multiple choice questions

1. MULTIPLE QUESTIONS (EXEMPLAR PROBLEMS)
1. Let R be a relationon the setN of natural numbersdefinedbynRm if n dividesm. thenR is
a. Reflexive andsymmetric
b. Transitive and symmetric
c. Equivalence
d. Reflexive,transitive butnot symmetric
2. Let N be the set of natural numbers and the function f : N  be definedby f(n) = 2n + 3  n  N. thenf is
a. Surjective b. Injective c. Biijective d. None of these
3. Set A has 3 elementsandthe setB has 4 elements.Thenthe number ofinjective mappings that can be
definedfromA to B is
a. 144 b. 12 c. 24 d. 64
4. Let f : R  R be definedbyf(x) = sin x and g : R R be definedbyg(x) = x,then f o g is
a. x2
sinx b. (sinx)2
c. sin x2
d.
𝒔𝒊𝒏 𝒙
𝒙 𝟐
5. Let f : R  R be definedbyf(x) = 3x – 4. Then f-1
(x) isgiven by
a.
𝒙+𝟒
𝟑
b.
𝒙
𝟑
- 4 c. 3x – 4 d. none of these
6. The maximumnumber of equivalence relationsonthe set A = {1, 2, 3} are
a. 1 b. 2 c. 3 d. 5
7. Let us define a relationR in R as aRb if a b. thenR is
a. An equivalence relation
b. Reflexive,transitive butnot symmetric
c. Symmetric,transitive but not reflexive
d. Neithertransitive nor reflexive butsymmetric.
8. The identityelementforthe binary operation  definedinQ  {0} as a  b=
𝒂𝒃
𝟐
 a, b  Q  {0} is
a. 1 b. 2 c. 0 d. none of these
9. Let A = {1, 2, 3,….., n} and B = {a, b}. thenthe numbersof surjections from A intoB is
a. n
P2 b. 2n
– 2 c. 2n
– 1 d. none of these
10. Let f : R  R be definedbyf(x) = 3x2
- 5 and g : R  R by g(x) =
𝒙
𝒙 𝟐+ 𝟏
.theng o f is
a.
𝟑 𝒙 𝟐− 𝟓
𝟗𝒙 𝟒− 𝟑𝟎𝒙 𝟐+ 𝟐𝟔
b.
𝟑 𝒙 𝟐− 𝟓
𝟗𝒙 𝟒− 𝟔𝒙 𝟐+ 𝟐𝟔
c.
𝟑 𝒙 𝟐
𝒙 𝟒+𝟐𝒙 𝟐− 𝟒
d.
𝟑 𝒙 𝟐
𝟗𝒙 𝟒+ 𝟑𝟎𝒙 𝟐− 𝟐
11. Whichof the followingfunctionsfromZ into Z are bijections?
a. f(x) = x3
b. f(x) = x + 2 c. f(x) = 2x + 1 d. f(x) = x2
+ 1
12. Let f : R  R be the functionsdefinedbyf(x) = x3
+ 5. Then f-1
(x) is
a. (𝒙 + 𝟓)
𝟏
𝟑 b. (𝒙 − 𝟓)
𝟏
𝟑 c. (𝟓 − 𝒙)
𝟏
𝟑 d. 5 – x
13. Let f : R - {
𝟑
𝟓
}  R be definedbyf(x) =
𝟑𝒙+𝟐
𝟓𝒙−𝟑
. Then

2. a. f-1
(x) = f(x) b. f-1
(x) = - f(x) c. (fo f) x = - x d. f-1
(x) =
𝟏
𝟏𝟗
f(x)
14. Let f : [2, )  Rbe the functiondefined byf(x) = x2
- 4x + 5, then the range of f is
a. R b. [1, ) c. [4, ) d. [5, )
15. Let f : R  R be givenby f(x) = tan x. Then f-1
(1) is
a.
𝝅
𝟒
b. { n  +
𝝅
𝟒
: n  Z} c. does not exist d. none of these
16. Let the relationR be definedinN by aRb if2a + 3b = 30. Then R = …………………….
17. The principal value branch of sec-1
is
a. [−
𝝅
𝟐
,
𝝅
𝟐
] - {0} b. [0, ]- {
𝝅
𝟐
} c. (0, ) d. (−
𝝅
𝟐
,
𝝅
𝟐
)
18. The value of 𝐬𝐢𝐧−𝟏 ( 𝒄𝒐𝒔(
𝟒𝟑 
𝟓
)) is
a.
𝟑𝝅
𝟓
b.
−𝟕𝝅
𝟓
c.
𝝅
𝟏𝟎
d. -
𝝅
𝟏𝟎
19. The principal value ofthe expressioncos-1
[cos (-6800
)]is
a.
𝟐𝝅
𝟗
b.
−𝟐𝝅
𝟗
c.
𝟑𝟒𝝅
𝟗
d.
𝝅
𝟗
20. If tan-1
x =
𝝅
𝟏𝟎
for some x  R, thenthe value of cot-1
x is
a.
𝝅
𝟓
b.
𝟐𝝅
𝟓
c.
𝟑𝝅
𝟓
d.
𝟒𝝅
𝟓
21. The principal value ofsin-1
(
−√ 𝟑
𝟐
) is
a. −
𝟐𝝅
𝟑
b. −
𝝅
𝟑
c.
𝟒𝝅
𝟑
d.
𝟓𝝅
𝟑
22. The greatestand leastvaluesof (sin-1
x)2
+ (cos-1
x)2
are respectively
a.
𝟓𝝅 𝟐
𝟒
𝒂𝒏𝒅
𝝅 𝟐
𝟖
b.
𝝅
𝟐
𝒂𝒏𝒅
−𝝅
𝟐
c.
𝝅 𝟐
𝟒
𝒂𝒏𝒅
−𝝅 𝟐
𝟒
d.
𝝅 𝟐
𝟒
𝒂𝒏𝒅 𝟎.
23. The value of sin (2 sin-1
(.6)) is
a. .48 b. .96 c. 1.2 d. sin 1.2
24. If sin-1
x + sin-1
y =

𝟐
, then value of cos-1
x + cos-1
y is
a.
𝝅
𝟐
b.  c. 0 d.
𝟐𝝅
𝟑
25. The value of the expressionsin[cot-1
(cos (tan-1
1))]is
a. 0 b. 1 c.
𝟏
√ 𝟑
d. √
𝟐
𝟑
26. Whichof the followingisthe principal value branch ofcosec-1
x?
a. [−
𝝅
𝟐
,
𝝅
𝟐
] b. (0, ) - {
𝝅
𝟐
} c. (−
𝝅
𝟐
,
𝝅
𝟐
) d. [−
𝝅
𝟐
,
𝝅
𝟐
] - {0}
27. If 3tan-1
x + cot-1
x = ,thenx equals
a. 0 b. 1 c. -1 d. ½
28. The value of sin-1
𝐬𝐢𝐧−𝟏 ( 𝒄𝒐𝒔(
𝟑𝟑 
𝟓
)) is
a.
𝟑𝝅
𝟓
b.
−𝟕𝝅
𝟓
c.
𝝅
𝟏𝟎
d. -
𝝅
𝟏𝟎

3. 29. If cos (𝐬𝐢𝐧−𝟏 𝟐
𝟓
+ 𝐜𝐨𝐬−𝟏 𝒙)= 0, thenx is equal to
a.
𝟏
𝟓
b.
𝟐
𝟓
c. 0 d. 1
30. The value of cos-1
(𝒄𝒐𝒔
𝟑𝝅
𝟐
)is equal to
a.
𝟑𝝅
𝟐
b.
𝟓𝝅
𝟐
c.
𝝅
𝟐
d.
𝟕𝝅
𝟐
31. The value of the expression2sec-1
2 + sin-1
(
𝟏
𝟐
) is
a.
𝝅
𝟔
b.
𝟓𝝅
𝟔
c. 1 d.
𝟕𝝅
𝟔
32. If sin-1
(
𝟐𝒂
𝟏+ 𝒂 𝟐
) + cos-1
(
𝟏− 𝒂 𝟐
𝟏+ 𝒂 𝟐
) = tan-1
(
𝟐𝒙
𝟏− 𝒙 𝟐
), where a, x  ]0, 1, then the value of x is
a. 0 b.
𝒂
𝟐
c. a d.
𝟐𝒂
𝟏− 𝒂 𝟐
33. The value of the expressiontan (
𝟏
𝟐
𝐜𝐨𝐬−𝟏 𝟐
√ 𝟓
) is
a. 2 + √ 𝟓 b. √ 𝟓 – 2 c. 5 + √ 𝟐 d.
√ 𝟓+ 𝟐
𝟐
34. If A = [
𝟐 −𝟏 𝟑
−𝟒 𝟓 𝟏
] 𝒂𝒏𝒅 𝑩 = [
𝟐 𝟑
𝟒 −𝟐
𝟏 𝟓
] , then
a. Only AB is defined
b. Only BA is defined
c. AB and BA both are defined
d. AB and BA both are not defined.
35. If A and B are two matrices of the order 3  m and 3  n. respectively,andm = n, then the order of matrix
(5A – 2B) is
a. m  3 b. 3  3 c. m  n d. 3  n
36. if A = [
𝟎 𝟏
𝟏 𝟎
] , thenA2
isequal to
a. [
𝟎 𝟏
𝟏 𝟎
] b. [
𝟏 𝟎
𝟏 𝟎
] c. [
𝟎 𝟏
𝟎 𝟏
] d. [
𝟏 𝟎
𝟎 𝟏
]
37. If matrix A = [ 𝒂𝒊𝒋]2  2 , where aij = 1 if i  j= 0 if i = j
a. I b. A c. 0 d. none of these
38. The matrix [
𝟏 𝟎 𝟎
𝟎 𝟐 𝟎
𝟎 𝟎 𝟒
]is a
a. Identitymatrix b. symmetric matrix c. skew symmetricmatrix d. none of these
39. If A is matrix of order m  n and B is a matrix such that AB and BAare both defined,thenorderof matrix B is
a. m  m b. n  n c. m  n d. n  m
40. if A and B are matrices ofsame order, then(AB - BA) isa
a. skew symmetricmatrix
b. null matrix

4. c. symmetricmatrix
d. unit matrix
41. If A is a square matrix such that A2
= I, then(A – I)3
+ (A + I)3
- 7A is equal to
a. A b. I – A c. I + A d. 3A
42. For any two matrices A and B, we have
a. AB = BA b. AB  BA c. AB= 0 d. none of these
43. If x , y  R, thenthe determinant  = |
𝒄𝒐𝒔 𝒙 −𝒔𝒊𝒏 𝒙 𝟏
𝒔𝒊𝒏 𝒙 𝒄𝒐𝒔 𝒙 𝟏
𝒄𝒐𝒔(𝒙 + 𝒚) −𝒔𝒊𝒏(𝒙 + 𝒚) 𝟎
| liesinthe interval
a. [- √ 𝟐, √ 𝟐] b. [-1, 1] c. [-√ 𝟐, 1] d. [-1. -√ 𝟐]
44. The value of determinant |
𝒂 − 𝒃 𝒃 + 𝒄 𝒂
𝒃 − 𝒂 𝒄 + 𝒂 𝒃
𝒄 − 𝒂 𝒂 + 𝒃 𝒄
|
a. a3
+ b3
+ c3
b. 3 bc c. a3
+ b3
+ c3
– 3abc d. none of these
45. the determinants |
𝒃 𝟐 − 𝒂𝒃 𝒃 − 𝒄 𝒃𝒄 − 𝒂𝒄
𝒂𝒃 − 𝒂 𝟐 𝒂 − 𝒃 𝒃 𝟐 − 𝒂𝒃
𝒃𝒄 − 𝒂𝒄 𝒄 − 𝒂 𝒂𝒃 − 𝒂 𝟐
| equals
a. abc(b-c) (c-a) (a-b) b. (b-c) (c-a) (a-b) c. (a+b+c)(b-c) (c-a) (a-b) d. None ofthese
46. the maximumvalue of ∆ = |
𝟏 𝟏 𝟏
𝟏 𝟏 + 𝒔𝒊𝒏 𝟏
𝟏 + 𝒄𝒐𝒔 𝟏 𝟏
| is ( is a real number)
a.
𝟏
𝟐
b.
√ 𝟑
𝟐
c. √ 𝟐 d.
𝟐√ 𝟑
𝟒
47. If A = [
𝟐  −𝟑
𝟎 𝟐 𝟓
𝟏 𝟏 𝟑
] , then A-1
existsif
a.  = 2 b.   2 c.   -2 d. none of these
48. If x, y, z are all differentfromzero and |
𝟏 + 𝒙 𝟏 𝟏
𝟏 𝟏 + 𝒚 𝟏
𝟏 𝟏 𝟏 + 𝒛
| = 0, thenvalue of x-1
+ y-1
+ z-1
is
a. x y z b. x-1
y-1
z-1
c. –x-y-z d. -1
49. The functionf(x) = {
𝒔𝒊𝒏𝒙
𝒙
+ 𝒄𝒐𝒔 𝒙, 𝒊𝒇 𝒙 ≠ 𝟎
𝒌, 𝒊𝒇 𝒙 = 𝟎
iscontinuous at x = 0, then the value of k is
a. 3 b. 1 c. 2 d. 1.5
50. The functionf(x) = [x] denotesthe greatestintegerfunction,continuous at
a. 4 b. -2 c. 1 d. 1.5
51. The functionf(x) = | 𝒙| + | 𝒙 + 𝟏| is
a. Continuousat x = 0 as well as x = 1
b. Continuousat x = 1 but not at x = 0
c. Discontinuousat x = 0 as well as at x = 1
d. Continuousat x = 0 but not at x = 1

5. 52. The value of k which makes the functiondefinedby f(x) = {
𝒔𝒊𝒏
𝟏
𝒙
, 𝒊𝒇 𝒙 ≠ 𝟎
𝒌, 𝒊𝒇 𝒙 = 𝟎
, continuose at x = 0 is
a. 8 b. 1 c. -1 d. none of these
53. If u = sin-1
(
𝟐𝒙
𝟏+ 𝒙 𝟐
) and y = tan-1
(
𝟐𝒙
𝟏− 𝒙 𝟐
) , then
𝒅𝒖
𝒅𝒗
is
a.
𝟏
𝟐
b. x c.
𝟏− 𝒙 𝟐
𝟏+ 𝒙 𝟐
d. 1
54. The value of c inMean value theoremfor the functionf(x) = x(x – 2), x  [1. 2] is
a.
𝟑
𝟐
b.
𝟐
𝟑
c.
𝟏
𝟐
d.
𝟑
𝟐
55. The functionf(x) =
𝟒− 𝒙 𝟐
𝟒𝒙− 𝒙 𝟑
is
a. Discontinuousat onlyone point
b. Discontinuousat exactlytwo point
c. Discontinuousat exactlythree point
d. None of these
56. The setof points where the functionf givenby f(x) = | 𝟐𝒙 − 𝟏| isdifferentiable is
a. R b. R - {
𝟏
𝟐
} c. (0,  ) d. none of these
57. If y = log (
𝟏− 𝒙 𝟐
𝟏+ 𝒙 𝟐
) , thenfind
𝒅𝒚
𝒅𝒙
is equal to
a.
𝟒𝒙 𝟑
𝟏− 𝒙 𝟒
b.
−𝟒𝒙
𝟏− 𝒙 𝟒
c.
𝟏
𝟒− 𝒙 𝟒
d.
−𝟒𝒙 𝟑
𝟏− 𝒙 𝟒
58. If x = t2
, y = t3
, then
𝒅 𝟐 𝒚
𝒅𝒙 𝟐
is
a.
𝟑
𝟐
b.
𝟑
𝟒𝒕
c.
𝟑
𝟐𝒕
d.
𝟑
𝟐𝒕
59. The value of c on Rolle’stheoremforthe functionf(x) = x3
- 3x in the interval [0, √ 𝟑] is
a. 1 b. -1 c.
𝟑
𝟐
d.
𝟏
𝟑
60. For the functionf(x) = x +
𝟏
𝒙
, x  [1, 3], the value ofc for mean value theoremis
a. 1 b. 2 c. √ 𝟑 d. none of these
61. The two curves x3
-3xy2
+ 2 = 0 and 3x2
y – y3
= 2
a. Touch each other
b. Cut at right angle
c. Cut an angle
𝝅
𝟑
d. Cut an angle
𝝅
𝟒
62. The tangent to the curve givenby x = et
. cost, y =et
. sin t at t =
𝝅
𝟒
makes withx– axis an angle :
a. 0 b.
𝝅
𝟒
c.
𝝅
𝟑
d.
𝝅
𝟐
63. The equationof the normal to the curve y = sin x at (0, 0) is:
a. x = 0 b. y = 0 c. x + y = 0 d. x – y = 0

6. 64. the pointon the curve y2
= x, where the tangentmakes an angle of
𝝅
𝟒
with x – axis is
a. (
𝟏
𝟐
,
𝟏
𝟒
) b. (
𝟏
𝟒
,
𝟏
𝟐
) c. (4, 2) d. (1, 1)
65. Minimumvalue of f if f(x) = sin x [
−𝝅
𝟐
,
𝝅
𝟐
] is…………………..
66. The sidesof an equilateral triangle are increasingat the ratio of 2 cm/sec. the rate at which the area
increases,whenside is 10 cm is:
a. 10 cm2
/s b. √ 𝟑cm2
/s c. 10 √ 𝟑cm2
/s d.
𝟏𝟎
𝟑
cm2
/s
67. The curve y = 𝒙
𝟏
𝟓 has at (0, 0)
a. A vertical tangent (parallel to y 0 axis)
b. A horizontal tangent(parallel to x – axis)
c. An oblique tangent
d. No tangent
68. The equationof normal to the curve 3x2
- y2
= 8 which isparallel to the line x + 3y = 8 is
a. 3x – y = 8 b. 3x + y + 8 = 0 c. x + 3y ±8 = 0 d. x + 3y = 0
69. If y = x4
– 10 and ifx changes from 2 to 1.99, what is the change in y
a. .32 b. .032 c. 5.68 d. 5.968
70. The pointsat which the tangents to the curve y = x3
- 12x + 18 are parallel to x – axisare:
a. (2, -2), (-2,-34) b. (2, 34), (-2,0) c. (0, 34), (-2, 0) d. (2,2), (-2, 34)
71. The two curves x3
- 3xy2
+ 2 = 0 and 3x2
y – y3
- 2= 0 intersectat an angle of
a.
𝝅
𝟒
b.
𝝅
𝟑
c.
𝝅
𝟐
d.
𝝅
𝟔
72. y = x (x – 3)2
decreasesforthe valuesof x givenby :
a. 1 < x< 3 b. x < 0 c. x > 0 d. 0< x <
𝟑
𝟐
73. Whichof the followingfunctionsisdecreasingon (𝟎,
𝝅
𝟐
)
a. sin 2x b. tan x c. cos x d. cos 3x
74. if x isreal, the minimumvalue of x2
- 8x + 17 is
a. -1 b. 0 c. 1 d. 2
75. The smallestvalue of the polynomial x3
– 18x2
+ 96x in [0,9] is
a. 126 b. 0 c. 135 d. 160
76. The functionf(x) = 2x3
– 3x2
- 12x + 4, has
a. Two points of local maximum
b. Two points of local minimum
c. One maximumand one minimum
d. No maximumor minimum
77. Maximumslope of the curve y = -x3
+ 3x2
+ 9x – 27 is:

7. a. 0 b. 12 c. 16 d. 32
78. ∫ 𝒆 𝒙 (𝒄𝒐𝒔 𝒙 − 𝒔𝒊𝒏 𝒙 )dx is equal to
a. 𝒆 𝒙 cos x + C b. 𝒆 𝒙 sin x + C c. -𝒆 𝒙 cos x + C d. -𝒆 𝒙 sin x + C
79. ∫
𝒅𝒙
𝒔𝒊𝒏 𝟐 𝒙 𝒄𝒐𝒔 𝟐 𝒙
is equal to
a. Tan x + cot x + C b. (tan x + cit x)2
+ C c. tan x – cot x+C d. (tan x – cot x)2
+ C
80. If ∫
𝟑𝒆 𝒙− 𝟓𝒆−𝒙
𝟒𝒆 𝒙+ 𝟓𝒆−𝒙
dx = ax + b log | 𝟒𝒆 𝒙 + 𝟓𝒆−𝒙|+ 𝑪 , then
a. a =
−𝟏
𝟖
, 𝒃 =
𝟕
𝟖
b. a =
𝟏
𝟖
, 𝒃 =
𝟕
𝟖
c. a =
−𝟏
𝟖
, 𝒃 =
−𝟕
𝟖
d. a =
𝟏
𝟖
, 𝒃 =
−𝟕
𝟖
81. if f and g are continuous functionsin [0, 1] satisfy f(x) = f(a – x) and g = (x) + g (a – x), then∫ 𝒇( 𝒙). 𝒈(𝒙)
𝒂
𝟎 dx is
equal to
a.
𝒂
𝟐
b.
𝒂
𝟐
∫ 𝒇(𝒙)
𝒂
𝟎 dx c. ∫ 𝒇(𝒙)
𝒂
𝟎 dx d. a ∫ 𝒇(𝒙)
𝒂
𝟎 dx
82. If ∫
𝒆 𝒕
𝟏+𝒕
𝒅𝒕 = 𝒂, 𝒕𝒉𝒆𝒏 ∫
𝒆 𝒕
( 𝟏+𝒕) 𝟐
𝟏
𝟎
𝟏
𝟎 dt is equal to
a. a – 1 +
𝒆
𝟐
b. a + 1 -
𝒆
𝟐
c. a – 1 -
𝒆
𝟐
d. a + 1 +
𝒆
𝟐
83. ∫ | 𝒙 𝒄𝒐𝒔 𝝅𝒙| 𝒅𝒙
𝟏
𝟎 is equal to
a.
𝟖
𝝅
b.
𝟒
𝝅
c.
𝟐
𝝅
d.
𝟏
𝝅
84. ∫
𝒄𝒐𝒔 𝟐𝒙−𝒄𝒐𝒔 𝟐
𝒄𝒐𝒔 𝒙−𝒄𝒐𝒔
dx is equal to
a. 2(sinx+ x cos  ) + C b. 2(sinx - x cos  ) + C c. 2(sinx + 2xcos  ) + C d. 2(sinx - 2xcos  ) + C
85. ∫
𝒅𝒙
𝒔𝒊𝒏 ( 𝒙−𝒂) 𝒔𝒊𝒏 (𝒙−𝒃)
is equal to
a. sin (b– a) log |
𝒔𝒊𝒏 (𝒙−𝒃)
𝒔𝒊𝒏 (𝒙−𝒂)
|+ C
b. cosec (b– a) log |
𝒔𝒊𝒏 (𝒙−𝒂)
𝒔𝒊𝒏 (𝒙−𝒃)
|+ C
c. cosec (b– a) log |
𝒔𝒊𝒏 (𝒙−𝒃)
𝒔𝒊𝒏 (𝒙−𝒂)
|+ C
d. sin (b– a) log |
𝒔𝒊𝒏 (𝒙−𝒂)
𝒔𝒊𝒏 (𝒙−𝒃)
|+ C
86. ∫ 𝐭𝐚𝐧−𝟏
√ 𝒙 dx is equal to
a. (x + 1) tan-1
√ 𝒙 - √ 𝒙 + C
b. x tan-1
√ 𝒙 - √ 𝒙 + C
c. √ 𝒙 - x tan-1
√ 𝒙 + C
d. √ 𝒙 – ( + 1) x tan-1
√ 𝒙 + C
87. ∫ 𝒆 𝒙 (
𝟏−𝒙
𝟏+ 𝒙 𝟐
)
𝟐
is equal to
a.
𝒆 𝒙
𝟏+ 𝒙 𝟐
+ C b.
−𝒆 𝒙
𝟏+ 𝒙 𝟐
+ C c.
𝒆 𝒙
( 𝟏+ 𝒙 𝟐) 𝟐
+ C d.
−𝒆 𝒙
( 𝟏+ 𝒙 𝟐) 𝟐
+ C
88. ∫
𝒙 𝟗
( 𝟒𝒙 𝟐+ 𝟏) 𝟔
dx is equal to

8. a.
𝟏
𝟓𝒙
(𝟒 −
𝟏
𝒙 𝟐
)
−𝟓
+ 𝑪
b.
𝟏
𝟓
(𝟒 +
𝟏
𝒙 𝟐
)
−𝟓
+ 𝑪
c.
𝟏
𝟏𝟎𝒙
( 𝟏 + 𝟒)−𝟓 + 𝑪
d.
𝟏
𝟏𝟎
(
𝟏
𝒙 𝟐
+ 𝟒)
−𝟓
+ 𝑪
89. If ∫
𝒅𝒙
( 𝒙+𝟐)(𝒙 𝟐+ 𝟏)
= 𝒂 𝒍𝒐𝒈| 𝟏+ 𝒙 𝟐| + 𝒃 𝐭𝐚𝐧−𝟏 𝒙 +
𝟏
𝟓
𝒍𝒐𝒈 | 𝒙 + 𝟐| + 𝑪, then
a. a =
−𝟏
𝟏𝟎
, 𝒃 =
−𝟐
𝟓
b. a =
𝟏
𝟏𝟎
, 𝒃 = −
𝟐
𝟓
c. a = 𝟏𝟎 , 𝒃 =
𝟐
𝟓
d. a =
𝟏
𝟏𝟎
, 𝒃 =
𝟐
𝟓
90. ∫
𝒙 𝟑
𝒙+𝟏
is equal to
a. x +
𝒙 𝟐
𝟐
+
𝒙 𝟑
𝟑
− 𝒍𝒐𝒈| 𝟏− 𝒙| + 𝑪
b. x +
𝒙 𝟐
𝟐
−
𝒙 𝟑
𝟑
− 𝒍𝒐𝒈| 𝟏− 𝒙| + 𝑪
c. x -
𝒙 𝟐
𝟐
−
𝒙 𝟑
𝟑
− 𝒍𝒐𝒈 | 𝟏+ 𝒙| + 𝑪
d. x -
𝒙 𝟐
𝟐
+
𝒙 𝟑
𝟑
− 𝒍𝒐𝒈 | 𝟏+ 𝒙| + 𝑪
91. If ∫
𝒙 𝟑 𝒅𝒙
√ 𝟏+ 𝒙 𝟐
= 𝒂( 𝟏 + 𝒙 𝟐)
𝟑
𝟐
+ 𝒃√ 𝟏 + 𝒙 𝟐 + C, then
a. a =
𝟏
𝟑
, b = 1 b. a =
−𝟏
𝟑
, b = 1 c. a =
−𝟏
𝟑
, b = -1 d. a =
𝟏
𝟑
, b = -1
92. ∫
𝒅𝒙
𝟏+𝒄𝒐𝒔 𝟐𝒙
𝝅
𝟒
−𝝅
𝟒
is equal to
a. 1 b. 2 c. 3 d. 4
93. ∫ √ 𝟏 − 𝒔𝒊𝒏 𝟐𝒙
𝝅
𝟐
𝟎 dx is equal to
a. 2√ 𝟐 b. 2 (√ 𝟐+ 1) c. 2 d. 2(√ 𝟐- 1)
94. The area enclosedby the circle x2
+ y2
= 2 is equal to
a. 4 sq unit b. 2 √ 𝟐 sq unit c. 42
sq unit d. 2 sq unit
95. The area enclosedby the ellipse
𝒙 𝟐
𝒂 𝟐
+
𝒚 𝟐
𝒃 𝟐
= 1 is equal to
a. 2
ab b.  ab c.  a2
b d. ab2
96. The area of the regionboundedby the curve y = x2
and the line y = 16
a.
𝟑𝟐
𝟑
b.
𝟐𝟓𝟔
𝟑
c.
𝟔𝟒
𝟑
d.
𝟏𝟐𝟖
𝟑
97. The area of the regionboundedby the y – axis,y = cos x and y = sinx,0 ≤ 𝒙 ≤
𝝅
𝟐
is

9. a. √ 𝟐sq unit b. (√ 𝟐+ 1) sq unit c. (√ 𝟐- 1) sq unit d. (2√ 𝟐- 1) sq unit
98. The area of the regionboundedby the curve y = √𝟏𝟔 − 𝒙 𝟐 and x – axis is
a. 8 sq unit b. 20  sq unit c. 16  squnit d. 256  squnit
99. Area of the regionboundedby the curve y = cos x betweenx= 0 and x =  is
a. 2 sq unit b. 4 sq unit c. 3 sq unit d. 1 sq unit
100. The area of the regionboundedby parabola y2
= x and the straight line 2y = x is
a.
𝟒
𝟑
sq unit b. 1 sq unit c.
𝟏
𝟑
sq unit d.
𝟐
𝟑
sq unit
101. The area of the regionboundedby the curve y = x + 1 and the linesx = 2 and x = 3 is
a. 2 sq unit b.  sq unit c. 3 sq unit d. 4  squnit
102. The area of the regionboundedby the curve y = x + 1 and the linesx = 2 and x = 3 is
a.
𝟕
𝟐
sq unit b.
𝟗
𝟐
sq unit c.
𝟏𝟏
𝟐
sq unit d.
𝟏𝟑
𝟐
sq unit
103. The degree ofthe differential equation(1+
𝒅𝒚
𝒅𝒙
)3
= (
𝒅 𝟐 𝒚
𝒅𝒙 𝟐
)2
is
a. 1 b. 2 c. 3 d. 4
104. The degree ofthe differential equation
𝒅 𝟐 𝒚
𝒅𝒙 𝟐
+ 𝟑 (
𝒅𝒚
𝒅𝒙
)
𝟐
= 𝒙 𝟐 𝒍𝒐𝒈 (
𝒅 𝟐 𝒚
𝒅𝒙 𝟐
) is
a. 1 b. 2 c. 3 d. not defined
105. The order and degree ofthe differential equation [ 𝟏+ (
𝒅𝒚
𝒅𝒙
)
𝟐
]
𝟐
=
𝒅 𝟐 𝒚
𝒅𝒙 𝟐
respectively,are
a. 1, 2 b. 2, 2 c. 2, 1 d. 4, 2
106. The solutionof the differential equation2x.
𝒅𝒚
𝒅𝒙
– y = 3 representsa familyof
a. Straight lines
b. Circles
c. Parabolas
d. Ellipses
107. The integratingfactor ofthe differential equation
𝒅𝒚
𝒅𝒙
(x log x) + y = 2 logx is
a. ex
b. log xc. log (logx) d. x
108. A solutionof the differential equation(
𝒅𝒚
𝒅𝒙
)2
– x
𝒅𝒚
𝒅𝒙
+ y = 0 is
a. y = 2 b. y = 2x c. y = 2x – 4 d. y = 2x2
– 4
109. Whichof the followingisnot a homogeneousfunctionof x and y.
a. x2
+ 2xyb. 2x – y c. cos2
( ) + d. sinx – cos y
110. Solutionof the differential equation
𝒅𝒚
𝒙
+
𝒅𝒚
𝒚
= 0 is
a. =
𝟏
𝒙
+
𝟏
𝒚
= 𝒄 b. log x . log y = c c. xy = c d. x + y = c
111. The solutionof the differential equationx + 2y = x2
is

10. a. y =
𝒙 𝟐+ 𝒄
𝟒𝒙 𝟐
b. y =
𝒙 𝟐
𝟒
+ c c. y =
𝒙 𝟐+ 𝒄
𝒙 𝟐
d. y =
𝒙 𝟐+ 𝒄
𝟒𝒙 𝟐
112. The degree ofthe differential equation[1 + (
𝒅𝒚
𝒅𝒙
)2
] =
𝒅 𝟐 𝒚
𝒅𝒙 𝟐
is
a. 4 b.
𝟑
𝟐
c. not defined d. 2
113. The order and degree ofthe differential equation
𝒅 𝟐 𝒚
𝒅𝒙 𝟐
+ (
𝒅𝒚
𝒅𝒙
)
𝟏
𝟒
+ 𝒙
𝟏
𝟓= 0, respectivelyare
a. 2 and not defined b. 2 and 2 c. 2 and 3 d. 3 and 3
114. Integrating factor of the differential equationcosx
𝒅𝒚
𝒅𝒙
+ y sin x = 1 is:
a. cos x b. tan xc. sec x d. sin x
115. Familyy = Ax + A3
of curvesis representedbythe differential equationofdegree:
a. 1 b. 2 c. 3 d. 4
116. Solutionof
𝒅𝒚
𝒅𝒙
- y = 1, y(0) = 1 isgiven by
a. xy = -ex
b. xy = -e-x
c. xy = -1 d. y = 2 ex
– 1
117. The numberof solutions of
𝒅𝒚
𝒅𝒙
=
𝒚+𝟏
𝒙−𝟏
when y(1) = 2 is:
a. None b. one c. two d. infinite
118. Integrating factor of the differential equation(1– x2
)
𝒅𝒚
𝒅𝒙
- xy = 1 is
a. –x b.
𝒙
𝟏+ 𝒙 𝟐
c.√𝟏 − 𝒙 𝟐 d. ½ log(1 – x2
)
119. The general solutionof ex
cos y dx – ex
siny dy = 0 is:
a. ex
cos y = k b. ex
sin y = k c. ex
= k cos y d. ex
= k siny
120. The solutionof the differential equation
𝒅𝒚
𝒅𝒙
=
𝟏+ 𝒚 𝟐
𝟏+ 𝒙 𝟐
is:
a. y = tan-1
x b. y-x = k (1 =xy) c. x = tan-1
y d. tan (xy) = k
121. The integratingfactor ofthe differential equation
𝒅𝒚
𝒅𝒙
+ 𝒚 =
𝟏+𝒚
𝒙
is:
a.
𝒙
𝒆 𝒙
b.
𝒆 𝒙
𝒙
c. xex
d. ex
122. The solutionof the differential equationcosx sin y dx + sinx cos y dy = 0 is:
a.
𝒔𝒊𝒏𝒙
𝒔𝒊𝒏 𝒚
= c b. sin x siny = c c. sin x + siny = c d. cos x cos y = c
123. The solutionof x
𝒅𝒚
𝒅𝒙
+ y = ex
is:
a. y =
𝒆 𝒙
𝒙
+
𝒌
𝒙
b. y = xex
+ cx c. y = xex
+ k d. x =
𝒆 𝒚
𝒚
+
𝒌
𝒚
124. The differential equationofthe familyof curves x2
+ y2
– 2ay = 0, where a isarbitrary constant, is:
a. (x2
– y2
)
𝒅𝒚
𝒅𝒙
= 2xy b. 2(x2
+ y2
)
𝒅𝒚
𝒅𝒙
= xy c. 2(x2
- y2
)
𝒅𝒚
𝒅𝒙
= xy d. (x2
+y2
)
𝒅𝒚
𝒅𝒙
= 2xy
125. The general solutionof
𝒅𝒚
𝒅𝒙
= 2x 𝒆 𝒙 𝟐− 𝒚 is:
a. 𝒆 𝒙 𝟐− 𝒚 = 𝒄 b. 𝒆−𝒚 + 𝒆 𝒙 𝟐
= 𝒄 c. 𝒆 𝒚 + 𝒆 𝒙 𝟐
= 𝒄 d. 𝒆 𝒙 𝟐+ 𝒚 = 𝒄
126. The general solutionof the differential equation
𝒅𝒚
𝒅𝒙
+ xy is:

11. a. y = c𝒆
−𝒙 𝟐
𝟐 b. y = c𝒆
𝒙 𝟐
𝟐 c. y = (x + c) 𝒆
𝒙 𝟐
𝟐 d. y = (c-x) 𝒆
𝒙 𝟐
𝟐
127. the solutionof the equation(2y – 1) dx– (2x + 3) dy = 0 is:
a.
𝟐𝒙−𝟏
𝟐𝒚+𝟑
= k b.
𝟐𝒚+ 𝟏
𝟐𝒙− 𝟑
= k c.
𝟐𝒙+𝟑
𝟐𝒚−𝟏
= k d.
𝟐𝒙−𝟏
𝟐𝒚−𝟏
= k
128. The solutionof
𝒅𝒚
𝒅𝒙
+ y = e-x
,y (0) = 0 is:
a. y = e-x
(x– 1) b. y = xex
c. y = xe-x
+ 1 d. y = xe-x
129. The order and degree ofthe differential equation (
𝒅 𝟑 𝒚
𝒅𝒙 𝟑
)
𝟐
− 𝟑
𝒅 𝟐 𝒚
𝒅𝒙 𝟐
+ 𝟐 (
𝒅𝒚
𝒅𝒙
)
𝟒
= 𝒚 𝟒 are:
a. 1, 4 b. 3, 4 c. 2, 4 d. 3, 2
130. The order and degree ofthe differential equation [ 𝟏+ (
𝒅𝒚
𝒅𝒙
)
𝟐
] =
𝒅 𝟐 𝒚
𝒅𝒙 𝟐
are:
a. 2,
𝟑
𝟐
b. 2, 3 . 2, 1 d. 3, 4
131. Whichof the followingisthe general solutionof
𝒅 𝟐 𝒚
𝒅𝒙 𝟐
−
𝒅𝒚
𝒅𝒙
𝟐 + y = 0?
a. y = (Ax+ B)ex
b. y = (Ax + B) e-x
c. y = Aex
+ Be-x
d. y = Acos x + B sinx
132. Solutionof the differential equation
𝒅𝒚
𝒅𝒙
+
𝒚
𝒙
= sinx is:
a. x (y + cos x) = sin x + c
b. x (y – cos x 0 = sin x + c
c. xy cos x = sin x + c
d. x (y + cos x) = cos x + c
133. The solutionof the differential equation
𝒅𝒚
𝒅𝒙
= ex – y
+ x2
e-y
is:
a. y = ex – y
– x2
e-y
+ c
b. ey
– ex
= + c
c. ex
+ ey
= + c
d. ex
– ey
=
𝒙 𝟑
𝟑
+ c
134. The magnitude of the vector6𝒊̂ + 𝟐𝒋̂ + 𝟑𝒌̂ is
a. 5 b. 7 c. 12 d. 1
135. The positionvector of the point which dividedthe joinof points withpositionvectors 𝒂⃗⃗ + 𝒃⃗⃗ and 2𝒂⃗⃗ − 𝒃⃗⃗
in the ratio 1 : 2 is
a.
𝟑𝒂⃗⃗ + 𝟐𝒃⃗⃗
𝟑
b. 𝒂⃗⃗ c.
𝟓𝒂⃗⃗ − 𝒃⃗⃗
𝟑
d.
𝟒𝒂⃗⃗ + 𝒃⃗⃗
𝟑
136. The vector with initial pointP(2, -3, 5) and terminal point Q(3, -4, 7) is
a. 𝒊̂ − 𝒋̂ + 𝟐𝒌̂ b. 𝟓𝒊̂ − 𝟕𝒋̂ + 𝟏𝟐𝒌̂ c. −𝒊̂ + 𝒋̂ − 𝟐𝒌̂ d. None of these
137. The angle betweenthe vectors 𝒊̂ − 𝒋̂ 𝒂𝒏𝒅 𝒋̂− 𝒌̂ is
a.
𝝅
𝟑
b.
𝟐𝝅
𝟑
c.
−𝝅
𝟑
d.
𝟓𝝅
𝟔
138. The value of  for whichthe two vectors 2𝒊̂ − 𝒋̂ + 𝟐𝒌̂ 𝒂𝒏𝒅 𝟑𝒊̂ +  𝒋̂+ 𝒌̂ are perpendicularis

12. a. 2 b. 4 c. 6 d. 8
139. The area of the parallelogramwhose adjacent sidesare 𝒊̂ + 𝒌̂ and 𝟐𝒊̂ + 𝒋̂ + 𝒌̂ is
a. 3 b. 4 c. √ 𝟐 d. √ 𝟑
140. If | 𝒂⃗⃗ | = 𝟖, | 𝒃⃗⃗ | = 𝟑 𝒂𝒏𝒅 | 𝒂⃗⃗ × 𝒃⃗⃗ | = 𝟏𝟐 , thenvalue of is
a. 6√ 𝟑 b. 8√ 𝟑 c. 12√ 𝟑 d. none of these
141. The 2 vector 𝒋̂ + 𝒌̂ and 3𝒊̂ − 𝒋̂ + 𝟒𝒌̂ representsthe two sidesAB and AC,respectivelyofa ABC . the length
of the medianthrough A is
a.
√ 𝟑𝟒
𝟐
b.
√ 𝟒𝟖
𝟐
c. √ 𝟏𝟖 d. None of these
142. The projectionof vector 𝒂⃗⃗ = 𝟐𝒊̂ − 𝒋̂ + 𝒌̂ along 𝒃⃗⃗ = 𝒊̂ + 𝟐𝒋̂ + 𝟐𝒌̂ is
a. 2 b. √ 𝟔 c.
𝟐
𝟑
d.
𝟏
𝟑
143. If 𝒂⃗⃗ and 𝒃⃗⃗ are unit vectors,then what is the angle between 𝒂⃗⃗ and 𝒃⃗⃗ for √ 𝟑 𝒂⃗⃗ and 𝒃⃗⃗ to be a unit vector?
a. 30o
b. 45o
c. 60o
d. 90o
144. The unit vector perpendiculartothe vectors 𝒊̂ − 𝒋̂ 𝒂𝒏𝒅 𝒊̂ + 𝒋̂ forming a right handedsystemis
a. 𝒌̂ b. -𝒌̂ c.
𝒊̂− 𝒋̂
√ 𝟐
d.
𝒊̂+ 𝒋̂
√ 𝟐
145. If | 𝒂⃗⃗ | = 𝟑 and -1  k  2 , then| 𝒌𝒂⃗⃗ | liesinthe interval
a. [0,6] b. [-3,6] c. [3,6] d. [1,2]
146. The vector in the directionof the vector 𝒊̂ − 𝟐𝒋̂ + 𝟐𝒌̂ that has magnitude 9 is
a. 𝒊̂ − 𝟐𝒋̂ + 𝟐𝒌̂ b.
𝒊̂− 𝟐𝒋̂+ 𝟐𝒌̂
𝟑
c. (𝒊̂ − 𝟐𝒋̂ + 𝟐𝒌̂ ) d. 9( 𝒊̂ − 𝟐𝒋̂ + 𝟐𝒌̂)
147. The angle betweentwo vectors 𝒂⃗⃗ and 𝒃⃗⃗ with magnitude √ 𝟑 and 4, respectively,and 𝒂⃗⃗ . 𝒃⃗⃗ = 2√ 𝟑 is
a.
𝝅
𝟔
b.
𝝅
𝟑
c.
𝝅
𝟐
d.
𝟓𝝅
𝟐
148. Findthe value of  such that the vectors 𝒂⃗⃗ = 𝟐𝒊̂ +  𝒋̂+ 𝒌̂ and 𝒃⃗⃗ = 𝒊̂ + 𝟐𝒋̂ + 𝟑𝒌̂ are orthogonal
a. 0 b. 1 c.
𝟑
𝟐
d.
−𝟓
𝟐
149. the value of  for which the vectors 𝟑𝒊̂ − 𝟔𝒋̂ + 𝒌̂ 𝒂𝒏𝒅 𝟐𝒊̂ − 𝟒𝒋̂ +  𝒌̂ are parallel is
a.
𝟐
𝟓
b.
𝟐
𝟑
c.
𝟑
𝟐
d.
𝟓
𝟐
150. For any vector 𝒂⃗⃗ , the value of ( 𝒂⃗⃗ × 𝒊̂) 𝟐 + ( 𝒂⃗⃗ × 𝒋̂) 𝟐 + ( 𝒂⃗⃗ × 𝒌̂)
𝟐
is equal to
a. 𝒂⃗⃗ 𝟐 b. 3 𝒂⃗⃗ 𝟐 c. 4 𝒂⃗⃗ 𝟐 d. 2𝒂⃗⃗ 𝟐
151. If | 𝒂⃗⃗ | = 𝟏𝟎,| 𝒃⃗⃗ | = 𝟐 𝒂𝒏𝒅 𝒂⃗⃗ . 𝒃⃗⃗ = 𝟏𝟐, 𝐭𝐡𝐞𝐧 𝐯𝐚𝐥𝐮𝐞 𝐨𝐟 | 𝒂⃗⃗ × 𝒃⃗⃗ |, is
a. 5 b. 10 c. 14 c. 16
152. If 𝒂⃗⃗ , 𝒃⃗⃗ , 𝒄⃗ are unit vectorssuch that 𝒂⃗⃗ + 𝒃⃗⃗ + 𝒄⃗ = 𝟎⃗⃗ , thenvalue of 𝒂⃗⃗ . 𝒃⃗⃗ + 𝒃⃗⃗ . 𝒄⃗ + 𝒂⃗⃗ . 𝒄⃗
a. 1 b. 3 c. -
𝟑
𝟐
d. none of these
153. Projectionvector of 𝒂⃗⃗ on 𝒃⃗⃗ is

13. a. (
𝒂⃗⃗ .𝒃⃗⃗
| 𝒃⃗⃗ |
𝟐) 𝒃⃗⃗ b.
𝒂⃗⃗ .𝒃⃗⃗
| 𝒃⃗⃗ |
c.
𝒂⃗⃗ .𝒃⃗⃗
| 𝒂⃗⃗ |
d. (
𝒂⃗⃗ .𝒃⃗⃗
| 𝒂⃗⃗ | 𝟐
) 𝒃⃗⃗
154. If 𝒂⃗⃗ , 𝒃⃗⃗ , 𝒄⃗ are three vectors such that 𝒂⃗⃗ + 𝒃⃗⃗ + 𝒄⃗ = 𝟎⃗⃗ and | 𝒂⃗⃗ | = 𝟐, | 𝒃⃗⃗ | = 𝟑 , | 𝒄⃗ | = 𝟓, then value of
𝒂⃗⃗ . 𝒃⃗⃗ + 𝒃⃗⃗ . 𝒄⃗ + 𝒂⃗⃗ . 𝒄⃗
a. 0 b. 1 c. -19 d. 38
155. P is a point on the line segmentjoiningthe points(3, 2, -1) and (6, 2, -2). If x co-ordinate of P is5, thenits y co-
ordinate is
a. 2 b. 1 c. -1 d. -2
156. If ,, are the anglesthat a line makes withthe positive directionofx, y, z axis, respectively,thenthe
directioncosinesof the line are:
a. sin,sin,sin b. cos,cos,cos c. tan,tan,tan d. cos2
,cos2
,cos2

157. The equationsof x – axis in space are
a. x = 0, y = 0 b. x = 0, z = 0 c. x = 0 d. y = 0, z = 0
158. If the directionscosinesofa line are k, k, k then
a. k > 0 b. 0 < k< 1 c. k = 1 d. k =
𝟏
√ 𝟑
𝒐𝒓 −
𝟏
√ 𝟑
159. The distance of the plane 𝒓⃗ . (
𝟐
𝟕
𝒊̂ +
𝟑
𝟕
𝒋̂ −
𝟔
𝟕
𝒌̂) = 1 from the originis
a. 1 b. 7 c.
𝟏
𝟕
d. none of these
160. The area of the quadrilateral ABCD, where A(0,4,1), b(2,3,-1),c(4,5,0) and D(2, 6, 2) is equal to
a. 9 sq unit b. 18 sq unit c. 27 sq unit d. 81 sq unit
161. The unit vector normal to the plane x + 2y + 3z – 6 = 0
𝟏
√ 𝟏𝟒
𝒊̂ +
𝟐
√ 𝟏𝟒
𝒋̂ +
𝟑
√ 𝟏𝟒
𝒌̂ .
162. The corner pointsof the feasible regiondeterminedbythe systemof linearconstraints are (0, 10), (5,5), (15,
150), (0, 20). Let Z = px + qy, where p, q > 0. Conditionon p and q so that the maximumof Z occurs at both the
points(15, 15) and (0, 20) is
a. p = q b. p = 2q c. q = 2p d. q = 3p
163. Feasible region(shaded) fora LPP isshown in the Fig.Minimumof Z = 4x + 3y occurs at the point
a. (0, 80 b. (2, 5 ) c. (4, 3) d. (9, 0)
164. The corner points of the feasible regiondeterminedbythe system oflinear constraints are (0,0), (0, 40), (20,
40), (60, 20), (60, 0). The objective functionis Z = 4x + 3y. Compare the quantity inColumn A and ColumnB
ColumnA ColumnB
Maximumof Z 325
a. The quantity in columnA is greater
b. The quantity in columnB is greater
c. The two quantitiesare equal

14. d. The relationshipcan not be determinedonthe basis of the informationsupplied.
165. Corner pointsof the feasible regionforan LPP are (0, 2), (3, 0), (6,0), (6, 8) and (0, 5). Let F = 4x + 6y be the
objective function.The minimum value of F occurs at
a. (0, 2) only
b. (3, 0) only
c. The mid pointof the line segmentjoiningthe points(0, 2) and (3, 0) only
d. Any point on the line segmentjoiningthe points(0, 20 and (3, 0).
166. Let A and B be two events.If P (A) = 0.2, P(B) = 0.4, P(A B) = 0.6, then P(AIB) is equal to
a. 0.8 b. 0.5 c. 0.3 d. 0
167. Let A and B be two eventssuch that P(A) = 0.6, P(B) = 0.2 and P(AIB) = 0.5. thenP(AIB) equals
a.
𝟏
𝟏𝟎
b.
𝟑
𝟏𝟎
c.
𝟑
𝟖
d.
𝟔
𝟕
168. If A and B are independenteventssuchthat 0 < P(A) < 1 and 0 <P(B) < 1 then whichof the followingisnot
correct?
a. A and B are mutually exclusive
b. A and B are independent
c. A and B are independent
d. A and B are independent
169. If P(A) =
𝟒
𝟓
, and P(A  B) =
𝟕
𝟏𝟎
, thenP(B I A) is equal to
a.
𝟏
𝟏𝟎
b.
𝟏
𝟏𝟖
c.
𝟕
𝟖
d.
𝟏𝟕
𝟐𝟎
170. If P(A  B) =
𝟕
𝟏𝟎
and P(B) =
𝟏𝟕
𝟐𝟎
, then P(B I A) is equal to
a.
𝟏𝟒
𝟏𝟕
b.
𝟏
𝟖
c.
𝟕
𝟖
d.
𝟏𝟕
𝟐𝟎
171. If P(A) =
𝟑
𝟏𝟎
, P(B) =
𝟐
𝟓
and P(A  B) =
𝟑
𝟓
, thenP(B I A) + P(AI B) is equal to
a.
𝟏
𝟒
b.
𝟏
𝟑
c.
𝟓
𝟏𝟐
d.
𝟕
𝟐
172. If P(A) =
𝟐
𝟓
, P(B) =
𝟑
𝟏𝟎
and P(A  B) =
𝟏
𝟓
, thenP(B I A) . P(A IB) is equal to
a.
𝟓
𝟔
b.
𝟓
𝟕
c.
𝟐𝟓
𝟒𝟐
d. 1
173. If A and B are two eventssuch that P(A) = ½, P(B) =
𝟏
𝟑
, P(A / B) =
𝟏
𝟒
, thenP(A  B) equals
a.
𝟏
𝟏𝟐
b.
𝟑
𝟒
c.
𝟏
𝟒
d.
𝟑
𝟏𝟔
174. If P(A) = 0.4, P(B) = 0.8 and P(B I A) = 0.6, thenP(A  B) is equal to
a. 0.24 b. 0.3 c. 0.48 d. 0.96
175. You are giventhat A and B are two eventssuch that P(B) =
𝟑
𝟓
, P(A I B) = ½ and P(A  B) =
𝟒
𝟓
, thenP(A) equals
a.
𝟑
𝟏𝟎
b.
𝟏
𝟐
c.
𝟏
𝟓
d.
𝟑
𝟓

15. 176. If P(B) =
𝟑
𝟓
, P(AIB) = ½ and P(A  B) =
𝟒
𝟓
, then P(A  B) + P(A  B) =
a.
𝟒
𝟓
b.
𝟏
𝟐
c.
𝟏
𝟓
d. 1
177. Let A and B be two eventssuch that P(A) =
𝟑
𝟖
, P(B) =
𝟓
𝟖
and thenP(A I B) . P(A IB) isequal to
a.
𝟐
𝟓
b.
𝟑
𝟖
c.
𝟑
𝟐𝟎
d.
𝟔
𝟐𝟓
178. If A and B are such eventsthat P(A) > 0 and P(B)  1, thenP(A I B) equals
a. 1 – P(A I B) b. 1 – P(A IB) c.
𝟏−𝑷(𝑨∪𝑩)
𝑷(𝑩′)
d. P(A) IP(B )
179. A bag contains5 red and 3 blue balls. If3 balls are drawn at random without replacementthe probabilityof
gettingexactly one red ball is
a.
𝟒𝟓
𝟏𝟗𝟔
b.
𝟏𝟑𝟓
𝟑𝟗𝟐
c.
𝟏𝟓
𝟓𝟔
d.
𝟏𝟓
𝟐𝟗
180. Assume that in a family,each child isequallylikelyto be or a girl.A family withthree childrenischosen at
random. The probabilitythat the eldestchildisa girl given that the familyhas at least one girl is
a.
𝟒
𝟕
b.
𝟏
𝟐
c.
𝟏
𝟑
d.
𝟐
𝟑
181. A die is thrown and a card is selectedatrandom from a deck of 52 playing cards. The probabilityof gettingan
evennumberon the die and a spade card is
a.
𝟑
𝟒
b.
𝟏
𝟐
c.
𝟏
𝟒
d.
𝟏
𝟖
182. A box contains 3 orange balls,3 greenballsand 2 blue balls. Three balls are drawn at random from the box
without replacement.The probabilityof drawing 2 greenballsand one blue ball is
a.
𝟑
𝟖
b.
𝟐
𝟐𝟏
c.
𝟏
𝟐𝟖
d.
𝟏𝟔𝟕
𝟏𝟔𝟖
183. Eight coins are tossedtogether.The probabilityof gettingexactly 3 headsis
a.
𝟏
𝟐𝟓𝟔
b.
𝟕
𝟑𝟐
c.
𝟓
𝟑𝟐
d.
𝟑
𝟑𝟐
184. In a college,30% studentsfail in physics,255 fail in mathematics and 10% fail in both. One studentis chosen
at random. The probabilitythat she failsin physicsif she has failedin mathematics is
a.
𝟏
𝟏𝟎
b.
𝟐
𝟓
c.
𝟏
𝟑
d.
𝟗
𝟐𝟎
185. The probabilitydistributionof a discrete random variable X is givenbelow:
X 2 3 4 5
P(X) 𝟓
𝒌
𝟕
𝒌
𝟗
𝒌
𝟏𝟏
𝒌
The value of k is
a. 8 b. 16 c. 32 d. 48