Part II – Mathematics
SECTION – I (Single Correct Choice Type)
This Section contains 6 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONLY ONE is correct.
1. For r = 0, 1, …, 10, let Ar, Br and Cr denote, respectively, the coefficient of xr in the expansions of (1 + x)10, (1 + x)20 and (1 + x)30. Then 
is equal to
(A) B10 – C10
(B) 
(C) 0
(D) C10 – B10
2. Let S = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of S is equal to
(A) 25
(B) 34
(C) 42
(D) 41
3. Let f be a real-valued function defined on the interval (−1, 1) such that 
for all x ∈ (−1, 1), and let f−1 be the inverse function of f. The (f1)ʹ (2) is equal to
(A) 1
(B) 1/3
(C) 1/2
(D) 1/e
4. If the distance of the point P (1, − 2, 1) from the plane x + 2y − 2z = α, where α > 0, is 5, then the foot of the perpendicular from P to the plane is
(A)
(B) 
(C) 
(D) 
24. Two adjacent sides of a parallelogram ABCD are given by
The side AD is rotated by an acute angle α in the plane of the parallelogram so that AD becomes ADʹ. If ADʹ makes a right with the side AB, then the cosine of the angle α is given by
(A) 8/9
(B) √17/9
(C) 1/9
(D) 4√5/9
6. A signal which can be green or red with probability 
respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is 3/4. If the signal received at station B is green. then the probability that the original signal was green is
(A) 3/5
(B) 6/7
(C) 20/23
(D) 9/20
SECTION – II (Integer Type)
This Section contains 5 questions. The answer to each question is a single-digit integer, ranging from 0 to 9.
7. Two parallel chords of a circle radius 2 are at a distance √3 + 1 apart. If the chords subtend at the center, angles of 
where k > 0, then the value of [k] is
[Note : [k] denotes the largest integer less than or equal to k]
8. Consider a triangle ABC and let a, b and c denote the lengths of the sides opposite to vertices A, B and C respectively. Suppose a = 6, b = 10 and the area of the triangle is 15√3 . If ∠ACB is obtuse and if r denotes the radius of the incircle of the triangle, then r2 is equal to
9. Let f be a function defined on R (the set of all real numbers) such that f′(x) = 2010 (x − 2009) (x − 2010)2 (x − 2011)3 (x − 2012)4, for all x ∈ If g is a function defined on R with values in the interval (0, ∞) such that f (x) = ln (g (x)), for all x ∈ R, then the number of points in R at which g has a local maximum is
10. Let a1, a2, a3, …, a11 be real numbers satisfying a1 = 15, 27 – 2a2 > 0 and ak = 2ak – 1 – ak–2 for k = 3, 4, …., 11. If 
then the value of 
is equal to
11. Let k be a positive real number and let 
and 
If det (adj A) + det(adj B) = 106, then [k] is equal to
SECTION – III (Paragraph Type)
This Section contains 2 paragraphs. Based upon each of the paragraphs 3 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 questions 12 to 14
Consider the polynomial f(x) = 1 + 2x + 3x2 + 4x3. Let s be the sum of all distinct real roots of f(x) and let t = |s|.
12. The real number s lies in the interval
(A) 
(B) 
(C) 
(D) 
13. The area bounded by the curve y = f(x) and the lines x = 0, y = 0 and x = t, lies in the interval
(A) 
(B) 
(C) 
(D) 
14. The function f′(x) is
(A) increasing in (−t, −1/4) and decreasing in (−1/4, t)
(B) decreasing in (−t, −1/4) and increasing in (−1/4, t)
(C) increasing (−t, t)
(D) decreasing in (−t, t)
Paragraph for questions 15 to 17
Tangents are drawn from the point P(3, 4) to the ellipse 
touching the ellipse at points A and B.
15. The coordinates of A and B are
(A) 
(B) 
(C) 
(D) 
16. The orthocenter of the triangle PAB is
(A) 
(B) 
(C) 
(D) 
17. The equation of the locus of the point whose distances from the point P and the line AB are equal, is
(A) 9x2 + y2 – 6xy – 54x – 62y + 241 = 0
(B) x2 + 9y2+ 6xy – 54x + 62y – 241 = 0
(C) 9x2 + 9y2 – 6xy – 54x – 62y – 241 = 0
(D) x2 + y2– 2xy + 27x + 31y – 120 = 0
SECTION – IV (Matrix Type)
This Section contains 2 questions. Each question has four statements (A, B, C and D) given in Column I and five statements (p, q, r, s and t) in Column II. Any given statement in Column I can have correct matching with one or more statement(s) given in Column II. For example, if for a given question, statement B matches with the statements given in q and r, then for that particular question, against statement B.
18. Match the statements in column-I with those in column-II.
[Note: Here z takes the values in the complex plane and Im z and Re z denote, respectively, the imaginary part and the real part of z]
(A) (A) – (q)
(B) (B) – (p)
(C) (C) – (r)
(D) (D) – (t)
19.
(A) (A) – (t)
(B) (B) – (p)
(C) (C) – (r)
(D) (D) – (q)
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