3. 01 Problem
2
The differential equation of all ‘Simple Harmonic Motions’ of given period n is :
x = a cos (nt + b)
d2 x
a. dt 2
+ nx = 0
d2 x
b. + n 2x = 0
dt 2
d2 x
c. - n2 x = 0
dt 2
d2 x 1
d. + x=0
dt 2 n2
4. 02 Problem
The elimination of constants A, B and C from y = A + Bx – Ce-x leads the
differential equation :
a. y'’ + y’’ = 0
b. y’’ – y’’’ = 0
c. y’ + ex = 0
d. y’’ + ex = 0
5. 03 Problem
The solution of the differential equation xdy + ydx = 0 is :
a. xy = c
b. x2 + y2 = 0
c. xy logx = 1
d. log xy = c
6. 04 Problem
dy
The solution of the equation +2y tan x = sin x, is :
dx
a. y sec2 x = sec x + c
b. y sec x = tan x + c
c. y = sec x + c sec2 x
d. none of these
7. 05 Problem
x
a is equal to:
dx
x
a x
a. c
log a
b. 2a x
c
log a
x
c. 2a .log a c
d. None of these
8. 06 Problem
1/2 dx
1/4
xx 2
is equal to :
a. 6
b. 3
c. 4
d. 0
9. 07 Problem
/2 1
The value of
0 3
1 tan x
dx is :
a. 0
b. 1
c. /4
d. 2
10. 08 Problem
Q dx
The value of
P x
, (P Q) is :
a. log (Q - P)
b. log P
c. log Q
d. log (P Q)
11. 09 Problem
For a biased dice the probabilities of different faces to turn up are given below :
Face: 1 2 3 4 5 6
Probabili 0.1 0.32 0.21 0.15 0.05 0.17
ty
The dice is tossed and you are told that either the face 1 or 2 has turned up. The
probability that it is face is :
a. 1
7
4
b. 7
5
c. 21
4
d. 21
12. 10 Problem
1 x /2
If A
1
tan x dx and B
x / 2
cot x dx then A + B is equals :
a. 0
b. 1
c. 2
d. 4
13. 11 Problem
If f’(x) = (x - a)2n (x - b)2p+1 where x and p are positive integers, then :
a. x = a is a point of minimum
b. x = a is a point of maximum
c. x is not a point of maximum or minimum
d. none of these
14. 12 Problem
the inverse of a symmetric matrix is :
a. diagnol matrix
b. skew symmetric
c. square matrix
d. a symmetric matrix
15. 13 Problem
a p
The matrix product equals :
b x[xyz ]x q
c
r
pqr abc
a.
xyz
b. xyz pqr
abc
c. pqr abc
xyz
d. none of these
16. 14 Problem
In a group of 52 persons, 16 drink tea but not coffee, while 33 drink tea. How
many persons drink coffee but not tea :
a. 17
b. 36
c. 23
d. 19
17. 15 Problem
If the length of major axis of an ellipse is three times the length of minor
axis, then the eccentricity is :
1
a. 3
1
b.
2
1
c.
3
2 2
d.
3
18. 16 Problem
A conic section with centricity e is a parabola if :
a. e = 0
b. e < 1
c. e > 1
d. e = 1
19. 17 Problem
For the ellipse 3x2 + 4y2 = 12 length of the latusrectum is :
a. 3
b. 4
3
c. 5
2
d. 5
20. 18 Problem
If (2, 0) is the vertex and y-axis is the direction of a parabola, then its focus is
a. (2, 0)
b. (-2, 0)
c. (4, 0)
d. (-4, 0)
21. 19 Problem
The length of tangent from the point (6,-7) to the circle : 3x2 + 3y2 – 7x – 6y = 12
is
a. 6
b. 9
c. 7
d. 13
22. 20 Problem
Which of the following lines is fathest from the origin ?
a. x – y + 1 = 0
b. 2x – y + 3 = 0
c. x + 2y – 2 = 0
d. x + y – 2 = 0
23. 21 Problem
The normal to a given curve is parallel to x-axis if :
dy
a. dx =0
dy
b. dx
=1
c. dx =0
dy
dx
d. =1
dy
24. 22 Problem
The distance between the lines 3x + 4y = 9 and 6x + 8y = 15 is :
3
a. 2
3
b.
10
c. 6
d. 9
4
25. 23 Problem
a line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-
intercept is :
1
a. 3
b. 2
3
c. 1
4
d.
3
26. 24 Problem
If the volume in cm3 and surface and surface area is cm2 of a sphere are
numerically equal Then the radius of the sphere in cm is :
a. 2
b. 3
c. 4
d. 5
27. 25 Problem
the locus of a point represented by x a t 1 y a t 1 is :
2 t
2 t
a. an ellipse
b. a circle
c. a pair of straight lines
d. none of these
28. 26 Problem
The equation of a tangent parallel to y = x drawn x2 y 2 is :
1
3 2
a. x – y + 1 = 0
b. x – y + 2 = 0
c. x + y – 1 = 0
d. x – y + 2 = 0
29. 27 Problem
Which of the following is not correct :
a. A function f such that f(x) = x – 1 for every integer x is binary operation
b. The binary operation positive on IR is commutative
c. The set{(a, b)/a I and b = - a} is a binary operation on I
d. The division on IR – [0] is not as associative operation
30. 28 Problem
If cos - 0.6 and 1800 < < 2700, then tan 4 is equal to :
a. 5
2
(1 5)
b. 2
c. (1 5)
2
( 5 1)
d.
2
31. 29 Problem
z1 z
Let z1 and z2 be two complex number such that 2 1. Then :
z2 z1
a. z1, z2 are collinear
b. z1, z2 and the origin form a right angled triangle
c. z1, z2 and the origin form an equilateral triangle
d. none of these
32. 30 Problem
1i 3
The amplitude of is :
3 1
a.
6
b. 4
c. 3
d. 2
33. 31 Problem
In a G.P. if (m + n)th terms is p and (m - 1)th term is q then mth term is :
p
a. q
q
b. p
c. Pq
d. pq
34. 32 Problem
In a G.P. of positive terms, any term is equal to the sum of the next two terms.
Then the common ratio of the G.P. is
a. 2 cos 180
b. sin 18
c. cos 18
d. 2 sin 18
35. 33 Problem
If n+2C8 n-2P4 = 57 : 16, then the value of x is :
a. 20
b. 19
c. 18
d. 17
36. 34 Problem
A library has a copies of one book, b copies of each of two books, c copies of each
of three books, and single copies of d books. The total number of ways in which
these books can be distributed is :
(a b c d)!
a. a! b ! c !
(a 2b 3c d)!
b. a !(b !)2 (c !)3
(a 2b 3c d)!
c.
a! b ! c !
d. None of these
37. 35 Problem
Three identical dice are rolled. The probability that the same number will appear
on each of them is :
1
a. 6
1
b. 36
1
c. 18
3
d. 28
38. 36 Problem
If a, b, c are real numbers a 0. If , is a root of a2x2 + bx + c = 0, is a
root of a2x2 – bx + c = 0 and 0 < a < b, then the equation a2x2 + 2bx + 2c = 0 has a
root that always satisfied :
a.
2
b.
2
c.
d.
39. 37 Problem
The solution of log7log5 ( x 5 x ) = 0 is :
a. 4
b. 5
c. 6
d. 9
40. 38 Problem
1
The most general value of , satisfying the two equations, cos = , tan 1
2
is
5
a. 2n
4
b. 2n
4
5
c. n
4
d. (2n 1)
4
41. 39 Problem
Three faces of a fair dice are yellow, two faces red and only one is blue. The dice
is tosses three times. The probability that the colour yellow, red and blue appears
in the first, second and the third tosses respectively is :
a. 1
30
b. 1
25
1
c. 36
1
d. 32
42. 40 Problem
If a2 + b2 + c2 = 1, then ab + bc + ca lies in the interval :
a. 1
2 ,2
b. [-1, 2]
1
c. 2 ,1
1
d. 1, 2
43. 41 Problem
The common tangent to the parabola y2 = 4ax and x2 = 4ay is :
a. x + y + a = 0
b. x + y - a = 0
c. x - y + a = 0
d. x - y - a = 0
44. 42 Problem
The solution of the equation cos2 + sin + 1 = 0, lies in the interval :
a. 4 , 4
3
b. 4 , 4
3 5
c. 4 , 4
5 7
d. 4 , 4
45. 43 Problem
If cosec - cot = q, then the value of cot is :
1 q2
a.
q
1 q2
b. q
q
c. 1 q2
q
d. 1 q2
46. 44 Problem
The length of the intercept on the normal at the point (at2, 2at) of the parabola y2
= 4ax made by the circle on the line joining the focus and p as diameter is :
a. a 1 t2
b. a 1 t2
c. a 1t
d. a 1t
47. 45 Problem
From the top of a light house 60 metres high with its base at the sea level, the
angle of depression of a boat is 150. The distance of the boat from the foot of the
light house is :
3 1
3 1
a. 60 metres
3 1
3 1
b. 60 metres
3 1
c.
3 1
metres
d. none of these
48. 46 Problem
The expression 1 sin 1 sin equal to :
for
1 sin 1 sin 2
2 cos
a.
cos2
2 sin
b.
sin2
2 cos
c.
sin2
d. none of these
49. 47 Problem
x2 y2
If m1 and m2 are the slopes of the tangent to the parabola 1 , which
15 16
passes through the point (6, 2) the value of (m1 + m2) is :
a. 14
11
4
b.
11
11
c. 4
24
d. 11
50. 48 Problem
If f’(x) = sin (log x) and y = f 2 x 3 , then dy equals :
3 2x
dx
1
a. sin (log x) . x log x
12 2x 3
b. 2
sin log
(3 2x) 3 2x
2x 3
c. sin log
3 2x
d. none of these
51. 49 Problem
The derivatives of tan-1 2x w.r.t. sin-1 2x is :
2
1 x 2
1 x
a. 2
b. 4
c. 1
d. 2
52. 50 Problem
x
m equals to :
lim 1
x
x
a. em
b. e-m
c. m-e
d. me
53. 51 Problem
x 2
If y f (x ) then :
x 1
a. f(1) = 3
b. 2 = f(4)
c. f is rational function of x
d. y increases with x for x < 1
54. 52 Problem
x
The co-ordinates of the points on the curve, f (x) 1 x where the tangent to
2
the curve has greatest slope is :
a. (0, 2)
b. (0, 0)
c. (0, 1)
d. (1, 1)
55. 53 Problem
the maximum value of xy subjected to x + y = 8 is :
a. 8
b. 16
c. 20
d. 24
56. 54 Problem
df
If f = xy then is equal to :
dx
a. x log f
y
b. xy . x
c. xy
log x
d. yx
57. 55 Problem
The curve y - exy + x = 0 has a vertical tangent at the point :
a. (1, 1)
b. at no point
c. (0, 1)
d. (1, 0)
58. 56 Problem
The sum of the series 1 1 :
1 ...
(1 2) (1 2 3)
a. 1
2
b. 1
c. 2
3
d. 2
59. 57 Problem
Let S be a finite set containing n elements. Then the total number of binary
operations on S is :
2
a. nn
b. nn
c. n2
2
d. 2n
60. 58 Problem
The ratio of the volume of a cube to that of the sphere which will fit inside the
cube :
a. 5 :
b. 7 :
c. 6 :
d. 3 :
61. 59 Problem
The function f(x) = cos (log (x + x2 1 )) is :
a. Even
b. Constant
c. Odd
d. None of these
62. 60 Problem
The perimeter of a given rectangle is x, its area will maximum of its sides are :
a. x x
,
2 2
x x
b. ,
3 3
x x
c. ,
6 3
x x
d. ,
4 4
63. 61 Problem
d 3y dy
Which of the following functions hold the property of 3
=0?
dx dx
a. y = ex
b. y = cos x
c. y = tan x
d. y = sin x
64. 62 Problem
0.5
the domain of definition of the function : x.1 2( x 4) + (x + 4)0.5 +
2 ( x 4)0.5
4(x + 4)0.5 is :
a. R
b. (- 4, 4)
c. R+
d. (-4, 0) (0, )
65. 63 Problem
If a, b, c, d are in H.P. then :
a. ab > cd
b. ad > bc
c. ac > bd
d. none of these
66. 64 Problem
If 1 x then the value of cos x is :
tan tan ,
2 1 x 2 1 x cos
a. sin x
b. cos x
c. cos
d. sin
67. 65 Problem
A round balloon of radius r subtends an angle at the eye of the observer while
the angle of elevation of its centre is , then the height of the centre of balloon
is :
a. r tan sin
b. r cosec 2
c. r tan and
d. r sin 2
cosec ,
68. 66 Problem
1 a 1 1
If a-1 + b-1 + c-1 = 0 such that 1 1 b 1
then the value of
1 1 1 c
is :
a. 0
b. - abc
c. abc
d. none of these
69. 67 Problem
4 5 2
If A = then adj. (A) equals :
5 4 2
2
2 8
36 36 18
a. 36 36 18
18 18
9
36 36 18
b.
36 36 18
18
18 9
0 0 0
c.
0 0 0
0 0 0
d. none of these
70. 68 Problem
The statement of k = 1 10 by 2, do remit in :
a. 3 cycles
b. 2 cycles
c. 8 cycles
d. 9 cycles
71. 69 Problem
The Newton’s method converges first of f’(x) is :
a. Small
b. mid
c. large
d. zero
72. 70 Problem
are unit vectors and is the angle between them then
If a and b a b
2
is :
a. sin 2
b. sin
c. 2 sin
d. sin 2
73. 71 Problem
The area of ABC with vertices A (1, -1, 2), B (2, 1, -1) and C (3, -1, 2) is :
a. 13 2 sq. units
b. 3 6 sq. units
c. 13 sq. units
d. 15 3 sq. units
74. 72 Problem
If p, q, r be three non-zero vectors, then equation p . q = p . r implies :
a. q = r
b. p is orthogonal to both q and r
c. p is orthogonal to q – r
d. either q = r or p is orthogonal to q – r.
75. 73 Problem
j ˆ i j ˆ i j ˆ
ˆ a2 ˆ a3 k, b b1ˆ b2 ˆ b3 k, c c1ˆ c2 ˆ c3 k, be three non-
Let a a1i
zero vectors such that is a unit vector perpendicular to both a and b . If the
a1 a2 a3
a and b is
angle between 6' then b1 b2 b3 is equal to :
c1 c2 c3
a. 0
b. 1
1 2 2
| a||b|
c. 4
3 2 2
d. | a||b|
4
76. 74 Problem
The scalar a.{(b c ) x (a b c )} equals :
a. 0
b. 2 [a b c ]
c. [a b c ]
d. none of these
77. 75 Problem
If f(x) = x1/x-1 for all positive x = 1 and if f is continuous at 1, then equals :
a. 0
1
b. e
c. e
d. e2
78. 76 Problem
the point equidistant from the four points (0, 0, 0), (3/2, 0, 0), (0,5/2, 0) and
(0, 0, 7/2) is :
2 1 2
a. 3' 3' 5
3
b. 3,2, 5
3 5 7
c. 4, 4, 4
1
2 , 0, 1
d.
79. 77 Problem
The area bounded by the curve y = x sin x and the x-axis between x = 0 and x = 2
is :
a. 2
b. 3
c. 4
d.
80. 78 Problem
The value of (i)I where i2 = -1 is :
a. Whole imaginary number
b. Whole real number
c. Zero
d. E /2
81. 79 Problem
The equations of the sphere with A (2, 3, 5) and B (4, 9, -3) as the ends of a
diameter is :
a. x2 + y2 + z2 – 6x – 12y – 2z + 20 = 0
b. 2x2 + 2y2 + 2z2 – x - y – z + 1 = 0
c. 3x2 + 3y2 + 3z2 – 2x – 2y –2z –1 = 0
d. none of these
82. 80 Problem
If (2, 0) is the vertex and y-axis the direction of a parabola, then its focus is :
a. (2, 0)
b. (-2, 0)
c. (4, 0)
d. (-4, 0)
83. 81 Problem
If f(x) = log (x + a), for x a , then the value of f’(x) is equal to :
1
a. x a
1
b. | x a|
1
c. | x a|
1
d. | x a|
84. 82 Problem
In a collection of 6 mathematics books and 4 physics books, the probability that 3
particular mathematics books will be together is :
1
a. 8
1
b. 10
1
c. 15
1
d. 20
85. 83 Problem
Find the mode from the data given below :
Marks 0-5 10-15 15-20 15-20 20-25 25-30
Obtained 18 20 25 30 16 14
a. 16.1
b. 16.2
c. 16.7
d. 16.3
86. 84 Problem
If the mean of four observations is 20 and when a constant c is added to each
observation, the mean becomes 22. The value of c is :
a. - 2
b. 2
c. 4
d. 6
87. 85 Problem
The area of surface generated by rotating the circle x = b = cos y = a + b sin ,
0 2 abut x-axis is :
a. ab sq. units
b. ab sq. units
x
c. 2 loge x c
d. ( x )a loge x c
88. 86 Problem
The eigin values of the matrix 5 4 are :
A
1 2
a. 2, 4
b. 1, 6
c. 3, -3
d. 1, -2
89. 87 Problem
x y 2 z 3
the length of perpendicular from the point P(3, -1, 11) to the line
2 3 4
is :
a. 2 13
b. 53
c. 2 14
d. 8
90. 88 Problem
the angle A, B, C of a triangle are in the ratio of 3 : 5 : 4, then a + i 2 is equal to :
a. 3b
b. 2b
c. 2b
d. 4b
91. 89 Problem
If the angle between the lines joining the foci of an ellipse to an extremity of the
minor axis is 900, the eccentricity of the ellipse is :
1
a.
8
11
b.
4
c. 3
2
1
d. 2
92. 90 Problem
If a man running at the rate of 15 km per hour crosses a bridge in 5 minutes, the
length of the bridge in metres is :
a. 7500
b. 1250
c. 1000
1
d. 1333 3
93. 91 Problem
If x y 15, x 2 y 2 49, xy 44 and n = 5, then byx is equal to :
1
a. - 3
2
b. - 3
c. - 1
4
1
d. -
2
94. 92 Problem
If (x + 1) is a factor of x4 + (p -3)x3 – (3p - 5)x2 + (2p -10) x + 5 then the value of p is
:
a. 2
b. 1
c. 3
d. 4
95. 93 Problem
Two finite sets have m and n elements. The total number of subsets of the first
set is 56 more than the total number of subsets of the second set. The values of
m and n are :
a. 7, 6
b. 6, 3
c. 5, 1
d. 8, 7
96. 94 Problem
The circle x2 = y2 – 2ax – 2ay + a2 = 0 touches :
a. x-axis only
b. y-axis only
c. both axis
d. none of these
97. 95 Problem
If |A| = 2 and A is a 2 x 2 matrix, what is the value of det, {adj. {adj. (adj. A2)}} is
equal to
a. 4
b. 16
c. 64
d. 128
98. 96 Problem
If y = (4x - 5) is a tangent to the curve y2 = px3 + q at (2, 3), then :
a. p = -2, q = -7
b. p = -2, q = 7
c. p = 2, q = -7
d. p = 2, q = 7
99. 97 Problem
given that x > 0, y > 0, x > y and z 0. the inequality which is not always correct is :
a. x + z > y + z
b. x – z > x – z
c. xz > y > z
x y
2
d. z 2
z
100. 98 Problem
Given that log10343 = 2.5353. The least n such that 7n > 105 is :
a. 4
b. 3
c. n
d. 6
101. 99 Problem
the smallest angle of the triangle whose sides are 6 12, 48, 24 is :
a. 4
b. 6
c.
3
d. none of these
102. 100 Problem
1 p
If f’(x) = (1 cos x) for 0 x 2 and that f(0) = 3, then f 2
lies in the interval :
a. 1
2 ,1
b. 4 , 2
c. 3, ,3
4
2
d. 3 ,3
4 2
