PART III : MATHEMATICS
SECTION I : (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 is correct.
1. For a > b > c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less than 2√2. Then
(A) a + b – c > 0
(B) a – b + c < 0
(C) a – b + c > 0
(D) a + b – c < 0
2. The area enclosed by the curves y = sin x + cos x and y = | cos x − sin x | over the interval 
is
(A) 4(√2 – 1)
(B) 2√2(√2 – 1)
(C) 2(√2 + 1)
(D) 2√2(√2 + 1)
3. The number of points in (−∞, ∞), for which x2 − x sin x − cos x = 0, is
(A) 6
(B) 4
(C) 2
(D) 0
4. The value of 
is
(A) 
(B) 
(C) 
(D) 
5. A curve passes through the point 
Let the slope of the curve at each point (x, y) be 
Then the equation of the curve is
(A) 
(B) 
(C) 
(D) 
6. Let 
(the set of all real numbers) be a positive, non-constant and differentiable function such that f ‘(x) < 2f(x) and 
Then the value of 
lies in the interval
(A) (2 e − 1, 2 e)
(B) (e − 1, 2e − 1)
(C) 
(D) 
7. Let 
and 
determine diagonals of a parallelogram PQRS and 
be another vector. Then the volume of parallelepiped determined by the vectors 
is
(A) 5
(B) 20
(C) 10
(D) 30
8. Perpendiculars are drawn from points on the line
to the plane x + y + z = 3. The feet of perpendiculars lie on the line
(A) 
(B) 
(C) 
(D) 
9. Four persons independently solve a certain problem correctly with probabilities 
Then the probability that the problem is solved correctly by at least one of them is
(A) 
(B) 
(C) 
(D) 
10. Let complex numbers 
lie on circles (x – x0)2 + (y – y0)2 = r2 and (x – x0)2 + (y – y0)2 = 4r2, respectively, If z0 = x0 + iy0 satisfies the equation 2|z0|2 = r2 + 2, then |α| =
(A) 
(B) 
(C) 
(D) 
SECTION II : (One or more options correct Type)
This section contains 5 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE are correct.
11. A line ℓ passing through the origin is perpendicular to the lines
Then, the coordinate (s) of the point (s) on ℓ2 at a distance of √17 from the point of intersection of ℓ and ℓ1 is (are)
(A) 
(B) (−1,−1,0)
(C) (1, 1, 1)
(D) 
12. Let f(x) = x sin πx, x > 0. Then for all natural numbers n, f ‘ (x) vanishes at
(A) a unique point in the interval 
(B) a unique point in the interval 
(C) a unique point in the interval (n, n + 1)
(D) two point in the interval (n, n + 1)
13. Let 
Then Sn can take value (S)
(A) 1056
(B) 1088
(C) 1120
(D) 1332
14. For 3 × 3 matrices M and N, which of the following statement (s) is (are) NOT correct?
(A) NT M N is symmetric or skew symmetric,, according as M is symmetric or skew symmetric
(B) MN – NM is skew symmetric for all symmetric matrices M and N
(C) M N is symmetric for all symmetric matrices M and N
(D) (adj M) (adj N) = adj (M N) for all invertible matrices M and N
15. A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are
(A) 24
(B) 32
(C) 45
(D) 60
SECTION III : (Integer value correct Type)
This section contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive).
16. Consider the set of eight vectors 
Three non-coplanar vectors can be chosen from V in 2p ways. Then p is,
17. Of the three independent events E1, E2 and E3, the probability that only E1 occurs is α, only E2 occurs is β and only E3 occurs is γ. Let the probability p that none of events E1, E2 or E3 occurs satisfy the equations (α − 2β) p = αβ and (β − 3γ) p = 2βγ. All the given probabilities are assumed to lie in the interval (0, 1).
Then 
18. The coefficients of three consecutive terms of (1 + x)n + 5 are in the ratio 5 : 10 : 14. Then n =
19. A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then k − 20 =
20. A vertical line passing through the point (h, 0) intersects the ellipse 
at the points P and Q. Let the tangents to the ellipse at P and Q meet at the point R.
If ∆(h) = area of the triangle PQR, 
and 
then 
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