| 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) <B> 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) 2»Cn

(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 <j) e R. [5]

( n y^ ( n sin x(3 + sin2 x) Q.23 If tan ~ + ~ ! = ta r r ~ + ~ then prove that s iny= 5 . [5] 2.) \ 4 J,) l + ^sin^x

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<5 . Z B equals (Q) 90

(B) In a triangle ABC is b = S , c = 1 andA= 30° (R) 120 then angle B equals

(C) In a A ABC if (a + b + c)(b + c - a) = 3bc (S) 75° 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

Column-I Column-II Q.3 (A) The sequence a, b, 10, c,d is an arithmetic progression. (P) 10

The value o fa + b + c + d

(B)' The sides of right triangle form a three term geometric (Q) 20 sequence. The shortest side has length 2. The length

of the hypotenuse is of the form a + Vb where a e N (R) 26 and 7b is a surd, then a2 + b2 equals

(C) The sum of first three consecutive numbers of an (S) 40 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 off forming an angle of 60°.

If the perimeter of the parallelogram is 2[-Ja + Vb) where a, b e N then (a + b) equals

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 /

<f 1

V I 1 _ J _ 1 71 Q.5 Prove that 2 ^ 3 n + 1 3 n + 2 j = ^ [9]

- S O

Q. 1 If cos A, cos B and cos C are the roots of the cubic x3 + ax2 + bx + c = 0 where A, B, C are the angles of a triangle then find the value of a2 - 2b - 2c. [4]

Q.2 Find all func t ions , / :R->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 ¥

J2 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 — cos—cos -Let a, b, c are non zero constant number then Lim —-— — equals

r-»co . b . C sin—sin -

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 —

Y—VtYl 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 = PAPT

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<5.

[6]

[6]

—, show that 2 (i-f) HI) f 1 — tan—1

I 2 J sin a + sin P + sin y - 1 —, show that 2 l + tan —

2 j [ i + t » | ]

( y^ l + tan —

I 2> 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 ==• < 0. {4]

V2x~- x 2 +8

K Q.5 In A ABC, suppose AB = 5 cm, AC = 7 cm, Z ABC :

(a) Find the length of the side BC.

(b) Find the area of A ABC. [4]

Q. 6 The sides of a triangle are n- \,n and n + 1 and the area is n-Jn • Determine n. [4] Q. 7 With usual notions, prove that in a triangle ABC,

r + r{ + r2 - r3 = 4R cos C. [5]

Q.8 Find the general solution of the equation, sin %x + cos nx = 0. Also find the sum of all solutions in [0,100], [5]

Q. 9 Find all negative values of'a' which makes the quadratic inequality sin2x + a cos x + a2 > 1 + cos x true for every x e R [5]

Q.10 Solveforx, si°g2*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 a2 - 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 <0. [2 + 4]

Q. 15 Given the product p of sines of the angles of a triangle & product q of their cosines, find the cubic equation, whose coefficients are functions o f p & q & whose roots are the tangents of the angles of the triangle. [6]

Q. 16 If each pair of the equations

x2 +pjX + qj = 0 x 2 + p 2 x + q 2 = 0 x 2 +p 3 x- i -q 3 = 0

has exactly one root in common then show that (p, + p2 + p3)2 = 4(pjp2 + p2p3 + p3pj - q, - q2 - q3). [6]

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 <= R then find the value of ( a -b ) . [4]

Q.5 For x > 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 2<n+1)-V2^Tn~. n-*>o

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 ec 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 x2

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 ec 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- * r—r 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 — > —2—7 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<x<2n which satisfy

V2 (cos 2x - sin x - 1 ) = 1 + 2 sin x, is (A) 2TT (B) 3n (C) 4ti (D) 671

n-»°o n=0 Q.24 The value of Lim

Q.25

v Tn

2 y is

(A) 1 (B) 2 (C) 4 (D) none

If sin(x + 2y) = 2x cos y, the the value of dy/dx at the point (0,71) must be ( A ) - l (B) - 3/2 (C)0 (D)2

| | | 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 ex - 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<x> 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

g(*) = e - 1 8x - 47t

(B) h has an irremovable discontinuity at x = 7t/2

(C) h has a removable discontinuity at x = tc/2 ( D ) / ( ( _ "N

% 71 2

= g 2

\ / V J Q. 2 Two balls are drawn from 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

(A) 94 (B) 47 (C) 38 (D) 19

Q.3 If A ABC if cosA, cosB, cosC areinA.P. then which ofthe following is also an A. P.?

A 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

Q. 4 If tan"1 (x) + tan"1 (2x) + t an 4 (3x) = n, then (A) x = ± 1 (B) x = 0 (C) x = 1 (D)xG(j) Q.5 The most general solutions of the equation x3 s i n 2 x + 2 = ^ is

(A)x = n7t + ( - l ) " - | ( B ) x = y - C - l ) " ^ (C)x = 0

where n e I

(D) x = nrc - ( - l ) n 71 12

Q. 6 The sum of the square of the length of the chords intercepted by the line x + y = n, n e N on the circle x2 + y2 = 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?

(A) Limf(x) = 0 x-»l

(B) Lim f (x) - 1 x-»0

(C) Lim f (x) exists at eveiy point x0 is ( -1 ,1) (D) Lim f (x) = 1 " X->1

y y=f(x) I

, . o \ y . . +Y \ I1 1" i /

a s i n b x - b sinax , Lim — — (a ^ b) is x-+o tan bx - tan ax

(A) 1 (B)

2x2

Lim is x->0 3 - 3 c o s x

2

a - b a + b (C)

a + b a~-b

(D)nonexistant

(A) 4

( a ) . 3

< C >4 CD)

Q. 10 Let / = Lim-x2 - 9

and m = Lim x 2 - 9

then X ^ 3 V X 2 + 7 - 4 " x ->-3yjx 2 + 7 - 4

(A)Um (B) I-2m (C) / = - m

Q. 11 If (2 - x2) < g (x) < 2 cosx for all x, then Lim g (x) is equal to x-»0

(A)l (B)2

Q 1 2 Le, + + 5 x-1

( A ) - 5

(C) 1/2

then Lim f (x) is x-»l

i Q. 13 Lim (/nx)x_e is

x—»e+

£ (A) e e

(B)2

(B)ee

( C ) - 6

(C) e-

(D) / = m

(D)0

(D) non existent

(D)e

Q.14 Lim (ex + x ) x

x->0

(A) e1

/n(x + l) Q. 15 Lim -7— is

X->QO log2 x

(A) log2e

(B)e

O)0

1/2 (C) e3/2

(C) /n2

(D) e2

(D) non existent

30 + 71 Q. 16 Let / = Lim — j ^ then [ I ] is, where [ ] indicates greatest integer function

e->--

(A) equal to 3

A 7 1

sin 0 + -I 3y

(B) equal to 2

Q.17 Lim x->0

e - sin x - e x

(A) e + 1 (B)e-3 + l

Q.l 8 Lim cos(tan_1 (sin (tan-1 x)]) is equal to

(C) equal to 1

(C) e3 + 1

(A) - 1 (B)V2 (C) 1

S

(D) none existent

(D) e3 - 1

Q. 19 Let f (x) = a X + , if Lim f (x) = 1 and Lim f (x) = 1 then f (2) + f (-2) is equal to X + 1 x-»0 x-»oo

(A) 1 (B) 2 (C)0 (D) 4 x n —3n

Q.20 If Lim x—>3 X - 3

(A) 3

SUBJECTIVE

108 (n e N) then the value of n is

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

_ , t a n 2 x - 2 s i n x Q. 21 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,

L i m l n Q + * 2 + x 4 )

(e x - l )>

4 4 4 4 71 tan -1 — + tan"1 — + tan-1 — + tan -1 — + 00 = — +cot_1 3.

PI

[3]

[3]

Jig BANSAL CLASSES MATHEMATICS vSTarget IIT JEE 2007 Daily Practice Problems CLASS: XII (ALL) DATE: 22-23/05/2006 TIME: 55 to 60 Min. DPP. NO.-15

Q. 1 Draw the graph of the function / (x) =

at x = 1.

3X , -1<X<1 4 - x , l<x<4

& discuss its continuity & defferentiability

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 crosses the y - axis.

Q. 3 Given/(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:

x + 1 (x +1) (2x +1) (2x+l)(3x + l) .00

l + /nt 3 + 2/nt dv f d v A

Q.5 If x = — 2 — and y = . Showthat y — = 2x — t t dx Vdx j

+1.

X X X x X X X Q.6 Let f (x) = tan — secx + tan T T sec — + tan sec t t + +tan ~ sec t t z t v / 2 2 2 2 2 2 2

and g (x) = f (x) + tan ~ where x e ^ 71 7CN

2 2 and n eN. Evaluate the following limits.

(a) Limit x-»0

' g ( x ) ^ V X y

(b) Limit x->0 v A y

(c) Limit x-^0 V X y

Q. 7 Without using L' Hospitals rule or series expansion evaluate: Lim x-»0

x + / n | V x 2 +1 - x

ill BANSAL CLASSES MATHEMATICS vSTarget BIT JEE 2007 Daily Practice Problems CLASS: XII (ALL) DATE: 24-25/05/2006 TIME: 55 to 60 Min. DPP. NO.-16

Q.l Suppose f(x) = tan (sin-1 (2x)) (a) Find the domain and range of / (b) Express f (x) as an algebaric function of x.

(c) Find f' I

v4y

Q. 2 Discuss the limit, continuity & differentiability of the function f (x) = X(3e1/x+4)

2 - e 1 / x ,XTK)

,x=0 atx=0.

In Q.3 Evaluate: Limit

x->0

tan r * a 7t —hax

4 ( b * 0 ) .

sin bx

Use of series expansion and L'Hospital's rule is not allowed.

Q.4 The function/is defined by y=f(x) . Wherex=2t- | 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.

Q.5 Evaluate U n } ' 1 1 A

/nx x - 1 Use of series expansion or L' Hospital's rule is not allowed.

Q. 6 If g is an inverse function of / & /'(x) = — 7 , prove that g'(x) = 1 + [g(x)]n. 1 ~h X

Q. 7 Given a real valued functions f(x) as follows

x2 + 2 c o s x - 2

f(x)

for x < 0 x"

p f o r x = 0 sinx - ^n(ex cosx)

6x2 for x > 0

Determine the value of p if possible, so that the function is continuous at x=0 . Use of power series or L'Hospital's rule is not allowed.

ill BANSAL CLASSES MATHEMATICS V g Target IIT JEE 2007 Daily Practice Problems CLASS: XII (ALL) DATE: 03-04/05/2006 TIME: 60 Min. DPP. NO. -11

t ^ 2

V Q 4

X 5

If 'a' is a fixed constant then Lim x-»0

ax-s inax

< B ) £ ( O -

I f / (x) = x2 + 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 ) 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 (B) V2P (C) 2-s/P (D)

Number of real solutions of the equation cos x + cos { [2 x) = 2, is (A) 0 (B) 1 (C) 2 (D) infinite A quadratic polynomial y = ax2+bx+c has its vertex at (4, -5) and has two x-intercept, one positive and one negative as shown. Which one of the following must be negative? (A) only a (B)onlyb (C) only c (D) only b and c If in a AABC, the altitudes from the vertices A, B, C on opposite 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 C2 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. Number of three term arithmetic progressions which exist in the set {1,2, 3, ,40} and common difference d 0, is (A)190 (B)200 (C)380 (D)400

x is equal to

_ 2 a2 a3 a 2 a 3

( D ) j

Q.6

Q 8

Q. 10

Q.ll

J & 2

r X 1 3

Y

\ O B \ 4 / A "

- 5 '

The smallest positive integer x so that tan

(A) 8 (B)9

_i x _t 1 tan — + tan 10

(C)7 x + 1

% : tan— is 4

(D)0

When (XV4 _ x2 /3)7 is multiplied out and simplified, one of its terms has the form kx3 where 'k' is a

constant which is equal to (A) 7 ( B ) - 7 (C) 35

„ - 1 + /V3 , - I - / V 3 I f x = and y = ,

where i2 - -1, then which of the following is not correct? (A) x5 + y5 = - 1 (B) x7 + y7 — - 1 (C)x9 + y9 = Number of solutions of the trigonometric equation

cos3x - 3 cos x sin^ = cos 3x where x e (0,1), is (A) 0 (B) 1 (C) 2 In which one of the following cases, limit does not tend to e?

i / 1 /

( D ) - 3 5

l (D) x11 + y n = - 1

(D)infhinte

(A) Lim xx~' X—>1

(B) Lim X—

r i 1 + -

V Xy (C) Lim

X—

x + 4 x + 2

NX+3

(D) Lim ( l+/(x))?oo when Lim / ( x ) 0 X->CC

Q. 14 The lines L and K are symmetric to each other with respect to the line y = x. If the equation of the line L * y is y = ax+b where a and b are non zero, then the equation 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

3 X - 4 X

Q. 15 Domain of definition of the function f (x) = is V x - 3 x - 4

(A) ( - o o , 0 ] (B) [0, oo) (C) ( - oo, - l ) u [ 0 , 4) ( D ) ( - o o , l ) u ( l , 4 ) Q. 16 The roots of x2 + bx + c = 0 are both real and greater 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

Q. 17 Which one of the following does not reduce to sin x for every x where the expressions are defined? • - 2 s m x ^ s ^ sin xsecx „ . A . (A) — 9 5— (B) csc x - cot x cos x (C) (D) all reduce to sin x v ' sec x - tan x w tanx

Q' 1 8 Let/(x) be a fiinction with two properties (a) for any two real number x and y, f ( x + y) = x + / ( y ) and (b) f (0) = 2. The value o f / (100) , is (A) 2 (B) 98 (C) 102 (D) 100

Q.19 Read the following statements carefully: I If a, b and c are positive numbers not equal to 1 and a < b, then logac < logbc. II The equation x2 - b = 0 has a real solution for x for any real number b. HI The sequence an defined by an = 3 (0.2)"n is a geometric sequence. IV cos(cos(x)) < 1/2, V x e R ' Now indicate the correct alternative. (A) exactly one is always true (B) exactly two are always true (C) exactly three are always true (D) exactly four are always true.

Q.20 If x = a + b/ is a complex number such that x2 = 3 + 4i and x3 = 2 + 11/ where i = J I \ , then (a + b) equal to (A) 2 (B)3 (C)4 (D)5

MORE THAN ONE ARE CORRECT Q 21 If x satisfies log2x + logx2 = 4, then log2x can be equal to

(A) t a n ~ ( B ) c o t y ( C ) t a n | ( D ) c o t ^

Q.22 In a triangle ABC, altitude from its vertex meet the opposite sides in D , E and F. Thenthe perimeter of the triangle DEF, is

abc 2A R(a + b + c) _ 2rs ( A ) - F ( B ) T ( C i - ^ — l ( D ) T

where A is the area of the triangle ABC and all other symbols have their usual meaning.

, Q 23 In a triangle ABC if Z B = 3 0°, b = 3 V2 - a/6 and c = 6 - t h e n

(A) the triangle ABC is an obtuse triangle (B) angle Z A can be 15° (C) there can be only one value for the side BC (D) the value of tanA tanC will be unique.

n Q.24 Let z=(0, l )eC. Where C is the set ofcomplex numbers, then the sum ^ z for n e N can be equal to

k=0

(A) 1 + i (B)i (C)0 ( D ) - l Q.25 Value of the expression log1/2(sin6° • sin42° • sin45° • sin 66° • sin 78°)

(A) lies between 4 and 5 (B) is rational which is not integral (C) is irrational which is a simple surd (D) is irrational which is a mixed surd.

i l l BANSAL CLASSES MATHEMATICS \8Target IIT JEE 2007 Daily Practice Problems CLASS:XII(ALL) DATE: 05-06/05/2006 TIME: 50Min. DPP. NO.-12

f 1 \l/x ' l + ax v

l + 2x , Ifa>0and Lim has the value equal to unity then'a'is equal to

x-»0+ 1 J- ^

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

Q 2 The first three 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 of x, is (A) 6 (B) 12 (C) 24 (D) 30

, y^f.3 The value of the expression sin2 1° + sin2 2° + sin2 3° + + sin2 90°, is

(A) 0 (B)45 (C) 45.5 (D)90

^JQA In a triangleABC with altitude AD, ZBAC = 45°, DB = 3 and CD = 2. The area of the triangle ABC is (A) 6 (B) 15 (C) 15/4 (D) 12

When the polynomial 5x3 + Mx + N is divided by x2 + 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 Find the equation of the circle which has its diameter the chord cut off on the line px+qy - 1 = 0 by the circle x2 + y2 = a2. [4]

, Q 9 Obtain a relation in a and b, if possible, so that the function

j . xn (a + sin(xn))+ (b - sin(xn )) / ( X ) = ^ (l + xn)sec(tan (xn +x"n)j - ^ i n u o u s a t x = 1. [6] n-»oo

QT10 The interior angle bisector of angle A for the triangle ABC whose coordinates of the vertices are A (-8, 5); B(- l5, -19) and C(1, - 7) has the equation ax + 2y + c = 0. Find 'a' and V. [6]

i Si BANSAL CLASSES Target I IT JEE 2007

MATHEMATICS Daily Practice Problems

CLASS: XII (ALL) DATE: 08-09/05/2006 TIME: 50Min. DPP. NO.-13

v/ Q. l

v / 4 2

v ^ 4

J*' .5

Q.6

In AABC (a + b)(a - b) = c(b + c), the measure of angle A, is (A) 30° (B) 60° (C) 90° (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)255°

The radius of the circle inscribed in a triangle with sides 12,3 5 and 37, is (A) 4 (B)5 (C)6 (D)7

Consider the equation 19z2 - 3/z - k = 9, where z is a complex variable and i2 - - 1. Which ofthe 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.

The set of values of x for which the function defined as

1 - x x < l

/ ( x ) = (1 - x)(2 - x) 1 < x < 2

3 - x x > 2

fails to be continuous or differentiable, is (A)(1) (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 of these numbers is the reciprocal of F, which is (A) A (B)B (C)C (D)D

A triangle has side a - , the opposite angle a = 69°, and the sum of the two other sides is (b + c) = 5. The ratio of the longest to the shortest side of the triangle, is

(A) 1 (B)V2 3

K)-, „# SUBJECTIVE

Q.8 Evaluate : Lim - n + V a ) ( a > o , n e N ) n—»oo Use of series expansion and L'Hospital's rule is not allowed. [4]

Q.9 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 3xlieonthel ine,y(m 1m 2 + m2m3 + m3m1 +3 ) = (mj +m2 + m3 + 3 m1m2m3)x.

[6]

^QTlO Find the equations of the circles which touch the co-ordinate axes and the line, 3x+4y = 12. [6]

Js BANSAL CLASSES MATH EMATICS V g Target ilT JEE 2007 Daily Practice Problems 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 of 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, (x2 + y2)3 = C2x2y2. |5]

Q.2 Evaluate: cosx / n ( x - a )

x->a ln(e" - e ) [51

Q.3 Let t,, t2 and t3 be the lengths of the tangents drawn from a point (x,, yj) to the circles x2 + y2 = a2, x2 + y2 = 2ax and x2 + y2 = 2ay respectively. The lengths satisfy the equation

ty = t2 t2 + a 4 . Show that locus of (xj, y^ consists of x + y = 0 and x2 + y2 = a(x + y) [5]

- 9 . T; 2cos0 + l Q.4 Let a = 2 c o s — - l then show that L i m(a,a,a,. . . .an ,an) = , 0 e R . n 9 " n~>oov 1 i i ii-l n-7 l ' [8]

Q. 5 Consider a function f: x —> x + a x — 1

; x e R - {1} where a is a real constant. If / is not a constant

function, find the following

(i) the range of / (ii) f~x, is it exist (iii)/ - / / / V V

1

W J J [8]

-9 f f x ) Q. 1 Given Lim - 2 then evaluate the following limits, giving explicit reasoning.

x-»0 x

(i) Lim [f(x)l ;(ii)Lim x->0 x-»0

f (x ) X

where [x] denotes greatest integer function. [5]

Q. 2 Find the sum to n terms of the series

Sn = c o t - ^ 2 2+ £ ) + c o r i f 2 3

+ ^

Also deduce that Limit Sn =cot _ 12 . n—

+ cor1 2*+' 23y

+ , upto n terms

[5]

Q. 3 The vertices of a triangle are A(x t, x}tan 6j), B(x2, x^an 02) & CCxg, x3tan03). If the circumcentre O of ' . . ^ X COS0, +COS0~ +COS0o 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 of ways in which the balls can be distributed in the boxes if no two adjacent boxes remain empty.

[8]

JF»3F»- ; l_ Q

Q. 1 Tangents are drawn from any point on the circle x2 + y2 = R2 to the circle x2 + y2 = r2 Show that if the line joining the points of intersection of these tangents with the first circle also touches the second, then R = 2r. [5]

- / n (2 -cos2x) f o r x < Q

/n (l + sin3x) Q.2 Let a functionf (x) be defined as f(x) =

sin 2 x _ i for X > 0

/n(l + tan9x)

Find whether it is possible to define f (0) so that ' f ' may be continuous at x = 0. [5]

Q. 3 Find all possible values of a and b so that f (x) is continuous for all x e R if

| ax+ 3 | , if x < - 1 13x + a ] , if - 1 < x < 0 *sia2*-2b, if 0 < x <7t

c o s 2 x - 3 , if x>7t

/ ( x ) [5]

Q.4 Prove that in a AABC, the median through A divides the angle Ainto two parts whose 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 Find the value of y = sin cot -1 cos tan -1x where x - cosec - i 2 cos J cos v 3 V

V6+1 2A/3 [8]

> y

1 a BANSAL CLASSES MATHEMATICS Target 11T JEE 2007 Daily Practice Problems

CLASS: XII (ALL) DATE: 24-25/04/2006 TIME: 55 to 60Min. DPP. NO.-7

Q.l Let T = {1, 2 ,3 ,4 , 5 }. A funct ion/:T-»Tis said to be one-to-one if tj * t2 implies that/(tj) ^/(t2). Obtain a one-to-one function such that t + / ( t ) is a perfect square for every t in T. [4]

C ab+l^i / b c + n / c a + l"] Q.2 If a > b > c > 0 then find the value of : cot -1 r + cor 1 t — + cot""1 — ~ . [4] \ a - b y v b - c y v c — a y

Q.3 Find the equation to the locus of the centre of all circles which touch the line x = 2a and cut the circle x2 + y2 = a2 orthogonally. [4]

Q.4 Le t / (x ) = ( * - 4 ) ( X 2 ~ 4 X - 5 ) , F i n d

(x - 2 x - 3 ) ( 4 - x ) (a) the domain o f f (x) (b) the roots o f f (x) (c) all x such t h a t / ( x ) > 0 (d) all x such tha t / (x) < 0 [6]

Q.5 The points ( -6 ,1) , (6,10), (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 ax2 + bx • Find the set of real values of'a' for which there is at least one positive real value

of 'b' for which the domain of / and the range of / are the same set. [6]

Q. 7 Two circles of different radii R and r touch each other externally. The three common tangents form a

2(RrW2

triangle. Show that the area of the triangle is — — - — . f8| R - r

3BANSAL c l a s s e s Target I IT JEE 2007

MATHEMATICS Daily Practice Problems

CLASS: XII (ALL)

The value of

2 - V 3

ZX4r£: 21-22/04/2006

sin 120° 16 cos 15° • cos3 0° • cos 120° • cos 240°

TIME: 60Min.

is

DPP NO.-6

(A) 8 (B)

V 3 - 1 (C)

2 - V 3 ( D ) 2 - V 3

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

(A)r (B) 2r

(C) 2 r - 3 (D) 5r

V Q.4 711 + 1511 when divided by 22 leaves the remainder (A) 0 (B) 1 (C) 7 (D)10

The coefficient of x in the expansion of (1 + x)(l + 2x)(l + 3x) (1 + lOOx), is (A) 4950 (B) 5000 (C) 5050 (D)5100

Suppose AABC is an equilateral triangle and P is a point interior to AABC. If the distance from P to sides AB, BC and AC is 6, 7 and 8 units respectively, then the area of the AABC, is

147V3 , N 21V3 441 (A) 147V3 (B) (C) (D)

^ M 7 The tune Twinkle Twinkle Little Star' has 7 notes in its first line, CCGGAAG 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

/ Q . 8

m (A) 72 (B) 105 (C) 210 (D)5040

If the graphs of y = cos x and y=tan x intersect at some value say 9 in the first quadrant. Then the value of sin 9 is

(A) -1 + V2

(B) -l + fi

(C) - 1 + V5

(D) - 1 ± V 5

V Q.9 If S= 1 + - + - + — + , then 1 + - + — H—— + equals 4 9 16 9 25 49

S ( A ) -

3S ( B ) t (C) > 4 C D ) S - i

Q 10 How many solutions are there for the equation cos2x - sin22x = 0 on [0, 2n]7 (A) 6 (B)4 (C)2 (D) l

Q. 11 Number of ways in which 7 people can be divided into two teams, each team having at least one member, is (A) 72 (B) 32 (C) 144 (D)63

Q. 12 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 (B) 2y = 2 - x (C)y = x - 2 (D)y = 2 x - 1

13 The positive value o fx that satisfies VlO = ex + e~x, is

( A ) | / n ( 4 - V l 5 ) (B)^ /n (4 + Vl5) ( Q ~/n(4 + Vl?) ( D ) ^ - J r i )

f ^ / Q . 1 4 Let / : R - {0} - > R be any function such that/(x) + 2 /

n — - 3x. The sum of the values of x for which v x ;

/ ( x ) = l , i s

(A)l (B)2 ( C ) - l ( D ) - 2

^ QTl 5 Let r j, r2, r3, r4 be the roots of the equation, x4 - 9x3 + ax2 + bx + 16 = 0 r, + r, + r3 + r. where a, b are constants. Then the difference between the arithmetic mean - — and the

4

geometric mean i ] ^ 2 h r 4 the roots, is equal to

(A) 5/2 (B) 1/2 (C) 17/4 (D) 1/4

16 Sum ofthe x and^ intercepts ofthe circle described on the line segment joining (-2, 1) and (1, 2) as diameter, is (A)l (B)2 (C)3 (D)4

1 1 1 2 4 Q. 17 If sinlf,a = —, then the value of H — h —I , is

5 cos a 1 + sin a 1 + sin a 1 + sin a (A) 4 (B)6 (C) 8 (D) 10

. Q. i 8 A variable circle touches the x-axis and also touches the circle with centre 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

1 1 19 The set of real numbers x satisfying > -3—— is

X 1 X 2* ( A ) x > 2 ( B ) x < - 1 ( C ) x < 2 a n d x < - 1 ( D ) - l < x < 2

\J$.20 A sequence of three real numbers forms an arithmetic progression with 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 geometric 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 (xl3 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 If the sum of the solutions of the equation sin2x - sin x = cos2 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 (B) 9 (C)10 (D) 11

Q. 24 Quadratic equation with real coefficients whose one root is (2 + /')(3 - z) where /' = ^ T J , is (A) x2 - 14x + 48 = 0 (B) x2 - 14x + 50 = 0 (C) x2 + 1 4 x - 4 8 = 0 (D) x2 + 14x + 49 = 0

Q 2 5 Which of the following is equal to sec(t) + tan(t) ?

[ t n (A) cot H + 4 (B) cot - + — 2 4 ( C ) - c o t - t+—

v 2

r ( D ) - c o t

71

\ ^ 2 6 If / ( x ) = x + 4 and g(f (x>) = 2x + 1, then the function g (x) is (A) 2x - 7 (B) 8x + 3 (C)2x + 9 (D)2x2 + 5x + 4

Q.27 In a triangle ABC, Z A = 7 2 ° , b = 2 a n d c = ^5 +1 thenthe triangle ABC is (A) obtuse isosceles (B) acute isosceles (C) right isosceles (D) not isosceles

Js BANSAL CLASSES MATH EMATICS V8Target I1T JEE 2007 Daily Practice Problems CLASS: XII (ALL) DATE: 19-20/04/2006 TIME: 50 to 60 Min. DPR NO.-5

Only one is correct. [6X3 = 18] There is NEGATIVE marking for each wrong answer 1 mark will be deducted.

1 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 + I 2 V 3 (B) 1 2 - 7 ^ 3 (C) 12 + 7^3 (E>) 4n

J%2 If 0 < 9 < 2n, then the intervals of values of 9 for which 2sin29 - 5sin9 + 2 > 0, is

(A)

(C)

0, f i 5n „ ^

— , 2 n J v 6

u

u 71 571

6'T

(B)

(D)

n 5tx 8 ' T

41TT 48

-,7t

w — wz Q.3 If w = a + ip where p and z ^ 1, safisfies the condition that — — — is purely real, then the set of

1 — z values ofzis (A) {z: |z| = l} (B) {z: z = z ) (C) { z : z * l } (D) {z : | z | = 1, z ^ 1}

x_ Q.4 Let a, b, c be the sides of triangle. No two of them are equal and X, e R. If the roots of the equation x2 + 2(a + b + c)x + 3X(ab + be + ca) = 0 are real, then

(A) ( B ) ^ > 3 ( C H ' 1 v 3 ' 3 y

(D) X € v 3 ' 3 y

^ QC5 If r, s, t are prime numbers and p, q are positive integers such that the LCM of p, q is r2t4s2, then the numbers of ordered pair of (p, q) is (A) 252 (B) 254 (C)225 (D)224

A$.6 Le tOe V 4y

and t, = (tan0)tan0, t2 = (tan9)cote, ^ = (cot9) tane, t4 = (cot0)cote, then

( A ) t j < t 2 < t 3 < t 4 (B) t 4 > t 3 > t j > t 2 ( C ) t 3 < t j < t 2 < t 4 ( D ) t 2 < t 3 < t 1 < t 4

One or more than one is/are correct. [1X5 = 5] There is NEGATIVE marking for each wrong answer 1 mark will be deducted.

\ JQ.1 Internal bisector of Z A of a triangle ABC meets side BC at D. A line drawn through D perpendicular to AD intersects the side AC at E and the side AB at F. If a, b, c represent sides of AABC then

(A) AE is harmonic mean of b and c

4bc . A ( C ) E F = — s i n -

b + c 2

2bc A (B) A D = cos— v 7 b + c 2

(D) the triangle AEF is isosceles

Only one is correct. [ 3 x 5 = 15] There is NEGATIVE marking for each wrong answer 2 marks will be deducted.

Comprehension Let ABCD be a square of side length 2 units. C2 is the circle through vertices A, B, C, D and C, is the circle touching all the sides of the square ABCD. L is a line through A

P A 2 + P B 2 + P C 2 + P D 2

, JQ. 8 I fP is a point on C, and Q in another point on C7, then 7-72 , , isequalto 1 t^A + v D + ' V*-'

(A) 3/4 (B) 3/2 (C) 1 (D)9

- Q. 9 A circle touches the line L and the circle C, externally such that both the circles are on the same side of the line, then the locus of centre of the circle is (A) ellipse (B) hyperbola (C) parabola (D) parts of straight line

Q. 10 Aline M through A is drawn parallel to BD. Point S moves such that its distances from the line BD and the vertex A are equal. If locus of S cuts M at T2 and T3 and AC at Tj, then area of ATjT2T3 is (A) 1 /2 sq. units (B) 2/3 sq. units (C) 1 sq. unit (D) 2 sq. units

SUBJECTIVE: [ 2 x 6 = 12] There is NO NEGATIVE marking.

"4 11 If roots of the equation x 2 - 1 Ocx - 1 Id = 0 are a, b and those of x2 - 1 Oax — 1 lb — 0, then find the

value of a + b + c + d. (a, b, c and d are distinct numbers) v3

f3" f

3' ^4;

Q.12 If A = — - — + — +• + ( - l ) n ' — and B = 1 - A , then find the minimum natural n A A I A V / 1 4 J n n

number n0 such that Bn > An. V n > nQ.

43BANSAL CLASSES MATHEMATICS 8Target I IT JE1 2007 DaiSy Practice Problems

CLASS : XI (P, Q, R, S) DATE: 26/01/2006 TIME: 45 Min. DPP. NO.-53

Q. 1 Which of the following sets does NOT represent a function? I {(x, y) | y = 2x + 1} II {(x, y) j x2 + y2 = 10, y > 0} III {(3, 1) (4, 1),(5, 2), (6, 2), (7, 3)} IV {(x, y) | y = 2X + 1} (A) I (B) II (C) III (D) IV (E) none

Q.2 I f f (x) is a function from R R, we say that f (x) has property I if f (f (x) ) = x for all real number x, and we say that f (x) has property 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) infinite

Q.3 The function f (x) is defined by f (x) = cos4x + K cos22x + sin4x, where K is a constant. If the function f (x) is a constant function, the value of k is

(A) - 1 (B) - 1/2 (C) 0 (D) 1/2 (E) 1

Q.4 Define the function f (n) where n is a non negative integer satisfying f (0) = 1 and f (n) is defined n-l

for n > 0 as f (n) = n • . Let 2m be the highest power of 2 that divides f (20). The value of i=0

m is (A) 18 (B) 19 (C) 20 (D) 21 (E)22

Direction for Q.5 and Q.6 The graph of a relation is (i) Symmetric with respect to the x-axis provided that whenever (a, b) is a point on the graph,

so is (a, - b) (ii) Symmetric with respect to the y-axis provided that whenever (a, b) is a point on the graph,

so is ( - a, b) (iii) Symmetric with respect to the origin provided that whenever (a, b) is a point on the graph,

so is ( - a, - b) (iv) Symmetric with respect to the line y = x, provided that whenever (a, b) is a point on the

graph, so is (b, a)

Q.5 The graph of the relation x4 + y3 = 1 is symmetric with respect to (A) the x-axis (B) the y-axis (C) the origin (D) the line y = x (E) both the x-axis and y-axis

Q.6 Suppose R is a relation whose graph is symmetric to both the x-axis and y-axis, and that the point (1,2) is on the graph of R. Which one of the following points is NOT necessarily on the graph of R? (A) ( -1 ,2) " (B) ( 1 , - 2 ) ( C ) ( - l , - 2 ) (D) (2, 1) (E) all of these points are on the graph of R.

Q.7 Suppose that f (n) is a real valued function whose domain is the set of positive integers and that f (n) satisfies the following 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 • qn - r, for n = 1, 2, then the value of p + q + r is (A) 16 (B) 17 (C) 20 (D) 26 (E)31

x r x Q.8 Let f (x) = —— and let g (x) = . Let S be the set of all real numbers 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

Q.9 Let f be a linear function with the properties that f (1) < f (2), f (3) > f (4), and f (5) = 5. Which of the following statements is true? (A) f (0) < 0 (B) f (0) = 0 ( C ) f ( l ) < f ( 0 ) < f « - 1 ) (D) f (0) = 5

f

1 - x 2 * Q. 10 If g (x) = 1 - x2 and f (g(x) ) = — w h e n x * 0. then f (1/2) equals

x

(A) 3/4 (B) 1 (C)3 (D)V2

Q. 11 A function f from integers to integers is defined as follows

- n + 3 if n is odd f (n) =

L n / 2 if n is even

Suppose k is odd and f ( f ( f (k))) = 27. The sum of the digit of k is (A) 3 (B) 6 (C) 9 (D) 12

Q.l2 Let f be the function defined by f (x) = ax2 - ^ 2 for s o m e positive a. If f | f (V2) )= - ^ 2 t h e n a

equals

(A) V2 (B) | (C) ^ (D) ^

ill BAN SAL CLASSES MATHEMATICS I S T a r g e t IIT J E E 2 0 0 7 Daily Practice Problems 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. Circles C p C2 and C3 are drawn with sides AB, BC and CAas their diameters. The radical axis between 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 (B) -3 /4 (C) 1/4 (D) 1/2

Q .3 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 , -2006) (D)(19 , -97)

Q.4 Which is the inverse of the function/(x) = - / n fx + V x 2 + l j ?

(A) 3(e3x + e"3x) (B) | (e3x + e-3x) (C) ^ (e~3x - e3x) (D) - (e3x - e~3x)

sin(a + P) p Q-? if

sin(a - p) q t a n a ' c o t P has the value equal to p + q p - q p + q p - q

(A) (B) ^ (C) (D) v ' p - q v ' p + q q 4

C6/ Number of seven digit whole numbers 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)53

t I f / (x ) = x4 + ax3 +bx2 + cx + d be polynomial with real coefficient and real roots. If | f ( i ) | = 1, where i = ^/CT > then a + b + c + dis equal to (A) - 1 (B) 1 (C)0 (D) can not be determined

Q-8- Let ABCDE is a regular pentagon with all sides equal to 4. Which one of the following is a correct solution for the length AC? (i) 2 csc(18°) (ii) 2 sec (72°)

(iii) V 3 2 - 3 2 c o s ( 1 0 8 ° ) (A) only (i) and (ii) are correct (B) only (ii) and (iii) are correct (C) only (iii) and (i) are correct (D) all are correct

9 A line x = k intersects the graph of y=log5x and the graph of y=log5(x+4). The distance between the

points of intersection is 0.5. Given 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 Which of the folio wing sets of restrictions is true for the function / ( x ) = ax2 + bx + c represented by the graph as shown (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

^yQ. 11 The radius of the circle passing through the vertices of the triangle ABC, is

/ 0

(A) 8y/l5

5

(C) 3^5

(B) 3y/l5

CD) 3V2

Q. 12 The area of the region consisting of all points (x, y) so that x2 + y2 < 1 < | x | +1 y |, is (A) n ( B ) n - l ( C ) t c - 2 (D)TC-3

1 3 5

j Q 13 7 9 11 A^.iJ 1 3 1 5 1 7 1 9

21 23 25 27 29 Consecutive odd numbers are arranged in rows as shown. If the rows are continued in the same pattern, then the middle number of row 51, is (A) 2601 (B) 2500 (C)2704 (D)2401

Q.14 The expression [x + (x3-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 ,13 }. The minimum possible value of (a + b + c + d)2 + (e + f + g + h)2 is (A) 30 (B)32 (C) 34 (D)40

J | BANSAL CLASSES MATHEMATICS ^BTarget IIT JEE 2007 Daily Practice Problems CLASS: XII (ALL) DATE: 12-13/04/2006 TIME: 35 to 45 Min. DPR NO.-2

_ Q.l If the circles x2 + y2 + 2ax + 2by + c = 0

and x2 + y2 + 2bx + 2ay + c = 0

where c > 0, have exactly one point in common then the value of ^ is 2c

(A)l (B)V2 (C) 2 (D) 1/2

( / Q.2 Suppose/ is a real function satisfying/(x +/(x)) - 4 / (x) and / ( I ) = 4. Then the value of / (21) is (A) 16 (B)21 (C) 64 (D) 105

100 Q.3 The value of equals (where i= )

n=0 (A) - 1 (B)i (C) 96 + i (D) 97+ i

>jQ. 4 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

n 2n (A) (B) (C)

71

18 (D) 71

8

, The value of b > 0 for which the region bounded by both the x-axis and y=-12x | + b has an area of 72, is (A) 12 (B) 36 (C) 6 V2 (D) 144

, 0(6 If 500! = 2m • N, where N is an odd positive integer, then m is equal to y (A) 452 (B) 494 (C)498 (D) none of these

Le t / be a linear function for which/(6) - / ( 2 ) = 12. The value o f / ( 1 2 ) - / ( 2 ) is equal to (A) 12 (B) 18 (C) 24 (D)30

v ^ r . 8 a = tan-1 A / 2 + 1

V2-1

(A) is equal to 1

- t a n rv2] 1 and P = tan~'(3) - sin-1

U s ) t ^ - —

pB-)

I 2 J and P = tan~'(3) - sin-1

I 5 J —§tn=i

W , then cot(a - p)

(B) is equal to 0 (C) is equal to J 2 - 1 (D) is non existent

v/Q.9 There are three teachers and six students. Number 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° for n = 2 ,4 ,6 , 180

(A) 1 (B)cot 1c (C)tan 1° (D)

A circle with center O is tangent to the coordinate axes and to the hypotenuse ofthe 30°-60°-90° triangle ABC as shown, where AB = 1. To the nearest hundredth, the radius ofthe circle, is (A) 2.37 (B) 2.24 C (C) 2.18 (D)2.41

^ 1 2 If s = l + + + ,then 1 - 1 - + I - - 1 - + equals

( B ) ^ - s ( C ) " - 2

1 B

(D) s — 1

^ jQ. 13 Find the value of x that satisfies the equation log ' x1 / x ^ X l/(x+l) 5050

(A)l (B)10 (C) 100 (D)1000

Q. 14 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 form y = Kx2 for some K which is equal to

(A) V

12 V5 (B)T

4 12

. 15 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 MATHEMATICS Target IIT JEE 2007 Daily Practice Problems

CLASS: XII (ALL) DATE; 14-15/04/2006 TIME: 35 to 45 Min. DPP. NO.-3

Q.l Let / be the function defined by/(x, y, z) :

( J3 -2

(x + y + z)(xy + xz + yz) xyz for all positive real numbers x,

y and z. The smallest possible value of / , is (A) 9 (B)8 (C)6 (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

^ 3 . 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 then/(50) is equal to (A) 10 (B)30 (C) 40 (D)50

Q.4 An equilateral triangle, with sides of 10 inches, is inscribed in a square ABCD in such a way that one vertex is at A, another vertex on BC and one on CD. The area of the square is

Q.5

^A-l

^ . 8

(A) 25(2-V3) (B) 25(2 + V3) (C) 25

The coefficient of x3 in the expansion of (1 + x + x2)12, is (A)352 (B)350 (C)342

100

(D) 332

The radius ofthe 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 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/(2004) is

(A) 1

1002 (B) 1

2004 (C) 2004 2005

(D)2004

The line (k + 1 )2x + ky - 2k2 - 2 = 0 passes through a point regardless 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

If the solution set for/(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) (-oo,+ 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 ZBCD - 100°, ZADB = 20°, AD = BD and BC = DC shown in figure. The product (AC) x (BD) is equal to D^

V3 2V3 C A ) V (B) —

4V3 (C) V3 (D)

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-interceptp 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 through A, B, C and D is a conic whose length of the latus rectum is

.2 a 2 - b 2 (A) — (B) 2aVi

a

(C) 2 a b V a 2 - b 2 (D) 2 V a 2 - b 2

v_X).13 Let / ( x ) = 1 if x is rational

0 if x is irrational

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

^ /Q. 14 How many of the 900 three digit number have at least one even digit? (A) 775 (B) 875 (C)450 (D)750

15 cot 10° + tan 5 0 equal to (A) sec 10° (B) sec 5° (C)cosec5° (D)cosecl0°

ill BAN SAL CLASSES MATHEMATICS Target IIT JEE 2007 Daily Practice Problems

CLASS: XII (ALL) DATE: 17-18/04/2006 TIME: 35 to 45 Min. DPP. NO.-4

(^A A sequence of equilateral triangles is drawn. The altitude of each is J 3 times the altitude of the preceding

triangle, the difference between the area of the first triangle and the sixth triangle is 968 square unit. The perimeter ofthe first triangle is (A) 10 (B) 12 (C) 16 (D) 18

Two circles with centres at A and B, touch at T. BD is the tangent at D and TC is a common 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 positive numbers x, and x2 are such that the line passing through (x,, - 1 ) and (x2,1) is tangent to C then (A) x,x2 = 1 (B) X j X 2

= — 1 ( C ) x j + x 2 = L ( D ) 4 x , x 2 = 1

Q.5 Suppose that (o and z are complex numbers such that both (1 + 20® and (1 + 2/)z are different real numbers. The slope of the line connecting © and z in the complex plane is ( A ) - 2 ( B ) - 1 / 2 (C) 2 (D) can not be determined

J * 6 If

(A)

xsec0 + y t a n 0 = 2cos0

x t a n 0 + ysec0 = cot0

cos 20 sin0

then y equals

(B)sin0 (C) cos 0 (D) sin 20

The locus of the point of intersection ofthe tangent to the circle x2 + y2 = a2, which include an angle of 45° is the curve (x2 + y2)2 = la2 (x2 + y2 - a2). The value of X is (A) 2 (B)4 (C) 8 (D) 16

Consider the circle x2 + y2 - 14x - 4y + 49 = 0. Let 1, and 12 be lines through the origin 'O' that are tangent to the circle at points 'A' and 'B'. If the measure of angle AOB is tan -1 (X) then X equal to

2 ( A ) ~

21 < B > «

28 (D)none

The value of the expression, 14 tan

(B)5

tan ' - + tan ' - + tan + tan + tan 1

13 which is equal to (A) 2 (C)7

21

(D)10

31 is an integer

10 If a, b are positive real numbers such that a - b = 2, then the smallest value of the constant L for which

V x 2 + a x - V x 2 + b x < L for all x > 0, is

(A) 1/2 (B) 1/V2 (C) l (D)2

^ Q . l l If every solution of the equation 3 cos2x - cos x - 1 = 0 is a solution of the equation a cos22x + 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=3x, 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 ) = x2 + kx ; k is a real number. The set of values of k for which the equation f (x) = 0 and

/ ( / ( x ) ) = 0 have exactly the same real solution set is (A) (0,4) (B) [0,4) (C)(0,4] (D)[0,4]

^ 1 4 If X'°83(4) = 27, then the value of x0 o g 3 4)2

(A) 4 (B) 16 (C) 64 (D) 81

Q^ l 5 If Q is the point on the circle x 2 +y 2 - 1 Ox+6y+29 = 0 which is farthest from the point P(- l , -6), then the distance from P to Q is

(A)2V5 (B) 2V7 (C)4V5 (P)4y/7

ill BAN SAL CLASSES MATHEMATICS V8Target IIT JEE 2007 Daily Practice Problems CLASS: XI (P, Q, R, S) DATE: 14/11/2005 TIME: 120 Min. DPP. NO.- 50

DPP OF THE WEEK This is the test paper of Class-XI (J-Batch) held on 13-11-2005. Take exactly 120 minutes. To be discuss on Friday (18-11-2005)

PART-A Only one alternative is correct. [20 x 1.5 = 30] 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 ) | ( B ) ^ ( C ) ^ ( D ) i

Q. 2 Number of real x satisfying the equation | x - l | = | x - 2 | + | x - 3 | i s

(A) 1 (B)2 (C) 3 (D) more than 3

Q.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) -- (B) 1 (C) ~ (D) can not be determined T1 Z

Q.4 Consider an A.P . t j j ^ tg , If 5 th, 9 th 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[ (B) tan"1 - I (C) tan-1! - (D) cot ^ _ - i f 3 ^

A) 2 .-l S3\

v2y

Q. 8 If logAB + logBA2 = 4 and B < A then the value of iogA8 equals

( A ) V 2 - 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 4 n, 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.l l IfthelinesL, : 2 x + y - 3 = 0 , L2 : 5 x + k y - 3 = OandL3: 3 x - y - 2 = 0, are concurrent, then the value o fk is ( A ) - 2 (B)5 ( C ) - 3 (D)3

Q. 12 Suppose x, y, z is a geometric series with a common ratio of'r' such that x^y . If x, 3y, 5z is an arithmetic sequence then the value of'r1 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 of integers 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 (VTj)2959 is

(A)l ( B ) - l (C)V^T ( D ) - V = l

Q. 16 The points Q = (9,14) and R = (a, b) are symmetric w.r.t. the point (5,3). The coordinates of the point R are

(A) ' 17^ v 2 , (B) (13,25) ( C ) ( l , - 8 ) (D)none

Q. 17 If F (x) = 3x3 - 2x2 + x - 3, then F(1 + /') has the value equal to (A) 8 + 3/ (B) 8 - 3 / (C) — 8 — 3/ ( D ) - 8 + 3/

Q. 18 If x2 + = 7 then the value of x X

(A) 18 (B)21

equals (x>0)

(C) 24 (D) 27

Q. 19 S et of all real x satisfying the inequality ! 4i—1 - log2x | > 5 is, where i = ^ p l .

(A) [4, oo) (B) r i rA i i ( l l

(C) 0, — I 16. (D)

I 16. u [4, °o)

Q.20 Let Xj and X2 are two realnumbers 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 resolution into two rational factors? [3]

Q.2 If one root of the quadratic equation x2 + mx - 24 = 0 is twice a root of the equation x2 - (m + 1 )x + m = 0 then find the value of m. [3]

Q.3 I fx is eliminated from the equation, sin(a+x) = 2b and sin(a-x) = 2c, then find the eliminant. [3]

Q.4 Solve the logarithmic inequality, logj r 2(x - 2) N

,(x + l ) (x-5) , P]

Q.5 Find allx such that ^Tk-x =20. [3] k=l

Q.6 Find the area of the convex quadrilateral whose vertices are (0,0); (4, 5); (9,21) and (-3,7).

P]

Q.7 Find the direction in which a straight line must be drawn through 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. [4]

Q. 8 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 the region between the new circle and the square. If the 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 prove 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 Daily Practice Problems CLASS: XI(P, Q, R, S) DATE: 10/10/2005 TIME: 50Min. DPP. NO.- 49

Q. 1 Identify whether the statement is True 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 than 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 - 8x2 + 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 10 10 10 ( A ) 4 , - y ( B ) - 6 , — (C)6, — ( D ) 6 , - y

A

Q. 4 Each side of triangle ABC is divided into 3 equal parts. The ratio of the area of hexagon UVWXYZ to the area of triangle ABC i s u,

5 2 (A) - (B) j

1 3 (C) 2 (D) 4

Q.5 If cos A, cos B and cos C are the roots of the cubic x3 + ax2 + bx + c = 0 where A, B, C are the angles of a triangle then (A) a2 - 2 b - 2c = 1 (B) a2 - 2b + 2c - 1 (C) a2 + 2 b - 2c = 1 (D) a2 + 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 sometimes 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 APBQ is

a/99 [99 199 (A)V99 (B) - f - (C) ^ f (D) ^

Q.9

Directions for Q.9 to Q. l l : A straight line 4x + 3y = 72 intersect the x and y axes at A and B respectively. Then

Distance between the incentre and the orthocentre of the triangle AOB is

(A)2V6 (B)3V6 (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. l l The radii of the excircles of the triangle AOB (in any order) fonn (A)anA.P. (B)aG.P. (C)anH.P. (D) none

Directions for Q.12 to Q.15: Consider two different infinite geometric progressions with their sums S j and S7 as

S ] = a + ar + ar2 + ar3 + 00 S2 = b + bR + bR2 + bR3 + 00

If Sj = S2 = 1. ar = bR and ar2 = — then answer the following:

Q.12 The sum of their common ratios is

(A) 1

( B ) 4 ( C ) l (D>2

Q. 13 The sum of their first terms is (A) l (B) 2 (C)3 (D)none

Q. 14 Common ratio of the first G.P. is

(A) 1

(B) l - x / 5

(C) V 5 - 1

4 (D)

V5+1

Q.15 Common ratio of the second G.P. is

(A) 3 + V5

(B) 3 - V 5

(C) (D)none

ill BAN SAL CLASSES MATHEMATICS V8Targe t I1T JEE 2007 Daily Practice Problems CLASS: XI (P, Q, R, S) 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. [20 x 1 = 20] There is NEGATIVE marking. For each wrong answer 0.5 mark will be deducted.

Q. 1 If n arithmetic means are between two quantities 'a' and 'b' then the /7th arithmetic mean is

b + na a + nb n ( b - a ) a + nb (A) (B) (C) p 2 (D) -v ' n + 1 v 7 n w n + 1 v ' n + 1

Q. 2 If logab + logbc + logca vanishes where a, b and c are positive reals different than unity then the value of (logab)3 + (logbc)3 + (logca)3 is (A) an odd prime (B) an even prime (C) an odd compo site (D) an irrational number

+ . Q. 3 Sum to n terms of the sequence + ^21 + T>4l

77(3" - 1 ) (C) — ^ 1 (D) none of these

Q.4 Ifthearcsofthe same length in two circles S t and S2 subtend angles 75° and 120° respectively at the

S, centre. The ratio — is equal to

S 2

, 1 81 64 25 (A) J C B ) - ( O - ( D ) -

Q. 5 Ifthe roots of the cubic, x3 + ax2 + bx + c = 0 are three consecutive positive integers. Then the value

a2

of is equal to b + 1

(A) 3 (B) 2 (C) l (D) none of these

2 3 2 3 2 3 Q.6 — + — + — + —r + —r + + 00 isequalto

5 5 5 5 5s 5

15 13 3 4 ( A ) ^ ( B ) ^ ( C ) ? ( D ) ?

Q. 7 Number of princip al solution of the equation tan 3x - tan 2x - tan x = 0, is

(A) 3 (B)5 (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.

1 I. sin765° = - ^ II. cosec(-1410°) = 2

1371 1 m . t a n — = ^ IV. cot

1571 = - 1 4 .

Indicate the correct order of sequence, where'T' stands for true and 'F' stands for false. (A) F T F T ( B ) F F T T ( C ) T F F F ( D ) F T F F

oo 2k+2

Q. 10 The sum ^T —— equal to k=i 3

(A) 12 (B) 8 (C) 6 (D)4

Q.l l The value of p which satisfies the equation 122p_1 = 5(3p -7p) is

/ n 5 - / n l 2 /n l2 + /n5 /n5 + /nl2 /n l2 / n 2 1 - / n l 2 ^ / n l 2 - / n 2 1 / n l 4 4 - / n 2 1 ^ / n l2 -5 /n21

Q. 12 If 0 is eliminated from the equations x = a cos(0 - a ) and y = b cos (0 - P) then

x 2 y2 2xy + JL cos(a - P) is equal to

a bl ab (A) sec2 ( a - P) (B) cosec2 ( a - P) ( C ) c o s 2 ( a - p ) ( D ) s i n 2 ( a - p )

Q. 13 Which of the following is the largest?

(A)21o8s6 (B) 3log65 (C) 3log56 (D)3

Q. 14 The quadratic equation X2 - 9X + 3 = 0 has roots r and s. If X2 + bX + c = 0 has roots r2 and s2, then (b, c) is (A) (75,9) (B) (-75,9) (C)(-87,4) (D)(-87,9)

n i r ^ . t an 2 20°-s in 2 20° . Q.15 The expression T ; simplifies to tan2 20°-sin 2 20°

(A) a rational which is not integral (B) a surd (C) a natural which is prime (D) a natural which is not composite

2024 571 971 Q.16 If sin2x= r r r r , where — < x < — , the value of the sin x - cos x is equal to

2025 ' 4 4 H

Q. 17 If a, b, c are real numbers such that a2 + 2b = 7, b2 + 4c = - 7 and c2 + 6a = - 14 then the value of a2 + b2 + c2 is (A) 14 (B)21 (C) 28 (D) 35

Q. 18 The value of x that satisfies the relation x = l - x + x 2 - x 3 + x 4 - x 5 + 00

(A) 2 cos36° (B) 2 cos 144° (C)2sinl8° (D)none Q.19 If sin 0 and cos 9 are the roots of the equation ax2 - bx + c = 0, then

(A) a2 - b2 ^ 2 a c (B)a2 + b2 = 2ac (C) a2 + b2 + 2ac = 0 ( D ) b 2 - a 2 = 2ac

Q.20 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. [5x2 = 10] There is NO negative marking.

Q.21 Thevalue(s) of 'p' for which the equation a x 2 - p x + 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 + b2 (B) a2 + b ( C ) a ( l + b ) ( D ) b ( l + a )

Q.22 If ax2 + b x + c = 0 , b * l be an equation with integral co-efficients and A > 0 be its discriminant, then the equation b2x2 - Ax - 4 ac = 0 has : (A) two integral roots (B) two rational roots (C) two irrational roots (D) one integral root independent of a, b, c.

Q.23 FortheAP. given by a t , a^, , an, , the equations satisfied are (A) aj+ 2a2 + % = 0 (B) ^ - + % = 0 (C) a, + 3a2 - 3a3 - a4 = 0 (D) aj - + - 4a4 + a5 = 0

V3sin(a + P) r - r T c o s ( a + P) • cosItc 6) Q. 24 It is known that sin P = — and 0 < P < % then the value of v ' ' is:

5 sin a

5

(A) independent of a for all p in (0,7t/2) (B) for tan p > 0

(7 + 24cota) (C) — - — — for tan P < 0 (D) none

Q. 25 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 the fourth and the third term is 54, is (A) 39 (B) -10 .5 (C) 27 ( D ) - 2 7

PART-B

Q. 1 If cos(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 roots of ax2 + bx + c = 0, find the value of (aa + b)~3 + (ap + b)~3. [3]

Q. 3 Find the largest integral value of x satisfying the inequality

log 2 ( 3 - 2 x ) > l . [3]

Q.4 If between any two positive quantities there be inserted two arithmetic means A p A^; two geometric means G t , G2 and two harmonic means Hj, F^, then show that GjG2 : H ,H 2 =A 1 +A2 : Hj + U2.

P] Q. 5 Find all the values of the parameter'm' for which every solution of the inequality 1 < x < 2 is a solution of

the inequality x2 - mx + 1 < 0. [3]

Q, 6 Find the general solution of the equation, sin42x+cos42x = sin 2x cos 2x. [3]

^ ^ I I Q. 7 Find the sum of the series, H + +——— — . [41

1.2.3 2.3.4 n(n + l)(n + 2)

Q. 8 Show that the triangle ABC is right angles if and only if sinA+ sinB + sinC = cosA+ cosB + cosC +1. [4]

Q. 9 Find the real solutions to the system of equations log10(2000xy) - log10x • log10y = 4 log 1 0 (2yz)- log 1 0 ylog 1 0 z=l

and log10(zx)-log10z-log10x = 0. [4]

i l l BANSAL CLASSES MATHEMATICS Target NT JEE 2007 Daily Practice Problems

CLASS: XI (P, Q, R, S) DATE: 26-27/09/2005 TIME: 60 Min. DPR NO.-47

O B J E C T I V E P R A C T I C E T E S T Select the correct alternative. Only one is correct. [3 x 20 = 60] For each wrong answer 1 mark will be deducted.

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 predecessor and two of its sides bisected by its predecessor's sides as shown. This process is repeated indefinitely. The total area enclosed by all the squares in limiting situation, is

5 79 (A) - sq. units (B) — sq. units

75 1 (C) — sq. units (D) — sq. units

1 1 1 1 Q.3 Thesum — — — — — — H — : — — — — — . + + . . is equal to

sm45 sin46° sin47°sin48° sin49°sin50° sin 133°sin 134° M

(A) sec (1)° (B) cosec (1)° (C)cot ( l ) 0 (D)none

8 _ Q.4 Number of real values of x e (0, n) for which — — — ^ 3 sin2x < 5, is

d sin x. sin J X

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

Q.5 If f (x) = x2 + 6x + c, where 'c' is an integer, then f (0) + f (-1) is (A) an even integer (B) an odd integer always divisible by 3 (C) an odd integer not 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 minimum value of a2 + b2 + c2 + d2 + ab + ac + ad + be + bd + cd is

(A) 6 (B) 10 (C) 12 (D) 20

Q.7 Minimum vertical distance between the graphs ofy = 2 + s inxandy = cosx is

(A) 2 (B) l (C)V2 ( D ) 2 - V 2

Q. 8 A square and an equilateral triangle have the same perimeter. Let Abe 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 3 27 3V6 (A). jg (B) - (C) - (D) - f

( l o g ] 0 n ) - l Q.9 Iflog10sinx + log 1 0cosx=- 1 and log10(sinx + cosx)= then the value of 'n'is

(A) 24 (B) 36 (C) 20 (D)12

Q. 10 Let f (x) = x2 +x4 + x6 + x8 + oo for all real x such that the sum converges. Number of real x for which the equation f (x) - x = 0 holds, is (A) 0 (B) 1 (C)2 (D)3

cos 96° +sin 96° Q. l l Find the smallest natural 'n' such that tan( 107n)° = — . — . ^ cos96 - s i n 9 6

(A) n = 2 (B) n = 3 (C)n = 4 (D)n = 5

Q. 12 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 71 571 ( A ) - < B ) 7 ( C ) j ( D ) ~

Q.13 Let a, b, c be the three roots of the equation x3 + x2 - 333x - 1002 = 0 then the value of a3 + b3 + c3. (A)2006 (B)2005 (C)2003 (D)2002

44

cosn

Q. 14 Let x = j y then the greatest integer that does not exceed 1 OOx is equal to 44 Z sin n° n=l

(A) 240 (B) 241 (C) 242 (D)243

Q. 15 The numbers b, c, d are all integers. The parabola y = x2 + bx + c and the line y = dx have exactly one point in common. With these assumption, which one of the following statement is necessarily True? (A) b = 0 (B) d - b is even (C) | a |2 = | b |2 (D)c = 0

Q. 16 The number of solutions to the system of equations y 2 - x y - | x | y + x | x | = 0 and x2 + y2 = 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 to each of the points P and Q in the same way, B is joined to R and 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 (B)y = x ( C ) y = | x ( D ) y = | x

Q.18 Equation of the line XY is (A) 3 x - 4 y = 0 (B)y = x + 1 ( C ) 4 x - 4 y + 3 = 0 (D)none

Q. 19 Radius of the circle inscribed in the triangle APS is

(A) 4 (B) 1 ( C ) j (D) 2

Q. 20 Distance between centroid and circumcentre of the triangle ABC is

J l 3 2 J 1 3 J u J u

J sBANSAL CLASSES 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 A B C

Q.l 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 (10x+12) + 4V2 sin(5x + 6) = 4.

Q.3 If p, q, r be the roots of x3 - ax2 + bx - c = 0, show that the area of the triangle whose sides are p, q & 1

r is — [a(4ab - a3 - 8c) ] m .

t an(a + f3-y) _ tany Q-4 I f tan (a - P + y) ~ tanp ' s h o w t h a t e i t h e r s i n ^ ~ ^ = 0 ' o r ' s i n 2 a + s i n 2 ^ + s i n 2 y = 0

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 the given circle forming a triangle ABC with sides BC = a, AC = b and AB = c. If A denotes the area of the triangle ABC, show that;

a b c a' + b' + c' , „ A ( a ' + b' + c ' ^ CO 77 ~ 17 = 77 = 7T~u~r md (a) a' b' c' a' + b' w A' V a' + b

PPP - 44 Q. 1 If log10(l 5) = a and log20(50) = b then find the value of log9(40)

Q. 2 Find the general and principal solution of the 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. Show that the ratio of the lengths of the bisectors of the interior angles

B & C is , 7 ( V 7 - 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), prove that cot 0 = cot A + cot B + cot C.

J a BANSAL CLASSES MATHEMATICS Target IIT JEE 2007 Daily Practice Problems

CLASS: XI(PQRS) 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, sl, s2 , s3 and the corresponding differences by d j , d 2 , d 3 , prove that,

dj d2 d3 + d, s2 s3 + d? s3 s, + d3 s, s2 = 0;

Q. 2 Find the general solution of the trigonometric equation cosec x - cosec 2x = cosec 4x

Q.3 Let the incircle ofthe A ABC touches its sides BC, CA&AB at A j , Bj & Cj respectively. If p j , p2 & p3 are the circum radii of the triangles, Bj IC j , Cj I A, and A, IB, respectively, then prove that, 2 p, p7 p3 = Rr2 where R is the circumradius and r is the inradius ofthe A ABC.

Q.4 If the area ofthe 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 equation

a b — + + c = 0, then prove that sin(a + P) + sin(P + y) + sin (y + a ) = 0

cosB sinQ

DPP - 46

Q. 1 If dp d2, d3 are diameters of the excircles of AABC, touching 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 circumradius)

Q.3 Find the least positive angle satisfying the equation cos 5a = cos5a.

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 intercepted between the axes.

Q.5 In a A ABC, if cosA+ cosB = 4sin2-y, prove that tan y . t a n ^ = ^ . Hence deduce that the sides of

the triangle are in A.R

J j B A N S A L CLASSES MATHEMATICS v S T a r g e t ||T JEE 2007 Daily Practice Problems CLASS: XI (PQRS) DATE: 09/09/2005 TIME: 60Min. DPP. NO.- 42 Q. 1 If a, b, c are positive real number such that

log a logb logc 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 - 4k2x — k2 + 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 ) ) x 2 + 2 x ( 1 + 1o& ph) - 2 ( ! + p f r ) > 0

is valid for all real x.

1 — — — 8 Q.4 If positive square root of , a* . (2a) 2 a . ( 4 a ) 4 a . (8a)8 a . . . . .... °° is — , find the value of 'a1.

1 2 4 2" 2 1 Q.5 Provethat - + - = — + — + + — = ——r^T"-;

x + 1 x + 1 x 4 + l (x2 +1) 1 - x 2

x Q.6 Find the general solution of the equation (1 + cosx) i j t an—- 2 + sinx = 2 cos x

Q.7 If p,q,r be the lengths of the bisectors of the angles of a triangle ABC from the angular pointsA,Band C respectively, prove that

1 A 1 B 1 C 1 1 1 pqr abc(a + b + c) cos—i— cos —- H— cos — = —l b— and nil ^ ^ = w p 2 q 2 r 2 a b c K > 4A (a + b)(b + c)(c + a)

Q.8 If x, y , z are perpendicular distances of the vertices of a A ABC from the opposite sides and A is the area of the triangle, then prove that

—7 + —r- + -r - = — (cotA + cotB + cotC) x2 y2 z2 Av

ill BANSAL CLASSES MATHEMATICS V S Targe* IIT JEE 2007 Daily Practice Problems CLASS : XI (PQRS) DATE: 05/09/2005 TIME: 60 Min. DPP. NO.- 41

Q. 1 Solve the inequality, ^j\ogy2x + 41og2 Vx < V2 (4 - log^x4).

Q. 2 Find the set of real values of 'a' for which there are distinct reals x, y satisfying x = a - y 2 and y = a - x 2 .

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, (x2 - 1) (x + 2).

Q.4 Find the general solution of the equation

sin6x + cos6x = — . 4

Q.5 If pj, p2 are the roots of the quadratic equation, ax2 + bx + c = 0 and q]5 q2 are the roots of the quadratic equation cx2 + bx + a = 0 such that Pj, qj, p2, q2 is an A.P. of distinct terms, then prove that a + c = 0 where a, b, c e R.

88 1 cosk Q.6 Let k = 1 t h e n prove that T \ ——: 7 — = —x—

^ cosnk • cos(n + l)k sin^k

Q. 7 S olve the equation, r 1 x

2

2 + V2 4 cos 2J J 1 UJ 2 y

cosx

Q. 8 Let al, a2, a3, a4 and b be real numbers such that 4 4

b + X a K = 8 ; b2 + Z 4 = 1 6 K=1 K=1

Find the maximum value of b.

?

J j S A N S A L CLASSES MATHEMATICS ^ B T a r g e t l i t JEE 2006 Daily Practice Problems CLASS: XI (PQRS) DATE: 29/08/2005 Max. Marks: 60 DPR NO.-40

P P P O F T H E W E E K

This is the test paper of Class-XI (J-Batch) held on 28-08-2005. Take exactly 120 minutes.

22 x Q. 1 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 -r + -r. [3] log4(2000)6 log5(2000)

1 1 1 1 Q.3 Prove that . • + ——r~ + ~—T~ + + . = cot x - cot 2nx for any natural number

sin2x sin2 x sm2 x sin2 x n and for all real x with sin 2r x ^ 0 where r = 1,2, n. [3]

Q.4 Let X = sin272° - sin260° and Y = cos248°-sin212° Find the value of XY. [3]

Q.5 If A + B + C = ^ then prove that

£ s i n 2 A + 2 ] ~ [ s i n A = 1- PI

Q. 6 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 ofk for which the graph of the quadratic polynomial P (x) = x2 + (2x + 3)k + 4(x + 2) + 3k - 5 intersects the axis of x at two distinct points. [3]

Q.8 Let u = 10x 3 - 13x2 + 7x and v = l l x 3 - 15x 2 -3 .

du dv Find the integral values of x satisfying the inequality, ™ > ^ • [3]

42. V6

Q. 9 Let a and b are two real numbers such that, sin a+sin b = - y and cos a+cos b = - - - . Find the value of

( i )cos(a-b) and (ii) sin(a + b). [3] Q.10 Let a and b be real numbers greater than 1 for which there exists a positive real number c,

different from 1, such that 2(logac + logbc) = 91ogabc

Find the largest possible value of logab. [5]

Q, 11 Find the product ofthe real roots of the equation

x2 + 18x + 30 = 2a/x2 +18x + 45 [5]

ix

Q.12 If a . p be two angles satisfying 0 < a, P < — and whose sum is a constant k„ find the maximum 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 o f the expressions

(cosec 10° - 73 sec 10°) and (0.5 cosec 10° - 2 sin70°) respectively. Also express the roots of this

(n A quadratic in terms of tangent of an angle lying in ~ . [6]

x + 2 x - 3 Q. 14 If y = —5 then find the interval in which y can lie for every x e R wherever defined. [6]

x + 2x — 8

Q.15 Prove the inequality,

1 1 s inx+ - s in2x+ - s i n 3 x > 0 f o r 0 < x < 1 8 0 ° [6]

4gBANSAL CLASSES MATHEMATICS B Target NT JEE 2007 Holiday Assignment

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 - 37 For 9 = 1°, prove that

2 sin20 + 4 sin40 + 6 sin60 + + 180 sinl 800 = 90 cot0

Q. 2 Find all the solutions of the equation - x) = Vcosx

which satisfy the condition x € [0, 2n]

Q.3 Solve the equation, l+ log x =[log!0(log,0 p ) - l ] logx10-V 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

( 1 + a ) x

VX2 + 1 ,

X 3 a —i H 4 a = 0 have real roots.

x2 + 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 of the first and last terms is 14 and the sum of the inner terms is 12.

DPP - 38 59

Q. 1 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. i

Q.2 Solve for '0' satisfying cos(0) • cos (7t0) = 1.

Q. 3 Find the solution set of the inequality 31x1-2 | x | - l

> 2

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 triangleand so on; in this way (n + 1) circles are described; if r, Xj, x2, , xn be the radii of the circles, show that, r = x t + x2 + x3 + + xn _} + 2 xn.

DPP - 39

Q. 1 If the equation sin4x + cos4x = a has real solutions then find the range of values of'a'. Find the general

1 solution of the equation when a :

2 '

Q.2 Find the complete set of real values of 'a' for which both roots ofthe quadratic equation

( a2 - 6a + 5) x2 - y a2 + 2a x + (6a - a2 - 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 <2 + x - 12.

Q.5 Show that tan f • \ 7 r s i n x

v 4 sin y y + tan ^TtCOSX ^

v 4 c o s y j 7Z TC 7t

> 1 for 0 < x < - and - < y < - . 2 6 i

ill BANSAL CLASSES MATHEMATICS Target I8T JEE 2007 Daily Practice Problems

CLASS:XI(P, Q, R, S) DATE: 10-13/08/2005 DPP. NO.-35

Take approx. 50 min. for each Dpp. DPP - 35

Q. 1 If sec(a - 2P), seca and sec (a + 2p) are in arithmetical progression, show that cos2a = 2 cos2|3 (P & nrc, n e l )

Q. 2 Show that the sum to n terms of the series :

. . sin2(n + l)a.sin2na n . sma cos 3 a + sin 3 a cos 5a + + sm(2n~ l)a.cos(2n+ l ) a = 2sin^a ~ ~2 s a

Q. 3 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.

cos0 1 + sinB 2 Q.4 Solve the equation :—— + — —.

l + sm0 cos0 cos0

Q. 5 Prove that if (ac)'08a b = c 2 , then the numbers logaN, logbN and logcN are three successive terms of an

arithmetic progression for any positive value o f N ^ 1.

x 2 - a x - 2 Q.6 Find all values of ' a ' for which —5 lies between -3 and 2 for all real values of x.

x + x +1

2(a -1 ) 2 Q.7 Solve the inequality for every a e R x < — (x + 1).

a 3a

DPP - 36 Q. 1 Let Aj , ^ , A3 An are the vertices of a regular n sided polygon inscribed in a circle of radius R.

If (Aj A2)2 + (Aj A3)2 + + (Aj An)2 = 14 R 2 , find the number of sides in the polygon.

3 + cosx Q.2 Showthat — — - — V x e R can not have any value between - 2 V 2 and 2V2 . What inference

sinx can you draw about the values of ?

3 + cosx

Q.3 Find the solution set of the equation, log _x2_6 x (sin 3 x + sinx) = log x2_6x (sin 2x). 10 10

Q.4 Find the set of values o fx satisfying the equation sin |x| tan5x = cosx

Q.5 Find the general solution of the equation, tan2(x + y) + cot2(x + y) - 1 - 2x - x2.

Q.6 If p, q are the roots of the quadratic equation x2 + 2bx + c = 0, prove that

2 log (i/y-~i) + \/y~~q) = log 2 + log f y + b + jy2 + 2by + c

Q. 7 Find all real values of x for which the expression Jlogl/2 [ ^ | is a real number.

JaBANSAL V S Target IIT JE JEE 2007

CLASSES

CLASS:XI(P, Q,R,S) DATE: 03-04/08/2005 TIME: 70Min. DPP. NO.-3 4

Q. 1 If (xj, yj) is the solution of the equation, log225(x) + log64(y) - 4 and (x2, y2) as the solution of logx(225) - logy(64) = 1 then show that the value of log30(x1y1x2y2) =12.

Q.2 Let P (x) = x2 + bx + c, where b and c are integer. If P (x) is a factor of both x4 + 6x2 + 25 and 3x4 + 4x2 + 28x + 5, find the value ofP( l ) .

Q.3 Given a, b, c are +ve integer forming an increasing geometric sequence, b - a is a perfect square, and log6a + log6b + log6c = 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 denominator is one less than the square of the numerator. If we add 2 to both numerator & denomenator, the quotient will exceed 1/3 & if we subtract 3 from numerator & denomenator, the quotient will lie between 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 310; the sum of the even terms is 340; 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 of x for which sin x° = sin (xc)

J | BANSAL CLASSES MATHEMATICS y S T a r g e t NT JEE 2007 Daily Practice Problems CLASS: XI (P, Q, R, S) DATE: 01-02/08/2005 TIME: 60 Min. DPP. NO.-33

O B J E C T I V E P R A C T I C E T E S T Select the correct alternative. Only one is correct. [3 x 25 = 75] For each wrong answer 1 mark will be deducted.

Q. 1 Solution set of the inequality, 2 - log, (x2 + 3x) > 0 is: (A) [ - 4 , - 3 ) u (0,1] ' " (B) [ - 4 , 1 ] (C) ( - 0 0 , - . 3 ) U (1 ,00) (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 x3 - px2 - r = 0 are tan a , tan (3 and tan y then the value of sec2a • sec2|3 • sec2y is (A) p2 + r2 - 2rp + 1 (B) p2 + r2 + 2rp + 1 (C) p 2 - r 2 - 2rp + 1 (D)None

^ „ 1 3 7 15 Q. 5 The sum to n terms of the series, 1S equal to:

(A) 2 n - n - 1 (B) 1 - 2 " n (C) 2~n + n - 1 ( D ) 2 n - 1

Q. 6 sinx - cos2x - 1 assumes the least value for the set of values of x given by: (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(x2 - 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 is:

(A)V2 (B)2V2-1 (C)2V2 (D)2

Q.9 If 2x+i) ^ ' 11) ^ 1 are in Harmonical Progression then

(A) x is a positive real (B) x is an integer (C) x is rational which is not integral (D) x is a negative real

Q. 10 The absolute term in tile quadratic expression

t l X 3k+ 1

' 1 ^ x-

A 3 k - 2 y as n —» oo is

2 1 (A) zero (B) 1 (C) - (D)

Q. 11 Given four positive number in A.P. If 5 , 6 , 9 and 15 are added respectively to these numbers, 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 The equation, sin2 9 - — r r — ' = 1 x H . sin3 0 - 1 has :

(A) no root (B) one root sin3 0 - 1

(C) two roots (D) infinite roots

Q.13 The equation (x e R) + 1 — , = x : 4 Vx?' - i

(A) has no root (B) exactly one root (C) two roots (D) four roots

Q.14 If x s in9 = y c o s 9 then —L— 4 — is equal to sec29 cosec29

" (A) x ( B ) y ( C ) x 2 (D)y 2

Q. 15 An H.M. is inserted between the number 1 /3 and an unknown 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 ) m i + j (B)

(C) mt + or 2 nn±n (D) ( 4 n + l ) ^ or (2n+l)n

where n e I

Q. 18 If x2 + Px + 1 is a factor of the expression ax3 + bx + c then (A) a2 + c2 = - ab (B) a2 - c2 = - ab (C) a2 - c2 = ab (D) none of these

Q. 19 The expression (tan49 + tan29) (1 - tan2 39 tan29 ) is identical to (A) 2 cot 39 . sec29 (B) 2 sec 39. tan29 (C)2tan39. sin29 (D) 2 tan39. sec29

x 2 ~ 3 x + c i Q.20 If the maximum and minimum values of y= are 7 and — respectively then the value of

X "i" i X H~ C /

c is equal to (A) 3 (B)4 (C)5 (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

(C) 2nTX -n

(D) 2mc + 7t

6

Q. 22 If the 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 is: (A) rp/q (B) r^P ( C ) r P ^ (D)rP + i

Q.23 The sum ofthe roots ofthe equation (x + 1) = 2 log2(2x + 3) - 2 log4(l 980 - 2"x) is (A) 3954 (B)log2ll (C)log23954 (D) indeterminate

Q.24 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:

(A) 'it 57r" r57i l b O f 5n llTl")

W 6 , u

U ' 6 J (B)

U ' 6 J /

(C) v

71 7T

6 ' 2 1 (5n 3T^

— —

J 1 4 2 J (D) V 6 /

u 571 5tc T ' T

u In

,2n

Q.25 Solution set of the inequality log3 x - log? x < : log, .4 is

(A) [3,9] (B) (H %/2j2)

u[9,oo) ( C ) f - o o , i u[9 , oo) (D) u (1,9]

J j B A N S A L CLASSES MATHEMATICS v B Target I IT JEE 2007 Daily Practice Problems CLASS: XI (P, Q, R, S) DATE: 28/07/2005 TIME: 60 Min. DPP. NO.-32

O B J E C T I V E P R A C T I C E T E S T Select the correct alternative. Only one is correct. [3 x 25 = 75] For each wrong answer 1 mark will be deducted.

Q. 1 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 (B) 2 (C) V2 (D) 2V2

Q.2 If 3 + ^ (3 + d) + ^ y (3 + 2d) + + upto 00 = 8, then the value of d is :

(A) 9 (B) 5 ( C ) l (D) none of these

Q.3 If in a A ABC, sin3A + sin3B + sin3C = 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 ^ + » + ^ + ^ is equal to (A) 4 (B) 6 (C)8 (D)2

Q.5 The set of angles btween 0 & 2n satisfying the equation 4 cos2 0 - 2 - ^ 2 cos 0 — 1 — 0 is

J j L ^L 2 3 n ] ! JL l2L ]2JL 2 3 n

( A ) I12 ' 12 ' ~12 ' T T j ( B ) 1 12 ' 12 ' 12 ' 12 f ^L l^ZL 1 9 7 t ] f 7C 1% 1771 2371

( C ) I 1 2 ' 12 ' 12 J

1 1 1 1 1 1 Q.6 Let x= + + + 00 and y = ~ + 7TT + T T 4 " °° then

1-4 4.7 7.10 ' 1-2 2-3 3-4

(A)y = 3x (B)y = 2x (C)x + >>=1 (D)x+y=^

Q.7 If cos a = ""cosP—- then tan ^1 cot^- has the value equal to, where(0 < a < n and 0 < B < 71) 2 - cosp 2 2

(A) 2 (B)V2 (C) 3 (D) Vs

Q.8 2m. 41/8. 81/16.161/32. 321/64 00 is equal to (A) 2 (B) 1 (C) 1/2 (D) 1/4

Q.9 If xsin0 = y s i n | e + y j = z sin^e + then:

(A) x + y + z = 0 (B)xy + yz + zx = 0 (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 xth arithmetic mean is (A) I- x (B)l+x (C)x2 — x + 1 (D)x

Q. 11 If a cos3 a + 3a cos a sin2 a = m and a sin3 a + 3a cos2 a sin a = n . Then (m + n)2/3 + (m - n)2/3 is equal to : (A) 2 a2 (B) 2 a1/3 (C) 2a2 / 3 (D) 2 a3

Q. 12 Consider the A.P. a t , s^ , , a n ; the G.P. b>, b2 , , bn

9 such that a1 = bj = 1 ; a9 = b9 and ]!Tar = 369 then

r = l (A) b6 = 27 (B) b? = 27 (C)bg = 81 ( D ) b 9 = 1 8

Q. 13 If tan A & tan B are the roots of the quadratic equation x2 - ax + b = 0, then the value of sin2 (A + B) is:

a2 a2 _ a2 a2

(A) , « (B ( Q - ^ - T (D) a2 + (1-b)2 V a2 + b2 (b + a)2 b2 (1 - a)2

Q. 14 If a, b, c are distinct positive reals in G. P., then; loga n , logb n , logc n (n > 0, n * 1) are in: (A) A. P. (B) G. P. (C)H.P. (D) none

A Q.15 If A=340° then 2 sin — is identical to v 2

(A) Vl + sinA + Vl - sin A (B) - V 1 + sinA - V l - s i n A

(C) Vl + sinA - -\/l - sin A ^ D ) - Vl + sinA + V l - s i n A

Q.16 Consider a decreasing G.P.: g1 ,g2 ,g3 , gn such that g1 + g2 + g3 = 13 and gj + g 2 + g 3 = 9 1

then which of the following does not hold? (A) The greatest term ofthe G.P. is 9. (B) 3g4 = g3

(C)g, = l (D)g2 = 3

*\/3 +1 "J3 Q. 17 Number of roots of the equation cos2 x + — - — sinx - — - 1 = 0 which lie in the interval [-71, tt] is

(A) 2 (B)4 (C) 6 (D) 8

Q.18 The sum ofthe first three terms of an increasing G.P. is 21 and the sum oftheir squares is 189.Then the sum of its first n terms is

r 1 \ 1

V 2 / (A) 3 ( 2 n - 1) (B)12 1 - ^ r (C)6 1 - ^ r ) ( D ) 6 ( 2 « - l )

1

\

Q.19 Ifs in (6 + a ) = a & sin(G + p) = b (0 < a , p9 0 < tc/2) then cos2 (a - (3) - 4 ab cos(a - P) = (A) 1 - a2 - b2 (B) 1 - 2a2 - 2b2 (C.) 2 + a2 + b2 ( D ) 2 - a 2 - b 2

Q.20 J . f S = 4 - + - r — r + + * r \ ' - , n = 1,2, 3, Then S„ is not greater than V n l3 1 +2 1 +2 +3 + + n n S

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

Q.21 The exact value of cos273° + cos247° + (cos73°. cos47°) is (A) 1/4 (B) 1/2 (C)3/4 (D) 1

Q.22 Let Sj , S2 , S3 be the sums of the first n , 2n and 3n terms of an A.P. respectively. If S3 = C (S2 - S,) then, 'C' is equal to (A) 4 " (B)3 (C)2 (D) l

\

V 9 e R, is Q.23 Maximum value of the expression cos6 • sin ® v 6 y

1 V3 1 (A) j ( B ) ^ (C) 4 (D) l

Q.24 The value of the expression (sinx + cosecx)2 + (cosx + secx)2 - (tanx + cotx)2 wherever defined is equal to (A) 0 (B)5 (C)7 (D) 9

Q.25 The roots of the equation 2 + cotx = cosec x always lie in the quadrant number (A) I only (B) I and II (C) II and IV (D) II only

i k BANSAL CLASSES MATHEMATICS ^STa rge t IIT JEE 2007 Dally Practice Problems CLASS: XI (P, Q, R, S) DATE: 24/07/2005 TIME: 60 Min. DPP. NO.-31

O B J E C T I V E P R A C T I C E T E S T Select the correct alternative. Only one is correct. [3 x 25 = 75] For each wrong answer 1 mark will be deducted.

J (x -8 ) (2-x) Q.l The set of values of x satisfying simultaneously the inequalities t— -y > 0 and

iogo.3 ("T (log2 5 - 1))

2X - 3 - 31 > 0 is : (A) a unit set (B) an empty set (C) an infinite set (D) a set consisting of exactly two elements.

Q.2 The roots of the equation (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 progression.

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 sector in radians is: (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 x2 - 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 log2x • log,xyz =10 log2y-log2xyz = 40 log2z • log2xyz = 50 where x > 0 ; y > 0 ; z > 0

then which of the following inequalities may be true? ( A ) x < y < z & z < x < y ( B ) x > y > z & z < x < y ( C ) x > y > z & x < y < z (D) x > z > y & z < x < y

Q 7 The quadratic equation whose roots are the A.M. and H.M. between the roots 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/2unitup, l/4unittotheright, 1/8 unit down, 1/16 unit to the right etc. The length of each move is half the length of the previous move 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 s U CJ 1/16 1 " v

0 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 - 12x+ 1 = 0

1 00 1 00 Q. 11 Let an be the nth term of a G.P. of positive numbers . Let X a2n = a & X = P such that

n = 1 n n = 1

a * p. Then the common ratio of the G.P. is :

W f C B ) £ ( C ) j f ( D ) j i p a \j p V a

Q. 12 Given a sequence a p a2, a3, an, in which the sum of the first m terms is Sm = m2 - 5m then which of the following is not true? (A) a5 = 0 (B) a5 = 4 (C)a6 = 6 (D)i t i sanAP.

Q.13 Given l°g2x + l°S4y + l ° g 4

z = 2 log3y + log9z + log9x = 2 log4z + log16x + log16y = 2

then which ofthe following is true? (A) y > z (B) x > y ( C ) x > y > z ( D ) x < y < z

Q. 14 The number of integral values of m, for which the roots of x2 - 2mx+m2 - 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 formed by the length of its altitudes. This process is repeated. This procedure being repeated n times. The limit of the sum of areas of all the triangle as n —> oo is

(A) 3a2 (B)V3a 2 (C) 2a2 ( D ) V ^ a 2

Q.16 Let y = r • • \2/3

2sinx + sin2x 1 - c o s x The interval in which y can lie for V x e R .

v 2 c o s x + s i n 2 x 1 - s i n x ^

(A) [l,oo) (B)(-a,, co) ( C ) [ 0 , 1 ] (D)[0,oo)

Q. 17 A horse is teethered to a stake by a rope 9m long. Ifthe horse moves along the circumference of a circle always keeping the rope tight then the distance traversed by the horse when the rope has raced an angle of 70°, is (Assume n = 22/7) f (A) 7 m ( B ) 9 m ( C ) l l m (D)22m

Q.18 If x g R, the numbers ( 51+x + 51"*), a/2, (25x + 25'x) form an A.P. then 'a' must lie in the interval (A) [6,o)) (B)[12,oo) (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 term then the 18th term of the A.P. is (A) 0 (B) 7 (C) 11 (D) 18

Q.20 ABC is a triangle such that, sin (2A+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) none of these

3 2 5

Q.21 Set ofintegral solution of the equation x +iog2X 4 _ ^ js

(A) 1 (B) 2 (C) 3 (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 (B) V? (C) l (D) none

. 1025 Q.23 If 10 i m i - P anc* ^ = Q then the value of log10 4100 in terms of p and q is equal to

(A)p + 9q (B) p + lOq (C)12p + q (D)p + 12q 2

Q.24 Number ofvalues of x satisfying the equality 2cos2x = 3.2C0S x - 4 and the inequality x2 < 30 is (A) 0 (B) 1 (C) 2 (D) 3

Q.25 Let a, P, y be the roots of the equation x3 + 3 ax2 + 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 CLASSES MATHEMATICS v B Target IIT JEE 2007 Daily Practice Problems CLASS: XI (P, Q, R, S) DATE: 18/07/2005 TIME: 60Min. DPP. NQ.-30

O B J E C T I V E P R A C T I C E T E S T Select the correct alternative. Only one is correct. [3 x 25 = 75] For each wrong answer 1 mark will be deducted.

Q. 1 If a > b > 0 are two real numbers, the value of,

ib + (a - b) -,/ab + (a - b) ^/ab + (a - b) -v/ab + is :

(A) independent of b (B) independent of a (C) independent of both a & b (D) dependent on both a & b .

Q. 2 If a , P are the roots of the equation ax2 + bx + c = 0 , then the roots of the equation ax2 + bx(x+l) + c (x +1)2 = 0 are

(A) a - 1 , 0 - 1 (B) a + 1, P + 1 (C) ~ , ^ (D) ^ , ^

VQ 3 The number of the integral solutions for the equation x+2y ~2xy is (A) 2 (B) 1 (C) 4 ' (D) infinitely many

^ Q.4 Given a, b, c are non negative real numbers and if a2 + b2 + c2 = 1, then the value of a + b + c is

(A) > 3 (B) > 2 ( C ) < V 3 (D)< V2

Q. 5 The value of (0.2)!°S^ & + * + " + i s equal to (A) 2 (B) 4 (C) 5 (D) none

Q. 6 The maximum value of the sum of the A.P. 50, 48, 4 6 , 4 4 , is (A)325 (B)648 (C)650 (D)652

Q. 7 Solution set of the inequation, glo§2x - 2x2 > 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 roots for all 8 (C) complex roots for all 0 (D) real or complex roots depending upon 0

Q. 9 If the roots of the quadratic equation ax2 + bx + c = 0 are imaginary then for all values of a, b, c and x e R , the expression a2x2 + abx + ac is (A) positive (B) non - negative (C) negative (D) may be positive, zero or negative

Q. 10 If \ + -V + 4" + u P t 0 00 = t h e n TT + \ + ~7 + equal to l2 22 32 6 I2 3 5

( A ) ~ (B) (C)~ (D) none

Q.l l 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)432 (D)none

Q.12 The sum ofthe first 100 terms common to the series 17,21,25, and 16,21,26, is (A) 101100 (B) 111000 (C) 110010 (D) 100101

Q.13 If x,y,z e N then the number of ordered triplets of (x,y, z) satisfying the equation x + y + z=102is (A) 4950 (B) 5050 (C) 5150 (D)None

Q. 14 Consider an A.P. with first term 'a' and the common difference d. Let Sk denote the sum of the first K

SjQJ terms. Let "77" is independent of x, then

(A) a = d/2 (B) a = d (C) a = 2d (D)none

Q. 15 If p, q, r in harmonic progression and p & r be different having same sign then the roots ofthe equation, px 2 + qx + r = 0 are : (A) real and equal (B) real and distinct (C) irrational (D) imaginary.

Q. 16 The numbers 1 , —-— & — - — consitute log32 log62 j log122

(A) an A. P. (B)aG.P. (C)aH.P. (D)None

Q. 17 If log2, l og (2 x - 1) and log(2X +3) are in A.P,, then xis equal to : 1 (A) 5/2 (B) log3 2 (C) log2 5 (D)none

Q. 18 Complete set of the values of x satisfying the inequality, log7 x - logx(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) sin2x - (k + 3) = 0 has a solution then k must lie in the interval: (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 of the terms of the progression is:

(A) — ( B ) S 2 (C) (D) S2 V ; 2 S - 1 V ; 2 S - 1 2 - S K '

sin(a + 0) - sin(a - 0) Q.21 The expression c o s ( p _ e ) _ c o s ( p + e ) is

(A) independent of a (B) independent of |3 (C) independent of 0 (D) independent of a and p

Q. 22 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 ofthe 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 V 3 - 1 • S - l _ 3(V3 - l)

Q.25 The values ofx smaller than 3 in absolutevalue which satisfy the inequality log(2a_x2 ( x - 2 a x ) > 1 for

a l l a > 5 is (A) - 2 < x < 3 (B) - 3 < x < 3 ( C ) - 3 < x < 0 ( D ) - 3 < x < - l

fi BAN SAL CLASSES Target IIT JEE 2007

MATHEMATICS Daily Practice Problems

CLASS : XI (P, Q, R, S) DATE: 16/07/2005 TIME: 60Min. DPR NO.-29

[3 x 25 = 75] O B J E C T I V E P R A C T I C E T E S T

Select the correct alternative. Only one is correct. For eacii wrong answer 1 mark will be deducted.

a 2 p2

Q. 1 If a , p are the roots of the equation ax2 + 3x + 2 = 0 (a < 0) then — + — is

(A) > 0 (B) > 1 (C) < 1 ( D ) < 0

Q.2 The value of f (x) = x2 + (p - q)x + p2 + pq + q2 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

For an increasing A.P. a,, a2, a3 ,an,.... if a! + a3 + a5 = - 12 ; a,a3a5 = 80 then which of the following does not hold?

Q.3

(A) a,= - 10 (B)a2 1 (C) a, = - 4 (D) a, = 2

Q. 4 The solution set of the inequality log< o 2 3 2x - x > 1 is

(A)

(C)

1 1 u

2 ' 4 u

U )

(B)

(D) - c o . \ f 3 ^

u J T ' 0 0

J

Q.5 If a , p are the roots of the quadratic equation (p2 + p + l)x2 + (p - 1 )x + p2 = 0 such that unity lies between the roots then the set of values of p is (A)* ( B ) p e O , - l ) U ( 0 , o o ) (C)p e ( -1 ,0 ) (D) ( -1 ,1)

Q.6 If cos9 + cos(|) = a and sinB + sincj) = b, then the value of cos9-cos(|) has the value equal to

I 2 - 4 a 2 | ( a 2+ b 2 ) ! 2 - 4 b 2 1 ( a 2 + b 2 )

2 1 - 4a2 | ( a 2 + b 2 ) |2 - 4 b 2

41 hb2) 2 ( a 2 + b 2 ) I ^ 2 ( a 2 + b 2 ) (U) " 41 ( a 2 + b 2 ) 1 (A)

Q.7 If both roots of the equation (3X + l)x2 - (21 + 3 p)x + 3 = 0 are inifinte then

1 2 1 (A) A, = p

1 3 (C)X = - - ; p = - (D)?i = --;n = - -

SPACE FOR ROUGH WORK

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)Vl6 (B)V64 (C)V80 (D) none of these

<x Q. 10 If the quadratic equation 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.l l The set of values of'p' for which the expression x2 - 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 5 l o g " X + 1 + 5 l o g o 25 X " 1 = y has

(A) no integral solution (B) only one rational solution (C) one irrational solution (D) two real solutions

Q. 13 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 cos20 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 log215 . log | /f)2. log31/6 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) None

Q.17 Given a2 + 2a + cosec2 ~ ( a + x) = 0 then, which of the following holds good? V.2 y

(A)a = l ; | e l (B)a = - l ; | e l

(C) a e R ; x e^ (D) a, x are finite but not possible to find SPACE FOR ROUGH WORK

Q.18 If a, P are the roots of the equation, x2 + (sin<j)- l)x— — cos2{j) = 0 then the maximum value of the

sum ofthe squares of the roots is : (A) 4 (B) 3 (C) 9/4 (D) 2

Q. 19 If In2 x + 3 In x - 4 is non negative then x must lie in the interval: (A) [e, co) (B) (-oo, e~4)u[e, °o) (C) ( l / e , e ) (D) none

Q.20 If the quadratic polynomial, y = (cot a)x2 + 2 (Vsin 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 , it) (B) a

Q. 21 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

X

P.T.O.

X

Q.22 The value of the expression, log4

(A) - 6 (B) - 5 v 4 y

- 2 log4 (4 x4) when x = - 2 is :

(C) - 4 (D) meaningless

1 Q.23 If x, and x2 are the roots of the equation x2 + px - = 0 , (p e R ) then the minimum value of

xj1 + x2 is equal to

(A)V2 ( B ) V 2 ( 2 - V 2 ) ( C ) 2 + V ^ (D) 2 + 2^/2

Q. 24 Number of integral values of x satisfying the inequality

(A) 6 (B) 7 (C) 8

{ ^ \6x+10-x2

v4y 27 .

< — is 64

(D) infinite

a a +1 Q.25 If the roots of the equation ax2 + bx + c = 0 are real and of the form r and then the value of

a - 1 a (a + b + c)2 is (A) b2 - 4ac (B) b2 - 2ac (C) b2 + 4ac (D) b2 + 2ac

X - X

Class - XI

Roll No.

ANSWER KEY Date: 16-07-2005 Max. Time: 1 Hr.

Only alternative is correct. There is NEGATIVE Marking. For each wrong answer 1 mark will be deducted.

Max. Marks: 75

[25x3 = 75]

A B c D A B c D • 1 O o o o 14 o o o o 2 O o o o 15 o o o o 3 o o o o 16 o o o o 4 o o o o 17 o o o o 5 o o o o 18 o o o o 6 o o o o 19 O o o o 7 o o o o 20 O o o o 8 o o o o 21 O o o o 9 o o o o 22 O o o o 10 o o o o 23 O o o o 11 o o o o 24 o o o o 12 o o o o 25 o o o o 13 o o o o •

X

J j B A N S A L CLASSES MATHEMATICS y S T a r g e t IIT JEE 2007 Daily Practice Problems CLASS: XI (P, Q, R, S) DATE: 08-09/07/2005 TIME: 40 Min. DPP. NO.-27

Select the correct alternative : (Only one is correct) Q. 1 Which of the statement is false.

(A) If 0 < p < n then the quadratic equation, (cos p - 1) x2 + cos px + sinp = 0 has real roots. (B) If 2a + b + c = 0 (c * 0) then thequadratic equation, ax2 + bx + c = 0 has no root in (0,2). (C) The necessary & sufficient condition for the quadratiic function f(x) = ax2 + bx + c to take both positive & negative values is, b2 > 4ac. (D) The sum of the roots of the equation cos2x = 1 which liei in the interval [0,314] is 49 5071.

Q.2 For every x e R, the polynomial x8 - x5 + x2 - x + 1 is :

(A) positive (B) never positive (C) positve as well as negative (D) negative

Q.3 If x ] 5 x 2 & x 3 are the three real solutions of the equation;

x l o g 20 x + l o g 1 0 x 3

+ 3 = 2 w h g r e X ] > X 2 > X 3 ; t h £ n

^x+T-i ,/x+i+i

(A) Xj + x3 = 2 x2 (B) x , . x3 = x22 (C) x2 = 2 X l * 2 (D) x f 1 + x f 1 = x f 1

X, + x2

Q.4 If exactly one root of the quadratic equation x2 - (a +1 )x + 2a = 0 lies in the interval (0,3) then the set of values 'a' is given by

(A) (-co , 0) u (6,oo) ( B ) ( - o o , 0 ] u ( 6 , o o )

(C) (-00 , 0] vj [6,oo) CD)(0 ,6 )

Q.5 Three roots of the equation, x4 - px3 + qx2 - rx + s = 0 are tan A, tanB & tan C where A, B, C are the angles of a triangle. The fourth root of the biquadratic is :

( A ) - P ^ _ ( B ) - ^ - ( C ) - ^ - (D) —PJlI— 1 - q + s 1 + q - s ' 1 - q + s y 1 + q - s

„ , . . sin 8x cos x - sin 6x cos 3x n . Q.6 The value of the expression when x = — is sin 3x sin 4x - cos x cos 2x 24

( A ) V 3 - 2 ( B ) ^ ( C ) V 2 - 1 ( D ) V 2 + 1

Q. 7 If the roots ofthe 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 known to hold for y = ax2+bx + c then the least value of 'a' is :

(A) - 1 / 4 (B) - 1 / 3 (C) 1/4 (D) 1/8

Q.9 Number of ordered pair(s) satisfying simultaneously, the system of equations,

2 r x + f y - 2 5 6 & log10A/xy - log 1 0 1 .5 = 1, i s : (A) zero (B) exactly one (C) exactly two (D) more than two

Q. 10 Find the values of 'a' for which one of the roots of the quadratic equation,x2 + (2 a + 1) x + (a2 + 2)=0 is twice the other root. 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 Daily Practice Problems CLASS: XI (P, Q, R, S) DATE: 11-12/07/2005 TIME: 40 Min. DPP. NO.-28

Select the correct alternative: (Only one is correct)

Q.l If a, b, p, q are non-zero real numbers, the two equations, 2 a2 x2 - 2 ab x + b2 = 0 and p2 x2 + 2 pq x + q2 = 0 have : (A) no common root (B) one common root if 2 a2 + b2 = p2 + q2

(C) two common roots if 3 pq = 2 ab (D) two common roots if 3 qb = 2 ap

Q. 2 The equations x3 + 5x 2 +px + q = 0 and x3 + 7 x 2 + p x + r = 0 have two roots in common. If the third root of each equation is represented by Xjand x2 respectively, then the ordered pair (x15x2) is: (A) ( - 5 , - 7 ) ( B ) ( 1 , - 1 ) ( C ) ( - 1 , 1 ) (D)(5 ,7)

Q.3 If the roots of the quadratic equation x2 + 6x + b = 0 are real and distinct and they differ by atmost 4 then the range of values of b is : (A) [ - 3 , 5 ] (B) [5,9) (C) [6,10] (D) none

Q.4 The expression sec4x - 4 tan3x + 4 tanx is always : (A) positive (B) negative (C) non-positive (D) non-negative

Q.5 The value of the biquadratic expression, x 4 - 8 x 3 + 1 8 x 2 - 8 x + 2 when x = 2 + V3 is (A) 1 (B) 2 (C) 0 (D) none

Q.6 If one root o f the quadratic equation px2 + 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 J a ( a - b ) a + , / a ( a - b ) Va - V a - b

Q. 7 The minimum value of cos (cos x) for every x e R is :

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

Q.8 The area of the circle in which a chord of length 2a makes an angle 9 at its centre is

(A) 7i a2 c o t 2 | (B) 2 7i a2 (l + cot21) ( Q % a2 (l + cot2 | J (D) 4 71 a2 (l + cot21)

Q.9 The equation a sinx + cos2x = 2a - 7 has a solution, if

(A) a > 2 (B) a < 2 ( C ) 2 < a < 6 ( D ) a < 2 o r a < 6

Subjective:

Q. 10 Solve the inequality, log2x (x2 - 5x + 6) < 1.

JjBANSAL CLASSES MATHEMATICS V B Target IIT JEE 2007 Daily Practice Problems CLASS: XI (P, Q, R, S) DATE: 06-07/07/2005 TIME: 40 Min. DPP. NO.-26

Select the correct alternative : (Only one is correct)

x2 + 2x + c Q. 1 If x is real, then —- can take all real values if: x + 4x + 3c

(A) 0 < c < 2 (B) 0 < c < 1 (C) - 1 < c < 1 (D) none

27c 3TT 671 9n 1 8TC 27 n Q.2 T h e exac t va lue o f c o s — cosec— + c o s — cosec— + c o s — cosec—— isequalto

28 28 28 28 28 28 (A) -1 /2 (B) 1/2 ( C ) l (D) 0

Q.3 If a,b,c are real numbers satisfying the condition a + b + c = 0thenthe roots ofthe quadratic equation 3ax2 + 5bx + 7c = 0 are : (A) positive (B) negative (C) real & distinct (D) imaginary

3 log _ 3

8l'°g59+3 ^ f l r-\ 2 , Q.4 Let, N = — . (V7 H 7 -125 log25 6 then log2N has the value =

y 409 (A) 0 (B) 1 (C) - 1 (D) none

^ , „ A . 271 . 47t . 8rc 2n 4TT 8TC Q.5 It A = sin — + s i n — + s m — and B = cos — + c o s — + c o s —

then 7 a 2 +B 2 e cIu a l to

(A)l (B) V2 (C)2 (D) V 3

x - 3 x - 2 Q.6 If — — - < then the most general values are :

ZX ~ I X 4" 4

(A) ( x < - 4 ) ( B > ( x > { l ( C ) ( " 4 < X < 1 ) (D) ( x < - 4 ) u ( x > |

Q.7 The equation | s i n x | = s inx+ 3 has in [0,2 71] : (A) no root (B) only one root (C) two roots (D) more than two roots.

Q. 8 The number of solution of the equation, log(- 2x) = 2 log (x + 1) is :

(A) zero (B) 1 (C) 2 (D) none

Q. 9 If A and B are complimentary angles, then :

(A) [l + tan | - ] [ l + t a n | j = 2 (B) [ l + c o t ^ j ( l + cot|j = 2

(C) + s e c y j ( l + c o s e c | j = 2 (D) f l - tan~j f l - t a n | j = 2

Subjective:

u 3 - u 5 U 5 - U 7 Q.10 I fu = sin"0 + cosn0, prove that = ~ n ^ u, u3

i l l BANSAL CLASSES MATHEMATICS ™ T a r g e t l i t JEE 2007 Daily Practice Problems CLASS: XI (P, Q, R, S) DATE: 04-05/07/2005 TIME: 40 Min. DPR NO.-25

Fill in the blank : Q. 1 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 x2 - 4x + 5 - sin y = 0, y e (0, 2n) then x = & y = .

Q.3 If the vectors, p =(log2x) i — 6 j — k and q =(log2x) i + 2 j +(log2x) k are perpendicular to each other, then the value of x is .

Select the correct alternative : (Only one is correct)

Q.4 The equation, 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 equation 25x2 + 5x - 12 = 0, - 1 < x < 0, then the value of sin 2a is : (A) 12/25 (B) -12 /25 (C) -24 /25 (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 + b2) - 1 holds true for all x e R are : (A) 1 (B) 2 (C) 3 (D) 4

Q.7 Let y = cos x (cos x - cos 3 x) . Then y is : (A) > 0 only when x > 0 (B) < 0 for all real x (C) > 0 for all real x (D) < 0 only when x < 0

x Q. 8 For V x e R , the difference between the greatest and the least value of y = 2 ^ is

(A)l (B)2 (C)3 ( D ) |

Q.9 In a triangle ABC, angle A = 36°, AB = AC = 1 & BC = x. If x = t h e n t h e o r d e r ed pair

(p, q) is :

( A ) ( l , - 5 ) (B) ( 1 , 5 ) ( C ) ( - l , 5 ) (D) (-1 , - 5 )

Subjective: n

Q.10 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 BANSAL CLASSES M A T H E M A T I C S V S Target I IT JEE 2007 Daily Practice Problems CLASS:XI(P, Q,R, S) DATE:20-21/06/2005 TIME:40Min. DPR NO.-22 Take approx. 40 min. for each Dpp.

PPP - 22 Q.l If 0 is eliminated from the equations asec0-xtan0=y and bsec0+ytan9=x then find the relation

between x and y, where a, b are constants. 2TT 4TI .6% 7 r . 3 7 i . 5 7 t

Q.2 Provethat: s in— + s in— - s i n - y = 4 s i n y s in— s in—

Q.3 If A, B, C denote the angles of a triangle ABC then prove that the triangle is right angled if and only if sin4A + sin4B + sin4C = 0.

1 1 1 1 Q.4 Solve the inequality: - — ^ —

X T I X X I Z

Q.5 Let p & q be the two roots ofthe equation, mx2 + x (2 - m) + 3 = 0. Let m t , m2 be the two values of m p q 2 nil m2 satisfying— + — =-.Determine the numerical value of — + —j . q p 3 m2 m i

17 Jgj Q.6 Find the value ofthe continued product ]~ [ s in—

k=i 1 8 $ $ $$ * # * * * * * * * * * * * # # * * * * * * * *

PPP - 23 Q.l If 15 sin4a + 10cos4a = 6, evaluate 8cosec6a + 27sec6a

Q.2 Prove that the function y = (x2 + x + l)/(x2 + 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

(tan2135°)x2 - (cosecl0° - V3 secl0°)x + tan2240° = 0 then prove that the quadratic equation whose roots are (2a + (3) and (a + 2P) is x 2 - 12x + 35=0.

Q.4 John has 'x' children by his first wife. Mary has x + 1 children by her first husband. They many and have children of their own. The whole family has 24 children. Assuming that the children ofthe same parents do not fight, find the maximum possible number of fights that can take place.

Q.5V Solve the following equation for x, 3x3 = [x2 + Vl8 x + a/32] [x2 - Vl8 x - V32] - 4x2, where x e R.

Q.6** If cosA = tanB, cosB = tanC and eosC=tanA, then prove that sinA = sinB = sinC=2 sin 18°. ** * * * t *** *** **** * * * * *

PPP - 2JU Q. 1 Find the minimum value of the expression 2 log10x - logx0.01 ; where x > 1.

Q.2 If x,y,z be all positive acute angle then find the least value of tanx (cot y + cot z)+tany (cot z + cot x) + tanz (cot x + cot y) r, ,, . ... sinx - 1 , 1 . 2 - sinx w „ Q.3 Prove the mequality + - > V x e R.

smx ~ 2 2 3 - sinx

Q.4 Prove that: 5 sin x = sin(x + 2y) =>2 tan(x + y) = 3 tan y.

Q.5 If cos 0 + cos <b = a and sin 0 + sin <b = b, prove that: tan—+tan— = „ ^v . 2 2 a + b + 2a

Q.6 Find the sum of (n - 1 ) terms of the series: . it . 2% 371 _

sin— + s i n — + s in— + Deduce the value of n if this sum is equal to 2 + J 3 . n n n n y

« a b a n s a l c l a s s e s Target IIT JEE 2007

MATHEMATICS Daily Practice Problems

CLASS: XI (P, Q, R, S) DATE: 17-18/06/2005 TIME: 40Min. DPP. NO.-21

Q.l

Ov)

(v) (vi)

Q.2

Q.3

Q.4

Q.5

Q.6

Q.7

Q.8

Q.9

Identify whether the statement is True or False. tan2 x sin2x — tan2 x - sin2x

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

(iii) If tan A VI & tanB = VI then tan (A - B) must be irrational. 4 - ^ 3 4 + S

If tanA = 1, tanB = 2 and tanC = 3 then A, B, C can not be the angles of a triangle.

If t anA= 1 c° sB , then tan2A=tanB. sinB '

There exists a value of 0 between 0 & 2n which satisfies the equation, sin4 0 - sin2 0 - 1 = 0 .

Select the correct alternative : (More than one are correct)

If x = sec (|) - tan <|) & y = cosec <j) + cot <j> then : ( B ) y - — (C) x = - — - (D) xy + x - y + 1 = 0

1 - x ' 1 (A) x = Z j t i w y - l v ' " l - x N ' y + l

If the sides of a right angled triangle are {cos2a + cos2p + 2cos(a + P)} and {sin2a+ sin2p + 2sin(a + p)}. then the length of the hypotenuse is:

(A)2[ l+cos (a -P) ] ( B ) 2 [ l - c o s ( a + P)] ( C ) 4 c o s 2 - ^ ^ (D)4sin2-a + f3

Which of the following functions have the maximum value unity ? sin2x - cos2x (A) s i n 2 x - c o s 2 x

( Q _ s i n 2 x - cos2x

(B)

(D)

V2 l . I sinx + —p^cosx

V3 M 5 ' For a positive integer n , let f n (0) = (2 cos0 + l) (2 cos0 - l ) (2 cos 2 0 - I) (2 cos 2 2 0 - l ) (2 cos 2n~10 - 1). Then: (A) f2 (ti/6) = 0 (B) f 3 (?t/8) = - 1 (C) f 4 (tt/32) = 1 (D) f5 (ti/1 28) = V2

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 . . . _ _ , 30 . 0 (A) 2 R sm — sin —

2 2

(C) (l + cos^J f l - 2 cos^j R

(B) 1 - cos Q

1 + 2 cos-1 R 2j

30 0 (D) 2 R sin -— sin —

4 4

Subjective Determine the smallest positive value ofx (in degrees) for which

tan(x + 100°) = tan(x + 50°) tan x tan (x - 50°).

Let s i n ( 9 - a ) = - & c o s ( 9 - ° 0 , — 4 then prove that cos (a - p) = a° ' b d

sin (0 - P) b cos (0 - P) d r ad + be

If X = sinf© + Y ^ j + s i n 0 -71

12 + sin 0 +

3tt 12

, Y=cos 0 + 7rc 12

+ cos 0 - 71

12 + COS 0 +

3tt 12

X Y then prove that -2 tan20. Y X

t fit BANSAL CLASSES Target I1T JEE 2007

M A T H E M A T I C S Daily Practice Problems

CLASS: XI (P, Q, R, S) DATE: 15-16/06/2005 TIME: 40Min. DPP. NQ.-20

Q.2

Q.4

Q.5

Q.6

Q.7

Q.9

Select the correct alternative : (Only one is correct)

Q. 1 The number of solutions ofthe equation cos

(A) more than 2 (B)2

c \ 71X

2V3 (C) l

: x 2 + 2A/3X + 4 is

(D)0

r r i l

The value of cot 7 — + tan 67 — - cot 67— 2 2 2

r -tan7— is:

2 (A) a rational number (B) irrational number (C) 2(3 + 2 v 3 )

1 ( D ) 2 ( 3 - V 3 )

Q.3 If t ana x2 - x

and tan p : 71

X - x + 1 the value equal to : (A)l ( B ) - l

2 -j 7 (X5fc0, l ) , w h e r e 0 < a , P < —, then tan (a + P) has ZX. ZX z

(C)2

The value of 4 cos — - 3 sec— - 2 tan — is equal to 10 10 10

(A) 1 (B) V5 - 1 (C) V5 + 1

(D) 3/4

(D) zero

The maximum value of ( 7 cosO + 24 sinO ) x ( 7 sinG - 24 cos9 ) for every 0 e R .

(A) 25 (B) 625 (C) 625 625

As shown in the figure AD is the altitude on BC and AD produced meets the circumcircle of AABC at P where DP = x. Similarly EQ = y and FR - z. If a, b, c respectively

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 + cosecB + cosecC

The graphs of y = sin x, y = cos x, y = tan x & y = cosec x are drawn on the same axes 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 ofthe line segment AB is:

(A) 1

Q.8 If tanB

(B) V 5 - 1

(C) V2 (D) V5 + 1

(A)

n s inAcosA

1 - n c o s 2 A sin A

then tan(A + B) equals

(n -1) cos A sin A (B) (C) (D)

sin A ( l - n ) c o s A K±JJ sinA v w ( n - l ) c o s A v ' (n + l)eosA

In a triangle ABC, angle A is greater than angle B. If the measures of angles A & B satisfy the equation, 3 sinx - 4 sin3 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

Q.10 The value of sin2 9 sin 9 +cos 9

, for all permissible vlaues of 9 s i n 9 - c o s 9 tan 9 - 1

(A) is less than - 1 (C) lies between - 1 and 1 including both

(B) is greater than 1 (D) lies between- J 2 a n d including both

Q. 11 The number of solution ofthe equation log3x (3/ x) + log2 x = l is (A) 3 (B)2 ( C ) l (D)0

4

ft BAN SAL CLASSES Target IIT JEE 2007

MATHEMATICS Paiiy Practice Problertis

CLASS: XI (P, Q, R, S) DATE: 13-14/06/2005 TIME: 40Min. DPP. NO.-19

Q.l

Q.3

Q.5

Q-9

Select the correct alternative : (Only one is correct)

5rc If — < x < 371, then the value of the expression

V l - s i n x + v l + sinx

V l - s i n x - VI + sinx is

(A) -cot-; x

( B ) c o t | ( Q t a n |

Q.2 The exact value of 9 6 s i n 8 0 ° s i n 6 5 ° s i n 3 5 ° is equal to sin20° + sin50° + sinllO0

(A) 12 (B) 24 (C)-12

The value of cot x + cot (60° + x) + cot (120° + x) is equal to

(A) cot3x (B) tan3x (C) 3tan3x

( D ) - t a n |

(D) 48

(D) 3 - 9tan x

3tanx - tan3x 3 + cot 76° cot 16°

is: Q.4 The value of cot 76 + cot 16

(A) cot 44° (B) tan 44° (C) tan 2° (D)cot46°

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

Q. 6 In A ABC, the minimum value of

(B) 2VTTk

2 > t

(C) 2Vk (D) 2k

2 A + 2 B —,cot — 2 2

n«* 2 A is

(A) 1

Q.7 The value of i 71 l+cos—

9y

(B)2

\ r 3tt l + cos

V 9 , l + cos

2

(C)3

571 9

l+cos 7tc

is

( B ) l ? ( C ) 1 2

(D) non existent

16 16 Q. 8 For each natural number k , let Ck denotes the circle with radius k centimeters and centre at the origin.

On the circle C k , a particle moves k centimeters in the counter- clockwise direction. After completing its motion on C k , the particle moves to Ck+1 in the radial direction. The motion of the particle continues in this manner .The particle starts at (1,0).lf the particle crosses the positive direction of the x- axis for the first time on the circle Cn then n equal to (A) 6 (B) 7 (C) 8 (D) 9

( n) ^ ( x - f ) The set of values of x satisfying 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

Q.10 If 0 = 3 a and sin 0 =

1 (A)

Va2 + b2

V a2 + b

(B) 2i/a2 + b2

. The value of the expression, a cosec a - b sec a is

(C) a + b (D) none

,|i BANSAL CLASSES MATHEMATICS gTarget i lT JEE 2007 Daily Practice Problems

CLASS: XI (P, Q, R, S) DATE: 08-09/06/2005 TIME: 50 Min. DPP. NO.-17 Fill in the blanks :

Q.l If log214 = athen log49 32 in terms of'a'is equal to .

1 ^ Q.2 The value of 10§ cosiht 9 7 is equal to . cos 6

27t 3

Q.3 The solution set of the system of equations, x + y = — , cos x + cos y = — , where

x & y are real, is .

Q.4 If a < s inx. s i n - xj . sin + xj < 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 x1/2 is rational II x2 and x5 are rational III x2 and x4 are rational

(A) I and II only (B) I and III only (C) II and III only (D) I, II and III

Q.6 If a3 + b3 and a + b * 0 then for all permissible values of a, b ; log (a + b) equals

1 1 (A) - (log a + log b + log 3) (B) - (loga + logb + log2)

(C) log(a2 - ab + b2) (D)log f 3

' a +b 3ab

Q.7 i he number of all possible triplets (a p a^ a3) such that a, + a2cos2x + a3sin2x = 0 for all x is : (A) 0 (B) 1 ~ (C) 3 (D) infinite (E)none

Q-8 T - L ^ r + cos 290° V3sin250°

(A) ~ (B) ^ (C) V3 (D) none

Q.9 The product cot 123°. cot 133°. cot 137°. cot 147°, when simplified is equal to : ( A ) - l (B) tan 37° (C) cot 33° (D) 1

Q. 10 V x e R the greatest and the least values of y = j cos 2x + sin x are respectively

3 1 3 3 1 3 3 ( A ) - , ~ (B) ( C ) - , - - (D) 1 , - -

Q. l l Given s inB= ~ sin (2A+B) then, tan(A+B) = ktanA, where k has the value equal to

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

( c \ C A B Q.12 If A + B + C = 7i & sin A + — = k s i n - , then tan— t a n - =

V 2 J 2 2 2

(A) (B) i l l (C) - A - (D) ^ k + 1 k - 1 k + 1 k

, | i BANSAL CLASSES MATHEMATICS Target NT JEE 2007 Daily Practice Problems

CLASS: XI (P, Q, R, S) DATE: 10-11/06/2005 TIME: 50 Min. DPR NO.-18 Q.l If tan A & tan B are the roots of the quadratic equation, ax2 + bx + c = 0 then evaluate

a sin2 (A + B) + b sin (A + B). cos (A + B) + c cos2 (A + B).

7i 3n 571 7TT Q.2 Find the exact value of tan2— + t a n 2 — + t a n 2 — +tan2 —

16 16 16 16

Q.3 If A + B + C = 7i:,provethat

X (tan A ) - 2 X (cot A). r tanA ^

v tanB.tanCy

Q.4 If a cos (x + y) = sin (y - x) then prove that,

1 1 2 l + as in2x l - a s i n 2 y 1 - a 2

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)

2 7t Q.7 If a = — , prove that,

sec a + sec2a + sec4a = - 4.

,|i BANSAL CLASSES MATHEMATICS 9Target 1IT JEE 2007 Daily Practice Problems

CLASS: XI (P, Q, R, S) 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 . (use n = 22/ 7)

Q.2 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 logarithms of whose reciprocals to the base 10 have the

i H characteristic (-3) then Iogio ~~ has the value equal to . Vn ) Q.3 ln(ab)- ln |b | simplifies to

Q.4 The least value of the expression c o t ^ x tan2x f o r Q < x < _ js 1 + sin - 8xj ^

Q. 5 The greatest value of the expression

sin2 f • - 4xj - sin2 • - 4xj for 0 < x < is .

Select the correct alternative : (Only one is correct) 10 9

Q.6 If £ kak = 22 and j ] pap = 32 then a ] 0 = k=l p=l

( A ) - 1 0 ( B ) - l (C)10 (D) 1 Q.7 Let x + y = 1 and x3 + y3 = 19 then the value of x2 + y2 is equal to

(A) 9 (B) 19 (C) 39 (D) none

Q. 8 The side of a regular dodecagon is 2 cm. The radius of the circumscribed circle in cms. is:

7 Fl ( A ) 4 ( V 6 - V 2 ) (B)V6 + V2 ( C ) V37T ( D ) V 6 - V 2

Q.9 In a triangle ABC, angle A is greater than angle B. Ifthe measures of angles A and B satisfy the equation 2 tanx - k(l+tan2x) = 0,where k e (0,1), then the measure of the angle C is

(A)

Q.10 Let

71 6 (B)f 571

(C)12 (D) 71 ~2

sin 30 cos20

= p where 0 e Us 2371 1

' 48 J 0 sin 3(3

& cos2p " where p e (I3n

t 48 1471

' 48 Then

(A) p > 0 and q > 0 ( B ) p > 0 a n d q < 0 ( C ) p < 0 a n d q < 0 ( D ) p < 0 a n d q > 0

Select the correct alternative : (More than one are correct)

Q.l l Which of the following statement(s) does/do not hold good ? (A) log 10 ((1.4)2 - 1 ) is positive ^ (B) log 1 + log 2 + log 3 = log (1 3)

3% f i V2'0637

(C) log0.1 c o t — is negative (D) If m = 4>°S47 and n = 8

T_ sin39 11 . 0 , , Q.12 If . ^ = — then tan — can have the value equal to :

smO 25 2 (A) 2 (B) 1/2 (C) - 2 ' ( D ) - 1 / 2

then n = m4.

BANSAL CLASSES MATHEMATICS 8 Target I IT JEE 2007 Daily Practice Problems

CLASS: XI (P, Q, R, S) DATE: 03-04/06/2005 TIME: 45 Min. DPP. NO.-15

Q.l If secA - tanA = p, p # 9, find the value of sinA.

2n-l

Q.2 Evaluate the product | " | t an ( r a ) where 4na = 7t. r=l

3 Q.3 If cos ( y - z ) + cos ( z - x ) + cos ( x - y ) = - ~ ,

prove that cosx + cosy + cosz = 0 = sinx + siny + sinz

sinx sin3x sin5x a j -2a ,+a 5 a 3 -3a j 0 4 If = = thenshowthat — — - — V ' a3 a5 a3 a}

Q.5 If 9 = , prove that 2" cos9 cos29 cos229 cos2n"19 = 1. What the value of the product At i 1

71 whould be if 9 = 7.

Q.6 Find the exact value of cosecl90 + cosec50°-cosec700 .

Q.7 Prove that from the equalities, * ^ + 2 ~ * > - y ( z + X " y )=

2 ( * + y ~ 2 > follows log x log y log z x y y X - y Z z y = z x x z

4g BAN SAL CLASSES MATHEMATICS 8Target i l l JEE 2007 Daily Practice Problems

CLASS: XI (P, Q, R, S) DATE: 01-02/06/2005 TIME: 50 Min. DPR NO. -14

cos4 a sin4 a cos4B sin4 3 Q.l i f — t w ^ - 2 d = 1 men rind the value o f — • ? • cos p sin P cos a sin a

Q.2 If [1 - sin (7t + a ) + cos (rc + a)]2 +

value of a and b.

' . (3n "l (3% 1 - sin| — + a j + C 0 S | /7,— a 2

= a + b sin 2a then find the

cos (A-B) cos(C + D) Q.3 If ^ ~ g7 + _ j)) = 0 then prove that tanA • tanB • tanC • tan D = - 1

tan 80 Q.4 Prove that = (1+ sec20) (1 + sec40) (1 + sec80)

Q.5 Express sin2a + sin2p - sin2 y + 2 sina sinP cosy as a product of two sines and two cosines.

Q.6 Find the solution set ofthe equation

5. 25

Y l n x sin2x + 4 5 C O S 2 X = (25) 2 where x e [ 0,2n]

Q.7 Show that x = J 2 cos 36° is the only solution of the equation

logx(x2 + l) = A/log^(x2(l + x 2 ) ) + 4

,|i BANSAL CLASSES MATHEMATICS pTerget SIT i i i 2007 Dally Practice Problems.

CLASS: XI (P, Q, R, S) DATE: 30-31/05/2005 TIME: SO Min. DPP. NO.-13 Fill in the blanks :

Q. 1 The exact value of cos4 9 + cos4 29 + cos4 30 + cos4 40 if 0 = 7t/3 is .

sin41 + cos41 - 1 Q.2 The expression — when simplified reduces to sin61 + cos61 - 1

sin240cos60-sin6°sin66Q

Q.3 The exact value of s i n 2i°Cos39°-cos510sin69° is.

f . 1 Q.4 cos — j log3 — + logjL 3j when simplified reduces to

Q.5 Exact value of tan200° (cot 10° - tan 10°) is .

— . TC 71 7t 7t 7t , Q.6 96 V3 sm — C O S48 C O S24 C0SL2 C0S"6 n a S v a * u e = •

1 1 Q.7 If cosa = and sin(3 = where a e 4th quadrant and p e 2nd quadrant then

tan a + p") f a - p

cot V 2

Select the correct alternative: (More than one are correct)

Q. 8 Identify the statements) which is/are incorrect

(A) Vl + s i n a - V l - s i n a =2sin— i f a e f e t 2 >

(% \ (n )

(B) sin2a + c o s ! y ~ a j • cos [—+aJ is independent of a .

u/(C) log, (cos2 (8 + <j>) + cos2 (0 - (j>) - cos 20-cos2<))) is equal to 1.

3 (D) If tana = — where a e 3rdQ then cos3a is positive. Q. 9 Which ofthe following when simplified reduces to unity?

, . l - 2 s i n 2 a „ sin(Ti-a) (A) T T 7 v (B) * J — + c o s ( 7 i - a ) 2cot —+ a cos — a s m a - c o s a tan —

U J U ) 2

4 sm a cos ' a 4 tan a (sma + cosa)

Q. 10 The equation logx+1 (x - 0.5) = logx_0s (x +1) has (A) no real solution (B) no prime solution (C) an irrational solution (D) no composite solution

BANSAL CLASSES MATHEMATICS P Target SIT J E E 2 0 0 7 D a l l y Practice Problems.

CLASS: XI (P, Qj R, S) DATE: 27-28/05/2005 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° i S £ q U a l t 0 '

Q.2 The logarithm of 32.5 to the base 10 is 1.5118834. The number whose logarithms to the same base is

4.5118834 is .

Q.3 If cos0 = log9log log log3273 then the set of values of 6 lying in [0, 2TC] is .

Q.4 Let a and (3 be the solution ofthe equation

logx2 • logx/162 = logx/642 where (a > (3) then a = & P =

Select the correct alternative ; (Only one is correct)

Q.5 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

Q.6 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 7T 71 3ir 7T 71 71 371 ( A > I & i ( B ) i & i ( c ) i & I < D ) i »

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 4sin5acos3acos2a is expressed as the sum of three sines then two of them are sin4a and sinl 0a.The third one is {A) sin 8a (B) sin 6a (C) sin 5a (D) sin 12a

Q. 9 Given a system of simultaneous equations 2x.5y = 1 and 5x+1.2y = 2. Then

(A) x = log105andy = log102 (B)x = log102andy = log105

(C) x - log1Q and y = log102 (D) x - logi05 and y = log10 ( j

Q.10 I f a e J then the expression . (

•\Asin4a + s in 2 2a + 4cos2( ^ - ^ J equals

(A) 2 (B) 2 - 4cosa ( C ) 2 - 4 s i n a (D)none

4! fit BANSAL CLASSES Target I1T JEE 2007

MATHEMATICS Daily Practice Problems

CLASS: XI (P, Q, R, S) DATE: 25-26/05/2005 TIME: 45 Min. DPP. NO.-ll

Select the correct alternative : (Only one is correct)

Q. 1 If cosQ = ^ ( a + t ^ 1 1 c o s t e r m s °f ' a ' =

(A) ( B ) j ( ^ (C)4(a 3 + - i (D)none

Q.2 The expression

reduces to: (A) 1

1 + sin 2a - sin 2a

cos (2a - 2%) . tan (a - 4r) 4

(B) 0 (C) sin2 (a/2)

a '3* cot— + cot 2 2 2 )_

when simplified

(D) sin2 a

Q.3 sin3 e - c o s 3 9 COS0 sinG - cosG ^ i + cot29

- 2 tan 0 cot 0 ~ — 1 if

( A ) 9 e | 0 , | ) ( B ) e e ( ^ ] ( 0 ) 0 6 ^ (D) 9 e

Q.4 Exact value of cos 20° + 2 sin2 55° - V2 sin 65° is :

(A) 1 (B) - jL . ( C ) V2 (D) zero

Q.5 If cos (0 + <j>) = m cos (0 - (j>), then tan 0 is equal to :

(A) f \ + m l , , f l - m) tan<)> (B) -

1 - my U + my tanij) (C) 1 - m

1 + my cot<j) (D)

f i , "N 1 + m 1 - my

COt<j)

Subjective:

Q.6 Solve the equation, 3 . 2 ,og*(3x ~ 2) + 2 . 3log*(3x " 2> = 5 . 6 '°8*2 (3x " 2)

Q.7 Prove the identity, c o s ^ y - + 4 a ] + sin (3tt - 8a) - sin (47t - 12a) = 4 cos 2a cos 4 a sin 6a.

Q.8 Solve the following equation for x : -i . aA - 3B = 9C

where A = log a x . log 1 0 a . log a 5 , B = log10(x/10) & C = log100x + log42.

cos5x + cos4x Q.9 Prove that: 2 c o s 3 x - 1 ~ ° ° S X + °°S 2 x '

Q. 10 Prove the identity, sin 2 a (1 + tan 2 a . tan a ) + * + s m a - = tan 2a + tan2 ^ + ^ ^ 1 - sma V4 2

JgBANSAL CLASSES MATHEMATICS Target l !T JEE 2007 Daily Practice Problems

CLASS: XI (P, Q, R, S) DATE: 23-24/05/2005 TIME: 45 Min. DPP. NO.-IO

Select the correct alternative : (Only one is correct) Q. 1 Exact value of cos2 73° + cos2 47° - sin2 43° + sin2107° is equal to :

(A) 1/2 (B) 3/4 (C) 1 (D) none

Q.2 Theexpression sin22° cos8° + cosl58°cos98° w h e n s i m p l i f i e d reduces to : sin23 cos7 + cos 157 cos97

(A) 1 (B) - 1 (C) 2 (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)7/25

Q.4 If s i n 2 a ~ s i n 3 a + s i n 4 a = tan ka is an identity then the value k is equal to: cos 2a - cos 3a + cos 4a

(A) 2 ( B ) 3 ( C ) 4 ( D ) 6

Q.5 log3fl + ) + l o g 3 f 1 + ~ ] + l o g 3 f l + ^ j + + log_5 1 + 2~42~ w ^ e n simplified has the value

equal to (A) 1 (B)3 (C)4 (D)5

Select the correct alternative : (More than one are correct)

5 99

Q.6 The sines of two angles of a triangle are equal to — & — . The cosine of the third angle is:

245 255 735 765 <A> T3T3 <B> a n <c> T3T3 ^ o i l 17 5

Q.7 If secA = — and cosecB = — then sec(A + B) can have the value equal to

(A) | ( B ) - f ( C ) - £ ( D ) |

Subjective:

Q. 8 If log6(l 5) = a and log12(l 8) = p, then compute the value of log25(24) in terms of a & p.

Q.9 Solve the equation : 2>x +11 - 2X = 12X - 11 + I.

Q.10 If sinx + cosx + tanx + cotx + secx + cosecx = 7 then sin 2x = a - W 7 where a, b e N. Find the ordered pair (a, b).

# fit BANSAL CLASSES Target I1T JEE 2007

MATHEMATICS Daily Practice Problems

CLASS : XI (P, Q, R, S) DATE: 20-21/05/2005 TIME: 40 Min. DPP. NO.-9

Fill in the blanks.

( 3« cosa 0 .1 If tana = 2 a n d a e ! I then the value ofthe expression — s 3— is equal to v 2 J r sin a + cos a

Q.2 If log,47 = a and log 14 = b then the value of log17556 is equal to

Q.3 logo.75 3%

logj cos — - l o g j cosec i ) 75 4

has the value equal to

Q.4 The value of (cos 15° - cos750)8 is.

Select the correct alternative ; (Only one is correct)

Q. 5 Number of roots of the equation tan2x + cot2x = 2 which lie in the interval (0,4n) is (A) 4 (B) 6 (C) 8 (D) 10

/-5„ \ 371 - e

hypotenuse is (A)l

Q.7 If f (x) = 3

(B)2

• 4 f 3n ) sm X

I 2 J + sin (3TC + X)

V 2 J (C) V2

f ~ ^ - 2 sin6 7C

— + X U J

and cosO - cos| — - 0 2 /

. The length of its

(D) some function of 0

+ sin (57t-x) then, for all permissible

values of x, f (x) is

( A ) - l (B) 0 (C) l

Select the correct alternative; (More than one are correct) Q. 8 Which ofthe following numbers are positive?

(D) not a constant function

(A) log log32l 2 (B) log. (C) log (log 9) (D) log (sinl25°)

Q. 9 Identify the correct statement (A) If f(x) - sinx - cosx then f( l) < 0

.371 . 5n sin s m —

w — £ — t COS H COS

4 3

7T 5n sec t a n —

have the same value 71 ^ 7C

cosec—+ cot— 4 4

( q 5 iog5 2+iog5 3 is equal to 6 (D) log:j5 + logs3 is greater than 2

Subjective:

Q.10 Solve the equation, | x - 1 | + | x + 2 | - | x - 3 | = 4.

4| BANSAL CLASSES MATHEMATICS pTorget SST JEE 2007 Daily Practice Problems

CLASS : XI (P, Q, R, S) DATE: 18-19/05/2005 TIME: 45 Min. DPR NO.-8

Select the correct alternative : (Only one is correct) Q.l Number of solutions of the equation. logC0,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

sinG sec70 + cos0 cosec79 is

( a + b ) 3 ( a 4 + b 4 ) (a + b)3(a4 - b 4) ( A ) (ab)7'2 (ab)7'2

(a + b)3(b4 - a 4 ) (a + b)3(a4 + b 4) ( C ) (ab)7'2 ( D ) (ab)7'2

Q.3 The solution set of the equation, 3 ^/log10 x + 2 log10 = 2 is :

(A) {10, 102} (B) {10,103} (C) {10, 104} (D) {10, 102, 104}

Select the correct alternative : (More than one are correct)

Q.4 If J -——— + - —-—, for all permissible values of A, then A belongs to Vl + sinA cos A cos A

(A) First Quadrant (B) Second Quadrant (C) Third Quadrant (D) Fourth Quadrant

( i > Q.5 The solution set of the system of equations, log12 x + log2 y = log2 x and

logx2 /

log2 x . (log3 (x + y)) = 3 log3 x is: (A) x = 6 ; y = 2 ( B ) x = 4 ; y = 3 ( C ) x = 2 ; y = 6 ( D ) x = 3 ; y = 4

Q.6 The equation ^l + logx V27 log3 x +1 = 0 has : (A) no integral solution (B) one irrational solution (C) two real solutions (D) no prime solution

Q. 7 Which of the following are correct ? (A) log319. log1/7 3 . log4 HI >2 (B) log5 (1/23) lies between - 2 & - 1

(C) log10 cosec (160°) is positive (D) log^ sinj — . log - 5 simplifies to an irrational number V5

Subjective:

Q. 8 If an equilateral triangl e and a regular hexagon have the same perimeter then find the ratio of their areas.

tan3 0 cot3 0 _ 1 - 2 s i n 2 Qcos2 0 Q.9 Prove the identity, + = sin 0 cos 0 '

Q.10 Solve the equation, | x - 1 | - 2 j x - 2 | + 3 | x - 3 | = 4.

4;BANSAL CLASSES MATHEMATICS B Target i l T JEE 2007 Dally Practice Problems

CLASS • XI (P, Q, R, S) 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 , then x has the value equal to .

Q.2 If log7 2 = m, then log49 28 in terms of m has the value equal to .

Q.3 3 5 1 0 8 , 5 + • 1 simplifies to . y V - l o g " (0.1)

Q.4 If x2 - 5x + 6 = 0 and log2 (x + y) = log425, 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 positive (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

i (A) a singleton \ (C) a set consisting of two elements

(B)<j) (C) a set consisting of more than 2 elements

Q.7 Number of values o f ' x ' in (-2%,2%) satisfying the equation 2 s in"x +4.2C0S x = 6 i s

(A) 8 (B)6 (C) 4 (D)2

2 2 Q.8 Solution set ofthe equation 32 x -2.3X + x + 6 + 3 2 ( x + 6 ) = 0 is

(A) { -3 ,2} (B) {6,-1} J C ) {—2, 3} ( D ) { l , - 6 }

Subjective:

Q.9 Find the set of values o f ' x ' satisfying the equation ^64 - ^2 3 x + 3 +12 = 0.

Q. 10 Find 'x' satisfying the equation 4logio X+1 - 6 logiox - 2.3logio x 2 + 2 - 0.

« &BANSAL CLASSES Target I IT JEE 2007

MATHEMATICS Daily Practice Problems

CLASS: XI (P, Q, R, S) DATE: 13-14/05/2005 DPR NO.-6

Time: Take approx. 40 min. Fill in the blanks.

^ . tan(l 80° - a)cos(l 80° - a)tan(90° - a ) , . . , . , . l 4 0.1 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 If tan 25° = a then the value of t a n 2 Q 5 t a n 1 1 5 in terms of 'a' is tan245° + tan335°

Q.4 The product, (log217) x (log1/5 2) * (log3 - ) lies between two successive integers which are _

and

Q.5 The value of the sum, — - — + —-— + -—-— + + - — i s log2 N log3 N log4 N

continued product of first 2000 natural numbers. l°g2oooN

, where N denotes the

Select the correct alternative : (Only one is correct) Q.6 If Xj and x2 are two solutions of the equation log3 j 2x — 7 j = 1 where Xj < x 2 , then the number of

integer(s) between Xj and x2 is/are: (A) 2 (B) 3 (C) 4 (D) 5

Q.7 The value of the expression, log4

( A ) - 6 ( B ) - 5

f X

V 4 . -2 log 4 (4x 4 )when x = - 2 is:

(C) - 4 (D) meaningless

Q.8 The number of real solution(s) of the equation, sin (2X) = nx + it x is: (A) 0 (B) 1 (C) 2 (D) none of these

tan Q.9 The expression

f TT^ 1Z (3n ^ . 3 (lit 1 X — .COS + x - s i n X

I 2) 1 2 ; I 2 J

f n) .tan fH ) COS x — .tan — + X I 2 j I 2 J

simplifies to

(A) ( 1 + cos2x) (B) sinhi

1 Q.10 Let y =

2 + -1

(C) - (1 + cos2x) (D) cos2x

, then the value of y is

3 + -2 + -

1 1

3 + ..

(A) V l 3 + 3

(B) V l 3 - 3

.(C) V15+3 (D)

V l 5 ~ 3

djBANSAL CLASSES MATHEMATICS 8 T a r g e t SIT JEE 2007 Daily Practice Problems

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 . cosec2 9 = sec2 0 + cosec2 9.

(b) There exist natural numbers, m & n such that m2 = n2 + 2002.

Fill in the blanks. Q.2 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 .

Q.3 . The positiveintegersp,q&r are all primes. If p 2 - q 2 = r then the set ofall possible values of ris .

Q.4 The solution set ofthe equation x loga x = (aK )log*x is , (where a > 0 & a * 1)

Select the correct alternative : (Only one is correct) Q.5 The number of real solution ofthe equation log10 (7x - 9)2 + log]0 (3x - 4)2 = 2 is

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

2log2,/4a_3iog27(a2

+i)3_2a

74iog49a_a_|

(A) a2 - a - 1 (B) a2 + a - 1 ( C ) a 2 - a + l ( D ) a 2 + a + l

Q.6 The ratio 4 f o ^ a _ 1 — " simplifies to:

Q. 7 Which one of the following denotes the greatest positive proper fraction?

/ | \ l ° S 2 6 /lY°g35

(A) 7 (B) - (C) 3 3 (D)8

i - log, 2

V-V

Select the correct alternative : (More than one are correct)

Q. 8 Which of the following when simplified, vanishes ?

1 2 3 (A) t — r + log3 2 log9 4 log27 8

(B) l o g ^ f j + log 4 ( j -

(C) - log8 log4 log216 (D) logjQ cot 1° + logjq cot 2° + logjQ cot 3° + + log10 cot 89°

Q. 9 Which ofthe following numbers are positive

(A) log9(2.7)-0-3 (B) log1/2(l/3) (C) logvT5 VlT (D) log1/2 ~ - 2

Subjective:

Q. 10 Compare the numbers log34 and log56.

c K BANSAL CLASSES Target SIT JEE 2007

M A I H f c t V l A S

Daily Practice Problems CLASS: XI (P, Q, R, S) DATE : 09-10/05/2005 DPP. NO.-4

Time : Take approx. 40 min. Fill in the blanks.

Q. 1 The expression -Jlog0 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 (a2 + b2)3 = (a3 + b3)2 and ab * 0 then the numerical value of — + — is equal to

b a

Q.3 A mixture 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

Q.5

Q.6

A polynomial in x of degree three which vanishes when x= 1 & x = - 2 , and has the values 4&28when x " — 1 and x = 2 respectively is .

The solution set ofthe equation 4!og9x - 6.xlog92 + 2log327 = 0 is .

The smallest natural number of the form 1 2 3 X 4 3 Y, which is exactly divisible by 6 where 0 < X, Y< 9, is .

Select the correct alternative : (Only one is correct)

. logx 0°S2 x+1) 1 + - log4

2(x4)+ 2"31og , '2( log2x) Q.7 The equation, log2 (2x2) + log2x . x ^

(A) exactly one real solution (B) two real solutions (C) 3 real solutions (D) no solution.

Select the correct alternative : (More than one are correct)

Q.8 The equation =3 has : (log8x)-

(A) no integral solution (C) two real solutions

Subjective

Q.9 Which is smaller ?

has

(B) one natural solution (D) one irrational solution

l o g , - - or log, u s + v:

8 ax Q. 10 It is known that x = 9 is a root ofthe equation log. (x2 + 15 a2) - log^ ( a -2 ) = log,, -—;-. Find the other

root(s) of this equation.

K BANSAL CLASSES Target SIT JEE 2007

MATHEMATICS Daily Practice Problems

CLASS: XI (P, Q, R, S) DATE: 02-03/05/2005 DPR iva-i

Time: Take approx. 30 min.

Fill in the blanks :

Q.l

Q.2

Q.3

Q-4

Q.5

Q.6

Q.7

Q.8

(i)

(Hi)

Q.9

The value of b satisfying log - b = 3 - is

The number of integral pair(s) (x ,y ) whose sum is equal to their product is

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

(C) 2(1+ 7 3 )

The equation = x _ 2 h a s

(A) two natural solution (C) no composite solution

(B) 3V3

( D ) 3 ( 3 - V 3 )

(B) one prime solution (D) one integral solution

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 with the winners. The total number of matches played in the tournament is (A) 115 (B) 53 (C) 232 (D)116

Subjective:

Prove that x4 + 4 is prime only for one value of x e N.

Establish tricotomy in each of this following pairs of numbers

3 log273and2 log42

4 and log 310 + log 10 81

Compute the value of ^ ^

(ii) log4 5 and logI/16 (1 / 25)

(iv) log1/5(l / 7) and !og1/7(l / 5)

log53 + 27 log 9 36

4

3 7 9

Q.l 0 The length of a common internal tangent to two circles is 7 and a common external tangent is 11, Compute the product of the radii of the two circles.

1st bpp OM rue mn-i of succee& J^f 1st

t fit BANSAL CLASSES Targe t I1T JEE 2 0 0 7

MATHEMATICS 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 :

1 1 1 — — + — + — has the value equal to log _ abc log ^ abc log abc

Q.l

OL

^Q-2

Q.3

J be v'ca Vab

(Assume all logarithms to be defined)

Solution set of the equation, log,20 x + log.0x2 = logj20 2 - 1 is

Q.4

log (o T) The expression (0.05) " v 2 0 ' ' ' is a perfect square of the natural number

(where o.T denotes 0.111111 oo)

The line AB is 6 meters in length and is tangent to the inner one of the two 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 expression, xlo»" ~ loez • y 1 ^ - log* • zlo°x - w h e n simplified reduces to

Select the correct alternatives : (More than one are correct) Q.6 If p, q G N satisfy the equation = jVxj then p & q are :

(A) relatively prime (B) twin prime (C) coprimc (D) if logq

p is defined then logpi is not & vice versa

Q.7 The expression, log log where p > 2, p e N, when simplified is : ^

n radical sign v(A) independent of p, but dependent on n (B) independent of n, but dependent on p _(€) dependent on both p & n j(D) negative.

Q. 8 Which ofthe following when simplified, reduces to unity ? 21og2 + log3 (A) log]05 . log1Q20 + log2

0 2

( C ) - l o g 5 l o g 3 ^

-1- logs 2 when simplified reduces to •

log48 - log4

Q.9 The number N :

(i + iog32) (A) a prime number (C) a real which is less than log,7t

Subjective :

Q.10 Given, log712 = a & log1224 = b . Show that, log54168 =

(B) an irrational number (D) a real which is greater than log7

6

1 + ab a (8 - 5 b)

J|BANSAL CLASSES MATHEMATICS HgTarget 1ST JEE 2007 Pally Practice Problems 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 - 1 , then the value of x3 + 3x - 14 is equal to

Q.3 Iflogx Iog1B(V2 + Vs) = - . Then the value of 1000 x is equal to .

Select the correct alternative : (Only one is correct)

Q. 4 Which one of the following when simplified does not reduce to an integer?

2log6 *°g2 3 2 log, 16-log, 4 ( 1N"2

Q. 5* Let m denotes the number of digits in 264 and n denotes the number of zeroes between decimal point and the first significant digit in 2 _ 6 4

5 then the ordered pair (m, n) is (you may use logi02 = 0.3) (A) (20.21) (B) (20.20) (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. Then, the ratio of area of the circle to the area ofthe square is

7i 11 3 7 (A) j (B) - (C) - (D) -

Q.7 V Let u = (logjx)2 - 6 log2x +12 where x is a real number. Then the equation xli - 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 1 1 ( 1 1 1 + r = + (x-y)2 (y-z)2 (z-x)2 Vx-y y - z z-x)

Q.9 Solve 3 X + 1 - 13* - 11 = 2 log5 j 6 - x j.

Q.10 If log)g36 = a & log2472 = b, then find the value of 4 (a+b) -5ab .

YJI K k . . !^ . IIQBANSAL CLASSES 7

1 8 T a r g e t SIT JEE 2 0 0 7

MATHEMATICS Daily Practice Problems

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

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

Q.l tan 9 = —j where 0 e (0,2%), find the possible value of 0. [2]

— 2 + 2 + '--oo

Q. 2 Find the sum of the solutions of the equation 2e 2 x ~ 5ex + 4 = 0. [2]

Q.3 Suppose that x and y are positive numbers for which log9x = log12y = log15(x + y). If the value of

- =2 cos 0, where 0 e (0,n/2) find 0. [3]

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

Limit Limit x->0 + n->«>

l2 (sinx)* ] + [22 (sinx)x ] + + [n2 (sinx)x ]

I n3 where [ . ] denotes the

greatest integer function. [4]

/ b - a . . 1 VT~ S1»2x I , Q.5 Consider f ( x ) = - j = , Va + b t a n x , f o r b > a > 0 & t h e functions g(x)&h(x)

Vb M v

\ 2 b - a . 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 discuss the continuity of 'g' & 'h' at x=— and x = 0 respectively. [5]

Q.6 J f ^ d x [ 5 ]

Q.7 Using substitution only, evaluate: jcosec3xdx. [5]

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

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

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

tanO 1 cot0 Q.3 If -———-—— = —, find the value of . [3]

tan 0 - t a n 30 3 cot0~cot30 1 J

Q.4

Q . 6

Q.7

If a A ABC is formed by the lines 2x + y - 3 = 0 ; x - y + 5 = 0 a n d 3 x - y + l = 0 , then obtain a cubic equation whose roots are the tangent of the interior angles of the triangle. |4]

r dx Q.5 Integrate: J / 2 2 x f ~ r ~

(a - t a n xK/b - tan2 x ( a > b )

J xsmxcosx (az cos^ x + bz sin2 x)2 dx

[5]

[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. [5]

SC 3$S 5J5 sj? ^ jjc Sjc ^

DATE : 03-04/07/2006 TIME; 50Min.

Q.2

Q. 1 Let x = (0.15)20. Find the characteristic and mantissa in the logarithm of x, to the base 10. Assume log in2 = 0.301 and Iog103 = 0.477. K O l o c / 3 f Z P I ^ o - M ^ Z - PI

Two circles of radii R & r are externally tangent. Find the radius ofthe third circle which is between them p 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. 5X ,5 ,5 then compute the value ofthe integral J( det A)dx.

[3] 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]

Q.5 Evaluate: J' dx - x + x Vx + X 2 + X 3

Q.6 If the three distinct points, ' a 3 a 2 - 3 f U3

a - 1 a - 1 \ /

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

b - 3 b - 1 ' b - 1

( „3

c — 1 c - 1

[5]

are collinear then

[5]

Q.7 Integrate: j \ / tanx dx

3 j V v ^ o t d &

2{.l AX*

1 4 - tG

[5j

2 vAV