2. SECTION – I Single Correct Answer Type This Section contains 75multiple choice questions. Each question has four choices A), B), C) and D) out of which ONLY ONE is correct.
3. 01 Problem If C is the mid point of AB and P is any point outside AB, then : a. b. c. d.
4. Problem 02 Let P be the point (1, 0) and Q a point on the locus y2 = 8x. The locus of mid point of PQ is : x2 – 4y + 2 = 0 x2 + 4y + 2 = 0 y2 + 4x + 2 = 0 y2 – 4x + 2 = 0
5. Problem 03 If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately : 24.0 25.5 20.5 22.0
6. Problem 04 Let R = {(3, 3), (6,6) (9, 9), (12,12), (6, 12)(3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12}. The relation is : Reflexive and symmetric only An equivalence relation Reflexive only Reflexive and transitive only
7. Problem 05 If A2 – A + I = 0, then the inverse of A is : I – A A – I A A + I
8. Problem 06 If the cube roots of unity are 1, , 2, then the roots of the equation (x - 1)3 + 8 = 0, are : -1, 1 + 2 , 1 + 2 2 - 1, 1 -2 , 1 – 2 2 -1, -1, -1 - 1, -1 + 2 , - 1 – 2 2
10. 08 Problem Area of the greatest rectangle that can be inscribed in the ellipse is : a. b. c. ab d. 2ab
11. Problem 09 The differential equation representing the family of curves , where c > 0, is a parameter, is of order and degree as follows: Order 2, degree 2 Order 1, degree 3 Order 1, degree 1 Order 1, degree 2
12. Problem 10 If in a p-n junction diode, a square input signal of 10V is applied as Shown Then the output signal across RL will be a. b. c. d.
13. Problem 11 Photon of frequency ν has a momentum associated with it. If c is the velocity of light, the momentum is ν/c hνc hν/c2 hν/c
14. 12 Problem ABC is a triangle Forces acting along IA, IB and IC respectively are in equilibrium, where I is the incentre of is cos A : cos B : cos C . . sin A : sin B : sin C
15. Problem 13 If the coefficients of rth, (r + 1)th and (r + 2)th terms in the binomial expansion of (1 + y)m are in A.P., then m and r satisfy the equation : m2 – m (4r - 1) + 4r2 + 2 = 0 m2 – m (4r + 1) + 4r2 – 2 = 0 m2 – m (4r + 1) + 4r2 + 2 = 0 m2 – m (4r - 1) + 4r2 – 2 = 0
16. Problem 14 In a triangle PQR, are the roots of ax2 + bx + c = 0, a 0 then : b = a + c b = c c = a + b a = b + c
17. Problem 15 If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the words SACHIN appears at serial number : 602 603 600 601
19. 17 Problem , then which one of the following holds for all , by he principle of mathematical induction ? An = 2n -1 A + (n - 1)I An = nA + (n - 1) I An = 2n-1 A – (n - 1) I An = nA – (n - 1) I
20. Problem 18 If the coefficient of x7 in equals the coefficient of x-7 in , then a and b satisfy the relation : ab = 1 = 1 a + b = 1 a – b = 1
21. Problem 19 Let f : (-1, 1) B, be a function defined by , then f is both one-one and onto when B is the interval : a. b. c. d.
22. Problem 20 If z1 and z2 are two non-zero complex numbers such that |z1 + z2| = |z1| + |z2|, then arg z1 – arg z2 is equal to : a. -π/2 b. 0 c. π d. π/2
23. Problem 21 If , then z lies on : a parabola a straight line a circle an ellipse
24. Problem 22 If a2 + b2 + c2 = -2 and then f(x) is a polynomial of degree : 2 3 0 1
25. Problem 23 The system of equations α x + y + z =α -1 x + α y + z = α -1 αx + y + α z = α - 1 has no solution, if α is : 1 Not -2 Either -2 or 1 - 2
26. Problem 24 The value of a for which the sum of the squares of the roots of the equation x2 – (a - 2)x – a – 1 = 0 assume the least value is : 2 3 0 1
27. Problem 25 If the roots of the equation x2 – bx + c = 0 be two consecutive integers, then b2 – 4c equals : 1 2 3 -2
28. Problem 26 Suppose f(x) is differentiable at x = 1 and then f’(1) equals : 6 5 4 3
29. Problem 27 Let f be differentiable for all x. If f(1) = -2 and f’ (x) ≥ 2 for x є[1, 6], then : f(6) = 5 f (6) < 5 f (6) < 8 f(6) ≥ 8
30. Problem 28 If f is a real-valued differentiable function satisfying |f(x) – f(y)| ≤ (x - y)2, x, y є R and f(0) = 0, then f(1) equals 1 2 0 -1
31. Problem 29 If x is so small that x3 and higher power of x may be neglected, then may be approximated as : a. b. c. d.
32. Problem 30 If where a, b, c are in A.P. and |a| < 1, |b| < 1, |c | < 1, then x, y, z are in : H.P. Arithmetic – Geometric Progression A.P. G.P.
33. Problem 31 In a triangle ABC, let , if r is the inradius and R is the circmradius of the triangle ABC, then 2 (r + R) equals : c + a a + b + c a + b b + c
34. Problem 32 If , then 4x2 – 4xy cos α + y2 is equal to : - 4 sin2α 4 sin2α 4 α 2 sin 2 α
35. Problem 33 If in a ∆ABC, the altitudes from the vertices A, B, C on opposite sides are in H.P., then sin A, sin B, sin C are in : H.P. Arithmetic- Geometric Progression A.P. G.P.
36. Problem 34 The normal to the curve x = a (cos θ + θ sin θ), y = a (sin θ - θ cos θ) at any point ‘θ’ is such that : It is at a constant distance from the origin It passes through (aπ/2, -a) It makes angle π/2 + θ with the x-axis It passes through he origin
37. Problem 35 A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched ? Interval Function (-∞, - 4] x3 + 6x2 + 6 (- ∞,1/3) 3x2 – 2x + 1 [2, ∞) 2x3 – 3x2 – 12x + 6 (- ∞, ∞) x3 – 3x2 + 3x + 3
38. Problem 36 Let α and β be the distinct rots of ax2 + bx + c = 0, then is equal to : a. b. c. 0 d.
39. Problem 37 If then the solution of the equation is : log = cy log 0 log = cy
40. Problem 38 The line parallel to the x-axis and passing through the intersection of the lines ax + 2y + 3b = 0 and bx – 2ay – 3a = 0, where (a, b) (0, 0) is : Above the x-axis at a distance of 2/3 from it Above the x-axis at a distance of 3/2 from it Below the x-axis a distance of 2/3 from it Below the x-axis at a distance of 3/2 from it
41. Problem 39 A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm3/min. When the thickness of ice is 5 cm, then the rate at which the thickness of ice decreases, is : a. b. c. d.
43. Problem 41 Let be a differentiable function having equals : 18 12 36 24
44. Problem 42 Let f(x) be a non-negative continous function such that the area bounded by the curve y = f(x), x – axis and the ordinates is is a. b. c. d.
46. Problem 44 The area enclosed between the cure y = loge (x + e) and the coordinate axes is : 4 3 2 1
47. Problem 45 The parabolas y2 = 4x and x2 = 4y divide the square region bounded by the lines x = 4, y = r and the coordinates axes. If S1, S2, S3 are respectively the areas of these parts numbered from top to bottom; then S1 : S2 : S3 is : 1 : 1 : 1 2 : 1 : 2 1 : 2 : 3 1 : 2 : 1
48. Problem 46 If the plane 2ax – 3ay + 4az + 6 = 0 passes through the midpoint of the line joining the centres of the spheres x2 + y2 + z2 + 6x – 8y – 2z = 13 and x2 + y2 + z2 – 10x + 4y – 2z = 8, then a equals : 2 - 2 1 - 1
49. Problem 47 The distance between the line and the plane is : a. b. c. d.
50. Problem 48 For any vector , the value of is equal to : 4 2 , 3
51. Problem 49 If non-zero numbers a, b, c are in H.P., then the straight line always passes through a fixed point. That point is : (1, -2) (-1, -2) (1,-1/2) (-1, 2)
52. Problem 50 If a vertex of a triangle is (1, 1) and the mid points of two sides through this vertex are (-1, 2) and (3, 2), then the centroid of the triangle is : a. b. c. d.
53. Problem 51 If the circles x2 + y2 + 2ax + cy + a = 0 and x2 + y2 – 3ax + dy – 1 = 0 intersect in two distinct points P and Q then the line 5x + by – a = 0 passes through P and Q for : Exactly two values of a Infinitely many values of a No value of a Exactly one value of a
54. Problem 52 A circle touches the x-axis and also touches the circle with centre at (0,3) and radius 2 the locus of the centre of the circle is : A parabola A hyperbola A circle An ellipse
55. Problem 53 If a circle passes through the point (a, b) and cus the circle x2 + y2 = p2 orthogonally, then the equation of the locus of its centre is : 2ax + 2by – (a2 + b2 + p2) = 0 x2 + y2 – 2ax – 3by + (a2 – b2 – p2) = 0 2ax + 2by – (a2 – b2 + p2) = 0 x2 + y2 – 3ax – 4by + (a2 + b2 – p2) = 0
56. Problem 54 An ellipse has OB as semi minor axis, F and F’ its foci and the angle FBF’ is a right angle. Then the eccentricity of the ellipse is : a. b. c. d.
57. Problem 55 The locus of a point moving under the condition that the line is a tangent to the hyperbola is : A hyperbola A parabola A circle An ellipse
58. Problem 56 If the angle θ between the line and the plane is such that sin θ =1/3 . The value of λ is : a. b. c. d.
59. Problem 57 The angle between the lines 2x = 3y = -z and 6x – y = - 4z is : 300 450 900 00
60. Problem 58 Let A and B be two events such that , where stands for complement of event A. Then events A and B are : Mutually exclusive and independent Independent but not equally likely Equally likely but not independent Equally likely and mutually exclusive
61. Problem 59 Three houses are available in a locality three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house, is : a. b. c. d.
62. Problem 60 A random variable X has Poisson distribution with mean 2. Then P(X > 1.5) equals : a. b. c. d. 0
63. Problem 61 Two points A and B move from rest along a straight line with constant acceleration and f’ respectively. If A takes m sec more than B and describes ‘n’ units more than B is acquiring the same speed, then : (f + f’)m2 = ff’n (f – f’) m2 = ff’n c. d.
64. Problem 62 A lizard, at an initial distance of 21 cm behind an insect, moves from rest with an acceleration of 2 cm/s2 and pursues the insect which is crawling uniformly along a straight line at a speed of 20 cm/s. Then the lizard will catch the insect after : 24 sec 21 sec 1 sec 20 sec
65. Problem 63 The resultant R of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is : 3 : 2 3 : 2 3 : 2 : 1
66. Problem 64 Let depends on : Neither x nor y Both x and y Only x Only y
67. Problem 65 Let a, b and c be distinct non-negative numbers. If the vectors lie in a plane, then c is : The harmonic mean of a and b Equal to zero The arithmetic mean of a and b The geometric mean of a and b
68. Problem 66 If are non-coplanar vectors and is real number then Exactly two values of λ Exactly three values of λ No value of λ Exactly one value of λ
69. Problem 67 A and B are two like parallel forces. A couple of moment H lies in the plane of A and B and is contained with them. The resultant of A and B after combining is displaced through a distance : a. b. c. d.
70. Problem 68 The sum of the series ……… ad inf. Is : a. b. c. d.
71. Problem 69 Let x1, x2, …., xn be n observations such that then a possible value of u at an angle of 600 with the horizontal. When it is moving in a direction at right angles to its direction at O, its velocity, then is given by : u/ 2u/3 u/2 u/3
72. Problem 70 If both the roots of the quadratic equation x2 – 2kx + k2 + k -5 = 0 are less than 5, then k lies in the interval : [4, 5] (- ∞, 4) (6,∞) (5, 6]
73. Problem 71 If a1, a2, a3, …, an, … are in G.P., then the determinant is equal to : 2 4 0 1
74. Problem 72 A real valued function f(x) satisfies the functional equation f(x - y) = f(x) f(y) –f (a - x) f(a + y) where a is a given constant and f (0) = 1, f (2a - x) is equal to : f(- x) f (a) + f (a - x) f (x) - f(x)
75. Problem 73 If the equation anxn + an-1 xn-1 + …. + a1x = 0, , has a positive root x = α, then the equation nanxn-1 + (n - 1) an-1xn-2 + …+ a1 = 0 has a positive root, which is : Equal to α Greater than or equal to α Smaller than α Greater thanα
