PART III : MATHEMATICS
Section-I
(Single Correct Answer Type)
This section contains 7 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.
1. Let (x0, y0) be the solution of the following equations
(2x)ln 2 = (3y)ln 3
3ln x = 2ln y.
Then x0 is
(A) 1/6
(B) 1/3
(C) 1/2
(D) 6
is
3. Let 
and 
(A) 
(B) 
(C) 
(D) 
4. Let P = {θ : sin θ – cos θ = √2 cos θ} and Q = {θ : sin θ + cos θ = √2 sin θ} be two sets. Then
(A) P ⊂ Q and Q – P ≠ ∅
(B) Q ⊄ P
(C) P ⊄ Q
(D) P = Q
5. Let the straight line x = b divide the area enclosed by y = (1 – x)2, y = 0, and x = 0 into two parts R1 (0 ≤ x ≤ b) and R2(b ≤ x ≤ 1) such that R1 – R2 = 1/4. Then b equals
(A) 3/4
(B) 1/2
(C) 1/3
(D) 1/4
6. Let α and β be the roots of x2 – 6x – 2 = 0, with α > β. If an = αn – βn for n ≥ then the value of 
is
(A) 1
(B) 2
(C) 3
(D) 4
7. A straight line L through the point (3, −2) is inclined at an angle 60° to the line √3x + y = 1. If L also interests the x-axis, then the equation of L is
(A) y + √3x + 2 – 3√3 = 0
(B) y – √3x + 2 + 3√3 = 0
(C) √3y – x + 3 + 2√3 = 0
(D) √3y + x – 3 + 2√3 = 0
Section-II
(Multiple Correct Answers Type)
This section contains 4 multiple choice questions. Each question has four choices (A), (B) and (D) out of which ONE or MORE may be correct.
8. The vector(s) which is/are coplanar with vectors 
and perpendicular to the vector 
is/are
(A) 
(B) 
(C) 
(D) 
9. Let M and N be two 3 × 3 non-singular skew-symmetric matrices such that MN = NM. If PT denotes the transpose of P, then M2N2 (MTN)−1 (MN−1)T is equal to
(A) M2
(B) −N2
(C) −M2
(D) MN
10. Let the eccentricity of the hyperbola 
be reciprocal to that of the ellipse x2 + 4y2 = 4. If the hyperbola passes through a focus of the ellipse, then
(A) the equation of the hyperbola is 
(B) a focus of the hyperbola is (2, 0)
(C) the eccentricity of the hyperbola is 
(D) the equation of the hyperbola is x2 – 3y2 = 3
11. Let f : ℝ → ℝ be a function such that
f(x + y) = f(x) + f(y), ∀x, y ∈ ℝ.
If f(x) is differentiable at x = 0, then
(A) f(x) is differentiable only in a finite interval containing zero
(B) f(x) is continuous ∀ x ∈ ℝ
(C) f ‘(x) is constant ∀ x ∈ ℝ
(D) f(x) is differentiable except at finitely many points
Section-III
(Paragraph Type)
This section contains 2 paragraphs. Based upon one of the paragraphs 3 multiple choice questions and based on the other paragraph 2 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct.
Paragraph for Question Nos. 12 to 14
Let a, b and c be three real numbers satisfying
12. If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a + b + c is
(A) 0
(B) 12
(C) 7
(D) 6
13. Let ω be a solution of x3 – 1 = 0 with Im(ω) > 0. If a = 2 with b and c satisfying (E), then the value of
is equal to
(A) −2
(B) 2
(C) 3
(D) −3
14. Let b = 6, with a and c satisfying (E). If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then
is
(A) 6
(B) 7
(C) 6/7
(D) ∞
Paragraph for Question Nos. 15 and 16
Let U1 and U2 be two urns such that U1 contains 3 white and 2 red balls, and U2 contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from U1 and put into U2. However, if tail appears then 2 balls are drawn at random from U1 and put into U2. Now 1 ball is drawn at random from U2.
15. The probability of the drawn ball from U2 being white is
(A) 13/30
(B) 23/30
(C) 19/30
(D) 11/303
16. Given that the drawn ball from U2 is white, the probability that head appeared on the coin is
(A) 17/23
(B) 11/23
(C) 15/23
(D) 12/23
Section-IV
(Integer Answer Type)
This section contains 7 questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9. The bubble corresponding to the correct answer is to be darkened in the ORS.
17. Consider the parabola y2 = 8x. Let ∆1 be the area of the triangle formed by the end points of its latus rectum and the point P(1/2, a) on the parabola, and ∆2 be the area of the triangle formed by drawing tangents at P and at the end points of the latus rectum. Then 
is
18. Let a1, a2, a3, …., a100 be an arithmetic progression with a1 = 3 and 
1 ≤ p ≤ 100. For any integer n with 1 ≤ n ≤ 20, let m = 5n. If 
does not depend on n, then a2 is
depend on n, then a2 is
19. The positive integer value of n > 3 satisfying the equation
is
20. Let f : [1, ∞) → [2, ∞) be a differentiable function such that f(1) = 2. If
for all x ≥ 1, then the value of f(2) is
21. If z is any complex number satisfying |z – 3 – 2i| ≤ 2, then the minimum value of |2z – 6 + 5i| is
22. The minimum value of the sum of real numbers a−5, a−4, 3a−3, 1, a8 and a10 with a > 0 is
23. Let 
where 
Then the value of 
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