - c) (b - c) + 4 a2 b2 = 0.

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.

- 1 9 Q. 1 Find the real solutions to the system of equations

log10(2000xy) - log10x log10y = 4 log10(2yz) - log10y log10z = 1

and log10(zx) - log1Qz log10x = 0

Q.2 Prove that, cos 1 1 - c o s x

12cosx + 13 = 71 - 2 cot 1

1 X - t a n 5 2 where x e (0, n).

Q.3 Compute the value of cos 1 _i 24 - t a n 1 4 7

Q.4 If g (x) = x3 + px2 + qx + r where 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 roots.

Q.5 Sum the series, c o r ' ( 2 a + a) + cot"1 (2a"1 + 3a) + cor ! (2a- 1 + 6a) + cor1(2a_ 1 + 10a) + + to ' n ' terms. Also find the sum of infinite terms, (a>0).

44 ^Tcosn

Q.6 Let x = 44 ^ s i n n c n=l

find the greatest integer that does not exceed 1 OOx.

l | | BANSAL CLASSES MATHEMATICS v B Target i lT JEE 2007 Daily Practice Problems CLASS: XII (ALL) DATE: 12-13/05/2006 TIME: 60 Min. DPP. NO.-14

This DPP will be discussed on Friday & Saturday.

Q.l

Q.8

Q.9

f(x) =

h(x)

2 cos x - sin 2x (7 t -2x) z '

; f (x) for x < 7t/2 = g (x) for x > 7t/2

then which of the following holds? (A) h is continuous at x = n!2