Part II : Mathematics
Section-I
Single Correct Choice Type
This section contains 4 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONLY ONE is correct.
1. If the sum of first n terms of an A.P. is cn2, then the sum of squares of these n terms is
(A) 
(B) 
(C) 
(D) 
2. A line with positive direction cosines passes through the point P(2, – 1, 2) and makes equal angles with the coordinate axes. The line meets the plane 2x + y + z = 9 at point Q. The length of the line segment PQ equals
(A) 1
(B) √2
(C) √3
(D) 2
3. The normal at a point P on the ellipse x2 + 4y2 = 16 meets the x-axis at Q. If M is the mid point of the line segment PQ, then the locus of M intersects the latus rectums of the given ellipse at the points
(A) 
(B) 
(C) 
(D) 
4. The locus of the orthocentre of the triangle formed by the lines (1 + p)x – py + p(1 + p) = 0, (1 + q)x – qy + q(1 + q) = 0 and y = 0, where p ≠ q, is
(A) a hyperbola
(B) a parabola
(C) an ellipse
(D) a straight line
SECTION−II
Multiple Correct Choice Type
This section contains 5 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONE OR MORE is/are correct.
5. If 
n = 0, 1, 2, …., then
(A) 
(B) 
(C) 
(D) 
6. An ellipse intersects the hyperbola 2x2 – 2y2 = 1 orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinates axes, then
(A) equation of ellipse is x2 + 2y2 = 2
(B) the foci of ellipse are (± 1, 0)
(C) equation of ellipse is x2 + 2y2 = 4
(D) the foci of ellipse are (± √2 , 0)
7. For function 
(A) for atleast one x in interval [1, ∞), f(x + 2) – f(x) < 2
(B) 
(C) for all x in the interval [1, ∞), f(x + 2) – f(x) > 2
(D) f′(x) is strictly decreasing in the interval [1, ∞)
8. The tangent PT and the normal PN to the parabola y2 = 4ax at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola whose
(A) vertex is (2a/3, 0)
(B) directrix is x = 0
(C) latus rectum is 2a/3
(D) focus is (a, 0)
9. For 
the solutions(s) of 
is (areo)
(A) π/4
(B) π/6
(C) π/12
(D) 5π/12
SECTION − III
Matrix − Match Type
This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statement in Column I are labelled A, B, C and D, while the statements in Column II are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement (s) in Column II.
10. Match the statements/expressions in Column I with the values given in Column II.
(A) (A) → (q, s) (B) → (p, r, s, t) (C) → (t) (D) → (r)
(B) (A) → (p, r) (B) → (s, p, t, r) (C) → (p) (D) → (q)
(C) (A) → (t, p) (B) → (q, p, r, s) (C) → (r) (D) → (p)
(D) (A) → (r, q) (B) → (q, s, p, r) (C) → (q) (D) → (t)
11. Match the statements/expressions in Column I with the values given in Column II.
(A) (A) → (q, s) (B) → (p, r, s, t) (C) → (t) (D) → (r)
(B) (A) → (p, r) (B) → (s, p, t, r) (C) → (p) (D) → (q)
(C) (A) → (t, p) (B) → (q, p, r, s) (C) → (r) (D) → (p)
(D) (A) → (p) (B) → (q, s) (C) → (q, r, s, t) (D) → (r)
SECTION – IV
Integer Answer Type
This section contains 8 questions. The answer to each of the questions is a single digit integer, ranging from 0 to 9.
12. The maximum value of the function f(x) = 2x3 − 15x2 + 36x − 48 on the set A = {x|x2 + 20 ≤ 9x|} is
13. Let (x, y, z) be points with integer coordinates satisfying the system of homogeneous equations:
3x − y − z = 0
− 3x + z = 0
−3x + 2y +z = 0.
Then the number of such points for which x2 + y2 + z2 ≤ 100 is
14. Let ABC and ABC′ be two non−congruent triangles with sides AB = 4, AC = AC′ = 2 √2 and angle B = 30°. The absolute value of the difference between the areas of these triangles is
15. Let p(x) be a polynomial of degree 4 having extremum at x = 1, 2 and 
Then the value of p(2) is
16. Let f: R → R be a continuous function which satisfies 
Then the value of f(ln5) is
17. The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C1 and C2 and C be a circle touching circles C1 and C2 If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then the radius of the circle C is
18. The smallest value of k, for which both the roots of the equation x2 − 8kx + 16(k2 − k + 1 ) = 0 are real, distinct and have values at least 4, is
19. If the function f(x) = x3 + ex/2 and g(x) = f−1(x), then the value of g′(1) is
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