2. SECTION-I
Single Correct Answer Type
There are five parts in this question. Four choices are given for each part and one of them is
correct. Indicate you choice of the correct answer for each part in your answer-book by
writing the letter (a), (b), (c) or (d) whichever is appropriate
3. 01 Problem
If the probability that A and B will die within a year are p and q
respectively, then the probability that only one of them will be alive at the
end of they year, is :
a. p+q
b. p + q – 2pq
c. p + q – pq
d. p + q + pq
4. 02 Problem
Ten different letters of alphabet are given. Words with five letters are formed
from these given letters. Then, the number of words which have atleast one
letter repeated, is :
a. 69760
b. 30240
c. 99748
d. none of these
5. 03 Problem
If (1 + x + x2)n = C0 + C1x + C2x2 + … then the value of C0C1- C1C2 + C2C3 –
a. 0
b. 3n
c. (-1)n
d. 2n
6. 04 Problem
Given positive integers r > 1, n > 2 and that the coefficient of (3r)th and (r +
2)th term in the binomial expansion of (1 + x)2n are equal, then :
a. n = 2r
b. n = 3r
c. n = 2r + 1
d. none of these
7. 05 Problem
A ball of mass 1 kg moving with velocity 7 m/s, overtakes and collides with a
ball of mass 2 kg moving with velocity 1 m/s in the same direction. If 2 =
¾, the velocity of lighter ball after impact is :
a. 6 m/s
3
b. 2
m/s
c. 1 m/s
d. 0 m/s
8. 06 Problem
A bullet of 0.05 kg moving with a speed of 120 m/s penetrates deeply into a fixed
target and is brought to rest in 0.01 s. The distance through which it penetrates is
:
a. 3 cm
b. 6 cm
c. 30 cm
d. 60 cm
9. 07 Problem
a, b, c are real a b, the roots of the equation (a - b)x2 –5 (a + b) x – 2 (a - b) =
0 are :
a. real and equal
b. complex
c. real and unequal
d. none of these
10. 08 Problem
/2
The value of
0
| sin x cos x | dx is equal to :
a. 0
b. 2( 2 -1)
c. 2 2
d. 2( 2 + 1)
11. 09 Problem
1
The value of
1
x | x | dx is equal to
a. 2
b. 1
c. 0
d. none of these
12. 10 Problem
x sin1 x
The value of 1 x2
dx is equal to :
a. (1 x 2 ) sin-1 x + c
b. x sin–1 x + c
c. x - (1 x 2 ) sin-1 x + c
d. (sin-1 x)2 + c
13. 11 Problem
cos3 2n 1 xdx has the value
2
For any integer n, the integral esin x
0
a. -1
b. 0
c. 1
d.
14. 12 Problem
The cube roots of unity when represented argand diagram from the vertices of :
a. an equilateral triangle
b. an isosceles triangle
c. a right angled triangle
d. none of the above
15. x
a
13 Problem
(a
x
The value of / x )dx is equal to :
x
a. a loge a + c
b. 2 a x log10 a + c
x
c. 2 a loga e + c
x
d. 2 a loge a + c
16. 14 Problem
If +b then
1
a. a = 3
2
b. a =
3
1
c. a = - 3
2
d. a = - 3
17. 15 Problem
a
If f(a - x) = f(x), then x f(x)d
0
x is equal to :
a a
2 0
a. f(x) dx
a
b. a
0
f(x)dx
a2 a
c.
2
0
f(x)dx
d. none of these
18. 16 Problem
1
If z 5 i Then z lies
, where | | 1.
z
a. a circle
b. a parabola
c. an ellipse
d. none of these
19. 17 Problem
If log 2, log (2x - 1), log (2x + 3) are in AP Then x is equal to :
a. 5/2
b. log2 5
c. log3 2
d. log5 3
20. 18 Problem
Points D, E are taken on the side BC of a triangle such that BD = DE = EC. If
BAD , DAE , EAC , then the value of sin sin is
sin sin
equal to :
a. 1
b. 2
c. 4
d. none of these
21. 19 Problem
The value of cos (2 cos-1 x + sin-1 x) at x = 1/5 is equal to :
2 6
a.
5
2 5
b.
6
2 6
c. -
5
2 5
d. - 6
22. 20 Problem
/2
If f(x) is an odd function of x, then
/ 2
f (cos x )dx is equal to :
a. 0
/2
b. 0
f (cos x )dx
/2
c.
0
f (cos x )dx
d. f cos x dx
23. 21 Problem
y = cos-1 [sin (1 x) / 2] + xx, then dy/dx at x = 1 is equal to :
a. 3/4
b. 0
c. 1/2
d. -1/2
24. 22 Problem
f(x) = (sin x + cos 2x), (x > 0) has minimum value for x is equal to :
n
a. 2
3
b. n 1
2
1
c. 2n 1
2
d. none of these
25. 23 Problem
The point P on curve y2 = 2x3 such that the tangent at P is perpendicular to the
line 4x – 3y + 2 = 0 is given by :
a. (2, 4)
b. (1, 2)
1 1
c. 2 , 2
1 1
d. ,
8 16
26. 24 Problem
Let I1 = 2 dx 2 dx then :
1
1 x2
and I2
1 x
,
a. I1 > I2
b. I2 > I1
c. I1 = I2
d. I1 > 2I2
27. 25 Problem
The value of the integral is equal to :
a. 0
1
b. 2
c. 1
2
d. none of these
28. 26 Problem
In ABC, the value of cosec A (sin B cos C + cos B sin C) is equal to :
c
a. a
b. a
c
c. 1
d. none of these
29. 27 Problem
The value of tan 90 – tan 270 – cot 270 + cot 90 is :
a. 2
b. 3
c. 4
d. none of these
30. 28 Problem
In ABC, 3 sin A = 6 sin B = 2 3 sin C, then the value of A is equal to :
a. 00
b. 450
c. 600
d. 900
31. 29 Problem
If the side of the triangle are 5k, 6k, 5k and radius of incircle is 6, then the
value of ‘k’ is
a. 4
b. 5
c. 6
d. 7
32. 30 Problem
The angle of depression of the top and the foot of the chimney as seen from the
top of second chimney which is 150 m high and standing on the same level as the
first are and respectively. The distance between their tops when tan
4 5 is equal to :
and tan
3 2
50 m
100 m
150 m
none of these
33. 31 Problem
The medians AD and BE of the triangle with vertices A (0, b), B (0, 0) and C
(a, 0) are mutually perpendicular, if :
a. b = 2 a
b. a = b 2
c. b = - 2 a
d. a = - 2b
34. 32 Problem
The distance between the chords of contact of the tangent to the circle x2 + y2
+ 2gx + 2fy + c = 0 from the origin and the point (g, f) is :
a. g2 + f2
b. (g2 + f2 + c)
g2 f 2 c
c.
2 g2 f 2
d. g2 f 2 c
2 g2 f 2
35. 33 Problem
A line is drawn through a fixed point p (h, k) to cut the circle x2 + y2 = a2 at Q
and R. Then PQ . PR is equal to :
a. (h + k)2 – a2
b. h2 + k2 – a2
c. (h – k)2 + a2
d. h2 + k2 + a2
36. 34 Problem
The locus of the mid point of a focal chord of a parabola is :
a. Circle
b. Parabola
c. Ellipse
d. Hyperbola
37. 35 Problem
2 2
The straight line x + y = c will be tangent to the ellipse x y 1 then c is equal
9 16
to :
a. 8
b. 5
c. 10
d. 6
38. 36 Problem
The length of the subnormal of the curve y2 = 2ax is equal to
a. a
b. 2a
c. a/2
d. -a
39. 37 Problem
2 2
If is the angle between the asymptotes of the hyperbola x y 1 with
a2 b2
eccentricity e, then sec 2 is equal to :
a. 0
b. e
c. e2
e
d. 2
40. 38 Problem
The latus rectum of the hyperbola 9x2 – 16y2 + 72x –32y –16 = 0 is :
9
a. 2
b. - 9
2
c. 32
3
d. - 32
3
41. 39 Problem
x2 1
The value of p and q from lim px q 0 are :
x 1
x
a. p = 0, q = 0
b. p = 1, q = -1
c. p = -1, q = 1
d. p = 2, q = - 1
42. 40 Problem
Which of the following functions is an even function ?
ax 1
a. f x x x
a 1
b. f(x) = tan x
ax a x
c. f x x x
a 1
ax 1
d. f x x
a 1
43. 41 Problem
x 2 sin1 / x, x 0
If f(x) = , then :
0, x 0
a. f and f’ are continuous at x = 0
b. f is derivable at x = 0
c. f is derivable at x = 0 and f’ is not continuous at x = 0
d. f is derivable at x = 0
44. 42 Problem
p q r
Let p, q, r be positive and not all equal, then the value of the determinant q r p
r p q
is equal to :
a. Positive
b. Negative
c. 0
d. none of these
45. 43 Problem
Suppose n 3 person are sitting in a row. Two them are selected at random.
Then probability that they are not together, is L:
a. 1- 2/n
b. 2/(n -1)
c. 1 – 1/n
d. none of these
46. 44 Problem
The minimum value of x2 – 3x + 3 in the interval (-3, 3/2) is equal to :
a. 3/4
b. 5
c. - 15
d. - 20
47. 45 Problem
100 100
Let tn be the nth term of the GP of positive numbers. Let t
n1
2n x and t2n1 y
n1
such that x y , then the common ratio is
x
a. y
x
b.
y
c. y
x
d. none of these
48. 46 Problem
If every element of third order determinant value of is multiplied by five, then
the value of the new determinant is :
a.
b. 5
c. 25
d. 125
49. 47 Problem
In the expansion of (1 + x)50, the sum of the coefficients of odd powers of x is :
a. 0
b. 249
c. 250
d. 251
50. 48 Problem
The equation sin6 cos6 a has a real solution, if :
a. 1/2 a 1
b. 1/4 a 1
c. -1 a 1
d. 0 a 1/2
51. 49 Problem
(1 x
2
) sin x cos2 x dx is equal to :
a. 0
b. - 3/3
c. 2 - 3
7
d. 2 3
2
52. 50 Problem
The derivative of f x x3 dt
(x 0) is equal to :
x2 log t
a. 1 1
3log x 2log x
1
b.
3log x
3x 2
c.
3 log x
x 1 x
d.
log x
53. 51 Problem
4 4 1
If f x dx 4 and [3 f (x)]dx 7 , then the value of f x dx is equal
1 2 2
to :
a. 2
b. - 3
c. - 5
d. none of these
54. 52 Problem
e [ (x) '(x)]dx is equal to :
x
a. ex '(x)dx
b. e x ( x ) c
c. ex '(x) c
d. none of these
55. 53 Problem
1
The value of | sin2 x |
0
dx is equal to :
a. 0
1
b. -
1
c.
2
d.
56. 54 Problem
A house has multi-storeys. The lowest storey is 20 ft high. A stone which is
dropped from the top of the house passes the lowest storey in 1/4 s, then
the height of the house is
a. 110.00 ft
b. 110.2 ft
c. 110.25 ft
d. none of these
57. 55 Problem
A particle was dropped from the top of the tower and at the same time
another body is thrown vertically upwards from the bottom of the tower with
such a velocity that it can just reach the top of the tower, then they will meet
at the height of :
a. h/4
b. 3h/4
c. h
d. h/2
58. 56 Problem
A particle starts with a velocity of 200 cm/s and moves in a straight line with
a retardation of 10 cm/s2. Then the time it takes to describe 1500 cm is :
a. 10 s, 30 s
b. 5 s, 15 s
c. 10 s
d. 30 s
59. 57 Problem
The area bounded by the curve y = x3, the x – axis and the ordinates x = -2 and
x = 1 is :
a. -9 sq unit
b. -15/4 sq unit
c. 15/4 sq unit
d. 17/4 sq unit
60. 58 Problem
Two balls are projected respectively from the same point in direction inclined
at 300 and 600 to the horizontal. If they attain the same height, then the ratio
of their velocities of projection is
a. 1 : 3
b. 3 :1
c. 1 : 1
d. 1 : 2
61. 59 Problem
A particle is projected under gravity (g = 9.81 m/s2) with a velocity of 29.43
m/s at an angle of 300. The time of flight in seconds to a height of 9.81 m are
:
a. 5, 1.5
b. 1, 2
c. 1.5, 2
d. 2, 3
62. 60 Problem
The path of a projectile in vacuum is a :
a. A straight line
b. Circle
c. Ellipse
d. Parabola
63. 61 Problem
A particle is projected with initial velocity u making an angle with the
horizontal, its time of flight will be given by :
a. 2u sin
g
b. 2u 2 sin
g
u sin
c.
g
u2 sin
d. g
64. 62 Problem
If x2 + px + 1 is a factor of x2 + bx + c, then :
a. a2 + c2 = - ab
b. a2 – c2 = - ab
c. a2 – c2 = ab
d. none of the above
65. 63 Problem
The probabilities of occurrence of two events E and F are 0.25 and 0.50
respectively. The probability of their simultaneous occurrence is 0.14. The
probability that neither E occurs nor F occurs :
a. 0.39
b. 0.25
c. 0.11
d. none of these
66. 64 Problem
A sphere impinges directly one an equal sphere which is at rest. Then the
original kinetic energy lost is equal to :
1 e2
a. 2
times the initial KE
1 e2
b.
2
1 e2
c. times the initial KE
2
d. none of these
67. 65 Problem
A hockey stick ball is at rest for 0.01s with an average force of 5 N. If the ball
weight 0.2 kg, then the velocity of the ball after being pushed is equal to :
a. 2.5 m/s
b. 2 m/s
c. 3.0 m/s
d. 5 m/s
68. 66 Problem
A given force is resovled into components P and Q is equally inclined to it, then :
a. P = 2Q
b. P = Q
c. 2P = Q
d. none of these
69. 67 Problem
If the forces of 12,5 and 13 unit weight balance at a point, two of them are
inclined at :
a. 300
b. 450
c. 900
d. 600
70. 68 Problem
ABC is a triangle. Forces P, Q, R act along the lines OA, OB and OC and are in
equilibrium, if O is incentre of ABC, then :
P Q R
A B C
a. cos cos cos
2 2 2
P Q R
b.
OA OB OC
P Q R
c.
A B C
sin sin sin
2 2 2
d. none of the above
71. 69 Problem
If two equal perfectly elastic balls impinges directly, after impact :
a. Their velocities are not effected
b. They interchange their velocities
c. Their velocities changes their direction
d. Their velocities get doubled
72. 70 Problem
In triangle ABC (sin A + sin B + sin C) (sin A + sin B – sin C) = 3 sin A sin B then :
a. A = 600
b. B = 600
c. C = 600
d. A = 900
73. 71 Problem
If sin A p, cos A q then :
sin B cos B
p q2 1
tan A
a. q 1 p2
2
b. tan A p q 1
2
q 1 p
q2 1
c. tan B
1 p2
d. all are correct
74. 72 Problem
In a triangle ABC, A and AD is median then :
3
a. 4AD2 = b2 + bc + c2
b. AD2 = b2 + bc + c2
c. 2AD2 = b2 + bc + c2
d. 4AD2 = b2 – bc + c2
75. 73 Problem
If the radius of the circumcircle of isoseceles triangle ABC is equal to AB =
AC, then angle A is equal to :
a. 300
b. 600
c. 900
d. 1200
76. 74 Problem
The focus of the parabola y2 – x – 2y + 2 = 0 is :
a. (1/4, 0)
b. (1/2)
c. (3/4, 1)
d. (5/4, 1)
77. 75 Problem
If A (3, 1), B (6, 5) and C (x , y) are three points such that the angel CAB is a right
angle and the area of CAB = 7, then the number of the point C is ;
a. 0
b. 1
c. 2
d. 4
79. 77 Problem
lim
Evaluate x 0 [sin (x + a) + sin (a - x) – 2sin a]/x sin x
80. 78 Problem
A rod is moveable in a vertical plane about a hinge at one end, another end is
fastened to a weight equal to half the weight of the rod, this end is fastened by
a string of length l to a pint at a height c vertically over the hinge find the
tension in of the string.
81. 79 Problem
Sun of infinity of the series
12 22 12 22 32
1 ........
2! 3!
82. 80 Problem
Evaluate, cos x dx
1 sin x 2 sin x
2
83. 81 Problem
Find the equation to the chord of the hyperbola 25x2 – 16y2 = 400 having mid
point at (6, 2).
84. 82 Problem
i j ˆ i j ˆ i j ˆ
Find the vector moment f the three vectors ˆ 2ˆ 3k,2ˆ 3ˆ 4k, ˆ ˆ k acting
on a particle at point P (0, 1, 2) about the point A (1, -2, 0).
85. 83 Problem
If the equation k(6x2+3) + rx+2x2 -1 =0 and 6k (2x2 +1 ) + px +4x 2 -2 =0, have
both the common roots find the value of (2r – p)
86. 84 Problem
Find the equation to the common tangnt to the parabola y2=2x and x2 = 16y
87. 85 Problem
x 5
Determine the value of ‘x’ in the expansion of x xlog
10 if the third term in
the expansion is 10,00,000.
88. 86 Problem
In parallelogram ABCD the interior bisectors of the consecutive angles B and C
intersect at P, then find . BPC
89. 87 Problem
Find the area bounded by the curve y = 2x – x2 and the straight line y = - x.
90. 88 Problem
b2 c 2 a2 a2
Find the value of determinant b2 c 2 a2 b2
c2 c2 a2 b2
91. 89 Problem
The probability of getting sum more than 15 in three dice will be 5/108. Prove
it.
92. 90 Problem
If tan 2 tan =1, then find the value of .
93. 91 Problem
If A = [1 2 3] and B = 5 4 0 , then find AB.
0 2 1
1 3 2
94. 92 Problem
Find the length of tangent of circle x2 + y2 + 6x – 4y – 3 = 0 from point (5, 1).
95. 93 Problem
Find the value of cos 200 cos 400 cos 600 cos 800.
96. 94 Problem
x 3 y 4 z 5
Find the distance from point (3, 4, 5) of that point where line 1
2
2
cut the plane x + y + z – 17 = 0.
97. 95 Problem
Find the point on the curve 9y2 = x3, where normal to the curve makes equal
intercepts with the axes.
98. 96 Problem
A ladder 15 m long leans against a wall 7 m high and a portion of the ladder
protrudes over the wall such that its projection along the vertical is 3 m. How fast
does the bottom start to slip away from the wall, if the ladder slides down along
the top edge of the wall at 2 m/s ?
99. 97 Problem
Solve the equation sin [2 cos-1 {cot (2 tan-1 x) }]= 0
100. 98 Problem
The side AB, BC, CD and DA of a quadrilateral have the equations x + 2y = 3, x =
1, x – 3y = 4, 5x + y + 12 = 0 respectively. Find the angle between the diagonals
AC and BD.
101. 99 Problem
The angles of top of the tower from the foot and top of a building are . Find the
height of tower.
102. 100 Problem
If lines px2 – pxy – y2 = 0, make the angle from x-axis, then find the value of tan .
