practice test papers 1 to 14 e

7/30/2019 Practice Test Papers 1 to 14 E

1/35

PRACTICE TEST PAPERS

TARGET IIT JEE 2012XIII (XY)MATHEMATICS

A-10, "GAURAV TOWER", Road No.-1, I.P.I.A., Kota-324005 (Raj.) INDIA

Tel.: 0744-2423738, 2423739, 2421097, 2424097 Fax: 0744-2436779

E-mail: [email protected] Website : www.bansal.ac.in

OTHER STUDY CENTERS:

Jaipur- BansalClasses, Pooja Tower, 3, Gopalpura, Gopalpura Bypass, Jaipur Tel.:0141-2721107,2545066, E-mail: [email protected]

Ajmer- BansalClasses, 92, LIC Colony, Vaishali Nagar, Ajmer (Raj.) Tel.: 0145-2633456, E-mail: [email protected]

Palanpur- C/o Vidyamandir School, Teleybaug (Vidya Mandir Campus-1) Palanpur-385001, Dist: Banaskantha, North Gujarat,

Tel.: 02742-258547, 250215 E-mail: [email protected]

Guwahati- C/o Gems International School, 5-B, Manik Nagar, Near Ganeshguri, RGB Road, Guwahati-781005

Tel.: 0361-2202878 Mobile: 84860-02472, 73, 74 E-mail: [email protected]

Meerut- C/o Guru Tegh Bahadur Public School, 227, West End Road, Meerut Cant-250001

Tel.: 0121-3294000 Mobile: +9196584-24000 E-mail: [email protected]

Nagpur- Bansal Classes, Saraf Chambers Annexe, Mount Road, Sadar Nagpur-1

Tel.: 0712-6565652, 6464642 E-mail: [email protected]

Dehradun- C/o SelaQui International School, Chakrata Road, Dehradun, Uttarakhand-248197

Tel.: 0135-30510 00 E-mail: [email protected]

I N D E XPRACTICE TEST PAPER-1...... ........... .......... ........... .......... ........... ....... Page-2

PRACTICE TEST PAPER-2...... ........... .......... ........... .......... ........... ....... Page-4








PRACTICE TEST PAPER-10 ..... ........... .......... ........... .......... ........... ..... Page-22

PRACTICE TEST PAPER-11 .......... .......... ........... .......... ........... .......... .. Page-24




ANSWER KEY ........... .......... ........... .......... ........... .......... ........... .......... .... Page-33
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://www.bansal.aci.n/mailto:[email protected]


2/35PAGE # 2

PRACTICE TEST PAPER-1

Time: 60 Min. M.M.: 56

[SINGLE CORRECT CHOICE TYPE]

Q.1 to Q.3 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. [3 3 = 9]

Q.1 If the sum of all values of [0, 4] satisfying the equation (2 + sin ) (3 + sin ) (4 + sin ) = 6is k, then k equals

(A) 6 (B) 5 (C) 4 (D) 2Q.2 There are six mathematics books, three geography books and seven history books on a bookshelf.

Number of different ways in which one can select four books, so that each selection must have

atleast one mathematics book, is (Assume that books of the same subject are different.)

(A) 720 (B) 1044 (C) 1610 (D) 1820

Q.3 A cubic polynomial y = f(x) is such that A(1, 3) and B (1,1) are the relative maximum and relative

minimum points respectively. The value of f(2) is

(A) 6 (B) 4 (C) 0 (D) 3

[PARAGRAPH TYPE]


Paragraph for Question 4 to 6

Let P(x) be a polynomial of degree 4, vanishes at x = 0. Given P(1) = 55 and P(x)

has relative maximum / relative minimum at x = 1, 2, 3.

Q.4 Area of triangle formed by extremum points of P(x), is

(A) 1/2 (B) 1/4 (C) 1/8 (D) 1

Q.5 The value of definite integral

1

1

dx)x(P)x(P , is

(A)

15

252(B)

15

452(C)

15

652(D)

15

752

Q.6 Which one of the following statement is correct?

(A) P (x) has two relative maximum points and one relativeminimum point.

(B) Range of P(x) contains 9 negative integers.

(C) Sum of real roots of P(x) = 0 is 5. (D) P (x) has exactly one inflection point.

[REASONING TYPE]

Q.7 & Q.8 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. [2 3 = 6]

Q.7 Statement-1: Let A and B be two non-zero square matrices of order 2 such that AB = O, then

det. (A) = 0 and det.(B) = 0.

Statement-2: If M and N are two square matrices of order 2 such that MN = O then it does not

imply that atleast one of the matrices M and N is a zero matrix.(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.

(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.

(C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true.

Q.8 Statement-1: The plane through the points A, B, C whose co-ordinates are (1, 1, 1), (1, 1, 1)

and (1, 3, 5) passes through the point (2, k, 4) for every k R.

Statement-2: The equation of plane passing through 3 non-collinear points P (x1, y

1, z

1), Q (x

2, y

2, z

2)

and R (x3, y

3, z

3) is given by

)zz()yy()xx(

)zz()yy()xx(

)zz()yy()xx(

131313

121212

111

= 0.


3/35PAGE # 3

(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.



[MULTIPLE CORRECT CHOICE TYPE]

Q.9 to Q.10 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct. [2 4 = 8]

Q.9 Consider the function g defined by g(x) =

1,0xif,0

1,0x,

1x

sin1x

x

sinx22

then which of the following statement(s) is/are correct ?

(A) g (x) is differentiable x R. (B) g'(x) is discontinuous at x = 0 but continuous at x = 1.(C) g'(x) is discontinuous at both x = 0 and x = 1.

(D) Rolle's theorem is applicable for g(x) in [0, 1].

Q.10 A vector which is coplanar with vectors kji2 and kji and orthogonal to k6j2i5 lies in the plane

(A) x + y + 3z = 5 (B) 2x + y + 3z = 5 (C) 3x + y + 3z = 5 (D) x + y + 4z = 5

PART-B

[MATRIX TYPE]

Q.1has three/fourstatements (A, B, C OR A, B, C, D) given in Column-I andfour/five statements (P, Q, R,

S ORP, Q, R, S) given in Column-II. Any given statement in Column-I can have correct matching with one or

more statement(s) given in Column-II. [3+3+3=9]

Q.1 Column-I Column-II

(A) Number of five digit numbers in which the sum of their digit is (P) 15

equal to the sum of the square of their digit, is

(B) If the value of the

b

1

4b

dx

1x

xLim equals L, then the value of

L120, is (Q) 16

(C) If the system of equations (R) 18

ax + 3y z = a

2ax y + z = 2 (S) none

bx 2y + z = 1 a

is inconsistent then the sum of all possible values of b (where a [1, 8], a I), is

PART-C

[INTEGER TYPE]

Q.1 to Q.3 are "Integer Type" questions. (The answer to each of the questions are upto 4 digits) [3 5 =15]

Q.1 If true set of values of a (a1, a2) satisfy the condition that the point of local minima and the point oflocal maxima is less than 4 and greater than 2 respectively for the function

f(x) = x33ax2 + 3(a21) x + 1, then find the value of (a12 + a

22).

Q.2 If P and Q are the images of the point R (3, 4) in the line mirror 2xy + 6 4x 3y = 0

then find the inradius of the triangle PQR.

Q.3 Let f be a differentiable function on R and satisfying f(x) = (x2x + 1) ex + x

0

yxdy)y('fe

If f(1) + f '(1) + f '' (1) = ke, where k N, then find k.


4/35PAGE # 4





Q.1 Let f and g be two differentiable functions defined from R R+. If f(x) has a local maximum at x = c

and g(x) has a local minimum at x = c, then h(x) =

)x(g

)x(f

(A) has a local maximum at x = c (B) has a local minimum at x = c(C) is monotonic at x = c (D) has a point of inflection at x = c

Q.2 If O (origin) is a point inside the trianglePQR such that 0ORkOQkOP21

, where k1, k2 are

constants such that 4OQRArea

PQRArea

, then the value of k1

+ k2

is

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

Q.3 Let matrix A =

211

321

zyx

where x, y, z N. If det.(adj.(adj. A)) = 28 34 then the number of such

matrices A, is [Note : adj. A denotes adjoint of square matrix A.](A) 91 (B) 45 (C) 55 (D) 110

[PARAGRAPH TYPE]


Paragraph for Question 4 to 6A line L with slope m > 0 is drawn through P (4, 3) to meet the lines L

1: 3x + 4y + 5 = 0

and L2

: 3x + 4y + 15 = 0 at A and B respectively. From A, a line perpendicular to L isdrawn meeting the line L

2at A

1. Similarly from B, a line perpendicular to L is drawn meeting

the line L1

at B1. A parallelogram AA

1BB

1is formed. The equation of line L is obtained so that

the area of the parallelogram AA1BB1 is least. The figure is given below.

B

B1

CA

D

A1

L2

L1

L

P(4, 3)

Q.4 The equation of line L is(A) 4x 3y 7 = 0 (B) x 7y + 17 = 0 (C) 3x 4y = 0 (D) 7x y 25 = 0

Q.5 If line L is orthogonal to the circle x2 + y26x + 4y 9 = 0, then equals(A) 1 (B) 1 (C) 2 (D) 2

Q.6 If line L is radical axis of two circles S and S' such that circle S has ends of its diameter at(0, 1) and (2, 0) and circle S' passes through (1, 2) then the equation of circle S' is(A) 4x2 + 4y2 + 4x 8y 32 = 0 (B) 4x2 + 4y2 + x 67y 11 = 0(C) 4x2 + 4y2 x + 67y 153 = 0 (D) 4x2 + 4y22x + 4y 10 = 0


5/35PAGE # 5



Q.7 If (2x 1)20(ax + b)20 = (x2 + px + q)10 holds true x R where a, b, p and q are real numbers,then which of the following is/are true?

(A) 2p + 3q = 1 (B) a + 2b = 0 (C)20 20 12a (D) 4q + p = 0

Q.8 Let f (x) =

16x

x12x32

2

, then which of the following statement(s) is(are) true?

(A) )x(fLim4x

does not exist. (B) f(x) is monotonic.

(C) The equation f (x) = k has no solution for exactly two values of k.(D) f(x) is discontinuous at exactly one point.

PART-B

[MATRIX TYPE]

Q.1has three/fourstatements (A, B, C OR A, B, C, D) given in Column-I andfour/five statements (P, Q, R,S ORP, Q, R, S) given in Column-II. Any given statement in Column-I can have correct matching with one ormore statement(s) given in Column-II. [3 + 3 + 3 + 3 = 12]


(A) The value of

0

x7

0

x9

dxex

dxex

2

2

is equal to (P) 4

(B) The maximum value of function f (x) = x33x subject to the condition (Q) 6

x4 + 36 13x2, is (R) 8

(C) A circle passes through the points (2, 2) and (9, 9) and touches the x-axis.

The absolute value of the difference of x-coordinate of the point of contact is (S) 12

(D) Let f(x) = cos1(3x 4x3) then f '

23 has the value equal to (T) 18

PART-C

[INTEGER TYPE]

Q.1 to Q.3 are "Integer Type" questions. (The answer to each of the questions are upto 4 digits) [3 5 = 15]

Q.1 Let f(x) be a differentiable function satisfying x

0

x

0

dtxttandtttan)t(f)x(f = 0

2

x .

Find the number of solutions of the equation f(x) = 0.

Q.2 Let f (x) be a cubic polynomial such that f '' (x) = 12x

4. If f (x) has a local minimum value 0 at x = 1,then find the x-intercept of normal to f (x) at point M whose abscissa is 2.

Q.3 Let f and g be two real-valued differentiable functions on R. If f '(x) = g(x) and g'(x) = f(x) x R

and f(3) = 5, f '(3) = 4 then find the value of )(g)(f 22 .


6/35PAGE # 6





Q.1 The expressions 210

102

9102

8102

2102

1102

010 CCC.......CCC equals

(A) 10! (B) 2

510 C (C) 10C5 (D) 10C5

Q.2 Let f(x) be a function satisfying f '(x) = f(x) with f(0) = 1, and g(x) satisfies f(x) + g(x) = ex (x + 1)2,

then the value of 1

0

x dx)x(g)x(fe is

(A) e (B) e 1 (C)2

e(D) e 2

Q.3 The value of definite integral

dx

xsin1xsin1

x

2

2

, is

(A)2

3 3 (B)3

3 (C)3

2 3 (D)6

3

[PARAGRAPH TYPE]


Paragraph for question nos. 4 to 6

Let kj2ixloga3

, i)x(logb3

+ ( log3

x) kj and c

= kji . Given angle between

a

and b

lies in the range

,2

for every x (0, ).

Q.4 For x = 3, the range of volume of tetrahedron formed by vectors b,a and c is

(A)

3

1,0 (B)

2

1,

3

1(C)

1,

2

1(D) [2, 3]

Q.5 If ca

= cb

, then the value of|ca|

|cb|

is

(A) 2 (B) 3 (C) 5 (D) 7

Q.6 If c)ba(

= b2a

, then the value of accbba

, is(A) 2 (B) 4 (C) 1 (D) 16



Q.7 Let f : R R be defined as f(x) =

x xe

1

e

1

22t1

dt

t1

dt, then

(A) f(x) is aperiodic. (B) )x(ff = f(x) x R.

(C) f(1) + f '(1) =2

. (D) f(x) is unbounded.


7/35PAGE # 7

Q.8 In a triangle ABC, if a = 4, b = 8 and C = 60, then which of the following relations is(are) correct?[Note: All symbols used have usual meaning in triangle ABC.]

(A) The area of triangle ABC is 38 . (B) The value of Asin2 = 2.

(C) Inradius of triangle ABC is33

32

. (D) The length of internal angle bisector of angle C is

3

4.

PART-B[MATRIX TYPE]

Q.1has three/fourstatements (A, B, C OR A, B, C, D) given in Column-I andfour/five statements (P, Q, R,S ORP, Q, R, S) given in Column-II. Any given statement in Column-I can have correct matching with one ormore statement(s) given in Column-II. [4 + 4 + 4 + 4 = 16 + 1 bonus]

Q.1 COLUMN-I COLUMN-II(A) If y = 4x 5 is a tangent to the curve C : y2 = px3 + q (P) 1

at M (2, 3) then the value of (p q) is

(B) The least value of the volume of parallelepiped formed by the vectors (Q) 3

1V = ji ,

2V = kj)cosec2(i and 3V = k

)cosec(2j

where ),0( , is

(C) Number of solutions of the equation 1x

xxcos

11

xxsin= 0, is (R) 5

(D) In a quadrilateral ABCD, if cot A = 4, cot B =2

3and cot C = 5, (S) 6

then the value of 3 Dtan is (T) 9

PART-C[INTEGER TYPE]


Q.1 Let A1 A2 A3 ............... An be a regular polygon of n sides. For some integer k < n, quadrilateralA

1A

2A

kA

k + 1is a rectangle of area 6. If the area of polygon is 60, then find the value of n .

Q.2 If the expression 2

bxtanarcaxtanarc8

x

x

1tanarc

(a, b R) is true x R0,

then find the value of 4(a2 + b2).

Q.3 In a triangle ABC, the internal angle bisector ofABC meets AC at K. If BC = 2, CK = 1 and

BK =2

23, then find the length of side AB.


8/35PAGE # 8





Q.1 The value of expression

8

03 10tan1

1equals

(A) 5 (B)4

21(C)

3

14(D)

2

9

Q.2 Let f be a differentiable function satisfying f '(x) = 2f(x) + 10 and f(0) = 0 then the number of roots

of the equation f(x) + 5 sec2x = 0 in (0, 2) is(A) 0 (B) 1 (C) 2 (D) 3

Q.3 The value of definite integral 2

2

12

1

dx1xx

xtanis equal to

(A)36

2(B)

33

2(C)

312

2(D)

34

2

[PARAGRAPH TYPE]


Paragraph for question nos. 4 and 5

Let 1

denotes the equation of the plane to which the vector 0,1,1 is normal and which contains the

line L whose equation is kjikjir

. 2

denotes the equation of the plane containing

the line L and a point with position vector 0,1,0 .

Q.4 Vector of magnitude 6 along the line of intersection of planes 1 and 2 and perpendicular to normalvector of plane

1, is

(A) 2 1,1,1 (B) 2 1,1,1 (C) 2 1,1,1 (D) 2 1,1,1

Q.5 The acute angle between 1

and 2, is

(A) 12tan1tan 11 (B) 32tan1tan 11

(C) 12tan1tan 11 (D) 32tan1tan 11



Q.6 Let g be a continuous function on [0, 1] such that g(0) = 1 and g(1) = 0.

Which of the following statement(s) is/are always correct?

(A) There exists a number h in [0, 1] such that g(h) g(x) for all x in [0, 1].(B) There exists a number h in [0, 1] such that g(h) = 1/2.

(C) There exists a number h in [0, 1] such that g(h) = 3/2.

(D) For all h in (0, 1), )h(g)x(gLimhx

.


9/35PAGE # 9

Q.7 The equation arc cos x = arc tan x has

(A) only one solution (B) exactly one solution in (0, 1)

(C) exactly two solution in (1, 1) (D) no solution in (1, 0).

[REASONING TYPE]

Q.8 & Q.9 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. [2 3 = 6]

Q.8 Statement 1: There exist exactly one real value of p, for which the equation x33x + p = 0

has two distinct roots in (0, 1).

Statement 2: If the function f(x) is differentiable in [a, b] and f(a) = f(b) then there exists some c in

(a, b) such that f '(c) = 0.



(C) Statement-1 is true, statement-2 is false.

(D) Statement-1 is false, statement-2 is true.

Q.9 Statement 1: Let A =

213

x51x

13x4x9x7

x4xx

2

43

2

and f(x) = tr.(A), where tr.(A) denotes trace

of matrix A , then minimum value of f(x) is 8 for x > 0.

Statement 2: For a, b > 0, ab2

ba

.



(C) Statement-1 is true, statement-2 is false.(D) Statement-1 is false, statement-2 is true.

PART-B

[MATRIX TYPE]



more statement(s) given in Column-II. [3 + 3 + 3 = 9]

Q.1 COLUMN-I COLUMN-II

(A) If curves C1

: y2 = 2ax (a > 0) and C2

: xy = 24 intersect (P) 0

orthogonally, then a equals

(B) If f : R [1, 4] is a differentiable function (Q) 1

such that )x('f)x(fLimx

= 3 then )x(fLimx

is (R) 2

(C) For each m R, the curve y = (m 1)x + (n + 2) always passes through (S) 3a fixed point P. If the ordinate of the point P is 3, then the value of n is


10/35PAGE # 10

PART-C

[INTEGER TYPE]


Q.1 Let K is a positive integer such that 36 + K, 300 + K, 596 + K are the squares of three consecutive

terms of an increasing arithmetic progression, then find the value of K.

Q.2 Let f (x) = x2 + ax + b. If x R, there exist a real value of y such that f (y) = f (x) + y,then find the maximum value of 100a.

Q.3 If the number of solutions satisfying the equation

(sin 1)(2 sin 1)(3 sin 1).......(n sin 1) = 0(where n N) in [0,] is 9, then find the number of ordered pairs (x, y) satisfying the equation

3 + cosec2x + ysin2

2 = n, where 0 x, y 4.

Q.4 Consider the graph of a real-valued continuous function f(x) defined on R (the set of all real numbers)

as shown below.

1 2 3 4 5x

1

1

2

2

345

1

2

3

4

5

O

y

(2,4) (4,4)

(1,2) (2,2)

Find the number of real solutions of the equation f(f(x))= 4.


11/35PAGE # 11





Q.1 If f(x) =

2x,1

2x,

2

x

}xsin{cos

where {k} represents the fractional part of k, then

(A) f(x) is continuous at x =2

. (B)

2x

Lim

f(x) exists, but f is not continuous at x =2

.

(C) )x(fLim

2x

does not exist. (D) 1)x(fLim

2x

.

Q.2 If 1, x, y is a geometric progression and x, y, 3 is an arithmetic progression, then the maximum value

of (x + y) is

(A) 0 (B)2

9(C)

4

15(D) 1

Q.3 Number of six digit numbers in which sum of the squares of the digits is 9 is

(A) 60 (B) 66 (C) 72 (D) 37

[PARAGRAPH TYPE]



Consider, M(x) =

x3

12xx

x2

1xxx

xxx4

1xx3

3

32

22

and P, Q, Rm

, Sk

are other matrices defined as

P = n32

n )0(M..............)0(M)0(M)0(MLim

Q = diag.

2

1,1,

3

1; R

m= M(x) (PQ)m , m N; S

k= M(x)

k

1r

rPQ .

[Note : Tr. (A) denotes trace of square matrix A and adj. A denotes adjoint of square matrix A.]

Q.4 Padjadj.Tr is equal to(A) 6 (B) 18 (C) 36 (D) 54


12/35PAGE # 12

Q.5 If Tr. (R100

) = f(x) then the value of

2

0

dx6

41)x(cosf)x(sinf , is

(A) 2 (B) 4 (C) 6 (D) 10

Q.6 If Tr. (S360

) = g(x) then the minimum value of g(x) is

(A) 570 (B) 690 (C) 36

19

(D) 36

23



Q.7 Let f(x) =22

x2

)x1(

e.)1x(

, then which of the following statement(s) is(are) correct ?

(A) f(x) is strictly increasing in (, 1 ).(B) f(x) is strictly decreasing in (1, ).(C) f(x) has two points of local extremum.

(D) f(x) has a point of local minimum at some x (1, 0).

Q.8 Which of the following conclusion(s) hold(s) true for a non-zero vector a

.

(A) ba

= ca

cb

(B) ba

= ca

cb

(C) ba

= ca

and ba

= ca

cb

(D) ba = ba ba = 0

PART-B

[MATRIX TYPE]

Q.1has three/fourstatements (A, B, C OR A, B, C, D) given in Column-I andfour/five statements (P, Q, R,S ORP, Q, R, S) given in Column-II. Any given statement in Column-I can have correct matching with one or

more statement(s) given in Column-II. [3 + 3 + 3 + 3 = 12]


(A) Let cos sin = sin2 and cos + sin = K cos . (P) 0If K + L = 11 then log

3L lies between two consecutive integers whose sum is

(B) The tangents drawn from origin to a variable circle of radius 2 are always (Q) 1

perpendicular. The locus of centre of the variable circle is a circle whose radius is

(C) If sin1

(1

2x) = 3

sin1

x, then the value of (12x

12x2

) is equal to (R) 2

(D) Given f is an odd function defined everywhere, periodic (S) 3

with period 2 and integrable on every interval.

Let g(x) = x

0

dt)t(f , then the value of g(10) is equal to (T) 5


13/35PAGE # 13

PART-C

[INTEGER TYPE]


Q.1 The circle C : given as x2 + y2 + kx + (k + 1)y (k + 1) = 0 always passes through two fixed points for

every real k. If the minimum value of the radius of the circle C isR

, find the value of R (R N).

Q.2 Let triangle ABC have altitudes ha, h

b, h

cfrom points A, B, C respectively. If h

a= 8, h

b= 8, h

c= 10

then the length of side AB can be expressed asq

p(where p, q are natural numbers).

Find the minimum value of (p + q).

Q.3 Let f(x) be a function satisfying f(x) =

x

100f x > 0. If

10

1

dxx

)x(f= 5 then find the value of

100

1

dxx

)x(f.





Q.1 In a triangle ABC, C = 90, BC = 3, AC = 4 and D is a point on AB so that BCD = 30, thenlength of CD equals

(A) )132(13

8 (B) )132(

9

8 (C) )334(

9

8 (D) )334(

13

8

Q.2 The value of !103

1

!3k!2k!1k

3k100

0k

is equal to

(A) 1 (B)2

1(C) 100 (D) 101

Q.3 If the system of equations x ky z = 0, kx y z = 0, x + y z = 0 has a non-zero solution,

then the possible values of k are(A) 1, 2 (B) 1, 2 (C) 1, 2 (D) 1, 1


14/35PAGE # 14

[PARAGRAPH TYPE]


Paragraph for Question 4 to 6

Let f(x) = x2 + bx + c be such that 1, b, c (taken in order) are in arithmetic progression and

2, 5b, 10c (taken in order) are in geometric progression, where b, c I (the set of all integers).

Given g(x) = (a2 + 1) x2

4a2 x 3 and h(x) = x2 (p 3) x + p, where a, p R

(the set of all real numbers).

Q.4 If M and m are maximum and minimum value of f(x) in interval x [0, 4] then (M + m) equals(A) 9 (B) 3 (C) 0 (D) 4

Q.5 Number of integral values of a for which g(x) < 0 is satisfied for atleast one real x, is

(A) atleast 7 (B) atmost 3 (C) 5 (D) 0

Q.6 If the range of function y = f(x) + h(x) is [0, ) then the true set of real values of p, is(A) {p | p R, 3 p < } (B) {p | p R, 2 p < }(C) {p | p R, < p 3} (D)

[REASONING TYPE]


Q.7 Statement-1 : For any arbitrary real values of a, b and c, the equation

a cos 3x + b cos 2x + c cos x + d sin x = 0 must have atleast one root in [0, 2].

Statement-2 : If q

p

dx)x(f (p < q) vanishes then the equation f(x) = 0 must have atleast one root

in [p, q].

(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.


Q.8 Statement 1: Number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is

empty is 9C3.

Statement 2: Number of ways of choosing any 3 places from 9 different places is 9C3.




Q.9 Statement-1: If the second term of an infinite geometric progression is x and its sum is 8 then the

range of x is (16, 2].Statement-2: Sum of an infinite geometric progression is finite provided 0 < | r | < 1 where r denotes

the common ratio of geometric progression.





15/35PAGE # 15


Q.10 to Q.11 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct.[2 4 = 8]

Q.10 Which of the following statement(s) is/are correct?

(A) The point where the function changes its monotonocity and concavity is always the point of

non-differentiability.

(B) If the derivative of the function at x = a is zero then f(x) can be increasing or decreasing

at that point.

(C) If f(x) is a twice differentiable function on R and has a relative maximum at x = c, then f "(c) is

negative.

(D) If f(x) is a differentiable function on R such that f '() = 0, then f(x) has an extremum at x = ( R).

Q.11 Let a1, a

2, a

3(in order) be three numbers in increasing arithmetic progression and g

1, g

2, g

3

(in order) be three numbers in geometric progression. Given a1

+ g1

= 85, a2

+ g2

= 76, a3

+ g3

= 84

and

3

1ii

a = 126. Then which of the following is(are) correct?

(A) Common difference of arithmetic progression is 25.

(B) Common ratio of geometric progression is41 .

(C) Common ratio of geometric progression is2

1.

(D) Common difference of arithmetic progression is 26.

PART-B

[MATRIX TYPE]



more statement(s) given in Column-II. [3+3+3= 9]


(A) Let A =

122212221

. If adj. A = kAT then the value of 'k' is (P) 2

(B) 'k' is the least positive integer for which the function (Q) 3

f (x) = (2x + 1)50 (3x 4)60 is increasing in [k, ). The value of 'k' is

(C) Let f(x) =

1|x|;bxa

1|x|;|x|

2

(R) 4

where a and b are constants.

If f(x) is differentiable at x = 1, then (2a + b) equals (S) 5


16/35PAGE # 16

PART-C

[INTEGER TYPE]

Q.1 to Q.2 are "Integer Type" questions. (The answer to each of the questions are upto 4 digits)[2 6 = 12]

Q.1 If Sn

denote the sum of n terms of the series 1 2 + 2 3 + 3 4 + ..............

and n 1

that to (n 1) terms of the series ...............6543

1

5432

1

4321

1

Find 118S 1nn .

Q.2 Let f(x) be a differentiable function satisfying

2

yxf =

2

)y(f)x(f x, y R

and f(0) = 0. If

2

0

2dxxsin)x(f is minimised, then find the value of f(42).





Q.1 Let m denotes the number of ways in which 5 boys and 5 girls can be arranged in a line alternately

and n denotes the number of of ways in which 5 boys and 5 girls can be arranged in a circle so that

no two boys are together. If m = kn then the value of k is

(A) 30 (B) 5 (C) 6 (D) 10

Q.2 Let kjia

and kzjyixr

be a variable vector such that ir

, jr

and kr

be

positive integers. If jr

3 and ar

12, then the total number of possible r

is equal to

(A)10

C3 (B)11

C3 (C)13

C4 (D)13

C9Q.3 Let f be an injective function with domain [a, b] and range [c, d]. If is a point in (a, b) such that f has

left hand derivative l and right hand derivative r at x = with both l and r non-zero different andnegative, then left hand derivative and right hand derivative of f1 at x = f() respectively, is

(A)l

1,

r

1(B) r, l (C)

r

1,

1

l(D) l, r



Q.4 Let f(x) =

x2

;xcos1

2x0;xsin

0x1;x10xx 23

then f(x) has

(A) local maximum at x =2

. (B) local minimum at x =

2

.

(C) absolute maximum at x = 0. (D) absolute maximum at x = 1.


17/35PAGE # 17

Q.5 Let f(x) =

xsinsin

2xcoscos

2

11where x [0, 2]. Then which of the following

statement(s) is(are) correct?

(A) f(x) is continuous and differentiable in [0, 2]. (B) Range of f(x) is

4,

4

22

.

(C) f (x) is strictly decreasing in [0, ]. (D)

2

0

dx)x(f =24

3 .

Q.6 In a triangle ABC with usual notation if13

a=

7

b=

15

cholds good then which of the following relations

is/are correct?

[Note: and s denotes area of triangle and semiperimeter of triangle respectively.]

(A) tan

2

CB=

11

34(B) Triangle is obtuse.

(C) r : r1 = 9 : 35 (D) : s2 = 33 : 1

PART-C

[INTEGER TYPE]


Q.1 If the line y = 2 x is tangent to the circle S at the point P(1, 1) and circle S is orthogonal to the

circle x2 + y2 + 2x + 2y 2 = 0, then find the length of tangent drawn from the point (2, 2) to circle S.

Q.2 In an arithmetic sequencen

a , let a1

> 0 and 3a8

= 5a13

. If Sn

be the sum of first n terms,

then find the value of n N for which Sn is maximum.


Q.3 A line y = x + 2 is drawn on the co-ordinate plane. This line is rotated by 90 clockwise about the

point (0, 2). A line y = 2x + 10 is drawn and a triangle is formed by these three lines. If the area of

the triangle is , then find the value [] where [k] denotes the greatest integer less than or equal to k.

Q.4 Leta i j k 3 2 4 ;

b i k 2 and

c i j k 4 2 3 .

If the equation xa y b z c

= x i y j z k has a non trivial solution, then find the sum of all

distinct possible values of

Q.5 Let ac2ycoscbxsinbar

, where c,b,a

are non-zero and non-coplanar vectors.

Ifr

is orthogonal to cba

, then find the minimum value of 220

(x2 + y2).

Q.6 Let f(x) = x2 + ax + 3 and g(x) = x + b, where F(x) =n2

n2

n x1

)x(gx)x(fLim

.

If F(x) is continuous at x = 1 and x = 1 then find the value of (a2 + b2).


18/35PAGE # 18





Q.1 Let A (z1), B (z

2) and C(z

3) be the vertices ofABC such that z

3+ iz

2= (1 + i) z

1, where ( 1) is

a cube root of unity, then ABC is

(A) equilateral (B) isosceles (C) scalene (D) right angled isosceles

Q.2 Two urns contain, respectively m1

and m2

white balls and n1

and n2

black balls. One ball is drawn

at random from each urn and then from the two drawn balls one is taken at random. The probability that

this ball will be white is

(A)

22

22

11

11

nm

nm

nm

nm

2

1(B)

22

2

11

1

nm

m

nm

m

2

1

(C)

22

12

11

21

nm

nm

nm

nm

2

1(D)

22

2

11

1

nm

n

nm

n

2

1

Q.3 If P(x) = (2013)x2012 (2012)x201116x + 8, then P(x) = 0 for

2011

1

8,0x has

(A) exactly one real root. (B) no real root.

(C) atleast one and at most two real roots. (D) atleast two real roots.

Q.4 The radius of circle touching parabola y2 = x at M(1, 1) and having directrix of y2 = x as its normal, is

(A)4

56(B)

4

57(C)

4

55(D)

4

53

[PARAGRAPH TYPE]Q.5 to Q.7 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. [3 3 = 09]

Paragraph for Question no. 5 to 7

The graph of a polynomial f(x) of degree 3 is as shown in the figure and slope of tangent at Q (0, 5) is 3.

x

y

Q 5

4

3

2

1

1 2 3O 1 2

P R

Q.5 Number of solutions of the equation |)x(|f = 3, is

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

Q.6 The equation of normal at the point where curve crosses y-axis, is

(A) 3x + y = 15 (B) x + 3y = 15 (C) x + 3y = 5 (D) 3x + y = 5


19/35PAGE # 19

Q.7 Area bounded by the curve y = f(x) with x axis and lines x + 1 = 0, x 1 = 0 is

(A)2

13(B)

2

15(C)

2

17(D)

2

19


Q.8 and Q.9 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct.[2 4 = 08]

Q.8 If f(x) = sin

12

2

x12

1x1

, then which of the following is(are) correct?

(A) f '(1) =4

1(B) Range of f (x) is

2

,0

(C) f '(x) is an odd function (D)x

)x(fLim

0x=

2

1

Q.9 The possible real values of m for which the simultaneous equations

y = mx + 3 and y = (2m 1)x + 4 are satisfied for atleast one pair of real numbers (x, y) is

(A) 0 (B) 1 (C) 2 (D) 3

PART-C

[INTEGER TYPE]


Q.1 A doctor assumes that a patient has one of three diseases d1, d

2or d

3. Before any test, he assumes an

equal probability for each disease. He carries out a test that will be positive with probability 0.8 if the

patient has d1, 0.6 if he has disease d

2, and 0.4 if he has disease d

3. Given that the outcome of the test

was positive, the probability that the patient has disease d1is

q

p. Find the minimum value of (p + q).

Q.2 If points P, Q and R have position vectors k

j

2i

3r1

, k

4j

3i

r2

and k

2j

i

2r3

respectively, relative to an origin O, then find the distance of P from the plane OQR.

Q.3 Let P1, P

2, ........, P

nbe the points on the ellipse 1

9

y

16

x2

and Q1, Q

2, ......., Q

n

are the corresponding points on the auxiliary circle of the ellipse. If the line joining C to Qi

(C is centre of ellipse) meets the normal at Pi

with respect to the given ellipse at Ki

and

n

1ii

KC = 175, then find the value of n.


20/35PAGE # 20





Q.1 The graph of f (x) = x2 and g (x) = cx3 intersect at two points. If the area of the region over the interval

c

1,0 is equal to

3

2, then the value of

2c

1

c

1, is

(A) 20 (B) 2 (C) 6 (D) 12

Q.2 If the chords of contact of tangents from two points (4, 2) and (2, 1) to the hyperbola 1b

y

a

x2

2

2

2

are at right angle, then the eccentricity of the hyperbola, is

(A)2

7(B)

3

5(C)

2

3(D) 2

Q.3 Suppose that an urn contains 3 balls, one black, one red and one white. A ball is drawn from the urn, itscolour was noted and the ball is replaced into the urn. This process was repeated 5 times and as a result

atleast two red and atleast two white balls were observed. If the probability of this event is243

kthen the

value of k is

(A) 50 (B) 40 (C) 25 (D) 20

Q.4 Let 1

=

333231

232221

131211

aaa

aaa

aaa

, 1 0

2

=

333231

232221

131211

bbb

bbb

bbb

where bij

is cofactor of aij i, j = 1, 2, 3

and 3

=

333231

232221

131211

ccc

ccc

ccc

where cij

is cofactor of bij i, j = 1, 2, 3.

then which one of the following is always correct.

(A) 1, 2, 3 are in A.P. (B) 1, 2, 3 are in G.P.

(C)2

321

(D)

1=

3

2


21/35PAGE # 21

[PARAGRAPH TYPE]


Paragraph for Question no. 5 to 7

Let C1

and C2

be the two curves on the complex plane defined as

C1

: z + z = 2 | z 1 |, C2

: arg (z + 1 + i) = where belongs to the interval (0, ) such that curves C

1and C

2touches each other at P(z

0).

Q.5 The value of | z0

| is

(A) 2 (B) 4 (C) 2 (D) 22

Q.6 A particle starts from a point P(z0). It moves horizontally away from origin by 2 units and then vertically

away from origin by 3 units to reach at a point Q(z1). If z

1= x

1+ i y

1, then (x

1+ y

1) equals

(A) 5 (B) 7 (C) 8 (D) 9

Q.7 If P(z0) is rotated about origin through an angle 2 in clockwise direction then the area bounded by the

C1

and the line joining P(z0) and Q(z

0') is (Where (z

0') is the new position of P(z

0) after rotation.)

(A)3

2sq. units (B) 1 sq. units (C)

3

4sq. units (D)

6

5sq. units

PART-B

[MATRIX TYPE]Q.1 has four statements (A, B, C, D) given in Column-I and five statements (P, Q, R, S, T) given in Column-II.

Any given statement in Column-Ican have correct matching with one or more statement(s) given in Column-II.

[3+3+3+3 = 12]


(A) If the circle passing through A(1, 2), B(2, 3) (P) 0

and having least possible perimeter intersects orthogonally

the circle x2 + y2 + 2x + 2ky = 26, then k equals (Q) 1

(B) If

a0x x

)xcos1cos(1cos1Lim

is finite then the value of a can be (R) 2

(C) If the matrix A =

1053

842

2p31

is non-singular, then p can be (S) 3

(D) If y (t) is a solution of 1tydt

dy)1t( , y (0) = 1, (T) 4

then 3)1(y2 equals

PART-C

[INTEGER TYPE]

Q.1 and .2 are "Integer Type" questions. (The answer to each of the questions are upto 4 digits)[2 5 = 10]

Q.1 Consider an isosceles ABC with AB = AC, BC = 4 and ABC = 30.Three points A

1, B

1and C

1on the incircle S

1ofABC having radius

r1

are taken (as shown in the figure) such that point A1is exactly below

A and A1B

1is parallel to AB and A

1C

1is parallel to that of AC.

The radius of incircle S2

ofA1B

1C

1is r

2. Find (2r

2+ 7r

1).

I

B C

A

B1

C1

A1

Q.2 If dx)x(g1)x(g

xsin8 2

0

2

= 6, where g(x) is a continuous positive function in (0, ),

then find the maximum value of g(x) in (0, ).


22/35PAGE # 22



PART-A



Q.1 If 6

12

C1r

2r 8

rn

n

0r

then the value of n equals(A) 4 (B) 5 (C) 6 (D) 7

Q.2 The value of definite integral dxxsincoscotxcossintan1

1

1111

is equal to

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

Q.3 If be the angle subtended at the focus by the chord which is normal at the point (, ), 0to the parabola y

2

= 4x, then the equation of line making angle with positive x-axisand passing through (1, 2) is

(A) y = 2 (B) x + 2y = 5 (C) x + y = 3 (D) x = 1

[PARAGRAPH TYPE]



There are eight delegates, 4 of them are Americans, 1 British, 1 Chinese, 1 Dutch and 1 Egyptian.

These delegates are paired randomly.

Q.4 The probability that no two delegates of the same country are paired is

(A)35

6(B)

35

8(C)

35

16(D)

35

24

Q.5 The probability that delegates of the same country form two pairs, is

(A)35

3(B)

35

6(C)

35

8(D)

35

2

Q.6 The probability that exactly two delegates of the same country are paired together, is

(A)35

5(B)

35

8(C)

35

24(D) none


23/35PAGE # 23

PART-B

[MATRIX TYPE]

Q.1 has four statements ( A, B, C, D) given in Column-I and five statements (P, Q, R, S, T) given in Column-II.


[3+3+3+3 = 12]


(A) Number of solutions of the equation 1iz

iz 4

, is 1i (P) 1

(B) A chord PQ is a normal to the parabola y2 = 4ax at P (Q) 2

and subtends a right angle at the vertex.

If SQ = SP where S is the focus then the value of , is (R) 3

(C) If 1

xsin

2 xsin1dt)t(ft ,

2

,0x then

2

1f is equal to (S) 4

(D) Number of ordered pairs (x, y) satisfying the equation

4y2 + 2 cos2x = 4y sin2x, where x, y [0, 2], is (T) 5PART-C

[INTEGER TYPE]


Q.1 Consider the locus of the complex number z in the argand plane given by Re(z) 2 = | z 7 + 2i |.

Let P(z1) and Q(z

2) be two complex numbers satisfying the given locus and also satisfying

)i2(z)i2(zarg

2

1 =2 ( R). Find the minimum value of PQ.

[Note: Re(z) denotes real part of complex number z and i2 = 1.]

Q.2 Let An

be the area bounded by the curve y = xn (n 1) and the line x = 0, y = 0 and x =2

1.

If3

1

n

A2n

1n

nn

then find the value of n.

Q.3 If w,v,u

are non-zero and non-coplanar vectors, then find the number of ordered pairs (p, q) so that

wpvpu3

uqwvp

uqvqw2

= 0 p, q R.


24/35PAGE # 24





Q.1 If the function f(x) = x3 ax2 + 2x satisfies the conditions of LMVT over the interval [0, 2]

and the tangent to the curve y = f(x) at x = 2

1

is parallel to the chord that joins the points of intersection

of the curve with ordinates at x = 0 and x = 2, then the value of a is

(A)4

9(B)

4

11(C)

4

13(D)

4

15

Q.2 The point of intersection of the plane 6)k2j5i3(r

with the straight line passing through the

origin and perpendicular to the plane 2x y z = 4, is (x0, y0, z0). The value of (2x03y0 + z0), is

(A) 0 (B) 2 (C) 3 (D) 4

Q.3 Let z and w be complex numbers such that z + w = 0 and z2

+ w2

= 1, then wz equals(A) 1 (B) 2 (C) 22 (D) 2

Q.4 If z be a complex number such that |z 2| + |z 4| = 5, where R+ always represents anellipse then the number of integral values of , is(A) 2 (B) 3 (C) 4 (D) 5

[PARAGRAPH TYPE]



Consider the real valued function f : R R defined as f(x) = x2 ex.

Q.5 Which one of the following statement is true?

(A) f(x) has a local maximum at x = 0 and a local minimum at x = 2.

(B) f(x) has a local minimum at x = 0 and a local maximum at x = 2.

(C) )x(fLimx

= 1.

(D) f(x) is an even function.

Q.6 Let g(x) =

xe2

0

2dt

t1

)t('f, then

(A) g(x) increases on (, 0) and decreases on (0, ).(B) g(x) has a local minimum at x = 0.

(C) g(x) decreases on (, 0) and increases on (0, ).(D) g(x) has neither maximum nor minimum at x = 0.

Q.7 Number of solutions of the equation 4x2 ex1 = 0, is

(A) 0 (B) 1 (C) 2 (D) 3


25/35PAGE # 25


Q.8 and Q.9 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct.[2 4 = 08]

Q.8 Which of the following statement(s) is/are true for any two events A and B?

(A) Suppose that P(A/B) = P(B/A), P(A B) = 1 and P(A B) > 0 then P(A) >2

1.

(B) If P(BC) =

4

1and P(A/B) =

2

1then maximum value of P(A) =

8

5.

(C) If P(A) =2

1and P(B) =

3

1then P(AC BC)C + P(AC BC)C =

3

2.

(D) If A is subset of B then P(B/A) is 1, where P(A) 0.

Q.9 Let first and second row vectors of matrix A be 311r1

and 112r2

and

let the third row vector be in the plane of 21 randr

and perpendicular to 2r

with magnitude 5 ,

then which of the following is/are true?

[Note : Tr. (P) denotes trace of matrix P.]

(A) Tr. (A) = 3(B) Volume of parallelopiped formed by 30equalsrrandr,r 3232

.

(C) Row vectors are linearly dependent.

(D) 133221rrrrrr

= 0

PART-C

[INTEGER TYPE]


Q.1 Find the value of k > 0 so that the area of the bounded region enclosed betwen the parabolas

y = x kx2 and y =k

x2

is maximum.

Q.2 The locus of the point P (3h 2, 3k) where (h, k) lies on the circle x2 + y2 2x 4y 4 = 0,

is another circle. Find its radius.

Q.3 Let an

(n 1) be the value of x for which x2

x

tdte

n

(x > 0) is maximum.

If L = )a(nLimn

nl

then find the value of eL.

Q.4 If A is a square matrix of order 3 such that det.(A) = 2, then find det.((adj. A1)1).

[Note: adj. P denotes adjoint of square matrix P.]


26/35PAGE # 26



PART-A



Q.1 The tangent at a point whose eccentric angle 60 on the ellipse 1b

y

a

x

2

2

2

2

(a > b)

meet the auxiliary circle at L and M. If LM subtends a right angle at the centre, then eccentricity of the

ellipse is

(A)7

1(B)

7

2(C)

7

3(D)

2

1

Q.2 A box contains 100 balls. All number of white or non white balls in the box are equally probable.

A white ball is dropped into the box and the box is shaken. Now a ball is drawn from the box.

The probability that the drawn ball is white, is

(A)101

51(B)

101

50(C)

100

51(D)

51

1

Q.3 If A, B and C are exhaustive events satisfying P(AB C ) =5

1, P(B C) P(AB C) =

15

1

and P(A C) =10

1, then P(C (A B)') is equal to

(A)30

17(B)

30

18(C)

30

19(D)

30

20

Q.4 Let be a complex cube root of unity with 0 < arg() < 2. A fair die is thrown three times.If a, b, c are the numbers obtained on the die, then probability that (a + b + c2) (a + b2 + c) = 1,is equal to

(A)18

1(B)

9

1(C)

36

5(D)

6

1

[PARAGRAPH TYPE]



Let L1

: kj2ikji2r

and L2

: kji2kj33

i8r

where , R

be two lines in space. A line L from the origin meets the lines L1

and L2

at P and Q respectively.

Q.5 Let M (1, 2, 3) be a point in space and N be a point on the line L1, then the value of for which vector

MN is parallel to the plane k3j4ir

= 1, is equal to

(A)4

3(B)

12

7(C)

8

5(D)

9

2


27/35PAGE # 27

Q.6 The equation of the plane passing through the points P and Q and perpendicular to the plane

kjir

+ 1 = 0, is

(A) j3i2r

= 0 (B) j2i3r

= 0 (C) jir

= 0 (D) jir

= 0

Q.7 The volume of tetrahedron OPAB (where O is origin) and A (0, 1, 3) and B (2, 0, 5) is equal to

(A)3

32(B)

2

51(C)

2

17(D)

3

13


Q.8 and Q.9 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct. [2 4 = 8]

Q.8 A coin is tossed three times. Consider the events :

A : Heads on the first toss

B : Tails on the second toss

C : Heads on the third toss

D : Exactly one head turns up

E : All 3 outcomes are same

which of the following statements are correct ?

(A) P(C), P(D) and P(E) (in that order) are in A.P. (B) A, B, C are exhaustive.(C) A, B, C are independent. (D) A, B, C are equally likely.

Q.9 Let > 0, > 0 be roots of the equation x2 + px + q = 0. Also,1

, are the roots of

x2 + p1x + q

1= 0 and ,

1are roots of x2 + p

2x + q

2= 0. Which of the following relations is(are)

correct?

(A) q1

q2

= 1 (B) p1

+ p2

=q

p(q + 1)

(C) qqpqq 22 = 0 (D) (qp1 qp2)2 = (p24q) (q + 1)2

PART-C

[INTEGER TYPE]


Q.1 Let ABC be an acute angled triangle such that a = 14, sin B =13

12and c, a, b (in that order)

form an A.P. Find the radius of the circle inscribed in triangle ABC.

[Note : All symbols used have usual meaning in triangle ABC]

Q.2 Let Sn =

0

n

1

n

+

1

n

2

n

+ ... +

1n

n

n

n

, where n N.

Ifn

1n

S

S =

4

15, then find the sum of all possible values of n.

Q.3 If planes )kji.(r

= 1, )kja2i.(r

= 2 and )kjaia.(r 2

= 3 intersect in a line,

then find the number of real values of a.


28/35PAGE # 28





Q.1 Let x, y R satisfying the equation tan1x + tan1y + tan1(xy) =12

11,

then the value ofdxdy at x = 1 is equal to

(A) 1+2

3(B) 1 +

2

3(C) 1

2

3(D) 1

2

3

Q.2 Let f(x) =

0x;0

0x;|x|x

1sin3

.

Then at x = 0, f has a

(A) local maximum (B) local minimum

(C) neither local maximum nor local minimum (D) point of discontinuity

Q.3 For the primitive integral equation, x dy = y (dx + y dy), y > 0, y (1) = 1 and y () = 3then is equal to(A) 5 (B) 5 (C) 15 (D) 15

Q.4 If M and m are maximum and minimum value of the function f (x) =xtan1

9xtan4xtan2

2

,

then (M + m) equals

(A) 20 (B) 14 (C) 10 (D) 8

[PARAGRAPH TYPE]



Let a hyperbola passes through the focus of the ellipse 16x2 + 25y2 = 400.

The transverse and conjugate axes of this hyperbola coincide with the major and minor axes

of the given ellipse. The eccentricity of the hyperbola is reciprocal of that the ellipse.

Q.5 Which one of the following statement is correct?

(A) Vertices of hyperbola are (3, 0). (B) Distance between foci of hyperbola is 6.

(C) Equation of directrices of hyperbola are x =

9

5 . (D) None

Q.6 Tangents are drawn from any point on the hyperbola to the auxiliary circle of the ellipse,

then the locus of mid-point of chord of contact is

(A)25

yx

16

y

9

x22

222

(B)

22222

25

yx

16

y

9

x

(C)25

yx

16

y

9

x2222

(D)

222

222

25

yx

16

y

9

x


29/35PAGE # 29

Q.7 The area of quadrilateral formed by joining foci of hyperbola and its conjugate hyperbola is

(A) 337 (B) 674 (C) 50 (D) 25


Q.8 and Q.9 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct. [2 4 = 8]

Q.8 A number is chosen at random from the set {1, 2, 3, 4,..., n} . Let E1

be the event that the number drawn

is divisible by 2 and E2

be the event that the number drawn is divisible by 3, then

(A) E1

and E2

are always independent

(B) E1 and E2 are independent if n = 6k (kN)(C) E

1and E

2are independent if n = 6k + 2 (kN)

(D) E1

and E2

are dependent if n = 10

Q.9 If 2xy dy = (x2 + y2 + 1)dx, y(1) = 0 and y (x0) = 3 , then x0 can be

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

PART-C

[INTEGER TYPE]


Q.1 A bag contains 18 coins. 13 of them are fair coins, and the remaining five of the coins are weighted and

are unfair in the way that they have2

1chance of landing head,

3

1chance of landing tail

and6

1chance of landing along its edge. A coin is drawn randomly from the bag and tossed three times.

The probability that it falls head wise on all the three occasions is equal toq

p, where p and q are

coprime. Find the value of (p + q).

Q.2 If the shortest distance between 2y2

2x + 1 = 0 and 2x2

2y + 1 = 0 is d, then find the number ofsolution of the equation | sin | = 22 d in the interval [, 2].

Q.3 A function y = f(x) satisfies x f '(x) 2f(x) = x4 f2(x), x > 0 and f(1) =6.

Find the value of

5

1

3'f .

Q.4 Consider the graph of y = x2. Let A be a point on the graph in the first quadrant.

Let B be the intersection point of the tangent on y = x2 at the point A and the x-axis. If the area of the

figure surrounded by the graph of y = x2 and the segment OA is

q

ptimes as large as the area of the

triangle OAB (where O is origin), then find the least value of (p + q) where p, q N.


30/35PAGE # 30





Q.1 A line L is common tangent to the circle x2 + y2 = 1 and the parabola y2 = 4x.

If is the angle which it makes with the positive x-axis, then tan2 is equal to(A) 2 sin 18 (B) 2 sin 15 (C) cos 36 (D) 2 cos 36

Q.2 Let f : R R be a twice differentiable function satisfying f(2) =1, f '(2) = 4 and

3

2

)x3( f ''(x) dx = 7, then the value of f(3) lies in the interval,

(A) (0, e) (B) (e, e2) (C) (e2, e3) (D) (e3, e4)

Q.3 Number of all possible symmetric matrices of order 3 3 with each entry 0 or 1 and whose

trace equals 1, is

(A) 24 (B) 48 (C) 192 (D) 512

Q.4 If z lies on the curve arg(z + i) =4

, then the minimum value of | z + 4 3i | + | z 4 + 3i | is

[Note : i2 = 1]

(A) 5 (B) 10 (C) 15 (D) 20

[REASONING TYPE]

Q.5 and Q.6 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. [2 3 = 06]

Q.5 Statement-1: Let k

3j

2i

a

and k

j

i

2b

then the vector x

satisfying baxa

and 0xa

is of length 10 .

Statement-2: If r,q,p

are non-zero distinct vectors such that rpqp

, then p

is parallel

to rq

.(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.




Q.6 Let f(x) = 1x sin (x).

Statement-1: f(x) is differentiable for all real x.Statement-2: f(x) has neither local maximum nor local minimum at x = 1.






31/35PAGE # 31



Q.7 Let A(1, 0) and B(2, 0) be two points on the x-axis. A point 'M' is moving in xy-plane

(other than x-axis) in such a way that MBA = 2MAB, then the point 'M' moves along a conicwhose

(A) eccentricity equals 2 (B) vertices (3, 0)

(C) length of latus-rectum equals 6 (D) equation of directrices are x = 2

1

Q.8 Let f(x) =

0x,0

1x0,x2cos1xx1.

If Rolle's theorem is applicable to f (x) for x [0, 1], then can be

(A) 2 (B) 1 (C)2

1(D) 1

Q.9 Let 'L' be the point (t, 2) and 'M ' be a point on the y-axis such that 'LM' has slope t,

then the locus of the midpoint of 'LM' , as t varies over real values, is a parabola, whose

(A) vertex is (0, 2) (B) lengths of latus-rectum is 2

(C) focus is

8

17,0 (D) equation of directrix is 8y 15 = 0

PART-B

[MATRIX TYPE]

Q.1 has three statements (A, B, C) given in Column-I and four statements (P, Q, R, S) given in Column-II.


[3+3+3+3 = 12]

Q.1 Column 1 Column-II

(A) Let z be a non-zero complex number and ( 1) be non-real (P) 2cube root of unity. If area of triangle formed by

A(z), B(z) and C (2z) is 348 , then z equals

(B) A tangent to the circle x2 + y2 = 4 intersects the hyperbola (Q) 4

x22y2 = 2 at P and Q. If locus of mid point of PQ is

(x22y2)2 = (x2 + 4y2), then equals(C) The length of perpendiculars from the foci S and S' on any (R) 6

tangent to ellipse 19

y

4

x22

are a and c respectively,,

then the value of

ac

ac

dx}x2{ is equal to (S) 8

[Note : {k} denotes fractional part of k.]


32/35PAGE # 32

PART-C

[INTEGER TYPE]


Q.1 Let B be a skew symmetric matrix of order 3 3 with real entries. Given I B and I + Bare non-singular matrices. If A = (I + B) (I B)1, where det.(A) > 0, then find the value ofdet.(2A) det. (adj A).[Note : det.(P) denotes determinant of square matrix P and det.(adj (P) denotes determinant

of adjoint of square matrix P respectively.]

Q.2 Let p be the perpendicular distance of point A(2, 3, 1) from the line passing through the point

B(3, 5, 2), which makes equal angles with positive direction of x, y and z axis.

Then find the value of 30p2.

Q.3 Let F(x) = max. (sin x, cos x). Find the value of

10

10

dx)x(F24

.


33/35PAGE # 33

ANSWER KEY

PRACTICE TEST PAPER-1PART-A

Q.1 B Q.2 C Q.3 D Q.4 D

Q.5 B Q.6 B Q.7 B Q.8 A

Q.9 ACD Q.10 ABC

PART-B

Q.1 (A) Q; (B) P; (C) R

PART-C

Q.1 0010 Q.2 0001 Q.3 0009


Q.1 A Q.2 B Q.3 C Q.4 B Q.5 A Q.6 C Q.7 BCD Q.8 BC

PART-B

Q.1 (A) P ; (B) T; (C) S; (D) Q

PART-C

Q.1 0001 Q.2 0086 Q.3 0009


Q.1 C Q.2 A Q.3 B Q.4 B Q.5 D

Q.6 B Q.7 BC Q.8 AB

PART-B

Q.1 (A) T (B) P (C) Q (D) R

PART-C

Q.1 0040 Q.2 0003 Q.3 0003


Q.1 A Q.2 A Q.3 A Q.4 A Q.5 B Q.6 ABD

Q.7 ABD Q.8 D Q.9 A

PART-B

Q.1 (A) R ; (B) S ; (C) Q

PART-C

Q.1 0925 Q.2 0050 Q.3 0020 Q.4 0005 or 0007


Q.1 C Q.2 C Q.3 D Q.4 C Q.5 B Q.6 A

Q.7 AC Q.8 CD

PART-B

Q.1 (A) T ; (B) R ; (C) Q ; (D) P

PART-C

Q.1 0008 Q.2 0061 Q.3 0010


34/35PAGE # 34


Q.1 D Q.2 B Q.3 D Q.4 B Q.5 C Q.6 D Q.7 C

Q.8 A Q.9 D Q.10 AB Q.11 AC

PART-B

Q.1 (A) Q ; (B) P ; (C) S

PART-C

Q.1 0002 Q.2 0003


Q.1 D Q.2 A Q.3 A Q.4 AD Q.5 BCD Q.6 BC

PART-C

Q.1 0002 Q.2 0020 Q.3 0021 Q.4 0007 Q.5 0025 Q.6 0017


PART-AQ.1 B Q.2 B Q.3 D Q.4 C Q.5 D

Q.6 B Q.7 D Q.8 AC Q.9 ACD

PART-C

Q.1 0013 Q.2 0003 Q.3 0025


Q.1 C Q.2 C Q.3 A Q.4 C Q.5 C

Q.6 B Q.7 A PART-B

Q.1 (A) S; (B) P, Q, R, S, T; (C) P,Q,R,S; (D) RPART-C

Q.1 0004 Q.2 0004


Q.1 B Q.2 D Q.3 D Q.4 B Q.5 A

Q.6 C

PART-BQ.1 (A) R ; (B) R ; (C) Q ; (D) Q

PART-C

Q.1 10 Q.2 0002 Q.3 1


Q.1 C Q.2 D Q.3 B Q.4 B Q.5 B

Q.6 A Q.7 D Q.8 ABD Q.9 BCDPART-C

Q.1 0001 Q.2 0009 Q.3 0002 Q.4 0004


35/35


Q.1 B Q.2 A Q.3 C Q.4 C Q.5 B

Q.6 D Q.7 C Q.8 ACD Q.9 ABCDPART-C

Q.1 0004 Q.2 0006 Q.3 0000


Q.1 C Q.2 B Q.3 D Q.4 C Q.5 A

Q.6 B Q.7 C Q.8 BCD Q.9 ABPART-C

Q.1 0009 Q.2 0003 Q.3 0008 Q.4 0005


Q.1 A Q.2 C Q.3 A Q.4 B Q.5 D

Q.6 B Q.7 ACD Q.8 CD Q.9 ACD

PART-B

Q.1 (A) S, (B) Q, (C) QPART-C

Q.1 0007 Q.2 0140 Q.3 0005

practice test papers 1 to 14 e

Documents