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MATHEMATICS
TARGET IIT JEE 2012 XIII (XY)
PRACTICE TEST PAPERS I N D E X PRACTICE TEST PAPER-1 .................................................................. Page-2 PRACTICE TEST PAPER-2 .................................................................. Page-4 PRACTICE TEST PAPER-3 .................................................................. Page-6 PRACTICE TEST PAPER-4 .................................................................. Page-8 PRACTICE TEST PAPER-5 .................................................................. Page-11 PRACTICE TEST PAPER-6 .................................................................. Page-13 PRACTICE TEST PAPER-7 .................................................................. Page-16 PRACTICE TEST PAPER-8 .................................................................. Page-18 PRACTICE TEST PAPER-9 .................................................................. Page-20 PRACTICE TEST PAPER-10 ............................................................... Page-22 PRACTICE TEST PAPER-11 ................................................................ Page-24 PRACTICE TEST PAPER-12 ............................................................... Page-26 PRACTICE TEST PAPER-13 ............................................................... Page-28 PRACTICE TEST PAPER-14 ............................................................... Page-30 ANSWER KEY ........................................................................................ Page-33
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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 ) = 6 is k, then k equals (A) 6 (B) 5 (C) 4 (D) 2
Q.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]
Q.4 to Q.6 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.
Q.4
[3 × 3 = 9]
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. Area of triangle formed by extremum points of P(x), is (A) 1/2 (B) 1/4 (C) 1/8 (D) 1 1
Q.5
The value of definite integral
P( x ) P(x ) dx , is
1
252 452 652 752 (B) (C) (D) 15 15 15 15 Which one of the following statement is correct? (A) P (x) has two relative maximum points and one relative minimum 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.
(A) Q.6
[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, y1, z1), Q (x2, y2, z2) ( x x1 ) ( y y1 ) ( z z1 ) and R (x3, y3, z3) is given by ( x 2 x1 ) ( y 2 y1 ) ( z 2 z1 ) = 0. ( x 3 x1 ) ( y3 y1 ) ( z 3 z1 )
PAGE # 2
(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. [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
Q.10
2 x sin x 12 sin , x 0, 1 Consider the function g defined by g(x) = 1 x x if x 0, 1 0, 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].
A vector which is coplanar with vectors 2ˆi ˆj kˆ and ˆi ˆj kˆ and orthogonal to 5ˆi 2ˆj 6kˆ 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.1 has three/four statements (A, B, C OR A, B, C, D) given in Column-I and four/five statements (P, Q, R, S OR P, 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 (A)
Column-I Number of five digit numbers in which the sum of their digit is equal to the sum of the square of their digit, is
(B)
If the value of the Lim
b
(C)
x
x 4 1 dx equals L, then the value of
Column-II (P) 15 120L , is
(Q)
16
If the system of equations (R) ax + 3y – z = a 2ax – y + z = 2 (S) bx – 2y + z = 1 – a is inconsistent then the sum of all possible values of b (where a [1, 8], a I), is
18
b
1
none
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 of local maxima is less than 4 and greater than – 2 respectively for the function f(x) = x3 – 3ax2 + 3(a2 – 1) x + 1, then find the value of (a12 + a22).
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. x xy 2 x · f ' ( y) dy Let f be a differentiable function on R and satisfying f(x) = – (x – x + 1) e + e 0 If f(1) + f '(1) + f '' (1) = ke, where k N, then find k.
Q.3
PAGE # 3
PRACTICE TEST PAPER-2 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
Let f and g be two differentiable functions defined from R R+. If f(x) has a local maximum at x = c f (x ) and g(x) has a local minimum at x = c, then h(x) = g( x ) (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 triangle PQR such that OP k1 OQ k 2 OR 0 , where k1, k2 are constants such that (A) 2
Area PQR 4 , then the value of k1 + k2 is Area OQR (B) 3 (C) 4
(D) 5
x y z Let matrix A = 1 2 3 where x, y, z N. If det.(adj.(adj. A)) = 28 · 34 then the number of such 1 1 2 matrices A, is [Note : adj. A denotes adjoint of square matrix A.] (A) 91 (B) 45 (C) 55 (D) 110 [PARAGRAPH TYPE] Q.4 to Q.6 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. [3 × 4 = 12] Q.3
Paragraph for Question 4 to 6 A line L with slope m > 0 is drawn through P (4, 3) to meet the lines L1 : 3x + 4y + 5 = 0 and L2 : 3x + 4y + 15 = 0 at A and B respectively. From A, a line perpendicular to L is drawn meeting the line L2 at A1. Similarly from B, a line perpendicular to L is drawn meeting the line L1 at B1. A parallelogram AA1BB1 is formed. The equation of line L is obtained so that the area of the parallelogram AA1BB1 is least. The figure is given below. L B D B1
A1
L2
C A
L1
Q.4 Q.5 Q.6
P(4, 3)
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 If line L is orthogonal to the circle x2 + y2 – 6x + 4y – 9 = 0, then equals (A) – 1 (B) 1 (C) – 2 (D) 2 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 2 2 (C) 4x + 4y – x + 67y – 153 = 0 (D) 4x2 + 4y2 – 2x + 4y – 10 = 0 PAGE # 4
[MULTIPLE CORRECT CHOICE TYPE] Q.7 to Q.8 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct. [2 ×4 = 8] Q.7
Q.8
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) a 20 220 1 (D) 4q + p = 0 3x 2 12 x , then which of the following statement(s) is(are) true? Let f (x) = 2 x 16
(A) Lim f ( x ) does not exist.
(B) f(x) is monotonic.
x4
(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.1 has three/four statements (A, B, C OR A, B, C, D) given in Column-I and four/five statements (P, Q, R, S OR P, 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] Q.1
Column-I
(A)
The value of
x
9
0
x
Column-II 2
· e x dx is equal to
7
(P)
4
2
· e x dx
0
(B)
The maximum value of function f (x) = x3 – 3x subject to the condition x4 + 36 13x2, is
(Q) (R)
6 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
3 has the value equal to Let f(x) = cos–1(3x – 4x3) then f ' (T) 18 2 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]
(D)
Q.1
Q.2 Q.3
x
x
0
0
Let f(x) be a differentiable function satisfying f ( x ) f ( t ) tan t dt tan t x dt = 0 x . 2
Find the number of solutions of the equation f(x) = 0. 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. 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 f 2 ( ) g 2 ( ) .
PAGE # 5
PRACTICE TEST PAPER-3 Time: 60 Min.
M.M.: 58
[SINGLE CORRECT CHOICE TYPE] Q.1 to Q.3 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. Q.1
The expressions
10
C0
2
10
10
1
C2
2
.......
2
10
(A) 10! Q.2
2
C (B) C
10
C8
2
10
C9
(C) – 10C5
5
2
10
C10
[3 × 3 = 9]
2
equals
(D) 10C5
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, 1
e
then the value of
x
f ( x ) g( x ) dx is
0
(A) e
(B) e – 1
(C) x2
Q.3
The value of definite integral
1 sin x 1 sin 2 x
(D) e – 2 dx , is
3 2 3 3 (C) (D) 3 3 6 [PARAGRAPH TYPE] Q.4 to Q.6 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.
(A)
33 2
e 2
(B)
[3 × 2 = 9]
Paragraph for question nos. 4 to 6 Let a log 3 x ˆi 2ˆj kˆ , b (log 3 x ) ˆi + ( log3 x) ˆj kˆ and c = ˆi ˆj kˆ . Given angle between
a and b lies in the range , for every x (0, ). 2 Q.4
For x = 3, the range of volume of tetrahedron formed by vectors a , b and c is 1 (A) 0, 3
Q.5
Q.6
1 (C) , 1 2
1 1 (B) , 3 2
(D) [2, 3]
|bc| If a c = b c , then the value of is |ac| (A) 2 (B) 3 (C) 5 (D) 7 If (a b) c = a 2b , then the value of a b b c c a , is (A) 2 (B) 4 (C) 1 (D) 16
[MULTIPLE CORRECT CHOICE TYPE] Q.7 to Q.8 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct. [2 × 4 = 8] ex
Q.7
dt Let f : R R be defined as f(x) = 2 1 t 1
(A) f(x) is aperiodic. (C) f(1) + f '(1) = . 2
e x
1
dt , then 1 t2
(B) f f ( x ) = f(x) x R. (D) f(x) is unbounded. PAGE # 6
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 8 3 . (B) The value of (C) Inradius of triangle ABC is
2 3 . 3 3
sin 2 A = 2.
(D) The length of internal angle bisector of angle C is
4 . 3
PART-B [MATRIX TYPE] Q.1 has three/four statements (A, B, C OR A, B, C, D) given in Column-I and four/five statements (P, Q, R, S OR P, 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. [4 + 4 + 4 + 4 = 16 + 1 bonus] Q.1 (A) (B)
COLUMN-I COLUMN-II 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 The least value of the volume of parallelepiped formed by the vectors (Q) 3 V = ˆi ˆj , V = ˆi ( 2 cosec ) ˆj kˆ and V = ˆj (2 cosec ) kˆ 1
2
3
where (0, ) , is sin x 1
x 1
(C)
Number of solutions of the equation
(D)
In a quadrilateral ABCD, if cot A = 4, cot B =
cos x x
x 1 = 0, is
3 and cot C = 5, 2
then the value of 3 tan D is
(R)
5
(S)
6
(T)
9
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
Q.2
Q.3
Let A1 A2 A3 ............... An be a regular polygon of n sides. For some integer k < n, quadrilateral A1 A2 Ak Ak + 1 is a rectangle of area 6. If the area of polygon is 60, then find the value of n . 1 x If the expression arc tan arc tan ax arc tan bx (a, b R) is true x R0, 8 2 x 2 2 then find the value of 4(a + b ).
In a triangle ABC, the internal angle bisector of ABC meets AC at K. If BC = 2, CK = 1 and BK =
3 2 , then find the length of side AB. 2
PAGE # 7
PRACTICE TEST PAPER-4 Time: 60 Min.
M.M.: 58 [SINGLE CORRECT CHOICE TYPE]
Q.1 to Q.3 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. 8
Q.1
The value of expression
0
1 tan 3 10
equals
21 14 9 (C) (D) 4 3 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
(A) 5 Q.2
1
[3 × 3 = 9]
(B)
2
Q.3
The value of definite integral
tan 1 x 1 x 2 x 1 dx is equal to 2
2 (A) 6 3
2 (B) 3 3
2 (C) 12 3
2 (D) 4 3
[PARAGRAPH TYPE] Q.4 to Q.5 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.
[2 × 3 = 6]
Paragraph for question nos. 4 and 5 Let 1 denotes the equation of the plane to which the vector 1, 1, 0 is normal and which contains the line L whose equation is r ˆi ˆj kˆ ˆi ˆj kˆ . 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 normal vector of plane 1, is (A) ± 2 1, 1, 1
Q.5
(B) ± 2 1, 1, 1
(C) ± 2 1, 1, 1
(D) ± 2 1, 1, 1
The acute angle between 1 and 2, is (A) tan 1 1 tan 1 (C) tan 1 1 tan 1
2 1 2 1
(B) tan 1 1 tan 1 2 3
2 3
(D) tan 1 1 tan 1
[MULTIPLE CORRECT CHOICE TYPE] Q.6 to Q.7 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct. [2 × 4 = 8] 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), Lim g ( x ) g (h ) . xh
PAGE # 8
Q.7
The equation arc cos x = arc tan x has (A) only one solution (C) exactly two solution in (–1, 1)
(B) exactly one solution in (0, 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. Q.8
[2 × 3 = 6]
Statement 1: There exist exactly one real value of p, for which the equation x3 – 3x + 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. (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.9
x x4 x2 9 3 4 x 4 13 and f(x) = tr.(A), where tr.(A) denotes trace Statement 1: Let A = 7 x x 2 5x 2 x 1 13
of matrix A , then minimum value of f(x) is 8 for x > 0. ab ab . 2 (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.
Statement 2: For a, b > 0,
PART-B [MATRIX TYPE] Q.1 has three/four statements (A, B, C OR A, B, C, D) given in Column-I and four/five statements (P, Q, R, S OR P, 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)
If curves C1 : y2 = 2ax (a > 0) and C2 : xy = 4 2 intersect orthogonally, then a equals
(P)
0
(B)
If f : R [1, 4] is a differentiable function
(Q)
1
such that Lim f ( x ) f ' ( x ) = 3 then Lim f ( x ) is
(R)
2
For each m R, the curve y = (m – 1)x + (n + 2) always passes through a fixed point P. If the ordinate of the point P is 3, then the value of n is
(S)
3
x
(C)
x
PAGE # 9
PART-C [INTEGER TYPE] Q.1 to Q.4 are "Integer Type" questions. (The answer to each of the questions are upto 4 digits) [4 × 5 = 20] 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 + 2sin
Q.4
2
y
= n, where 0 x, y 4.
Consider the graph of a real-valued continuous function f(x) defined on R (the set of all real numbers) as shown below. y 5 (–2,4)
(4,4)
4 3
(–1,2)
2
(2,2)
1 –5
–4
–3
–2
x
–1 O
1
2
3
4
5
–1 –2
Find the number of real solutions of the equation f (f(x)) = 4.
PAGE # 10
PRACTICE TEST PAPER-5 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.
Q.1
sin{cos x} , x If f(x) = 2 1,
x
2
x
2
[3 × 3 = 9]
where {k} represents the fractional part of k, then (A) f(x) is continuous at x = . (B) Lim f(x) exists, but f is not continuous at x = . 2 2 x 2
(C) Lim f ( x ) does not exist. x
Q.2
x
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 9 15 (C) (D) 1 2 4 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
(A) 0 Q.3
(D) Lim f ( x ) 1 .
2
(B)
[PARAGRAPH TYPE] Q.4 to Q.6 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.
[3 × 4 = 12]
Paragraph for question nos. 4 to 6
2 1 3 x x 4 2 Consider, M(x) = x x x
x2 x x
1 2
x3
x3 1 2 x 3 x
and P, Q, Rm, Sk are other matrices defined as P = Lim M (0) M (0) 2 M (0) 3 .............. M (0) n n
k r Rm = M(x) , m N; Sk = M(x) PQ . r 1 [Note : Tr. (A) denotes trace of square matrix A and adj. A denotes adjoint of square matrix A.] 1 1 Q = diag. , 1, ; 3 2
Q.4
Tr. adjadj P is equal to (A) 6 (B) 18
(PQ)m
(C) 36
(D) 54 PAGE # 11
2
Q.5
If Tr. (R100) = f(x) then the value of
f (sin x ) f (cos x )
0
(A) 2 Q.6
(B) 4
(C) 6
41 dx , is 6 (D) 10
If Tr. (S360) = g(x) then the minimum value of g(x) is (A) 570
(B) 690
(C)
19 36
(D)
23 36
[MULTIPLE CORRECT CHOICE TYPE] Q.7 to Q.8 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct. [2 × 4 = 8] Q.7
( x 1) 2 .e x Let f(x) = , then which of the following statement(s) is(are) correct ? (1 x 2 ) 2
(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) a · b = a · c b c (B) a b = a c b c (C) a · b = a · c and a b = a c b c (D) aˆ bˆ = aˆ bˆ aˆ · bˆ = 0
PART-B [MATRIX TYPE] Q.1 has three/four statements (A, B, C OR A, B, C, D) given in Column-I and four/five statements (P, Q, R, S OR P, 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] Q.1
Column-I
Column-II
(A)
Let cos – sin = 2 sin and cos + sin = K cos . (P) If K + L = 11 then log3L lies between two consecutive integers whose sum is
0
(B)
The tangents drawn from origin to a variable circle of radius 2 are always (Q) perpendicular. The locus of centre of the variable circle is a circle whose radius is
1
(C)
If sin–1 (1 – 2x) =
(D)
Given f is an odd function defined everywhere, periodic
– sin–1 x, then the value of (12x – 12x2) is equal to 3
(R)
2
(S)
3
(T)
5
with period 2 and integrable on every interval. x
Let g(x) = f ( t ) dt , then the value of g(10) is equal to 0
PAGE # 12
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
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 is
Q.2
, find the value of R (R N). R
Let triangle ABC have altitudes ha, hb, hc from points A, B, C respectively. If ha = 8, hb = 8, hc = 10 then the length of side AB can be expressed as
p q
(where p, q are natural numbers).
Find the minimum value of (p + q). Q.3
100 Let f(x) be a function satisfying f(x) = f x > 0. If x 100
1
10
f (x ) dx = 5 then find the value of x 1
f (x) dx . x
PRACTICE TEST PAPER-6 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. Q.1
In a triangle ABC, C = 90º, BC = 3, AC = 4 and D is a point on AB so that BCD = 30°, then length of CD equals (A)
Q.2
8 ( 2 3 1) 13
(B)
8 ( 2 3 1) 9
(C)
8 ( 4 3 3) 9
(D)
8 ( 4 3 3) 13
100 k 3 1 is equal to The value of k 0 k 1! k 2! k 3! 103 ! (A) 1
Q.3
[3 × 3 = 9]
(B)
1 2
(C) 100
(D) 101
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
PAGE # 13
[PARAGRAPH TYPE] Q.4 to Q.6 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.
[3 × 3 = 9]
Paragraph for Question 4 to 6 Let f(x) = + 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). x2
2 Given g(x) = (a2 + 1) x2 – a 4 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 to Q.9 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. Q.7
[3 × 3 = 9]
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]. q
Statement-2 : If
f ( x ) dx
(p < q) vanishes then the equation f(x) = 0 must have atleast one root
p
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. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true. 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. (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.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. (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.
PAGE # 14
[MULTIPLE CORRECT CHOICE TYPE] 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, a2, a3 (in order) be three numbers in increasing arithmetic progression and g1, g2, g3 (in order) be three numbers in geometric progression. Given a1 + g1 = 85, a2 + g2 = 76, a3 + g3 = 84 3
and
a i = 126. Then which of the following is(are) correct? i 1
(A) Common difference of arithmetic progression is 25. (B) Common ratio of geometric progression is
1 . 4
1 . 2 (D) Common difference of arithmetic progression is 26.
(C) Common ratio of geometric progression is
PART-B [MATRIX TYPE] Q.1 has three/four statements (A, B, C OR A, B, C, D) given in Column-I and four/five statements (P, Q, R, S OR P, 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 (A) (B)
(C)
1 2 2 1 2 . If adj. A = kAT then the value of 'k' is Let A = 2 2 2 1 'k' is the least positive integer for which the function f (x) = (2x + 1)50 (3x – 4)60 is increasing in [k, ). The value of 'k' is 2 | x | 1 ; Let f(x) = | x | a bx ; | x | 1
where a and b are constants. If f(x) is differentiable at x = 1, then (2a + b) equals
Column-II (P)
2
(Q)
3
(R)
4
(S)
5
PAGE # 15
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
1 1 1 ............... 1· 2 · 3 · 4 2·3· 4·5 3·4·5·6
Find Sn 18 n 1 1 . Q.2
xy Let f(x) be a differentiable function satisfying f = 2 2
and f(0) = 0. If
f (x) sin x
2
f ( x ) f ( y) x, y R 2
dx is minimised, then find the value of f(– 42).
0
PRACTICE TEST PAPER-7 Time: 60 Min.
M.M.: 55 [SINGLE CORRECT CHOICE TYPE]
Q.1 to Q.3 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. Q.1
Q.2
[3 × 4 = 12]
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 Let a ˆi ˆj kˆ and r xˆi yˆj zkˆ be a variable vector such that r · ˆi , r · ˆj and r · kˆ be positive integers. If r · ˆj 3 and r · a 12, then the total number of possible r is equal to (A) 10C3 (B) 11C3 (C) 13C4 (D) 13C9
Q.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 and negative, then left hand derivative and right hand derivative of f–1 at x = f() respectively, is 1 1 (A) , r l
(B) r, l
1 1 (C) , l r
(D) l, r
[MULTIPLE CORRECT CHOICE TYPE] Q.4 to Q.6 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct. [3 × 5 = 15]
Q.4
x 3 x 2 10 x ; Let f(x) = sin x ; 1 cos x ;
1 x 0 0x 2 x 2
then f(x) has . 2 (C) absolute maximum at x = 0.
(A) local maximum at x =
. 2 (D) absolute maximum at x = – 1.
(B) local minimum at x =
PAGE # 16
Q.5
Let f(x) = cos 1 cos x sin 1 sin x where x [0, 2]. Then which of the following 2 2 statement(s) is(are) correct? 2 2 , . (B) Range of f(x) is 4 4
(A) f(x) is continuous and differentiable in [0, 2].
2
3 (D) f ( x ) dx = . 24 0
(C) f (x) is strictly decreasing in [0, ].
Q.6
In a triangle ABC with usual notation if
a b c = = holds good then which of the following relations 13 7 15
is/are correct? [Note: and s denotes area of triangle and semiperimeter of triangle respectively.] BC 4 3 (A) tan = 11 2
(B) Triangle is obtuse.
(C) r : r1 = 9 : 35
(D) : s2 = 3 3 : 1 PART-C [INTEGER TYPE]
Q.1 to Q.6 are "Integer Type" questions. (The answer to each of the questions are upto 4 digits) [2 × 4 = 8] 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 sequence a n , 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 to Q.6 are "Integer Type" questions. (The answer to each of the questions are upto 4 digits)[4 × 5 = 20] Q.3
Q.4
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. Let a 3 i 2 j 4 k ; b 2 i k and c 4 i 2 j 3 k .
If the equation x a y b z c = x i y j z k has a non trivial solution, then find the sum of all
Q.5
distinct possible values of Let r a b sin x b c cos y 2 c a , where a , b, c are non-zero and non-coplanar vectors.
20 If r is orthogonal to a b c , then find the minimum value of 2 (x2 + y2). Q.6
Let f(x) = x2 + ax + 3 and g(x) = x + b, where F(x) = Lim
f (x ) x 2n g(x )
1 x 2n If F(x) is continuous at x = 1 and x = – 1 then find the value of (a2 + b2). n
.
PAGE # 17
PRACTICE TEST PAPER-8 Time: 60 Min.
M.M.: 44 [SINGLE CORRECT CHOICE TYPE]
Q.1 to Q.4 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.
[4 × 3 = 12]
Q.1
Let A (z1), B (z2) and C(z3) be the vertices of ABC such that z3 + iz2 = (1 + i) z1, 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
Q.3
Q.4
m2 n 2 1 m1n1 (A) 2 m n m n 2 2 1 1
m2 1 m1 (B) 2 m n m n 1 1 2 2
m 2 n1 1 m1n 2 (C) 2 m n m n 1 1 2 2
n2 1 n1 (D) 2 m n m n 1 1 2 2
1 2011 has If P(x) = – – 16x + 8, then P(x) = 0 for x 0, 8 (A) exactly one real root. (B) no real root. (C) atleast one and at most two real roots. (D) atleast two real roots.
(2013)x2012
(2012)x2011
The radius of circle touching parabola y2 = x at M(1, 1) and having directrix of y2 = x as its normal, is (A)
6 5 4
7 5 5 5 (C) 4 4 [PARAGRAPH TYPE]
(B)
(D)
3 5 4
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. y
P
Q 5 4 3 2 1
–2–1
O1 2
R
x
3
Q.5
Number of solutions of the equation f (| x |) = 3, is
Q.6
(A) 1 (B) 2 (C) 3 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) 4 (D) 3x + y = 5 PAGE # 18
Q.7
Area bounded by the curve y = f(x) with x axis and lines x + 1 = 0, x – 1 = 0 is (A)
13 2
(B)
15 2
(C)
17 2
(D)
19 2
[MULTIPLE CORRECT CHOICE TYPE] 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–1
(A) f '(–1) =
1 x 2 1 2 1 x2
, then which of the following is(are) correct?
1 4
(C) f '(x) is an odd function Q.9
(B) Range of f (x) is 0, 2
(D) Lim x0
f (x) 1 = x 2
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 to Q.3 are "Integer Type" questions. (The answer to each of the questions are upto 4 digits) [3 × 5 = 15] Q.1
A doctor assumes that a patient has one of three diseases d1, d2 or d3. 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 d2, and 0.4 if he has disease d3. Given that the outcome of the test p was positive, the probability that the patient has disease d1 is . Find the minimum value of (p + q). q
Q.2
If points P, Q and R have position vectors r1 3ˆi 2ˆj kˆ , r2 ˆi 3ˆj 4kˆ and r3 2ˆi ˆj 2kˆ respectively, relative to an origin O, then find the distance of P from the plane OQR.
Q.3
x y2 1 and Q1, Q2, ......., Qn 16 9 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
Let P 1, P 2, ........, P n be the points on the ellipse
n
C Ki
= 175, then find the value of n.
i 1
PAGE # 19
PRACTICE TEST PAPER-9 Time: 60 Min.
M.M.: 43 [SINGLE CORRECT CHOICE TYPE]
Q.1 to Q.4 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. 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 1 1 2 1 0, c is equal to 3 , then the value of c c 2 , is (A) 20 (B) 2 (C) 6
Q.2
[4 × 3 = 12]
(D) 12
If the chords of contact of tangents from two points (– 4, 2) and (2, 1) to the hyperbola
x2 a2
y2 b2
1
are at right angle, then the eccentricity of the hyperbola, is (A) Q.3
7 2
(B)
5 3
(C)
3 2
(D)
2
Suppose that an urn contains 3 balls, one black, one red and one white. A ball is drawn from the urn, its colour 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 is value of k is (A) 50
Q.4
Let
and
(B) 40
a11 a12 1 = a 21 a 22 a 31 a 32
(C) 25
k then the 243
(D) 20
a13 a 23 , 1 0 a 33
b11 b12 b 2 = 21 b 22 b 31 b 32
b13 b 23 where bij is cofactor of aij i, j = 1, 2, 3 b 33
c11 c12 3 = c 21 c 22 c 31 c 32
c13 c 23 c 33
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. 2 (C) 1
3 2
(D) 1 =
2 3
PAGE # 20
[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 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 C1 and C2 touches each other at P(z0). Q.5
The value of | z0 | is (D) 2 2 2 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 z1 = x1 + i y1, then (x1 + y1) equals (A) 5 (B) 7 (C) 8 (D) 9 (A) 2
Q.6
Q.7
(B) 4
(C)
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(z0') is (Where (z0') is the new position of P(z0) after rotation.) (A)
2 sq. units 3
4 sq. units 3 PART-B [MATRIX TYPE]
(B) 1 sq. units
(C)
(D)
5 sq. units 6
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-I can have correct matching with one or more statement(s) given in Column-II. Q.1
[3+3+3+3 = 12] Column-II (P) 0
Column-I If the circle passing through A(1, 2), B(2, 3) and having least possible perimeter intersects orthogonally the circle x2 + y2 + 2x + 2ky = 26, then k equals
(A)
1 cos1 cos(1 cos x ) is finite then the value of a can be x 0 xa 1 3 p 2 8 is non-singular, then p can be If the matrix A = 2 4 3 5 10 If Lim
(B)
(C) (D)
If y (t) is a solution of ( t 1)
dy ty 1 , y (0) = – 1, dt
(Q)
1
(R)
2
(S)
3
(T)
4
then 2 y(1) 3 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
A1 I B1 B
C1 C
8 sin 2 x 1 g ( x ) 2 dx = 6, where g(x) is a continuous positive function in (0, ), If g(x ) 0 then find the maximum value of g(x) in (0, ).
Q.2
A
Consider an isosceles ABC with AB = AC, BC = 4 and ABC = 30°. Three points A1, B1 and C1 on the incircle S1 of ABC having radius r1 are taken (as shown in the figure) such that point A1 is exactly below A and A1B1 is parallel to AB and A1C1 is parallel to that of AC. The radius of incircle S2 of A1B1C1 is r2. Find (2r2 + 7r1).
PAGE # 21
PRACTICE TEST PAPER-10 Time: 60 Min.
M.M.: 45 PART-A [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 = 09]
28 1 r2 n C then the value of n equals r 6 r 0 r 1 n
Q.1
If
(A) 4
(B) 5
(C) 6
(D) 7
1
Q.2
The value of definite integral
tan
1
1
1
1
sin cos x cot cossin x dx is equal to
1
(A) 0 Q.3
(B) – 1
(C) 1
(D)
If be the angle subtended at the focus by the chord which is normal at the point (, ), 0 to the parabola y2 = 4x, then the equation of line making angle with positive x-axis and passing through (1, 2) is (A) y = 2 (B) x + 2y = 5 (C) x + y = 3 (D) x = 1 [PARAGRAPH TYPE]
Q.4 to Q.6 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.
[3 × 3 = 09]
Paragraph for question nos. 4 to 6 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)
Q.5
(B)
8 35
(C)
16 35
(D)
24 35
(D)
2 35
The probability that delegates of the same country form two pairs, is (A)
Q.6
6 35
3 35
(B)
6 35
(C)
8 35
The probability that exactly two delegates of the same country are paired together, is (A)
5 35
(B)
8 35
(C)
24 35
(D) none
PAGE # 22
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-I can have correct matching with one or more statement(s) given in Column-II. Q.1
[3+3+3+3 = 12] Column-II
Column-I 4
zi Number of solutions of the equation 1 , is i 1 z i
(A)
A chord PQ is a normal to the parabola y2 = 4ax at P and subtends a right angle at the vertex. If SQ = SP where S is the focus then the value of , is
(B)
1
(C)
t
If
sin x
2
1 f ( t ) dt 1 sin x , x 0, then f is equal to 2 2
(P)
1
(Q)
2
(R)
3
(S)
4
(D)
Number of ordered pairs (x, y) satisfying the equation 4y2 + 2 cos2x = 4y – sin2x, where x, y [0, 2], is (T) 5 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
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(z2) be two complex numbers satisfying the given locus and also satisfying z ( 2 i) = arg 1 2 ( R). Find the minimum value of PQ. z ( 2 i ) 2 [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 = n
If
n 1
Q.3
2n A n n
1 . 2
1 then find the value of n. 3
If u , v, w are non-zero and non-coplanar vectors, then find the number of ordered pairs (p, q) so that 3u pv pw – pv w qu – 2w qv qu = 0 p, q R.
PAGE # 23
PRACTICE TEST PAPER-11 Time: 60 Min.
M.M.: 49 [SINGLE CORRECT CHOICE TYPE]
Q.1 to Q.4 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. Q.1
[4 × 3 = 12]
If the function f(x) = x3 – ax2 + 2x satisfies the conditions of LMVT over the interval [0, 2] 1 is parallel to the chord that joins the points of intersection 2 of the curve with ordinates at x = 0 and x = 2, then the value of a is
and the tangent to the curve y = f(x) at x =
(A)
9 4
(B)
11 4
(C)
13 4
15 4
(D)
Q.2
The point of intersection of the plane r · (3ˆi 5ˆj 2kˆ ) 6 with the straight line passing through the origin and perpendicular to the plane 2x – y – z = 4, is (x0, y0, z0). The value of (2x0 – 3y0 + 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 z w (A) 1
Q.4
(B)
2
(C) 2 2
equals
(D) 2
If z be a complex number such that |z – 2| + |z – 4| = 5, where R+ always represents an ellipse then the number of integral values of , is (A) 2 (B) 3 (C) 4 (D) 5 [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 nos. 5 to 7 Consider the real valued function f : R R defined as f(x) = x2 e–x. 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) Lim f ( x ) = 1. x
(D) f(x) is an even function. 2e x
Q.6
Let g(x) =
0
f ' (t ) dt , then 1 t2
(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 e–x – 1 = 0, is (A) 0 (B) 1 (C) 2
(D) 3
PAGE # 24
[MULTIPLE CORRECT CHOICE TYPE] 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) > (B) If P(BC) =
1 . 2
1 1 5 and P(A/B) = then maximum value of P(A) = . 4 2 8
1 1 2 and P(B) = then P(AC BC)C + P(AC BC)C = . 2 3 3 (D) If A is subset of B then P(B/A) is 1, where P(A) 0.
(C) If P(A) =
Q.9
Let first and second row vectors of matrix A be r1 1 1 3 and r2 2 1 1 and let the third row vector be in the plane of r1 and r2 and perpendicular to r2 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 r2 , r3 and r2 r3 equals 30 . (C) Row vectors are linearly dependent. (D) r1 r2 r2 r3 r3 r1 = 0
PART-C [INTEGER TYPE] Q.1 to Q.4 are "Integer Type" questions. (The answer to each of the questions are upto 4 digits) [4 × 5 = 20] 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 =
x2 is maximum. k
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
2x
e
tn
dt (x > 0) is maximum.
x
If L = Lim ln (a n ) then find the value of e– L. n
Q.4
If A is a square matrix of order 3 such that det.(A) = 2, then find det.((adj. A–1)–1). [Note: adj. P denotes adjoint of square matrix P.]
PAGE # 25
PRACTICE TEST PAPER-12 Time: 60 Min.
M.M.: 44 PART-A [SINGLE CORRECT CHOICE TYPE]
Q.1 to Q.4 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. Q.1
Q.2
x 2 y2 The tangent at a point whose eccentric angle 60° on the ellipse 2 2 1 (a > b) 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 1 2 3 1 (A) (B) (C) (D) 7 7 7 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)
Q.3
[4 × 3 = 12]
51 101
(B)
50 101
(C)
51 100
If A, B and C are exhaustive events satisfying P(A B C ) = and P(A C) =
(D)
1 51
1 1 , P(B C) – P(A B C) = 5 15
1 , then P(C (A B)' ) is equal to 10
17 18 19 20 (B) (C) (D) 30 30 30 30 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) Q.4
(A)
1 18
(B)
1 9
(C)
5 36
(D)
1 6
[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 nos. 5 to 7 8ˆi Let L1 : r 2ˆi ˆj kˆ ˆi 2ˆj kˆ and L2 : r 3ˆj kˆ 2 ˆi ˆj kˆ where , R 3 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 r · ˆi 4ˆj 3kˆ = 1, is equal to
(A)
3 4
(B)
7 12
(C)
5 8
(D)
2 9
PAGE # 26
Q.6
The equation of the plane passing through the points P and Q and perpendicular to the plane ˆ ˆ ˆ r · i j k + 1 = 0, is (A) r · 2ˆi 3ˆj = 0 (B) r · 3ˆi 2ˆj = 0 (C) r · ˆi ˆj = 0 (D) r · ˆi ˆj = 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)
32 3
(B)
51 2
(C)
17 2
(D)
13 3
[MULTIPLE CORRECT CHOICE TYPE] 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
Q.9
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. (C) A, B, C are independent.
(B) A, B, C are exhaustive. (D) A, B, C are equally likely. 1 Let > 0, > 0 be roots of the equation x2 + px + q = 0. Also, , are the roots of 1 x2 + p1x + q1 = 0 and , are roots of x2 + p2x + q2 = 0. Which of the following relations is(are) correct? p (q + 1) q (D) (qp1 – qp2)2 = (p2 – 4q) (q + 1)2
(A) q1 q2 = 1 (C)
(B) p1 + p2 =
q q2 p q2 q = 0
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 ABC be an acute angled triangle such that a = 14, sin B =
12 and c, a, b (in that order) 13
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
n Let Sn = 0
If Q.3
n n 1 + 1
n 2 + ... +
n n 1
n n , where n N.
Sn 1 15 Sn = 4 , then find the sum of all possible values of n.
If planes r .(ˆi ˆj kˆ ) = 1, r .(ˆi 2aˆj kˆ ) = 2 and r .(aˆi a 2ˆj kˆ ) = 3 intersect in a line,
then find the number of real values of a.
PAGE # 27
PRACTICE TEST PAPER-13 Time: 60 Min.
M.M.: 49 [SINGLE CORRECT CHOICE TYPE]
Q.1 to Q.4 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. Q.1
L et x, y R satisfying the equation tan–1x + tan–1y + tan–1(xy) =
then the value of (A) 1+
Q.2
Q.3
Q.4
3 2
dy at x = 1 is equal to dx 3 (B) –1 + 2
1 3 sin | x | ; x 0 x Let f(x) = . 0 ; x0 Then at x = 0, f has a (A) local maximum (C) neither local maximum nor local minimum
(C) –1 –
3 2
[4 × 3 = 12]
11 , 12
3 2
(D) 1 –
(B) local minimum (D) point of discontinuity
For the primitive integral equation, x dy = y (dx + y dy), y > 0, y (1) = 1 and y () = – 3 then is equal to (A) – 5 (B) 5 (C) 15 (D) –15 tan 2 x 4 tan x 9 If M and m are maximum and minimum value of the function f (x) = , 1 tan 2 x then (M + m) equals (A) 20 (B) 14 (C) 10 (D) 8 [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 nos. 5 to 7 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.
5 (C) Equation of directrices of hyperbola are x = . (D) None 9
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 2
2
x 2 y2 x 2 y2 (A) 25 9 16
x2 y 2 x 2 y 2 (B) 16 25 9
x2 y 2 x 2 y 2 (C) 9 16 25
x2 y 2 x 2 y 2 (D) 16 25 9
2
2
PAGE # 28
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
[MULTIPLE CORRECT CHOICE TYPE] 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 (k N) (C) E1 and E2 are independent if n = 6k + 2 (k N) (D) E1 and E2 are dependent if n = 10
Q.9
If 2xy dy = (x2 + y2 + 1)dx, y(1) = 0 and y (x0) = (A) 2
(B) – 2
(C) 3
3 , then x0 can be (D) – 3
PART-C [INTEGER TYPE] Q.1 to Q.4 are "Integer Type" questions. (The answer to each of the questions are upto 4 digits) [4 × 5 = 20] 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 have
1 1 chance of landing head, chance of landing tail 2 3
1 chance of landing along its edge. A coin is drawn randomly from the bag and tossed three times. 6 p The probability that it falls head wise on all the three occasions is equal to , where p and q are q coprime. Find the value of (p + q). If the shortest distance between 2y2 – 2x + 1 = 0 and 2x2 – 2y + 1 = 0 is d, then find the number of
and
Q.2
solution of the equation | sin | = 2 2 d in the interval [– , 2]. Q.3
A function y = f(x) satisfies x f '(x) – 2f(x) = x4 f 2(x), x > 0 and f(1) = – 6.
1 Find the value of f ' 35 . 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 p figure surrounded by the graph of y = x2 and the segment OA is times as large as the area of the q
triangle OAB (where O is origin), then find the least value of (p + q) where p, q N.
PAGE # 29
PRACTICE TEST PAPER-14 Time: 60 Min.
M.M.: 57 [SINGLE CORRECT CHOICE TYPE]
Q.1 to Q.4 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct. 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
[4 × 3 = 12]
3
(3 x ) f ''(x) dx = 7, then the value of f(3) lies in the interval, 2
(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) = [Note : i2 = – 1] (A) 5
(B) 10
, then the minimum value of | z + 4 – 3i | + | z – 4 + 3i | is 4
(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 a ˆi 2ˆj 3kˆ and b 2ˆi ˆj kˆ then the vector x satisfying a x a b and a · x 0 is of length 10 . Statement-2: If p, q, r are non-zero distinct vectors such that p q p r , then p is parallel to q r . (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.6
Let f(x) = x 1 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. (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.
PAGE # 30
[MULTIPLE CORRECT CHOICE TYPE] Q.7 to Q.9 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct. [3 × 4 = 12] 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 conic whose (A) eccentricity equals 2 (B) vertices (±3, 0) (C) length of latus-rectum equals 6
Q.8
1 2
1 x · x · 1 cos 2x , 0 x 1 Let f(x) = . x0 0, If Rolle's theorem is applicable to f (x) for x [0, 1], then can be
(A) – 2
Q.9
(D) equation of directrices are x = ±
(B) – 1
(C)
1 2
(D) 1
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 17 (C) focus is 0, 8
(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. Any given statement in Column-I can have correct matching with one or more statement(s) given in Column-II. Q.1 (A)
Column 1 Let z be a non-zero complex number and ( 1) be non-real cube root of unity. If area of triangle formed by
[3+3+3+3 = 12] Column-II (P) 2
A(z), B(z) and C (2z) is 48 3 , then z equals (B)
A tangent to the circle x2 + y2 = 4 intersects the hyperbola x2 – 2y2 = 2 at P and Q. If locus of mid point of PQ is (x2 – 2y2)2 = (x2 + 4y2), then equals
(Q)
4
(C)
The length of perpendiculars from the foci S and S' on any
(R)
6
(S)
8
2
2
x y 1 are a and c respectively,, 4 9
tangent to ellipse ac
then the value of
{2x}dx
is equal to
ac
[Note : {k} denotes fractional part of k.] PAGE # 31
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 B be a skew symmetric matrix of order 3 × 3 with real entries. Given I – B and I + B are non-singular matrices. If A = (I + B) (I – B)–1, where det.(A) > 0, then find the value of det.(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
4 2
10
F(x ) dx
.
10
PAGE # 32
ANSWER
Q.1 Q.5 Q.9 Q.1 Q.1
KEY
PRACTICE TEST PAPER-1 PART-A B Q.2 C Q.3 D Q.4 D B Q.6 B Q.7 B Q.8 A ACD Q.10 ABC PART-B (A) Q; (B) P; (C) R PART-C 0010 Q.2 0001 Q.3 0009
Q.1
PRACTICE TEST PAPER-2 PART-A A Q.2 B Q.3 C Q.4 B Q.5 A Q.6 C Q.7 BCD Q.8 BC PART-B (A) P ; (B) T; (C) S; (D) Q
Q.1
0001
Q.2
C B
PRACTICE TEST PAPER-3 PART-A A Q.3 B Q.4 B Q.5 D BC Q.8 AB
Q.1
Q.1 Q.6
Q.2 Q.7
0086
PART-C Q.3 0009
PART-B Q.1
(A) T (B) P (C) Q (D) R
Q.1
0040
Q.1 Q.7 Q.1 Q.1
Q.2
0003
PART-C Q.3 0003
PRACTICE TEST PAPER-4 PART-A A Q.2 A Q.3 A Q.4 A Q.5 B Q.6 ABD Q.8 D Q.9 A PART-B (A) R ; (B) S ; (C) Q PART-C 0925 Q.2 0050 Q.3 0020 Q.4 0005 or 0007
Q.2 Q.8
C CD
PRACTICE TEST PAPER-5 PART-A Q.3 D Q.4 C Q.5 B Q.6
Q.1 Q.7
C AC
Q.1
(A) T ; (B) R ; (C) Q ; (D) P
ABD
A
PART-B
Q.1
0008
Q.2
0061
PART-C Q.3 0010 PAGE # 33
Q.1 Q.8 Q.1
PRACTICE TEST PAPER-6 PART-A D Q.2 B Q.3 D Q.4 B Q.5 C Q.6 A Q.9 D Q.10 AB Q.11 AC PART-B (A) Q ; (B) P ; (C) S
D
Q.7
C
Q.5
D
C
Q.5
C
PRACTICE TEST PAPER-10 PART-A Q.2 D Q.3 D Q.4 B
Q.5
A
Q.5
B
PART-C Q.1
Q.1 Q.1
0002
D 0002
Q.1 Q.6
B B
Q.1
0013
Q.2
Q.2 Q.2
A
0003
PRACTICE TEST PAPER-7 PART-A Q.3 A Q.4 AD Q.5 BCD Q.6
0020
Q.3
PART-C 0021 Q.4 0007
Q.5
0025
Q.6
PRACTICE TEST PAPER-8 PART-A Q.2 B Q.3 D Q.4 C Q.7 D Q.8 AC Q.9 ACD PART-C Q.2 0003 Q.3 0025
BC 0017
PRACTICE TEST PAPER-9 PART-A Q.1 Q.6
C B
Q.2 Q.7
C A
Q.3
A
Q.4
PART-B Q.1 Q.1
Q.1 Q.6
(A) S; (B) P, Q, R, S, T; (C) P,Q,R,S; (D) R PART-C 0004 Q.2 0004
B C
PART-B Q.1
(A) R ; (B) R ; (C) Q ; (D) Q PART-C
Q.1
10
Q.1 Q.6
C A
Q.1
0001
Q.2
0002
Q.3
1
PRACTICE TEST PAPER-11 PART-A Q.2 D Q.3 B Q.4 B Q.7 D Q.8 ABD Q.9 BCD PART-C Q.2
0009
Q.3
0002
Q.4
0004
PAGE # 34
Q.1 Q.6
B D
Q.1
0004
Q.1 Q.6
C B
Q.1
0009
Q.1 Q.6 Q.1 Q.1
PRACTICE TEST PAPER-12 PART-A Q.2 A Q.3 C Q.4 C Q.7 C Q.8 ACD Q.9 ABCD PART-C Q.2 0006 Q.3 0000
PRACTICE TEST PAPER-13 PART-A Q.2 B Q.3 D Q.4 C Q.7 C Q.8 BCD Q.9 AB PART-C Q.2 0003 Q.3 0008 Q.4 0005
PRACTICE TEST PAPER-14 PART-A A Q.2 C Q.3 A Q.4 B B Q.7 ACD Q.8 CD Q.9 ACD PART-B (A) S, (B) Q, (C) Q PART-C 0007 Q.2 0140 Q.3 0005
Q.5
B
Q.5
A
Q.5
D
PAGE # 35
