MATHEMATICS

SECTION – I

This section contains 8 multiple choice questions. Each question has 4 choices (a), (b), (c) and (d), out of which only one is correct

39. The normal a 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

40. 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

41. The locus of the orthocenter 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

42. If the sum of first n terms of an A.P. is cn², then the sum of these n terms is

SECTION – II

This section contains 4 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

43. The tangent PT and the normal PN to the parabola y² = 4as 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) a,d          (B) b, d           (C) a, c              (D) c, d

(A) a, d         (B) c, d              (C) b, d             (D) a, b

(A) a, c, d            (B) a, c, d              (C) b, c, d            (D) a, b, d

46. 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 coordinate axes, then

(A) a, c            (B) a, d           (C) b, d           (D) a, b

(A) a, b, c              (B) a, c, d            (C) a, b, d              (D) b, c, d

SECTION – III

Matrix Match Type

This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statements in column I are labeled A, B, C and D, while the statements in column II are labeled 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.

48. Match the statements/expressions given in Column – I with the values given in Column – II

(A) (a: qs), (b: prst), (c: t), (d:r)

(B) (a: prst), (b: t), (c: dr), (d: qs)

(C) (a: t), (b: dr), (c: qr), (d: qrst)

(D) (a:dr), (b: qr), (c: qrst), (d: t)

49. Match the statements/expressions given in Column – I with the values given in Column – II

(A) (a: qs), (b: qrst), (c: r), (d: p)

(B) (a: qrst), (b: r), (c: p), (d: qs)

(C) (a: r), (b: p), (c: qs), (d: qrst)

(D) (a: p), (b: qs), (c: qrst), (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.

50. The smallest value of k, for which both the roots of the equation

(A) 0     (B) 2        (C) 4      (D) 5

X² – 8kx + 16(k² – k + 1) = 0 are real, distinct and have values at least 4, is

(A) 2            (B) 5           (C) 0             (D) 9

52. Let p(x) be a polynomial of degree 4 having extremum at x = 1, 2 and

Then the value of p(2) is

(A) 0        (B) 2       (C) 5           (D) 3

(A) 2            (B) 0             (C) 3           (D) 4

54. The centers of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let N be the mid point of the line segment joining the centers of C1 and C2 and C be a circle touching circles C1 and C2 externally. If a common tangent to C1 and C passing through N is also a common tangent to C2 and C, then the radius of the circle C is

(A) 8          (B) 9            (C) 7        (D) 6

55. Let (x, y, z) be points with integer coordinates satisfying the system of homogeneous equations:

(A) 8           (B) 9           (C) 7          (D) 6

3x – y – z = 0; - 3x + z = 0; - 3x + 2y + z = 0

(A) 1                   (B) 0        (C) 3           (D) 2

(A) 6              (B) 7                 (C) 8           (D) 9