1.The number of ways four boys can be seated around a round-table in four chairs of different colours is
(A) 24 (B) 12 (C) 23 (D) 64
2. If one root of the equation x2 + (1 - 3i)x - 2(1 + i) = 0 is -1 + i, then the other root is
(A) -1 - i (B) [tex]{{( - 1 - i)} \over 2}[/tex] (C) i (D) 2i
3. Three sets A, B, C are such that [tex]A = B \cap C[/tex] and [tex]B = C \cap A[/tex], then
(A) [tex]A \subset B[/tex] (B) [tex]A \supset B[/tex]
(C) A B (D) A B
4. The sum of the infinite series [tex]{\left( {{1 \over 3}} \right)^2} + {1 \over 3}{\left( {{1 \over 3}} \right)^4} + {1 \over 5}{\left( {{1 \over 3}} \right)^6} + ............[/tex] is
(A) [tex]{1 \over 4}{\log _e}2[/tex] (B) [tex]{1 \over 2}{\log _e}2[/tex] (C) [tex]{1 \over 6}{\log _e}2[/tex] (D) [tex]{1 \over 4}{\log _e}{3 \over 2}[/tex]
5. The values of x for which the given matrix will be non singular are
(A) -2 x < 2 (B) for all x other than 2 and -2
(C) x 2 . (D) x -2
6. If tan = cot then
(A) (B)
(C) (D) n is an integer.
7. The principal value of sin -1tan is
(A) (B)
(C) (D)
8. The value of is
(A) (B)
(C) 1 (D) 0
9. If a, b, c be in Arithmetic progression, then, the value of
(a + 2b-c)(2b + c - a)(a + 2b + c) is
(A) 16abc (B) 4abc
(C) 8abc (D) 3abc
10. The equation x2 - 3|x| + 2 = 0 has
(A) No real root (B) One real root ,
(C) Two real roots (D) Four real roots
11. The principal amplitude of (sin40° + icos40°)5 is
(A) 70° (B) -110°
(C) 110° (D) -70°
12. If log5 log5 Iog2 x = 0, then value of x is
(A) 32 (B) 125
(C) 625 (D) 25
13. A person draws out two balls successively from a bag containing 6 red and 4 white balls. The probability that at least one of them will be red is
` (A) (B)
(C) (D)
14. If three real numbers a, b, c are in Harmonic Progression, then which of the following is true ?
(A) aremA.P, (B) are in H.P.
(C) ab, be, ca are in A.P. (D) are in H.P.
15. A mapping f : N N where N is the set of natural numbers is defined as
f(n) = n2 for n odd
f(n) = 2n + 1 for n even .
for n N.
Then f is
(A) Surjective but not injective
(B) Injective but not surjective
(C) Bijective
(D) Neither injective nor surjective
16. If the magnitude of the coefficient of x7 in the expansion of
, where a, b are positive numbers, is equal to the
magnitude of the coefficient of x-7 in the expansion of then a and b are connected by the relation
(A) ab = 1 (B) ab = 2
(C) a2b = 1 . (D) ab2 = 2
17. The mapping f : N N given by f(n) = 1 + n2, n N where N is the set of natural numbers, is
(A) One to one and onto (B) Onto but not one-to-one
(C) One-to-one but not onto (D) Neither one-to-one nor onto
18. A and B are two points on the A rgand plane such that the segment AB is bisected at the point (0,0). If the point A, which is in the third quadrant has principal amplitude 6, then the principal amplitude of the point B is
(A) (B)
(C) (D)
19. A function f : A -> B, where A = {x / -1 < x < 1} and B = {y / 1 < y < 2} s defined by the rule y = f(x) = 1 + x2. Whi
ch of the following tatements is then true ?
(A) f is injective but not surjective
(B) f is surjective but not injective ,
(C) f is both injective and surjective
(D) f is neither injective nor surjective
20. The function f(x) which satisfies f(x) = f(-x) = is given by
(A) (B) f(x) =
(C) f(x) = (P) f(x) =
1-x2, for x < 1
21. A function f(x) is defined as follows for real x f(x) = 0, for x = 1
1+x2, for x > 1
(A) f(x) is not continuous at x = 1
(B) f(x) is continuous but not differentiable at x = 1
(C) f(x) is both continuous and differentiable at x = 1
(D) f(x) is continuous everywhere but differentiable nowhere.
22. Select the correct statement from (A), (B), (C), (D). The function
f(x) =
(A) strictly increases in the interval
(B) increases in the interval (0, °°)
(C) decreases in the interval (0, 2)
(D) strictly decreases in the interval (1, °°)
23. The equation ex + x - 1 = 0 has, apart from x = 0
(A) One other real root (B) Two real roots
(C) No other real root (D) Infinite number of real roots
24. The function f(x) = eax + e-ax, a > 0 is monotonically increasing for
(A) -1 < x < 1 (B) x < -1 •
(C) x > -1 (D) x > 0
25. For two complex numbers z1, z2 the relation |z1+ z2| = |z1| + |z2| holds if
(A) arg(z1) = arg(z2) (B) arg(z1 ) + arg(z2) =
(C) z1z2 = 1 (D) |z1| = |z2|
26. If 16Cr = l6Cr+1 , then the value of rPr-31 , is
(A) 31 (B) 120
(C) 210 (D) 840
27. The coefficient of x-10 in
(A) -252 (B) -210
(C) -(5!) (D) -120 .
28. If the matrix is commutative with the matrix then
(A) a = 0, b = c (B) b = 0, c = d
(C) c = X d = a ' (D) d = 0, a = b
29. If l are cube roots of unity, then has value
(A) 0 (B)
(C) (D)
30. Let A = {1, 2, 3} and B = {2, 3, 4}, then which of the following relations is a function from A to B ?
(A) {(1, 2), (2, 3), (3, 4), (2, 2)}.
(B) {(1, 2), (2, 3), (1,3)}
(C) {(1, 3), (2, 3), (3, 3)}
(D) {(1, 1), (2, 3), (3, 4)}
31. One possible condition for the three points (a, b), (b, a) and (a2, -b2) to be collinear is
(A) a - b = 2 (B) a + b = 2
(C) a = 1 + b (D) a = 1 - b
32. If the mth term and the nth term of an A.P. are respectively and
then the (mn) th term of the A.P. is
(A) (B)
(C) 1 (D)
33 x cos5 x dx equals
(A) (B) 20
(C) 0 (D)
34. The function f(x) = log satisfies the equation
(A) f(x + 2) - 2f(x + 1) + f(x) = 0
(B) f(x) + f(x+ l) = f(x(x+ 1))
(C) f(x) + f(y) = f
(D) f(x+ y) = f(x) f(y)
35. If I = dx, then I equals
(A) (B)
(C) (D)
36. If h(x) = dt, then h (x + ) equals
(A) (B) h(x)h( )
(C) h(x)-h( ) (D) h(x) + h( )
37. The value of where and are the complex cube roots of unity is .
(A) 0 (B) 32
(C) -32 . (D) 32
38. The degree of the differential equation is
(A) 1 (B) 5
(C) (D) 3
39. The differential equation of all parabolas whose axes are parallel to y-axis is
(A) (B)
(C) (D)
40. The solution of the differential equation = ey+x +ey-x is
(A) e-y = e x - e-x + c, c integrating constant
(B) e-y = e-x - ex + c, c integrating constant
(C) e-y = ex + e-x + c, c integrating constant
(D) e-y + ex + e-x = c, c integrating constant
41. The value of the integral is
(A) 0 (B) 2
(C) (D) -2
42. If x = e'sint, y = e'cost then at x = % is
(A) (B)
(C) (D)
43. The value of at x = where y is given by y = xsinx + is
(A) (B) 1
(C) (D)
44. The value of is
(A) (B) 2
(C) (D)
45. The value of where a, b, c, k are constants,
depends only on
(A) a and k (B) a and b
(C) a, b and c (D) k
46. The value of the integral is
(A) (B) 0
(C) (D) a
47. The value of the is
(A) log 2 (B) log 6
(C) 1 (D) log 3
48. The order and degee of the following differential equation are respectively
(A) 3, 2 (B) 3, 10
(C) 2, 3 (D) 3, 5
49. The differential equation of the family of circles passing through the fixed points (a, 0) and (-a, 0) is
(A) y1(y2 - x2) + 2xy +a2 = 0 (B) y1y2 + xy + a2x2 = 0
(C) y1 (y2 - x2 + a2) + 2xy = 0 (D) y1(y2 + x2) - 2xy + a2 = 0
50. The differential equation of the family of curves y - e2x (acosx + bsinx), where a and b are arbitrary constants, is given by
(A) y2 - 4y1 + 5y = 0 (B) 2y2 - y1 + 5y = 0 •
(C) y2+ 4y1 - 5y = 0 (D) y2 - 2y1 + 5y = 0
51.
(A) (B) = loge 2
(C) = loge a (D) = a
52. Rolle's theorem is not applicable to the function f(x) = |x| for -2 < x < 2 because
(A) f is continuous for -2 < x < 2
(B) f is not derivable for x = 0
(C) f(-2) = f(2)
(D) f is not a constant function
53. The equation of the circle which passes through the points of intersection of the circles x2 + y2 - 6x = 0 and x2 + y2 - 6y = 0, and has its centre at is
(A) x2 + y2 + 3x + 3y + 9 = 0
(B) x2 + y2 + 3x + 3y = 0
(C) x2 + y2- 3x- 3y = 0
(D) x2 + y2-3x- 3y + 9 = 0
54. If 2y = x and 3y + 4x = 0 are the equations of a pair of conjugate diameters of an ellipse, then the eccentricity of the ellipse is
(A) (B)
(C) (D)
55. The area enclosed between the curve y = 1 + x2, the y-axis, and the straight line y = 5 is given by
(A) square units (B) square units
(C) 5 square units (D) square units]
56. If t is a parameter, then x = a y = b represents
(A) An ellipse (B) A circle
(C) A pair of straight lines (D) A hyperbola
57. The line which is parallel to x-axis and crosses the curve y = at an angle 45° is
(A) y = (B) y =
(C) y = l (D) y = 4
58. The distance between the lines 5x - 12y + 65 = 0 and 5x - 12y -39 = 0 is
(A) 4 (B) 16
(C) 2 (D) 8
59. The co-ordinates of the foot of perpendicular from (a, 0) on the line
y = mx + are
(A) (B)
(C) (D)
60. The equation (x - x1)(x - x2) + (y - y1)(y - y2) = 0 represents a circle whose centre is
(A) (B)
(C) (x1 , y1) (D) (x2 , y2)
61. The circles x2 + y2 + 6x + 6y = 0 and x2 + y2 - 12x - 12y = 0
(A) cut orthogonally (B) touch each other internally
(C) intersect in two points (D) touch each other externally
62. The two parabolas x2 = 4y and y2 = 4x meet in two distinct points. One of these is the origin and the other is
(A) (2, 2) (B) (4, -4)
(C) (4, 4) (D) (-2, 2)
63. The vertex of the parabola x2 + 2y = 8x - 7 is
(A) (B)
(C) (D)
64. If P(at2, 2at) be one end of a focal chord of the parabola y2 = 4ax, then the length of the chord is
(A) (B)
(C) (D)
65. The length of the common chord of ihe parabolas y2 = x and x2 = y is
(A) (B) l (C) (D)
66. The equation of the ellipse having vertices at (±5, 0) and foci (±4, 0) is
(A) (B) 9x2 + 25y2 = 225
(C) (D)4x2 + 5y2 = 20
67. The area included between the parabolas y2 = 4x and x2 = 4y is
(A) sq. units (B) 8 sq. units
(C) sq. units (D) 12 sq. units
68. The locus of the centres of the circles which touch both the axes is given by
(A) x2 - y2 - 0 (B) x2 + y2 = 0 ,
(C) x2 - y2 = 1 (D) x2 + y2 = 1
69. The sum of the series (1 + 2) + (1 + 2 + 22) + (1 + 2 + 22 + 23) + ....up to n terms is
(A) 2n+2 – n - 4 (B) 2( 2n - 1) - n
(C) 2n+1 - n (D) 2n+1 - 1
70. The 5th term of the series .....is
(A) (B) 1 (C) (D)
71. The equation sinx + cosx = 4 has
(A) infinitely many solutions (B) no solution
(C) two solutions (D) only one solution
72. The value of tan + 2tan(2 ) + 4tan(4 ) + ...... + 2n-1 tan(2n-1 ) +
2ncot(2n ) is
(A) cot(2n ) (B) 2ntan(2n )
(C) 0 (D) cot
73. Out of 8 given points, 3 are collinear. How many different straight lines can be drawn by joining any two points from those 8 points ?
(A) 26 (B) 28 (C) 27 (D) 25
74. How many odd numbers of six significant digits can be formed with the digits 0, 1, 2, 5, 6, 7 when no digit is repeated ?
(A) 120 (B) 96 (C) 360 (D) 288
75. Let be the roots of x2 - 2xcos + 1 = 0, then the equation whose roots are is
(A) x2 - 2x cos n -1= 0 (B) x2 - 2x cos n +1=0
(C) x2 - 2x sin n + 1=0 (D) x2 + 2x sin n -1=0
76. The latus rectum of an ellipse is equal to one-half of its minor axis. The eccentricity of the ellipse is
(A) (B) (C) (D)
77. A particle is projected vertically upwards and is at a height h after t1 seconds and again after t2 seconds then
(A) h = gt1t2 (B) h = gt1t2
(C) (D) h =
78. The value of the limit
(A) (B) 3 (C) - 3 (D) -1
79. The limit
(A) 10 (B) (C) (D) Does not exist
80. The range of the function f(x) = loge is given by
(A) (0, ) (B) ( )
(C) ( , loge2) (D) (loge2, )
