Mathematics
SECTION – 1 : (Only One Option Correct Type)
This section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE option is correct.
1. The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has
(A) only purely imaginary roots
(B) all real roots
(C) two real and two purely imaginary roots
(D) neither real nor purely imaginary roots
2. Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is
(A) 1/2
(B) 1/3
(C) 2/3
(D) 3/4
3. Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is
(A) 264
(B) 265
(C) 53
(D) 67
4. In a triangle the sum of two sides is x and the product of the same two sides is y. If x2 – c2 = y, where c is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is
(A) 
(B) 
(C) 
(D) 
5. The common tangents to the circle x2 + y2 = 2 and the parabola y2 = 8x touch the circle at the points P, Q and the parabola at the points R, S. Then the area of the quadrilateral PQRS is
(A) 3
(B) 6
(C) 9
(D) 15
6. The function y = f(x) is the solution of the differential equation 
in (−1, 1) satisfying f(0) = 0. Then 
(A) 
(B) 
(C) 
(D) 
7. Let f : [0, 2] → R be a function which is continuous on [0, 2] and is differentiable on (0, 2) with f(0) = 1. Let 
for x ∈ [0, 2]. If F′(x) = for all x ∈ (0, 2), then F(2) equals
(A) 
(B) 
(C) 
(D) 
8. Coefficient of x11 in the expansion of (1 + x2)4 (1 + x3)7 (1 + x4)12 is
(A) 1051
(B) 1106
(C) 1113
(D) 1120
9. For x (0, π ), the equation sin x + 2 sin 2x – sin 3x = 3 has
(A) infinitely many solutions
(B) three solutions
(C) one solution
(D) no solution
10. The following integral 
is equal to
(A) 
(B) 
(C) 
(D) 
SECTION – 2 : Comprehension Type (Only One Option Correct)
This section contains 3 paragraphs, each describing theory, experiments, data etc. Six questions relate to the three paragraphs with two questions on each paragraph. Each question has only one correct answer among the four given options (A),(B),(C) and (D).
Paragraph For Questions 11 and 12
Box 1 contains three cards bearing numbers 1, 2, 3; box 2 contains five cards bearing numbers 1, 2, 3, 4, 5; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let xi be the number on the card drawn from the ith box, i = 1, 2, 3.
11. The probability that x1 + x2 + x3 is odd, is
(A) 
(B) 
(C) 
(D) 
12. The probability that x1, x2, x3 are in an arithmetic progression, is
(A) 
(B) 
(C) 
(D) 
Paragraph For Questions 13 and 14
Let a, r, s, t be nonzero real numbers. Let P(at2, 2at), Q, R (ar2, 2ar) and (as2, 2as) be distinct points on the parabola y2 = 4ax. Suppose that PQ is the focal chord and lines QR and PK are parallel, where K is the point (2a, 0)
13. The value of r is
(A) 
(B) 
(C) 
(D) 
14. If st = 1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is
(A) 
(B) 
(C) 
(D) 
Paragraph For Questions 15 and 16
Given that for each a ∈ (0, 1)
exists. Let this limit be g(a). In addition, it is given that the function g(a) is differentiable on (0, 1).
15. The value of 
is
(A) π
(B) 2π
(C) π/2
(D) π/4
16. The value of 
is
(A) π/2
(B) π
(C) − π/2
(D) 0
SECTION – 3 : Matching List Type (Only One Option Correct)
This section contains four questions, each having two matching lists. Choices for the correct combination of elements from List-I and List-II are given as options (A),(B),(C) and (D), out of which ONE is correct.
17.
List I List-II
P. The number of polynomials f(x) with non-negative 1. 8
integer coefficients of degree ≤ 2, satisfying f(0) = 0
and 
is
Q. The number of points in the interval 2. 2
at which f(x) = sin(x2) + cos(x2)
attains its maximum value, is
R. 
equals 3. 4
S. 
equals 4. 0
(A) P-3, Q-2, R-4, S-1
(B) P-2, Q-3, R-4, S-1
(C) P-3, Q-2, R-1, S-4
(D) P-2, Q-3, R-1, S-4
18.
List-I List-II
P. Let y(x) = cos(3cos−1x), x ∈ [−1, 1], 1. 1
Then 
equals
Q. Let A1, A2, ……, An (n > 2) be the vertices of a 2. 2
regular polygon of n sides with its its centre at
the origin. Let 
be the position vector of the
point Ak, k = 1, 2, ….. , n. If 
R. If the normal from the point P(h, 1) on the ellipse 3. 8
is perpendicular to the line x + y = 8,
then the value of h is
S. Number of positive solutions satisfying the equation 4. 9
is
(A) P-4, Q-3, R-2, S-1
(B) P-2, Q-4, R-3, S-1
(C) P-4, Q-3, R-1, S-2
(D) P-2, Q-4, R-1, S-3
19. Let f1 : R → R, f2 : [0, ∞) → R, f3 : R → R and f4 : R → [0, ∞) be defined by
List – I List-II
P. f4 is 1. onto but not one-one
Q. f3 is 2. neither continuous nor one-one
R. f2 o f1 is 3. differentiable but not one-one
S. f2 is 4. continuous and one-one
(A) P-3, Q-1, R-4, S-2
(B) P-1, Q-3, R-4, S-2
(C) P-3, Q-1, R-2, S-4
(D) P-1, Q-3, R-2, S-4
20. Let 
k = 1, 2, …9.
List-I List-II
P. For each zk there exists a zj such 1. True
that zk . zj = 1
Q. There exists a k ∈ {1, 2,……,9} such 2. False
that z1 . z = zk has no solution z in the
set of complex numbers.
R. 
equals 3. 1
S. 
equals 4. 2
(A) P-1, Q-2, R-4, S-3
(B) P-2, Q-1, R-3, S-4
(C) P-1, Q-2, R-3, S-4
(D) P-2, Q-1, R-4, S-3
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