Part III : Mathematics
This section contains 10 multiple questions. Each question has four choices (A), (B) (C) and (D) out of which ONLY ONE is correct.
1. The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is
(A) 75
(B) 150
(C) 210
(D) 243
2. Let 
then f is
(A) differentiable both at x = 0 but not differentiable at x = 2
(B) differentiable at x = 0 but not differentiable at x = 2
(C) not differentiable at x = 0 but differentiable at x = 2
(D) differentiable neither at x = 0 nor at x = 2
3. The function f : [0, 3] → [1, 29], defined by f(x) = 2x3 – 15x2 + 36x + 1, is
(A) one-one and onto.
(B) onto but not one-one.
(C) one-one but not onto.
(D) neither one-one nor onto.
4. If 
then
(A) a = 1, b = 4
(B) a = 1, b = −4
(C) a = 2, b = −3
(D) a = 2, b = 3
5. Let z be a complex number such that the imaginary part of z is nonzero and a = z2 + z + 1 is real. Then a cannot take the value
(A) −1
(B) 1/3
(C) 1/2
(D) 3/4
6. The ellipse 
is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E2 passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse E2 is
(A) √2/2
(B) √3/2
(C) 1/2
(D) 3/4
equals (for some arbitrary constant K)
10. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x – 5y = 20 to the circle x2 + y2 = 9 is
(A) 20(x2 + y2) – 36x + 45y = 0
(B) 20(x2 + y2) + 36x – 45y = 0
(C) 36(x2 + y2) – 20x + 45y = 0
(D) 36(x2 + y2) + 20x – 45y = 0
Section II : Multiple Correct Answer(s) Type
The section contains 5 multiple choice questions. Each question has hour choices (A), (B), (C) and (D) out of which ONE or MORE are correct.
11. Let θ, φ ∈[0, 2π] be such that
Then φ cannot satisfy
(A) 
(B) 
(C) 
(D) 
12. Let S be the area of the region enclosed by 
y = 0, x = 0, and x = 1. Then
(A) 
(B) 
(C) 
(D) 
13. A ship is fitted with three engines E1, E2 and E3. The engines function independently of each other with respective probabilities 
For the ship to be operational at least two of its engines must function. Let X denote the event that the ship is operational and let X1, X2 and X3 denote respectively the events that the engines E1, E2 and E3 are functioning. Which of the following is (are) true ?
(A) 
(B) P[Exactly two engines of the ship are functioning] 
(C) 
(D) 
14. Tangents are drawn to the hyperbola 
parallel to the straight line 2x – y = 1. The points of contact of the tangents on the hyperbola are
(A) 
(B) 
(C) 
(D) 
15. If y(x) satisfies the differential equation y’ – y tan x = 2x sec x and y(0) = 0, then
(A) 
(B) 
(C) 
(D) 
SECTION III : Integer Answer Type
This section contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive).
16. Let f : IR → IR be defined as f(x) = |x| + |x2 – 1|. The total number of points at which f attains either a local maximum or a local minimum is
17. The value of 
is
18. Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and local minimum at x = 3. If p(1) = 6 and p(3) = 2, then p'(0) is
19. If 
are unit vectors satisfying 
then 
is
20. Let S be the focus of the parabola y2 = 8x and let PQ be the common chord of the circle x2 + y2 – 2x – 4y = 0 and the given parabola. The area of the triangle PQS is
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