PART III : MATHEMATICS
SECTION 1 : (One or more options correct Type)
This section contains 8 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.
1. Let ω be a complex cube root of unity with ω ≠ 1 and P = [pij] be a n × n matrix with pij = ωi + j. Then P2 ≠ 0, when n =
(A) 57
(B) 55
(C) 58
(D) 56
2. The function f(x) = 2 | x | + | x + 2 | − | | x + 2 | − 2 | x | | has a local minimum or a local maximum at x =
(A) −2
(B) −2/3
(C) 2
(D) 2/3
3. Let 
and P = {wn = 1, 2, 3, …}. Further 
and 
where ℂ is the set of all complex numbers. If z1 ∈ P ∩ H1, z2 ∈ P ∩ H2 and O represents the origin, then ∠z1 O z2 =
(A) π/2
(B) π/6
(C) 2π/3
(D) 5π/6
4. If 3x = 4x – 1 , then x =
(A) 
(B) 
(C) 
(D) 
5. Two lines 
are coplanar. Then α can take values (s)
(A) 1
(B) 2
(C) 3
(D) 4
6. In a triangle PQR, P is the largest angle and cos P = 1/3. Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)
(A) 16
(B) 18
(C) 24
(D) 22
7. For a ∈ R (the set of all real numbers), a ≠ −1, 
Then a =
(A) 5
(B) 7
(C) −15/2
(D) −17/2
8. Circle(s) touching x−axis at a distance 3 from the origin and having an intercept of length 2√7 on y-axis is (are)
(A) x2 + y2 – 6x + 8y + 9 = 0
(B) x2 + y2 – 6x +7y + 9 = 0
(C) x2 + y2 – 6x – 8y + 9 = 0
(D) x2 + y2 – 6x – 7y + 9 = 0
SECTION 2 : (Paragraph Type)
This section contains 4 paragraphs each describing theory, experiment, data etc. Eight questions relate to four paragraphs with two questions on each paragraph. Each question of a paragraph has only one correct answer among the four choices (A), (B), (C) and (D).
Paragraph for Question 9 and 10
9. Let f : [0, 1] → R (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f(0) = f(1)= and satisfies f ″(x) – 2f ‘(x) + f(x) ≥ ex, x ∈ [0,1].
Which of the following is true for 0 < x < 1 ?
(A) 0 < f(x) < ∞
(B) 
(C) 
(D) −∞ < f(x) < 0
10. If the function e−x f(x) assumes its minimum in the interval [0, 1] at x = 1/4, which of the following is true?
(A) 
(B) 
(C) 
(D) 
Paragraph for Question 11 and 12
Let PQ be a focal chord of the parabola y2 = 4ax. The tangents to the parabola at P and meet at a point lying on the line y = 2x + a, a > 0.
11. Length of chord PQ is
(A) 7a
(B) 5a
(C) 2a
(D) 3a
12. If chord PQ subtends an angle θ at the vertex of y2 = 4ax, then tan θ =
(A) 
(B) 
(C) 
(D) 
Paragraph for Question 13 and 14
Let S = S1 ∩ S2 ∩ S3, where S1 = {z ∈ C : |z| < 4}, 
and S3 = {z ∈ C : Re z > 0}
13. Area of S =
(A) 
(B) 
(C) 
(D) 
14. 
(A) 
(B) 
(C) 
(D) 
Paragraph for Question 15 and 16
A box B1 contains 1 white ball, 3 red balls and 2 black balls. Another box B2 contains 2 white balls, 3 red balls and 4 black balls. A third box B3 contains 3 white balls, 4 red balls and 5 black balls.
15. If 1 ball is drawn from each of the boxes B1, B2 and B3, the probability that all 3 drawn balls are of the same colour is
(A) 
(B) 
(C) 
(D) 
16. If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these 2 balls are drawn from box B2 is
(A) 
(B) 
(C) 
(D) 
SECTION 3 : (Matching List Type)
This section contains 4 multiple choice questions. Each question has matching lists. The codes for the lists have choices (A), (B), (C) and (D) out of which ONLY ONE is correct
57. Match List − I with List − II and select the correct answer using the code given below the lists :
(A) P – 4 ; Q – 2 ; R – 3 ; S – 1
(B) P – 2 ; Q – 3 ; R – 1 ; S – 4
(C) P – 3 ; Q – 4 ; R – 1 ; S – 2
(D) P – 1 ; Q – 4 ; R – 3 ; S – 2
58. Consider the lines 
and the planes P1 : 7x + y +2z = 3, P2 : 3x + 5y – 6z = 4. Let ax + by + cz = d be the equation of the plane passing through the point of intersection of lines L1 and L2, and perpendicular to planes P1 and P2.
Match List – I with List – II and select the correct answer using the code given below the lists :
List I List II
P. a = 1. 13
Q. b = 2. −3
R. c = 3. 1
S. d = 4. −2
(A) P – 3 ; Q – 2 ; R – 4 ; S – 1
(B) P – 1 ; Q – 3 ; R – 4 ; S – 2
(C) P – 3 ; Q – 2 ; R – 1 ; S – 4
(D) P – 2 ; Q – 4 ; R – 1 ; S – 3
19. Match List I with List II and select the correct answer using the code given below the lists :
(A) P – 4 ; Q – 3 ; R – 1 ; S – 2
(B) P – 4 ; Q – 3 ; R – 2 ; S – 1
(C) P – 3 ; Q – 4 ; R – 2 ; S – 1
(D) P – 3 ; Q – 4 ; R – 1 ; S – 2
60. A line L : y = mx + 3 meets y − axis at E (0, 3) and the arc of the parabola y2 = 16x, 0 ≤ y ≤ 6 at the point F(x0, y0). The tangent to the parabola at F(x0, y0) intersects the y−axis at G(0, y1). The slope m of the line L is chosen such that the area of the triangle EFG has a local maximum.
Match List I and List II and select the correct answer using the code given below the lists :
List I List II
P. m = 1. 1/2
Q. Maximum area of ∆EFG is 2. 4
R. y0 = 3. 2
S. y1 = 4. 1
(A) P – 4 ; Q – 1 ; R – 2 ; S – 3
(B) P – 3 ; Q – 4 ; R – 1 ; S – 2
(C) P – 1 ; Q – 3 ; R – 2 ; S – 4
(D) P – 1 ; Q – 3 ; R – 4 ; S – 2
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