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**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) = 2x^{3} – 15x^{2} + 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 = z^{2} + 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 E_{2} passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse E_{2} is

(A) √2/2

(B) √3/2

(C) 1/2

(D) 3/4

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 x^{2} + y^{2} = 9 is

(A) 20(x^{2} + y^{2}) – 36x + 45y = 0

(B) 20(x^{2} + y^{2}) + 36x – 45y = 0

(C) 36(x^{2} + y^{2}) – 20x + 45y = 0

(D) 36(x^{2} + y^{2}) + 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 E_{1}, E_{2} and E_{3}. 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 X_{1}, X_{2} and X_{3} denote respectively the events that the engines E_{1}, E_{2} and E_{3} 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| + |x^{2} – 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 y^{2} = 8x and let PQ be the common chord of the circle x^{2} + y^{2} – 2x – 4y = 0 and the given parabola. The area of the triangle PQS is