Indicative question bank for entrance test for Actuarial ... · Indicative question bank for...

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Indicative question bank for entrance test for Actuarial Science & Analytics

Subjective /objective questions

Algebra:

1. If f(x), a polynomial in x is such that, f(0) = f`(0) = f"(0) =1. Find an expression

for it in its simplest form.

2. If X>0; Y>0 and X+Y=1; what is the maximum value of XY?

3. If -1, 1, -1, -1, 7 are the consecutive terms of a series, what is the next term?

4. If 8 identical balls are to be inserted in to 5 numbered cells so that no cell is empty,

in how many ways can it be done?

5. What is the greatest common divisor of two different positive integers which

are less than 144?

6. A cube was painted red on all its six surfaces. It was then cut into 8 equal

cubes. What percentage area of each of the eight cubes was painted red?

7. If 2n-1 is prime, prove that n is prime?

8. Find the smallest value of x for which

212

36 4 0Xx

9. Let the angle A, B, C of a triangle ABC be in A.P and let B: C = 3: 2. Find the

Angle A?

10. How many words can be formed from the letters of COURTESY? How many of

them will begin with C and end with Y?

11. How many number can be formed between 10 and 1000 with the help of digits

2,3,4,0,8,9?

12. How many three digit numbers can be formed with the digits 0,1,2,3,4,5,6,7,8,9

if no two digits are same?

13. Find the number of permutations of the letters of the word “ SIGNAL” such that

the vowels may occupy only odd positions?

14. How many ways 12 persons may divided into three groups of 2,4 and 6 persons?

15.

16. If x -2

– 2x - 1

= 8, find x

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17. If the sum of n terms of an A.P. is , 1

( 1)2

nP n n Q where P and Q are

constants, find the common difference.

18. Find the sum to n terms of the A.P., whose kth

term is 5k + 1.

19. A man starts repaying a loan as first installment of Rs. 100. If he increases the installment

by Rs 5 every month, what amount he will pay in the 30th instalment?

20. In a G.P., the 3rd term is 24 and the 6th term is 192.Find the 10th term

21. The sum of first three terms of a G.P. is 13/12 and their product is – 1. Find the common

ratio and the terms.

22. The pth

, qth

and rth

terms of an A.P. are a, b, c, respectively. Show that

(q – r )a + (r – p )b + (p – q )c = 0

23. Solve the system of inequalities:

3x – 7 < 5 + x , 11– 5 x ≤ 1

and represent the solutions on the number line.

24. Find the number of permutations of the letters of the word ALLAHABAD.

25. How many 4-digit numbers can be formed by using the digits 1 to 9 if repetition of digits

is not allowed?

26. A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In

how many ways can this be done? How many of these committees would consist of 1

man and 2 women?

27. Expand

4

2 3, 0x x

x

28. Find the coefficient of x6 y3 in the expansion of (x + 2y) 9 .

Calculus:

29. Use the limit definition to compute the derivative, f'(x), for

( ) 4 3f x x


1 3

( )2 5

f x x


2( ) 3f x x x

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32. Find two nonnegative numbers whose sum is 9 and so that the product of one number

and the square of the other number is a maximum.

33. Integrate xxe dx

34. Integrate 3 ln5x x dx

35. Integrate 2

2 3

9

xdx

x

36. Integrate 1

1xdx

e

37. If X>0; Y>0 and X+Y=1; what is the maximum value of XY?

i) ( ) .A B A AB B ii) ( )A B A B

Which of the above equations is correct? If incorrect, write down the correct Equation.

38. Find the smallest value of x for which

21236 4 0X

x

.

39. A speaks truth in 75 percent cases and B in 80 percent of cases. In what percent of cases are

they likely to contradict each other in narrating the same incident?

40. If n(A-B) = 18, n(A B)=70 and n(A B)=25, then find n(B).

41. There are 35 students in art class and 57 students in dance class. Find the number of students

who are either in art class or in dance class.

42. In a group of 100 persons, 72 people can speak in English and 43 can speak French. How many

can speak English only? How many can speak French only and how many can speak both English

and French?

43. Find the inverse of the following functions, if it exists, if the function does not have an inverse,

explain why?

(a) f(x) = 3x+2 (b) f(x) = (2x+3)/2 (c) f(x) = 13 (d) f(x) = x

44. Two functions f(x) = 3x-1 and g(x) = 3x3-1 then evaluate f(6)+g(2)

45. f(x) = { (6,8), (2,-4), (10,-2), (4,4)}. Find (fof-1)(5)

46. Find z

x

and

z

y

for each of the following functions.

(a) x3+z2-5xy5z = x2+y3 (b) x2 (2y-5z) = 1+y (6zx)

47. Find the partial derivatives fx and fy if f(x,y) is given by

f(x,y) = x2y + 2x+y

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48. Solve the system of inequalities:

3x – 7 < 5 + x , 11 – 5 x ≤ 1

and represent the solutions on the number line.

49. If &x y are two sets such that 17, 23n x n y and 38n x y ; Find n x y

50. List of the subsets of the set 1,0,1

51. In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked

product C. If 14 people liked products A & B, 12 people liked products C & A, 14 people

liked product B & C and 8 liked all the three products. Find how many liked product C

only.

52. Let 1,1 2,3 0, 1 1, 3f be a function form Z to Z defined by

, ,f x ax b a b f determine a, b.

53. Let 1,2,3,4 ; 1,5,9,11,15,16A B

54. 1,5 2,9 3,1 4,5 2,11f are the following true

(i) f is relation from A to B (ii) f is a function from A to B

Justify your answer in each case

55. Find the Domain of 2

2

2 3

6

x xf x

x x

56. Find the Range of 2

22

xf x x

x

57. If f form R into R defined by 3 1f x x then find 1 2,0,7f

58. ,0 ,2 , 3f a b c , , 1 ,1 ,2g a b c then find 2 3f g

59. 2 2, 5 6f x x g x x x then find

2 3 0

0 1 2

g g g

f f f

60. Find the derivative of 3/4 5/62 3x x x

61. If 2 1001 .....f x x x x , then 1 1f

62. Find the derivative of log

7log 0x

x w.r.t x

63. Find the derivative xxx

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64. Find the derivative of sin xe w.r.t sin x

65. Evaluate 2x xe dx

66. Evaluate 3

6

1

1

xdx

x

67. Evaluate

sin

cos

xdx

x a

68. Evaluate sin cos

cos sin

x xdx

x x

69. Evaluate log xdx

Statistics

70. The mean of 40 observations was 160. It was detected on rechecking that the value of 165 was

wrongly copied as 125 for computation of mean. Find the correct mean.

71. Write the class size in each of the following:

(a) 0 – 4, 5 – 9, 10 – 14 (b) 20 – 29, 30 – 39, (c) 5 – 5.01, 5.01 – 5.02

72. Write the class size and class limits in each of the following if the class marks are:

(a) 104, 114, 124, 134, 144, 154 (b) 47, 52, 57, 62, 67, 72, 77

(c) 12.5, 17.5, 22.5, 27.5, 32.5, 37.5

73. The monthly wages of 30 workers in a factory age given below:

830, 835, 890, 810, 835, 836, 869, 894, 898, 890, 820, 860, 832, 833, 855, 845,

804, 808, 812, 840, 885, 835, 836, 878, 840, 868, 890, 806, 840, 890.

(a) Form frequency distribution table with class size 10

(b) Find cumulative frequency table

(c) Draw histogram and frequency polygon

74. The class marks of a distribution are 26, 31, 41, 36, 46, 51, 56, 61, 66, 71.Find the

true class limits.

75. The marks obtained by 35 student in an examination are given below;

125, 130, 130, 120, 141, 146, 162, 163, 169, 173, 179, 188, 192, 195, 199. Form a cumulative frequency table with class interval of length 20.

76. The mean of 10 numbers is 20. If 8 is subtracted from every number, what will be

the mew mean?

77. Draw histogram and frequency polygon to the following data:

Marks 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70

No. of Students 5 10 4 6 7 3 2

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78. Construct simple bar diagram to the following scores of two groups of class IX students in a test.

Scores 50 – 52 47 – 49 44 – 46 41 – 43 38 – 40 35 – 37 32 – 34 Total

Group A 4 10 15 18 20 12 13 92

Group B 2 3 4 8 12 17 22 68

79. The mean monthly salary of 10 members of a group is Rs. 1445, one more member whose

monthly salary is Rs. 1500 has joined the group. Find the mean monthly salary of 11 members.

80. The mean of the following distribution is 6, find the value of P.

X 2 4 6 8 P+5

F 3 2 3 1 2

81. The median of the following observations arranged in ascending order, is 25. Find x.

11, 13, 15, 19, x+2, x+4, 30, 35, 39, 46

82. The mean of 1, 7, 5, 3, 4 and 4 is m, the numbers 3, 2, 4, 2, 3, 3 and p have mean m-1 and

median q. find p and q.

83. The following groups of data, explain whether the mean or the median best describes the data:

a)6, 4, 2, 12, 2 (b) 31, 28, 24, 9, 23 (c) 45, 51, 47, 65, 36 (d) 10, 20, 30, 100, 9

84. Find the median of the first 10 natural numbers. Is it equal to their mean?

85. If X is the mean of n observation x1, x2, …….., xn then find the mean of

x1-a, x2-a, ........xn-a , is , where a is any real number.

86. The following data has been arranged in ascending order:

24, 27, 28, 31, x, 37, 40, 42, 45

If the median of the data is 35, find x.

In the above data, if 45 is changed to 33, find the new median.

87. For what value of x, the mode of the following data is 5?

2, 4, 3, 5, 6, 4, x, 7, 5

88. A boy scored the following marks in various class tests during a term, each test being marked

out of 20:

16, 10, 7, 17, 9, 16, 14, 19, 20, 18, 12

Find mean, median and mode marks. Verify if the following relation holds true.

Mean – Mode = 3(Mean – Median )

89. Find the arithmetic mean of the first 6 natural numbers.

90. Find out the range of the following : 5, 10, 15, 20, 25, 30.

91. Find out the mode of the following : 5, 4, 3, 5, 6, 6, 6, 5, 4, 5, 5, 3, 2, 1

92. The mean of 16 numbers is 8. If z is added to every number, what will be new mean?

93. There are 50 students in a class of which 40 are boys and rest girls. The average weight of the

class is 44kg. and the average weight of the girls is 40kg. Find the average weight of the boys.

94. The mean of 100 items was found to be 300. If at the time of calculation tow items were

wrongly taken as 32 and 12 instead of 23 and 11, find the correct mean.

95. The average score of girls in class examination in a school is 67 and that of boys is 63. The

average score for the whole class is 64.5 find the percentage of girls and boys in the class.

96. Find, to the nearest tenth to the mean absolute deviation for the set {2, 5, 7, 9, 1, 3, 4, 2, 6, 7,

11, 5, 8, 2, 4}.

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97. The Math test marks of a class are as follows 52, 45, 25, 75, 63, 86, 72, 85, 55, 65, 70, 82, 90, 48, 68, 86, 65, 64, 78, 75, 32, 42.

Find the inter quartile range. 98. Find the standard deviation of the data set 5,10,15, 20, 25, 30, 35, 40, 45, 50.

99. Calculate the variance and also standard deviation for the following values: 1, 3, 5, 6, 6,

8, 9, and 10.

100. Measure the square of standard deviation for the given data. 350, 310, 325, 319,

101. Three coins are tossed find the following probabilities:

i) All heads ii)Two heads one tail iii)Two tails one head iv)Atleast two heads

vi) Atmost one head vii) Atleast one head viii)Exactly two heads

102. If two dice are thrown the find the probability of getting 5 on none of them.

103. A cubical die has six faces with 1, -1, 2, -2, 3, 0 is thrown 3 times. Find the

probability of getting a sum 6.

104. A family having 4 children then find the probability of having at least one girl in

the family if the probability of a child is boy or girl is 0.5.

105. Two persons A and B toss a coin. The person who first throws a head wins. If A

starts the game then what are the respective probabilities of their winning?

106. Suppose 50 fair coins are tossed. X denotes the number of heads. If P(X=r) is maximum then find “r”.

107. Three dice are thrown find the following probabilities: (i). sum on the faces is 8 ii). sum on the faces is 18

108. Two cards are drawn from a pack of 52 cards at random. What is the probability that the cards are aces?

109. Three cards are drawn from a pack of cards find the probability that the cards an ace, a king, and a queen.

110. A bag contains five white and three black balls. Four balls are drawn at a time without replacement. What is the probability that the balls are alternatively of different colors.

111. A bag contains 3white, 2black and 4red balls. Find the probability of drawing a W, a B and a R ball in succession in that order without replacement.

112. If 8 identical balls are to be inserted in to 5 numbered cells so that no cell is empty, in how many ways can it be done?

113. The probability of rain on any given day in a city is50%. What is the probability

that it rains exactly on three consecutive days in a 5 day period?

114. A speaks truth in 75 percent cases and B in 80 percent of cases. In what percent

of cases are they likely to contradict each other in narrating the same incident?

115. An urn contains 3 red, 2 black balls and second urn contains 2 red, 3 black balls.

A ball is transferred from first urn to the second find the probability that

composition remains unaltered.

116. Find the mean deviation about the mean for the data 4, 7, 8, 9, 10, 12, 13, 17

117. Find the mean deviation about the median for the data 13, 17, 16, 14, 11, 13,

10, 16, 11, 18, 12, 17

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118. Find the mean deviation about the mean for the data

5 10 15 20 25

7 4 6 3 5

i

i

x

f

119. Find the mean and variance for each of the data 6, 7, 10, 12, 13, 4, 8, 12

120. Find the mean and variance for each of the data

92 93 97 98 102 104 109

3 2 3 2 6 3 3

i

i

x

f

121. Events A & B are such that 1 7

;2 12

P A P B & P (Not A or Not B) 1

4

then state whether A & B are independent

122. A die is tossed thrice, find probability of getting an odd number at least once.

123. A random variable X takes the values – 1, 0, 1. Its mean 0.6 & If 0 0.2P x ,

then find 1P x .

124. Two cards are drawn successively with replacement from a well shuffled pack of

52 cards. The probability distribution of the numbers of kings is formed. The

mean of random variable.

125. Two coins whose faces are marked 2 & 3 are thrown, then find the area of total

value of the numbers turns on the faces.

126. The range of random variable 1,2,3,.....X and the probability are given by

3

!

CK

P X kK

& C is constant, then find C

127. The mean marks obtained by 300 students in the subject of Statistics are 45. The mean

of the top 100 of them was found to be 70 and the mean of the last 100 was known to

be 20. What is the mean of the remaining 100 students?

128. Find the variance of the number obtained on a throw of an unbiased die.

129. Given three identical boxes I, II and III, each containing two coins. In box I, both coins

are gold coins, in box II, both are silver coins and in the box III, there is one gold and one

silver coin. A person chooses a box at random and takes out a coin. If the coin is of

gold, what is the probability that the other coin in the box is also of gold?

130. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck

drivers. The probability of an accidents are 0.01, 0.03 and 0.1 respectively. One of the

insured persons meets with an accident. What is the probability that he is a scooter

driver?

131. Two cards are drawn simultaneously (or successively without replacement) from

a well shuffled pack of 52 cards. Find the mean, variance and standard deviation

of the number of kings.

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Objective type questions:

Algebra:

1. If 1,3,5,7,9,11,13,15,17A , 2,4,.....,18B and N is the universal set, then

' 'A U A B B is

1) A 2) N 3) B 4) None of these

2. If X and Y are two sets, then 'X Y X equals

1) X 2) Y 3) f 4) None of these

3. Let : is a multiple of 3A x x and : is a multiple of 5B x x . Then A B is given by

1) {3, 6, 9….} 2) {5, 10, 15,20,….} 3) {15, 30,45…..} 4) None of these

4. Which of the following is the empty set?

1) 2/ is real number and x 1 0x x 2) 2/ is real number and x 1 0x x

3) 2/ is real number and x 9 0x x 4) 2/ is real number and x 2x x x

5. Let A and B have 3 and 6 elements respectively. What can be minimum number of

elements in A B ?

1) 3 2) 6 3) 9 4) 18

6. Two finite sets have m and n elements. The total number of subsets of the first set is 56

more than total number of subsets of second set. The values of m and n are

1) 7, 6 2) 6, 3 3) 5, 1 4) 8, 7

7. The composite mapping fog, of the maps 2: , sin ; : ,f R R f x x g R R g x x is

1) 2sin x x 2) 2

sin x 3) 2sin x 4) 2

sin x

x

8. The function :f N N (N is set of natural numbers) defined by 2 3f n n is

1) surjective 2) not surjective 3) injective 4) none of these

9. The solutions of 8 6x (mod 14) are

1) 8 , 6 2) 8 , 4 3) 6 , 13 4) 8 , 4 , 16

10. If :f R R , defined by 2 1f x x , then the values of 1 17f and 1 3f

respectively are

1) , 4, 4 2) 3, 3 , 3) , 3, 3 4) 4, 4 ,

11. The geometric mean G of the product of n series of data with geometric means

1 2, ........., nG G G respectively, then

1) 1 2..... nG G G G 2) 1 2...... nG G G G 3) 1 2..... nG G G G 4) None of these

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12. A motor car when travelling from rest travels the first twentieth of a mile at 6 mph the

next three twentieths of the mile at 8, 12, 24 mph respectively. But its average speed over

the first one-fifth of a mile is not 12.5 mph, then the correct average is

1) 9.6 mph 2) 19.6 mph 3) 6.6 mph 4) None of these

13. The average rate of (i) motion in the case of person who rides the first mile at 10 mph, the

next mile at 8 mph, and the third mile at 6 mph, (ii) increase in population which in the

first decade has increased 20%, in the next 25% and in the third 44% are, (where, log 20

= 1.3010, log 25 = 1.3979, log 28.02 = 1.4475, log 44 = 1.6435)

1) 7.66, 1.7 2) 7.5, 2.9 3) 7.66, 28.02 4) None of these

14. A man motors from A to B. In motoring a distance uphill, he gets a mileage of only 10

miles per gallon of gasoline. On the return trip, he makes 15 miles per gallon. Then the

harmonic mean of his mileage (Verify that this is the proper average to be used here,

assuming that the distance from A to B is 60 miles) is

1) 12 2) 11 3) 10 4) None of these

15. If a variate takes values 2 1, , ,.........., na ar ar ar which of the relation between means hold?

1) 2AH G 2) 2

A HG

3) A G H 4) A G H

16. An aeroplane files around a squares, the sides of which measure 100 miles each. The

aeroplane covers at a speed of 100 mph the first side, at 200 mph the second side, at 300

mph the third side and 400 mph the fourth side, The average speed of the aeroplane

around the square is

1) 190 mph 2) 195 mph 3) 192 mph 4) 200 mph

17. If the variable takes values 0,1,2,3,....,n with frequencies proportional to

0 1 2, , .......n n n n

nC C C C respectively, the variance is

1) 4

n 2)

3

n 3)

2

5

n 4) None of these

18. Solve 4x + 3 < 6x +7

1) (–2, ∞) 2) (-2, 2) 3) (–∞, ∞) 4) None

19. Solve – 8 ≤ 5x – 3 < 7.

1) 0 ≤ x < 6 2) 0 ≤ x < 2 3) –5 ≤ x < 3 4) –1 ≤ x < 2

20. Two matrices A and B with order 2x3 and 3x4 respectively. Then the order of the

matrix AxB is

1) 2x3 2) 3x4 3) 2x4 4) none

21. Find the determinant of matrix A =

1)2 2)-1 3) 1 4) 0

22. A function f(x) = 3x-2, where x= {2,4,6,8,…….}then its range is

1){ 0,1,2,3,…..} 2) { 2,4,6,8,…..} 3) {1,3,5,7,…..} 4) none

23. If a set contains 5 elements then its power set contains --------------- number of elements

1) 25 2) 35 3) 55 4) None of these

24. In how many ways can the letters in the word BEACH be rearranged?

1) 120 2)60 3)30 4) 5

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25. Sum of infinite Geometric series 1, 2, 4 , 8, 16 ,32………..

1)2 2) -1 3) 1 4) 0

26. In a arithmetic progression a = -2 , d =4 and sn = 160 , find n

1)8 2) 10 3) 6 4) none

Calculus:

27. If 1,3,5,7,9,11,13,15,17A , 2,4,.....,18B and N is the universal set, then

' 'A U A B B is

1) A 2) N 3) B 4) None of these

27. If X and Y are two sets, then 'X Y X equals

1) X 2) Y 3) f 4) None of these

29. Let : is a multiple of 3A x x and : is a multiple of 5B x x . Then A B is given by

1) {3, 6, 9….} 2) {5, 10, 15,20,….} 3) {15, 30,45…..} 4) None of these

30. Which of the following is the empty set?

1) 2/ is real number and x 1 0x x 2) 2/ is real number and x 1 0x x

3) 2/ is real number and x 9 0x x 4) 2/ is real number and x 2x x x

31. Let A and B have 3 and 6 elements respectively. What can be minimum number of elements in

A B ?

1) 3 2) 6 3) 9 4) 18

32. Two finite sets have m and n elements. The total number of subsets of the first set is 56 more

than total number of subsets of second set. The values of m and n are

1) 7, 6 2) 6, 3 3) 5, 1 4) 8, 7

33. The composite mapping fog, of the maps 2: , sin ; : ,f R R f x x g R R g x x is

1) 2sin x x 2) 2

sin x 3) 2sin x 4) 2

sin x

x

34. The function :f N N (N is set of natural numbers) defined by 2 3f n n is

1) surjective 2) not surjective 3) injective 4) none of these

35. The solutions of 8 6x (mod 14) are

1) 8 , 6 2) 8 , 4 3) 6 , 13 4) 8 , 4 , 16

36. If :f R R , defined by 2 1f x x , then the values of 1 17f and 1 3f

respectively are

1) , 4, 4 2) 3, 3 , 3) , 3, 3 4) 4, 4 ,

37. 2

21

1

3 4lim

x

x

x x

1)1/5 2)2/5 3)3/5 4)4/5

38. What is the derivative with respect to x of (x + 1)3 – x

3?

1) 3x+6 2) 3x-3 3) 6x-3 4)6x+3

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39. Find the second derivative of y by implicit differentiation from the equation

4x2 + 8y

2 =36

1) 64x2 2) (– 9/4) y3 3) 32xy 4) (- 16/9) y3

40. Which of the following is the indefinite integral of

1) 2x+c 2) 𝑥3+7x 3) 𝑥3

2+ 7𝑥 4)

𝑥3

3+ 7𝑥 + 𝑐

41. Find the value of 15 10

5 5

(5 )x y

x y dy dx

1) 2875 2) 3250 3) 2812.5 4) 1406.2

42. A function f(x) = 3x-2, where x= {2,4,6,8,…….} then its range is

1){ 0,1,2,3,…..} 2) { 2,4,6,8,…..} 3) {1,3,5,7,…..} 4) none

43. If a set contains 5 elements then its power set contains --------------- number of elements

1)25 2) 35 3) 55 4) none

44. 2 16

dx

x

---------

1) (x-4/x+4) 2) log(x-4/x+4) + c 3) 1/8 log (x-4/x+4) + c 4) none

45. 2

4( )2 5)

d x

dx x = ------------

1)2

4

2 5

x

x 2) 4x 3) 2x2-5 4) none

46. ∫logx = -------------

1)Xlogx-x + c 2) xlogx+ x +c 3) xlogx 4) none

Statistics;

46. Two die are thrown simultaneously to get the coordinates of a point on x y plane. Then

the probability that this point lies inside or on the region rebounded by 3x y is

1) 3

14 2)

2

3 3)

1

12 4)

4

14

47. Suppose 3 2f x x ax bx c , where , ,a b c are chosen respectively by throwing a die

three times. Then the probability that f x is an increasing function is

1) 4

9 2)

3

8 3)

2

5 4)

16

34

48. Pal’s gardener is not dependable, the probability that he will forget to water the rose bush

is 2/3. The rose bush is in questionable condition. Any how if watered, the probability of

its withering is ½, if not watered, the probability of its withering is ¾. Pal went out of

station and upon returning, he finds that the rose bush has withered. Then the chance that

the gardener did not water the rose bush, is

1) 2

5 2)

1

2 3)

1

3 4)

3

4

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49. An urn contains m white and n black balls. A ball is drawn at random and is put back into

the urn along with k additional balls of the same colour as that of the ball drawn. A ball is

again drawn at random. Then the probability that the ball drawn now is white, is

1) n

m n 2)

m

m n 3)

2n

m n 4) None of these

50. An unbiased die, with faces numbered 1, 2, 3, 4, 5, 6 is thrown n times and the list of n

numbers shown up is noted. Then the probability that, among the numbers 1, 2, 3, 4, 5, 6,

only three numbers appear in this list, is

1) 6

33 .

6

n

n

C 2)

6

33 3.2 3 .

6

n n

n

C 3)

6

33 3

6

n

n

C 4) None of these

51. A box contains N coins, m of which are fair and the rest are biased. The probability of

getting a head when a fair coin is tossed is 1

2, while it is

2

3 when a biased coin is tossed.

A coin is drawn from the box at random and is tossed twice. Then the probability that the

coin drawn is fair, is

1) 9

8

m

N m 2)

9

8

m

N m 3)

9

8

m

m N 4)

9

8

m

m N

52. For a student to qualify, he must pass at least two out of three exams. The probability that

he will pass the 1st exam is p. If he fails in one of the exams, then the probability of his

passing in the next exam is 2

p, otherwise it remains the same. Then the probability that

he will quality, is

1) 22p p 2) 22 2p p 3) 2 32p p 4) None of these

53. A is targeting to B, B and C are targeting to A. Probability of hitting the target by A, B

and C are 2 1

,3 2

and 1

3 respectively. If A is hit then the probability that B hits the target

and C does not, is

1) 1

2 2)

3

4 3)

2

3 4) None of these

54. A is one of 6 horses entered for a race, and is to be ridden by one of two jockeys B and C.

It is 2 to 1 that B rides A, in which case all the horses are equally likely to win. If C rides

A, his chance of winning is trebled. What are the odds against winning of A?

1) 5:13 2) 5:18 3) 13:5 4) None of these

55. If in a distribution, probability of a random variable X taking the particular value x is n x n x

xC p q where 1p q , and 0,1,2,.....x n , then its mean is

1) np 2) npq 3) nq 4) np(1+p)

of 15

56. 0<P<1 which of the following has least value.

1) 2

1

p 2)

1

p 3)

1

( 1)P 4)

2

1

( 1)p 5)

2

1

1p

57. In a frequency distribution the class intervals are 0 – 9 10 – 19 20 – 29 30 – 39 then the length of the class interval is

1)9 2) 10 3) cannot determined 4) none

58. In a frequency distribution mean is 15 and median is 18 then its mode is:∫1

𝑥2 𝑑𝑥∞

1

1)1 2) 0 3) 2 4) none

59. Two events A and B are said to be independent if :

1) P(A ∩ B) = P(A) × P(B) 2) P(A ∩ B) = 0 3) P(A ∩ B) = 1 4) none

60. In a arithmetic progression a = -2 , d =4 and sn = 160 , find n

1)8 2) 10 3) 6 4) none

61. In a group of 100 persons, 72 people can speak English and 43 can speak French. How many can

speak English only?

1)15 2) 57 3) 28 4) none

62. The mean of 6 observations is 2 and the mean of another 8 observations is 6 then their

combined mean is:

1)4.3 2) 4.4 3) 4.2 4) none

63. If P(A) = 7 13 , P(B) = 9 13 and P(A ∩ B) = 4 13 , then P(A|B) = ?

1)0 2) 1 3) 3/9 4) 4/9

64. The pmf of a discrete random variable is :

X=x 0 1 2 3

P(x) a 2a 3a 4a

Then the value of a is:

1)9 2) 1/10 3) cannot be determined 4) none

65. The mean and variance of Binomial distribution are 5 and 3 respectively. Then how many trials

were there in this?

1)25/2 2) 25 3) 15 4) none

66. A and B are mutually exclusive if

1)A ∩ B = 1 2) A ∩ B = 0 3) A ∩ B = φ 4) none

67. Which of the following distributions has Mean = Variance ?

1)Binomial 2) Poisson 3) Normal 4) none

68. What is the probability of getting a sum 9 from two throws of a dice?

1)1/6 2) 1/8 3) 1/9 4) 1/12

69. A and B are two events such that P(A) = 0.54, P(B) = 0.69 and P(A ∩ B) = 0.35. then P(A ∪ B)

1) 0.88 2) 0.90 3) 0.5 4) none

of 15

69. The probability of success in a binomial distribution with 15 trials is 0.6 . Then its mean and

variances are:

1)5 and 2 2) 2 and 5 3) cannot be determined 4) none

70. The parameter of a Poisson distribution is 2.6 then its mean and variance are

1)2.6 and 2 2) 2 and 2.6 3) 2.6 and 2.6 4) none

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