ACET 2021 Question Paper (Jan)

Institute of Actuaries of India
ACET January 2021
Mathematics

1. The value of ∑ 𝑖𝑗 where the summation is over all 𝑖, 𝑗 such that 0 ≤ 𝑖, 𝑗 ≤ 10 is
A. 3025.
B. 55.
C. 110.
D. 6050.

2. Let 𝑅 be the relation defined on the set 𝐴 = {1,2,3,4,5,6} as 𝑅 = {(𝑥, 𝑦): 𝑦 = 𝑥 + 1}
then, 𝑅 is
A. Reflexive.
B. Symmetric.
C. Transitive.
D. neither reflexive nor symmetric nor transitive.

3. If 𝑓(𝑥) = [𝑥], the greatest integer less than or equal to 𝑥 and 𝑔(𝑥) = |𝑥 − 1| , then
𝑓°𝑔(− ) is
A. 2.
B. 3.
C. 4.
D. 5.
4. Let 𝑓: 𝑅 → 𝑅 defined as
−1 𝑖𝑓 𝑥 < 0,
𝑓(𝑥) = 0 𝑖𝑓 𝑥 = 0,
+1 𝑖𝑓 𝑥 < 0.
Then 𝑓(𝑥) is
A. one to one.
B. onto.

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C. one to one and onto.
D. neither one to one nor onto.

5. A root of 𝑥 − 2𝑥 + 5 = 0 lies in the interval
A. (−1,0).
B. (−2, −1).
C. (−3, −2).
D. (−4, −3).

6. The set 𝑆 of real numbers such that |1 − 2𝑥| − 2𝑥 ≥ 0 is
A. 𝑆 = −∞, .

B. 𝑆 = − , .
C. 𝑆 = (−∞, 0).
D. 𝑆 = −∞, .
7. Suppose 𝑓(𝑥) is a function having values as given in the following table.
x 1 2 3 4 5
f(x) 2 5 7 - 32

Based on the above information the value of 𝑓(𝑥) at 𝑥 = 4, estimated from a cubic
polynomial approximation, is
A. 10.
B. 14.
C. 16.
D. 20.

8. The sum of squares of the zeroes of the polynomial expression 𝑝(𝑥) = 𝑥 + 𝑏𝑥 − 2 is 5.
Then, the possible values of 𝑏 are
A. ±1.
B. ±2.

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C. ±3.
D. ±4.

9. The constant term in the expansion of − is

A. − .

B. − .

C. − .

D. − .
10. If 0 < 𝑥 < 𝜋 and 16 + 16 = 10, then tan 𝑥 is equal to
A. 0 or ∞.
B. 1.
C. or .

D. or √3.
√

3 marks

11. lim → is

A. 2.
B. −2.
C. 0.
D. .

12. If 𝑦 = (log 𝑥) , then is equal to

A. (log 𝑥) − sin 𝑥 log (log 𝑥) .

B. (log 𝑥) + sin 𝑥 log (log 𝑥) .

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C. (log 𝑥) − cos 𝑥 log (log 𝑥) .

D. (log 𝑥) + cos 𝑥 log (log 𝑥) .

13. Let 𝑓(𝑥) = 𝑥 − 𝑥 + 1. Then, 𝑓(𝑥) is
A. strictly increasing in (−1,1).
B. strictly decreasing in (−1,1).
C. neither strictly increasing nor strictly decreasing in (−1,1).
D. a constant function in (−1,1).

14. The value of ∫ 𝑑𝑥 is
( )( )

A. log | |
+ 𝐶.
( )
B. log | |
+ 𝐶.
( )
C. log + 𝐶.
( )

D. log + 𝐶.

15. The value of 𝑑𝑥 is
∫

A. .
B. 𝑒.
C. .

D. .
3 marks
16. If 𝑎⃗ + 𝑏⃗ = √29 and 𝑎⃗ ∙ 𝑏⃗ = 5, then 𝑎⃗ − 𝑏⃗ is
A. −3.
B. 4.
C. √3.
D. 3.

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17. The area of the parallelogram whose adjacent sides are determined by the vectors 𝑎⃗ =
𝚤⃗ − 𝚥⃗ + 3𝑘⃗ and 𝑏⃗ = 2𝚤⃗ − 7𝚥⃗ + 𝑘⃗ is
A. 15√2 sq. units.
B. 450 sq. units.
C. √20 sq. units.
D. 65 sq. units.

18. If 𝑃 is a non-singular matrix of order 𝑛, then 𝑑𝑒𝑡(𝑘𝑃) for 𝑘 ≠ 𝑛 is
A. 𝑘 det(𝑃).
B. 𝑛 det(𝑃).
C. 𝑘 det(𝑃).
D. det(𝑃).
19. If 𝑀 is the matrix such that|𝑀| ≠ 0 and 𝑀 − 𝑀 + 𝐼 = 0, then 𝑀 is
A. 𝐼 + 𝑀.
B. 𝐼 − 𝑀.
C. 𝑀 − 𝐼.
D. 𝑀 + 𝐼.

20. If
cos 𝑥 − sin 𝑥 0
𝑃(𝑥) = sin 𝑥 cos 𝑥 0 ,
0 0 1
then 𝑃(𝑥)𝑃(𝑦) is equal to
A. 𝑃(𝑥𝑦).
B. 𝑃(𝑥 − 𝑦).
C. 𝑃(𝑥 + 𝑦).
D. the identity matrix.

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Statistics

21. In a class of 56 students, 28 opted for Mathematics, 30 opted for Biology and 22 opted
for both Mathematics and Biology. If one of these students is selected at random, then
probability that the student has opted neither Mathematics nor Biology is
A. .

B. .

C. .

D. .

22. A man decided to visit four cities 𝐶 , 𝐶 , 𝐶 and 𝐶 in a random order. The probability
that he visits 𝐶 before 𝐶 and 𝐶 before 𝐶 is
𝟏
A. .
𝟔
𝟏
B. .
𝟖
𝟏
C. .
𝟑
𝟏
D. .
𝟏𝟐

23. If 𝐴 and 𝐵 are two events such that (𝐴 ∩ 𝐵) = , 𝑃(𝐵) = . Then 𝑃(𝐴 ∪ 𝐵 )

A. is .

B. is .

C. is .
D. cannot be computed from the given information.

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24. For two events 𝐸 and 𝐹, 𝑃(𝐸|𝐹) > 𝑃(𝐸), then which of the following is correct?
A. 𝑃(𝐹|𝐸) < 𝑃(𝐹).
B. 𝑃(𝐹|𝐸) = 𝑃(𝐹).
C. 𝑃(𝐸|𝐹) + 𝑃(𝐹|𝐸) < 𝑃(𝐸) + 𝑃(𝐹).
D. 𝑃(𝐹|𝐸) > 𝑃(𝐹).

25. Let 𝐴 , 𝐴 , 𝐴 be independent events with probabilities , , , respectively. Then
𝑃(𝐴 ∪ 𝐴 ∪ 𝐴 ) equals
A. .

B. .

C. .
D. 1.

26. A man is known to speak truth 80% of the times. He throws a die and reports that the
number appeared is greater than 4. The probability that it is actually a number greater
than 4 is
A. .

B. .

C. .

D. .
3 marks
27. The stem and leaf plot of lifetimes in days of a set of 40 light bulbs is shown below.
Stem Leaf
2 13
3 023578
4 01234556789
5 0346789
6 23478
7 0479
8 146
9 32
The median lifetime of light bulbs is
A. 49 days.

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B. 49.5 days.
C. 50 days.
D. 51.5 days.

28. Suppose there are 𝑛 observations on the variable 𝑥, out of which 𝑛 observations are one
and remaining are zero. Let 𝑦 = 𝑥 , then the variance of the values of 𝑦 is
A. 𝑛 ⁄𝑛 .
B. 𝑛 /𝑛 .
C. 𝑛 (𝑛 − 𝑛 )/𝑛 .
D. 𝑛 (𝑛 − 𝑛 ) /𝑛 .

29. Suppose there are three departments 𝐷 , 𝐷 and 𝐷 in an organization. The numbers of
workers employed in these departments are 50, 50 and 100, respectively. The mean
earnings in a certain month per worker in the three departments are Rs. 15000, Rs. 13000
and Rs. 18500, respectively. The mean earning per worker of the whole organization is
A. Rs. 15500.
B. Rs. 16250.
C. Rs. 16500.
D. Rs. 17000.
30. Consider the frequency distribution of weights (in kg) of 250 persons.
Weights in Kg Number of persons
35.5 – 40.5 8
40.5 – 45.5 12
45.5 – 50.5 25
50.5 – 55.5 40
55.5 – 60.5 45
60.5 – 65.5 60
65.5 – 70.5 46
70.5 – 75.5 12
75.5 – 80.5 2
Which of the following statement is true for the distribution of weight?
A. Mode of the distribution is 60.5.
B. The distribution is positively skewed.
C. The median value lies in the class interval 60.5 – 65.5.

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D. The coefficient of skewness is negative.

31. If the range of a set of observations is 20, which of the following statements is true?
If each observation is multiplied by -3, then the range of the new observations
A.
becomes 60.
If each observation is increased by 5, then the range of the new observations
B.
becomes 25.
If each observation is multiplied by 3 and then 5 added to it, then the range of the
C.
new observations becomes 65.
If each observation is multiplied by 5 and then 3 subtracted from it, the range of
D.
the new observations will be 97.
32. A discrete random variable 𝑋 follows uniform distribution over the values 7, 12, 15, 20,
23, 25, 30. Then 𝑃(𝑋 > 18|𝑋 < 26) equals
A. .

B. .

C. .

D. .

33. A vendor supplies an electronic item in lots of 25. A buyer, before taking a lot, tests a
random sample of 4 items and if all are good, he accepts the lot. The probability that he
accepts a lot containing 5 defective items is
A. 0.2.
B. 0.8.
C. 0.0016.
D. 0.4096.

34. The number of misprints per page of a book (𝑋) follows the Poisson distribution such
that 𝑃(𝑋 = 1) = 𝑃(𝑋 = 2). If the book contains 500 pages, the expected number of
pages containing at most one misprint is
A. 500𝑒 .
B. 1000𝑒 .
C. 1500𝑒 .

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D. 500(1 − 3𝑒 ).
35. Let 𝑋 be a continuous random variable with probability density function
( )
𝑓(𝑥) = 𝜆𝑒 , 𝑥≥𝛼
0, 𝑥<𝛼
The mean of the distribution is
A. 𝜆 + 𝛼.
B. .

C. .

D. .

36. If the 90th percentile of the standard normal distribution is 1.65. Then the 90 th percentile
of the normal distribution with mean 20 and variance 4 is
A. 21.65.
B. 23.30.
C. 26.60.
D. 36.30.

37. Let 𝑋 be a continuous random variable with pdf 𝑓(𝑥) = 0.5, − 1 ≤ 𝑥 ≤ 1. The
skewness of the distribution is
A. 1.
B. 0.5.
C. −1.
D. 0.
38. The marginal distributions of 𝑋 and 𝑌 are given in the following table.
x 1 2 Total
y
0 0.7
1 𝑝 0.3
Total 0.7 0.3 1

The value of 𝑝 for which 𝑋 and 𝑌 have correlation 0.1 is
A. 0.111.

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B. 0.09.
C. 0.15.
D. 0.22.
3 marks

39. Let 𝜌 be the correlation coefficient between 𝑋 and 𝑌. If 𝑈 = 𝑎𝑋 + 𝑏 and 𝑉 = 𝑐𝑌 + 𝑑,
where 𝑎, 𝑏, 𝑐 and 𝑑 are constants. Let 𝜌 be the correlation coefficient between 𝑋 and
𝑉. Then
A. 𝜌 = 𝜌 if 𝑐 > 0.
B. 𝜌 = 𝜌 if 𝑐 < 0.
C. 𝜌 = 𝑐𝜌 .
D. 𝑐𝜌 =𝜌 .

40. For the two variables 𝑥 and 𝑦 the equations of two regression lines are 4𝑥 − 5𝑦 + 𝑐 = 0
and 20𝑥 − 9𝑦 − 𝑑 = 0, where 𝑐 and 𝑑 are some constants. The ratio of variances of 𝑦
and 𝑥 is
A. 4𝑐/𝑑.
B. 𝑑/𝑐.
C. 16/9.
D. 25/16.

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Data Interpretation

Answer Questions 41 and 42 based on the data given in the following table. The table shows the
numbers (𝑓) of candidates obtaining marks 𝑥 or higher in a certain examination, where the total
number of candidates is 250.
𝑥 25 30 40 50 60 70 80 90 100
𝑓 250 240 200 150 120 80 20 10 0

41. The number of candidates who have scored less than 60 is
A. 100.
B. 120.
C. 130.
D. 170.

42. Suppose the pass mark is 40. Then the percentage of candidates who have passed the
examination is
A. 40.
B. 65.
C. 70.
D. 80.
Answer Questions 43-45 based on the data given the following table. The table shows the number
of students appeared in class X board examination in a state for the years from 2011 to 2019.
Year Male Female Total
(in lakhs) (in lakhs) (in lakhs)
2011 3.40 3.20 6.60
2012 3.38 3.32 6.70
2013 3.45 3.40 6.85
2014 3.53 3.52 7.05
2015 3.55 3.65 7.20
2016 3.60 3.85 7.45
2017 3.70 4.00 7.70
2018 3.85 4.20 8.05
2019 3.95 4.25 8.20

43. The annual percentage increase of candidates was the largest from
A. 2013 to 2014.

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B. 2015 to 2016.
C. 2016 to 2017.
D. 2017 to 2018.

44. More than 52% of the students appeared for the exam were female in the year
A. 2016.
B. 2017.
C. 2018.
D. 2019.

45. The number of female candidates surpassed the number of male candidates by 20000 or
more in the years
A. 2016, 2017, 2018, 2019.
B. 2017, 2018, 2019.
C. 2016, 2017, 2018.
D. 2018, 2019.
Answer Questions 46-48 based on the data in the Venn diagram given below. A group of students
were surveyed to find out which of the three types of food 𝐹 , 𝐹 and 𝐹 , they liked.

𝐹 𝐹
5 4
6
7
3
4

12 5 𝐹

46. The number of students who liked exactly one type of food is
A. 34.
B. 22.
C. 7.
D. 15.

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47. The number of students who liked 𝐹 and 𝐹 but not 𝐹 is
A. 5.
B. 7.
C. 9.
D. 12.

48. The number of students who liked 𝐹 or 𝐹 is
A. 41.
B. 30.
C. 19.
D. 11.
Answer Questions 49 and 50 based on the pie charts given below. The pie charts show export of
four products for the years 2010-11 and 2011-12. The total export in 2010-11 was Rs. 5000 crore
and increased by 7% in 2011-12.

2010-11 2011-12

10%
12%
15%
40% 16% 40%

32%
35%

P1 P2 P3 P4 P1 P2 P3 P4

49. Total export of the products P3 and P4 in two years is
A. Rs. 2846 crores.
B. Rs. 2748 crores.
C. Rs. 2737 crores.
D. Rs. 2825 crores.

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50. From 2010-11 to 2011-12, total value of export of the products P1 and P2
A. increased by 2.72%.
B. increased by 1.25%.
C. decreased by 3.00%.
D. decreased by 2.50%.
51. The line diagram shows annual sales (in crores of rupees) of a company from 2006 to 2019.
250

200
Annual Sales

150

100

50

0
2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019
Year

The number of years in which annual sales decreased over those of the corresponding
previous years is

A. 2.
B. 3.
C. 4.
D. 5.

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English

52. Agnostic is a person who
A. is a non-believer in the existence of God.
B. is not sure about the existence of God.
C. is a firm believer in the existence of God.
D. leads a life full of agony.
53. Fanatic is a person who
A. has excessive zeal for a cause.
B. acts against religion.
C. cannot express himself freely.
D. is mentally ill.

54. A person who studies the elections and trends in voting is called
A. an electologist.
B. a psephologist.
C. an electorate.
D. a statistician.

55. A hidden storage of items of the same type is called a
A. locker.
B. vault.
C. burrow.
D. cache.
56. A synonym of “Acumen” is
A. Preference.
B. Perfection.
C. Accuracy.
D. Sharpness.

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57. A synonym of “Endeavour” is
A. Effort.
B. Ending.
C. Endurance.
D. Cease.

58. An antonym of “Precarious” is
A. Post facto.
B. Confused.
C. Stable.
D. Unsure.

59. An antonym of “Quarantine” is
A. Proliferate.
B. Celebrate.
C. Desegregate.
D. Entertain.
60. Select the incorrect sentence.
A. I have seen him yesterday.
B. I saw him yesterday.
C. I have seen him.
D. He was seen by me yesterday.

61. Select the incorrect sentence.
A. Let us not pluck the flowers.
B. Let the flowers not be plucked.
C. Let the flowers not been plucked.
D. Let the flowers be plucked.

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Read the passage below and answer Question No. 11.
Coronavirus is a group of Ribonucleic Acid (RNA) viruses that cause diseases in mammals and
birds. These viruses cause respiratory tract infections. Mild illnesses include some cases of the
common cold, while more lethal varieties can cause SARS, MERS and COVID-19. Symptoms
in different species vary: in chickens, they cause an upper respiratory tract disease, while in cows
and pigs they cause diarrhea. The name "coronavirus" is derived from Latin corona, meaning
"crown" or "wreath". Coronaviruses were first discovered in the 1930s when an acute respiratory
infection of domesticated chickens was shown to be caused by the infectious bronchitis virus.
The infection of new-born chicks was characterized by gasping and listlessness. Human
coronaviruses were discovered in the 1960s. Infected carriers are able to shed viruses into the
environment. The interaction of the coronavirus spike protein with its complementary cell
receptor is central in determining the tissue tropism, infectivity, and species range of the released
virus. They are transmitted from one host to another host, depending on the coronavirus species,
by either an aerosol, fomite, or fecal-oral route. Human coronaviruses infect the epithelial cells
of the respiratory tract, while animal coronaviruses generally infect the epithelial cells of the
digestive tract. For example, SARS coronavirus infects, via an aerosol route, the human epithelial
cells of the lungs. Transmissible gastroenteritis coronavirus (TGEV) infects, via a fecal-oral
route, the pig epithelial cells of the digestive tract by binding to the alanine aminopeptidase
receptor.

I. COVID-19 and SARS are caused by
i. The same Coronavirus
ii. RNA viruses
iii. Acute Respiratory Infection

II. SARS infects
i. human epithelial cells
ii. pig epithelial cells
iii. both

III. Coronavirus may cause
i. diarrhea in pigs
ii. respiratory tract disease in cows
iii. both

62. The correct answers to I, II and III are
A. i, ii, iii, respectively.
B. ii, i, i, respectively.
C. ii, i, iii, respectively.
D. i, iii, iii, respectively.
3 marks

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Logical Reasoning

63. A man says to a woman, "You are the only child of the father- in- law of my father's only
child." How is the woman related to the man?
A. Sister in law.
B. Granddaughter.
C. Wife.
D. Daughter.

64. In a certain substitution code language, "NUMBER" is written as "156897" and
"BARREN" is written as "847791". How is "RUBBER" written in that
A. 756697.
B. 758897.
C. 795957.
D. 795579.

65. Statements:
All Forests are Cars.
All Lights are Cars.
All Vegetables are Lights.
Conclusions:
I. Some Vegetables are Forests
II. Some Cars are Lights
III. Some Cars are Forests
IV. All Vegetables are Cars

A. All conclusions follow.
B. Only II, III and IV follow.
C. Only I, II and III follow.
D. Only I, II and IV follow.

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66. Identify a meaningful order of the following.
1. Index
2. Bibliography
3. Chapters
4. Preface
A. 1, 3, 2, 4.
B. 3, 1, 4, 2.
C. 4, 3, 2, 1.
D. 4, 1, 3, 2.

67. A cube of side n is painted brown on all its sides. It is then cut into n3 identical cubes.
How many of the smaller cubes will have three of its sides painted?
A. 4.
B. 4𝑛.
C. 8.
D. 8𝑛.

68. Year 1895 is having a same calendar as that of the year X. Which of the following is a
possible value of X?
A. 1900.
B. 1901.
C. 1902.
D. 1903.
69. Six persons are sitting around a circular table. S is facing R who is to the left of A and to
the right of P. A is to the left of D. Y is to the left of P. If D exchanges his seat with Y
and P exchanges with R, who will be sitting to the left of D?
A. Y.
B. R.
C. A.
D. S.

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70. Wasim, Aaqib and Imran belong to three different tribes of people. The three tribes are
Sachhas (those who always speak the truth), Jhootas (who always lie) and Badlas (those
who alternatively speak the truth and lie).
Wasim said that Aaqib is not a Badla.
Imran said that Wasim is a Badla.
Who among the three is a Jhoota?
A. Aaqib.
B. Wasim.
C. Imran.
D. Cannot be determined.

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