Page 1 of 15 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 2 12 36 4 0 X x 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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Page 1 of 15
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
30. Use the limit definition to compute the derivative, f'(x), for
1 3
( )2 5
f x x
31. Use the limit definition to compute the derivative, f'(x), for
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,
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
Page 10 of 15
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