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
200
j
The coefficient of x100 in the expansion of 1 x is :
j 0
200
a. 100
201
b. 102
200
c. 101
201
d. 100
4. 02 Problem
The circles x2 + y2 – 10x + 16 = 0 and x2 + y2 = r2 intersect each other at two
distinct points, if :
a. r < 2
b. r > 8
c. 2 < r < 8
d. 2 r 8
5. 03 Problem
Three numbers are in AP such that their sum is 18 and sum of their equares is
158. The greatest number among them is :
a. 10
b. 11
c. 12
d. none of these
6. 04 Problem
Let
be three vectors. Then, scalar triple product [a b c] is equal to :
a, b and c
a. [ b a c]
b. [a c b]
c.
[ c b a]
d. [b c a]
7. 05 Problem
The roots of the equation x4 – 2x3 + x = 380 are :
a. 5, - 4,1 5 3
2
1 5 3
b. -5, 4, - 2
1 5 3
c. 5, 4, 2
1 5 3
d. -5, -4, 2
8. 06 Problem
Let y x .... dy is equal to :
xx , then
dx
a. yxy-1
y2
b. x 1 y log x
y
c.
x 1 y log x
d. none of these
9. 07 Problem
3
1
tan x is equal to :
dx
1 x2
a. 3 (tan-1 x)2 + c
4
b. tan 1 x
c
4
c. (tan-1 x)4 + c
d. none of these
10. 08 Problem
‘X’ speaks truth in 60% and ‘Y’ in 50% of the cases. The probability that they
contradict each other narrating the same incident is :
a. ¼
b. 1/3
c. ½
d. 2/3
11. 09 Problem
A set contains 2n + 1 elements. The number of subsets of this set containing
more than n elements is equal to :
a. 2n-1
b. 2n
c. 2n+1
d. 22n
12. 1
3
10 Problem
The area between the parabola y = x2 and the line y = x is :
1
a. 6
sq unit
1
b. sq unit
3
1
c. 2
sq unit
d. none of these
13. 11 Problem
The eccentricity of the hyperbola 5x2 – 4y2 + 20 x + 8y = 4 is :
a. 2
3
b.
2
c. 2
d. 3
14. 12 Problem
e x esin x is equal to :
lim
x 0 x sin x
a. -1
b. 0
c. 1
d. none of these
15. 13 Problem
x dx is equal to :
0 1 sin x
a. - π
b. π/2
c. π
d. none of these
16. 14 Problem
A man of mass 80 kg is traveling in a lift. The reaction between the floor of the lift
and the man when the lift is accelerating upwards at 4 m/s2 and the acceleration
due to gravity g = 9.81 m/s2, is equal to :
a. 884.8 N
b. 784.8 N
c. 464 N
d. 1104.8 N
17. 15 Problem
The argument of 1 i 3 / 1 i 3 is :
a. 600
b. 1200
c. 2100
d. 2400
18. 16 Problem
The points z1, z2, z3, z4 in a complex plane are vertices of a parallelogram taken in
order, then :
a. z1 + z4 = z2 + z3
b. z1 + z3 = z2 + z4
c. z1 + z2 = z3 + z4
d. none of these
19. 17 Problem
2
The harmonic mean between two numbers is 14 5 and the geometric mean is
24. The greater number between them is :
a. 72
b. 54
c. 36
d. none of these
20. 18 Problem
The angle between two forces each equal to P when their resultant is also equal
to P is :
a. 2π/3
b. π/3
c. π
d. π/2
21. 19 Problem
The solution of the differential equation sec2 x and y dx + sec2y tan x dy = 0 is :
a. tan y tan x = c
tan y
b. tan x c
tan2 x
c. c
tan y
d. none of these
22. 20 Problem
The real roots of the equation x2/3 + x1/3 – 2 = 0 are :
a. 1, 8
b. - 1, -8
c. - 1, 8
d. 1, -8
23. 21 Problem
Let f (x) = g(x) = ex. Then, (gof)’(0) is :
a. 1
b. -1
c. 0
d. none of these
24. 22 Problem
Cosine of the angle between two diagonals of a cube is equal to :
2
a. 6
b. 1
3
1
c.
2
d. none of these
25. 23 Problem
In a certain population 10% of the people are rich, 5% are famous and 3% are rich
and famous. The probability that a person picked at random from the population
is either famous or rich but not both, is equal to :
a. 0.7
b. 0.08
c. 0.09
d. 0.12
26. 24 Problem
Three numbers are in GP such that their sum is 38 and their product is 1728. The
greatest number among them is :
a. 18
b. 16
c. 14
d. none of these
27. 25 Problem
The equation of the circle touching x = 0, y = 0 and x = 4 is :
a. x2 + y2 – 4x – 4y + 16 = 0
b. x2 + y2 – 8x – 8y + 16 = 0
c. x2 + y2 + 4x + 4y - 4 = 0
d. x2 + y2 – 4x – 4y + 4 = 0
28. 26 Problem
1, when x is rational
Let f(x) = f x then lim f (x ) is :
0, when x is irraitonal ' x 0
a. 0
b. 1
c. 1/2
d. none of these
29. 27 Problem
| a | 4,| b | 4,|c | 2 and
a, b and c are three vectors with magnitude such that a
is perpendicular a b c ,b is perpendicular to (c a) and c is perpendicular to
(a b) . It follows that |a b c | is equal to :
a. 9
b. 6
c. 5
d. 4
30. 28 Problem
Let z1 and z2 be complex numbers, then |z1 + z2|2 + |z1 – z2|2 is equal to :
a. |z1|2 + |z2|2
b. 2 (|z1|2 + |z2|2)
c. 2(z12 + z22)
d. 4z1z2
31. 29 Problem
2
If tan tan
3
tan
3
3, then :
tan 2θ = 1
tan 3 θ = 1
tan2 θ = 1
tan3 θ = 1
32. 30 Problem
d2 y
Let y = t10 + 1 and x = t8 + 1, then 2 is equal to :
dx
5
a. 2 t
b. 20t8
5
c.
16t 6
d. none of these
33. 31 Problem
The vectors AB i j ˆ
3ˆ 5ˆ 4k and AC i j ˆ
5ˆ 5ˆ 2k are the side of a triangle ABC.
The length of the median through A is :
a. 13 unit
b. 2 5 unit
c. 5 unit
d. 10 unit
34. 32 Problem
If a, b, c are three non-coplanar vectors, then (a b c).[(a b) x(a c)] is
a. 0
b. 2 [a b c]
c. – [a b c]
d. [a b c]
35. 33 Problem
dx is equal to :
x( x 5 1)
1 5 5
a. 5 log x (x 1) c
1 x5 1
log c
b. 5 x5
1 x5
c. log 5 c
5 x 1
d. none of the above
36. 34 Problem
A function f on R into itself is continuous at a point a in R, iff for each > 0, there
exists, > 0 such that :
a. | f (x) f (a) | |x a|
b. | f (x) f (a) | |x a|
c. | x a| | f (x) f (a) |
d. | x a| | f (x) f (a) |
37. 35 Problem
x sin x is equal to :
dx
1 cos x
x
a. x tan +c
2
x
b. x sec2 2 + c
x
c. log cos
2
d. none of these
38. 36 Problem
A straight line through the point (1, 1) meets the x-axis at ‘A’ and the y-axis at ‘B’.
The locus of the mid point of AB is :
a. 2xy + x + y = 0
b. x + y – 2xy = 0
c. x + y + 2 = 0
d. x + y – 2 = 0
39. 37 Problem
2 4 5
If A 4 8 10 , then rank of A is equal to :
6 12 15
a. 0
b. 1
c. 2
d. 3
40. 38 Problem
A bag contains 8 red and 7 black balls. Two balls are drawn at random. The
probability that both the balls are of the same colour, is :
14
a.
15
11
b. 15
c. 7
15
4
d.
15
41. 39 Problem
2 x2 y2 1
If sin , then x must be :
2x
a. - 3
b. - 2
c. 1
d. none of these
42. 40 Problem
The solution of equation cos2 θ + sin θ + 1 = 0 lies in the interval :
a. ,
4 4
b. 3
,
4 4
c. 3 , 5
4 4
5 7
d. ,
4 4
43. 41 Problem
Coefficient of x19 in the polynomial (x –1) (x - 2) ……..(x - 20) is equal to :
a. 210
b. - 210
c. 20!
d. None of these
44. 42 Problem
Two pillars of equal height stand on either side of a road way which is 60 m wide.
The a point in the road way between the pillars, the elevation of the top of pillars
are 600 and 300. The height of the pillars is :
a. 15 3m
15
b. 3 m
c. 15 m
d. 20 m
45. 43 Problem
A light string passing over a light smooth pulley carries masses of 3 kg and 5kg at
its ends. If the string is allowed to move from the rest, the acceleration of the
motion is equal to :
a. (g/2)m/s2
b. (g/4)m/s2
c. 2g m/s2
d. 4g m/s2
46. 44 Problem
The equation of the directrix of the parabola x2 + 8y – 2x = 7 is :
a. y = 3
b. y = -3
c. y = 2
d. y = 0
47. 45 Problem
If iz4 + 1 = 0, the z can take the value :
1 i
a.
2
b. cos i sin
8 8
1
c. 4i
d. i
48. 46 Problem
If a i j ˆ
ˆ ˆ k, b i j ˆ
2ˆ 3ˆ k, and c ˆ
i ˆ
j are coplanar vectors, the value of a is :
a. - 4
3
3
b. 4
4
c. 3
d. 2
49. 47 Problem
The equation of the tangent parallel to y – x + 5 = 0 drawn to
a. x – y – 1= 0
b. x – y + 2 = 0
c. x + y – 1 = 0
d. x + y + 2 = 0
50. 48 Problem
The equation y2 – x2 + 2x – 1 = 0 represents :
a. A hyperbola
b. An ellipse
c. A pair of straight lines
d. A rectangular hyperbola
51. 49 Problem
The minimum value of 3 sin θ + 4 cos θ is :
a. 5
b. 1
c. 3
d. - 5
52. 50 Problem
A man in swimming with the uniform velocity of 6 km/h straight across a river
which is flowing at the rate of 2 km/h. If the breadth of the river is 300 m, the
distance between the point and the man is initially directed to and the point it
will reach on the opposite bank of the river is equal to :
a. 100 m
b. 200 m
c. 300 m
d. 400 m
53. 51 Problem
A ball is thrown vertically upwards from the ground with velocity 15 m/s and
rebounds from the ground with one-third of its striking velocity. The ratio of its
greatest heights before and after striking the ground is equal to :
a. 4 : 1
b. 9 : 1
c. 5 : 1
d. 3 : 1
54. 52 Problem
If the position vectors of the vertices A, B, C of a triangle ABC are
j ˆ i j ˆ
7ˆ 10k, ˆ 6ˆ 6k and i j ˆ
4ˆ 9ˆ 6k respectively, the triangle is :
a. Equilateral
b. Isosceles
c. Scalene
d. Right angled and isosceles also
55. 53 Problem
The number of solutions of the equation 2 cos (ex) = 5x + 5-x, are :
a. No solution
b. One solution
c. Two solution
d. Infinitely many solutions
56. 54 Problem
Probability of throwing 16 in one throw with three dice is ;
a. 1
36
1
b.
18
c. 1
72
1
d.
9
57. 55 Problem
The differential equation of all straight lines passing through origin is :
dy
a. y x
dx
dy
b. dx
=y+x
dy
c. dx
=y–x
d. Nome of these
58. 56 Problem
Ifa, b, c are three unit vectors such that a b c 0 where 0 is null
vector, thenb c c a
a b is :
a. - 3
b. - 2
3
c. - 2
d. 0
59. 57 Problem
The expression equal to :
a. -1
b. 0
c. 1
d. none of these
60. 58 Problem
is equal to :
a.
4
b.
6
c. 3
2
d.
3
61. 59 Problem
If f(x) = (a - xn)1/n, where a > 0 and n N, then f0f(x) is equal to :
a. a
b. x
c. xn
d. an
62. 60 Problem
The number of reflexive relations of a set with four elements is equal to :
a. 216
b. 212
c. 28
d. 24
63. 61 Problem
The maximum horizontal range of a ball projected with a velocity of 39.2 m/s is
(take g = 9.8 m/s2)
a. 100 m
b. 127 m
c. 157 m
d. 177 m
64. 62 Problem
Maximum value of sin x – cos x is equal to :
a. 2
b. 1
c. 0
d. none of these
65. 63 Problem
The equation of the bisector of the acute angles between the lines 3x – 4y + 7 = 0
and 12x + 5y -2 = 0 is :
a. 99x – 27y – 81 = 0
b. 11x -3y + 9 = 0
c. 21x + 77y – 101 = 0
d. 21 x + 77y + 101 = 0
66. 64 Problem
To reduce the differential equation + P(x)y = Q(x).yn to linear form, the subsitution
is
a. v =
b. v =
c. v = yn
d. v = yn – 1
67. 65 Problem
A particle possess two velocities simultaneously at an angle of tan-1 to each other.
Their resultant is 15 m/s. If one velocity is 13 m/s, then the other will be :
a. 5 m/s
b. 4 m/s
c. 12 m/s
d. 13 m/s
68. 66 Problem
If in the expansion of (1 + x)21, the coefficients of xr and xr + 1 be equal, then r is
equal to :
a. 9
b. 10
c. 11
d. 12
69. 67 Problem
A train is running at 5 m/s and a man jumps out of it with a velocity 10 m/s in a
direction making an angle of 600 with the direction of the train. The velocity of
the man relative to the ground is equal to :
a. 12.24 m/s
b. 11.25 m/s
c. 14.23 m/s
d. 13.23 m/s
70. 68 Problem
A ball is projected vertically upward with a velocity 112 m/s. The time taken by it
to return to the point of projection is (g = 10 m/s2) :
a. 11 s
b. 33 s
c. 5.5 s
d. 22 s
71. 69 Problem
If the sides of triangle are 4, 5 and 6 cm, then area of the triangle is equal to :
15
a. 4
cm2
15
b. 7 cm2
4
4
c. 7 cm2
15
d. none of these
72. 70 Problem
The volume of a spherical cap of height h cut off from a sphere of radius a is
equal to :
a. 3 h2 (3a - h)
b. π (a - h)(2a2 – h2 - ah)
4
c. h3
3
d. none of the above
73. 71 Problem
The eccentricity of the hyperbola conjugate to x2 –3y2 = 2x + 8 is :
2
a.
3
b. 3
c. 2
d. none of these
74. 72 Problem
The area of the parallelogram whose adjacent sides are is :
a. 2
b. 4
c. 17
d. 2 13
75. 73 Problem
Integrating factor of the differential equation dy is :
P(x)y Q(x)
dx
a. P dx
b. Q dx
c. P dx
e
d. e Q dx
76. 74 Problem
Angle of intersection of the curves r = sin + cos and r = 2 sin is equal to :
a. 2
b. 3
c.
4
d. none of these
77. 75 Problem
1
Define f on R into itself by x sin , when x 0, then :
f (x) x
0, when x 0
a. f is continuous at 0 but not differentiable at 0
b. f is both continuous and differentiable at 0
c. f is differentiable but not continuous at 0
d. none of the above
