ACET Dec 2023 Question paper

Institute of Actuaries of India
ACET December 2023
Mathematics
1. The solution(s) of the equation |2𝑥 + 23|= |𝑥 + 11| is (are)
A. −14.
34
B. − 3 and 4.
34
C. −12 and − 3 .

D. 6 and − 34.
3

2. If 𝑓(𝑥) = 4(𝑥 − 3) and 𝑔(𝑥) = 𝑥 2 − 7𝑥 + 12, then the value of 𝑥 for which the ratio 𝑔(𝑥)
𝑓(𝑥)
is undefined is
A. 1.
B. 2.
C. 3.
D. 4.

3. For any two sets 𝐴 and 𝐵 if 𝐴 ∩ 𝐵 = 𝐴 ∪ 𝐵, then
A. 𝐴 = 𝐵.
B. 𝐴 = Ω (the universal set).
C. 𝐴 = ∅ (the null set).
D. 𝐴 = 𝐵 𝑐 .

4. If 𝑓(𝑥) = 2𝑥 2 + 5𝑥 − 24 and 𝑔(𝑥) = 3𝑥 + 2 (𝑥 real), then the roots of 𝑓(𝑥) +
𝑔(𝑥) are
A. real and equal.
B. real and distinct.
C. complex conjugate pair of non-real roots.
D. one is real and the other is non- real.

5. The value of (√3 + 1)4 − (√3 − 1)4 is
A. 32√3.
B. 16√3.

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C. 8√3.
D. 96.
4 1
6. The value of cos −1 5 + tan−1 7 is (assuming all the angles to be in the first quadrant)
𝜋
A. .
2
𝜋
B. .
3
𝜋
C. .
4
𝜋
D. .
6

7. The value of ∑𝑛𝑗=1(3𝑗 − 1)2 is
𝑛
A. [6𝑛2 − 3𝑛 + 1].
2
𝑛
B. [6𝑛2 + 3𝑛 + 1].
2
𝑛
C. [6𝑛2 − 3𝑛 − 1].
2
𝑛
D. [6𝑛2 + 3𝑛 − 1].
2

(1+2𝑖)(2−𝑖)
8. The conjugate of is
(3−2𝑖)(2+3𝑖)

63 16
A. − 𝑖 169.
169
63 16
B. + 𝑖 169.
169
63 16
C. − 169 − 𝑖 169.

D. − 63 + 𝑖 16 .
169 169

9. Let
7 if 𝑥 ≤ 3,
𝑓(𝑥) = {𝑚𝑥 + 𝑛 if 3 < 𝑥 < 12,
18 if 𝑥 ≥ 12,
𝑚, 𝑛 real. The function 𝑓(𝑥) is continuous everywhere if (𝑚, 𝑛) is
11 10
A. ( , ).
10 3
11 10
B. ( , ).
9 3
11 1
C. ( , ).
9 3

D. (11 , − 10).
9 3
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𝑑𝑦
10. If 𝑥 3 + 𝑦 3 = cos 𝑥 + 𝑦 , then is equal to
𝑑𝑥

3𝑥 2 −sin 𝑥
A. .
1−3𝑦 2

3𝑥 2 −sin 𝑥
B. .
1+3𝑦 2

3𝑥 2 +sin 𝑥
C. .
1+3𝑦 2

D. 3𝑥 2 +sin 𝑥
.
1−3𝑦 2

11. Let 𝑓(𝑥) = −2𝑥 3 − 9𝑥 2 − 12𝑥 + 1 be a real function. The interval in which 𝑓(𝑥) is
increasing in 𝑥 is
A. (−∞, −2).
B. (−1, ∞).
C. (−2, −1).
D. (−2,0).
3 marks

𝜋⁄5
12. The value of ∫−𝜋⁄5 7 sin 𝑥 𝑑𝑥 is
5+cos 𝑥

A. 0.
B. 5.
C. log 𝑒 2.
D. 𝜋.

𝑥2
13. The area bounded by the curve 𝑦 = 2 , the lines 𝑦 = 0, 𝑥 = 1 and 𝑥 = 3 is

A. 13.
1
B. 4 3.
2
C. 18 3.

D. 4.

14. The value of the integral ∫ 𝑥 log 3𝑥 𝑑𝑥 is
𝑥2 𝑥2
A. log 3𝑥 − 2 + 𝐶 .
2

𝑥2 𝑥2
B. log 3𝑥 + 4 + 𝐶 .
2

𝑥2
C. 𝑥 2 log 3𝑥 − + 𝐶 .
4

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𝑥2 𝑥2
D. log 3𝑥 − 4 + 𝐶 .
2

15. If
1 cos 𝜃 sin 𝜃
𝑀 = [− cos 𝜃 −1 1 ],
sin 𝜃 1 1
then |𝑀| is equal to
A. 1.
B. −2.
C. −1.
D. 0.

16. If
𝑥𝑦 6 1 16 𝑎 1
[𝑧 + 4 𝑥+𝑦 2] = [−8 12 𝑏 ],
3 1 5 𝑐 1 𝑑
then 𝑥 + 𝑦 + 𝑧 is equal to
A. −16.
B. 0.
C. 16.
D. 24.

17. The approximate positive root of 𝑥 2 − 2 = 0, obtained after three iterations by Newton-
Raphson method starting from the initial value 𝑥0 = 1, is
A. 1.41426.
B. 1.92326.
C. 1.81526.
D. 1.41667.

18. The following table gives values of 𝑓(𝑥) for some value of 𝑥.
𝑥 50 60 70 80

𝑓(𝑥) 205 225 248 274

The value of 𝑓(54) by the Newton’s forward interpolation formula is
A. 202.23.
B. 227.78.

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C. 207.12.
D. 212.64.
3 marks

19. Let 𝐴⃑ = 𝑥 ⃑⃑ = − 𝑥
⃑⃑⃑⃑ + 𝑦⃑ + 𝑧⃑ , 𝐵 ⃑⃑⃑⃑ + 𝑦⃑ − 𝑧⃑ and 𝐶⃑ = 𝑦⃑ − 𝑧⃑ . The area of the triangle ∆𝐴𝐵𝐶 (in
sq.units) is
A. 3.
B. 2.
C. 1.
D. 0.

20. Let 𝐴⃑ = 2𝑥 ⃑⃑ = 3 𝑥
⃑⃑⃑⃑ + 4𝑦⃑ + 𝑧⃑ , 𝐵 ⃑⃑⃑⃑ − 𝑦⃑ + 2𝑧⃑ . The angle between 𝐴⃑ and 𝐵
⃑⃑ is
4
A. cos−1 7√3.
4
B. cos−1 7√6.
4
C. cos−1 √6.
4
D. cos−1 7√5.

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Statistics

21. The mean of 20 observations is 25. On checking it was found that two observations
are wrongly recorded as 15 and 17, while the correct observations were 25 and 27.
Then the correct mean is
A. 30.
B. 27.
C. 26.
D. 25.5.

22. The weighted arithmetic mean of 12, 16 and 20 with weights 𝑤1 , 𝑤2 and 𝑤3 ,
respectively, is known to be 18. If the weights are changed to 3𝑤1 , 3𝑤2 and 3𝑤3, then
the weighted arithmetic mean will be
A. 6.
B. 18.
C. 24.
D. 54.

23. The standard deviation of 20 observations is 2.1. Suppose each observation is first
multiplied by −0.5 and then 1.5 is added after multiplication. The standard deviation
of the new observations is
A. 0.45.
B. 0.525.
C. 1.05.
D. 2.55.

24. If the relationship between two variables 𝑥 and 𝑦 is 3𝑥 − 5𝑦 = 4 and mode of 𝑥 is
13, then the mode of 𝑦 is
A. 7.
B. 10.
C. 13.
D. −2.

25. If for a symmetrical distribution first quartile and median are 30 and 45, respectively,
then third quartile of the distribution is
A. 15.
B. 52.5.

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C. 60.
D. 75.

26. If the letters of the word “ENGINEERING” are written down at random, what is the
probability that all the 𝐸s occur together?
1
A. .
110
1
B. .
495
9
C. .
11
3
D. .
55

1
27. Probabilities of solving a specific problem independently by students 𝑆1 and 𝑆2 are 3
1
and 2, respectively. If both try to solve the problem independently, the probability that
exactly one of them solve the problem is
1
A. .
2
5
B. .
6
1
C. .
6
2
D. .
3

28. A bag contains 5 white, 6 red and 4 black balls. Three balls are drawn one by one with
replacement. What is the probability that at least one ball is red?
53
A. .
65
117
B. .
125
27
C. .
125
98
D. .
125

29. A box of chips contains 15 chips of which 4 are defective. If 3 of the chips are selected
at random and removed from the box in succession without replacement what is the
probability that all three chips are defective?
4
A. .
455
8
B. .
1125
25
C. .
546

D. 1
.
3375

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30. A construction company employs three sales engineers. Engineers 1, 2, and 3 estimates
the costs of 30%, 20%, and 50%, respectively, of all jobs bid by the company. Let 𝐸𝑖
be the event that a job is estimated by engineer 𝑖, 𝑖 = 1, 2, 3. The probabilities that
engineers make error in estimating the cost are 𝑃(error|𝐸1 ) = 0.01, 𝑃(error|𝐸2 ) =
0.03 and 𝑃(error|𝐸3 ) = 0.02. If a particular bid results in error in estimating job cost,
what is the probability that the error was made by engineer 1?
1
A. .
19
3
B. .
19
6
C. .
19
3
D. .
190

31. Suppose 𝑋 uniformly distributed on {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5}. Then 𝐸(|𝑋|)
is
15
A. .
11

B. 3.
30
C. .
11

D. 0.

32. The number of accidents per month at a busy intersection follows Poisson distribution
with average 𝜆 = 5. Suppose each accident costs local government Rs. 15,000 for clean-
up. On average, the accidents costs to the local government over a year time is
A. Rs. 36,000.
B. Rs. 75,000.
C. Rs. 180,000.
D. Rs. 900,000.

33. Suppose 𝑋 follows Poisson distribution with mean 8. Find the value of 𝑘 such that
𝑃(𝑋 = 𝑘) = 𝑃(𝑋 = 𝑘 + 1).
A. 6.
B. 7.
C. 8.
D. 9.

34. Suppose the research department of a steel manufacturer believes that one of the
company’s rolling machine is producing sheets of steel of varying thickness. The

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thickness 𝑋 is a Uniform random variable with values between 150 and 200 millimeters.
Any sheets less than 160 milimeters thick must be scrapped, since they are unacceptable
to buyers. The probability that steel sheets produced by this machine to be scrapped is
A. 0.10.
B. 0.15.
C. 0.20.
D. 0.25.

35. Let 𝑋 be a random variable with probability density function
𝑒 −𝑥 , for 𝑥 > 0
𝑓(𝑥) = {
0, elsewhere

The expected value of 𝑒 0.75𝑋 is
A. 4.
B. 0.75.
C. 𝑒 0.75 .
D. 𝑒 0.25 .

36. Suppose 𝑋 ∼ 𝑁(𝜇, 𝜎 2 ). Then which one of the following is true?

A. 𝑃(𝑋 ≥ 𝜇) = 0.25.
2
1 1 −𝑧
B. 𝑃(𝜇 − 𝜎 ≤ 𝑋 ≤ 𝜇 + 𝜎) = ∫
√2 𝜋 −1
𝑒 2 𝑑𝑧.
C. 𝑃(𝑋 ≤ 𝜇 + 𝜎) = 0.50.
D. 𝑃(𝑋 ≥ 𝜇 − 𝜎) < 0.50.

37. Let 𝑋 be a random variable with probability density function

𝑓(𝑥) = 𝜃𝑒 −𝜃𝑥 , 𝑥 > 0, 𝜃 > 0.

Suppose the median of the distribution is 𝑀. Then the mean of the distribution is
A. 𝑀.
B. 2𝑀.

C. − ln(0.5).
𝑀
𝑀
D. − ln(0.5).

38. Suppose 𝑋1 ∼ 𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙 (𝑛1 , 𝑝1 ) independent of 𝑋2 ∼ 𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙 (𝑛2 , 𝑝2 ). Then 𝑃(𝑋1 +
𝑋2 = 1) equals to
A. 𝑝1 (1 − 𝑝2 ) + 𝑝2 (1 − 𝑝1 ).

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B. 𝑝1 (1 − 𝑝2 )𝑛2−1 + 𝑝2 (1 − 𝑝1 )𝑛1−1.
C. [𝑛1 𝑝1 (1 − 𝑝2 ) + 𝑛2 𝑝2 (1 − 𝑝1 )](1 − 𝑝1 )𝑛1−1 (1 − 𝑝2 )𝑛2−1.
D. 𝑛1 𝑛2 𝑝1 𝑝2 (1 − 𝑝1 )𝑛1−1 (1 − 𝑝2 )𝑛2−1 .

39. Suppose 𝑋 ∼ 𝑁(5, 22 ) is independent of 𝑌 ∼ 𝑁(0, 1). Then correlation between 𝑌 and
𝑋𝑌 is equal to
A. 0.
B. 5.
C. 5
.
29

D. 5
.
√29

3 marks

40. If 2𝑢 = 𝑥 + 5 and 6𝑣 = 2𝑦 − 7 and the regression coefficient of 𝑥 on 𝑦 is 4, then
regression coefficient of 𝑢 on 𝑣 is
A. 6.
B. 4
.
3

C. 2.8.
D. 1.
3 marks

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Data Interpretation

The following frequency distribution represents the marks in a subject of 200 students in a class.
Answer questions 41-43 based on the information given in this table.

Marks Number of students
10-19 6
20-29 12
30-39 16
40-49 40
50-59 60
60-69 30
70-79 16
80-89 14
90-100 6

41. The minimum possible value of the mean of the actual marks summarized in the above
frequency distribution is
A. 10.0.
B. 54.5.
C. 49.8.
D. 50.0.

42. The percentage of students scoring less than 80% but not less than 60%
A. 23.
B. 46.
C. 30.
D. 16.

43. If the pass mark is 40%, the percentage of students who failed in the subject is
A. 74.
B. 37.
C. 34.
D. 17.

Answer questions 44-45 based on the following bar diagram on medal counts of eight countries
in Asian Games 2023.

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Medal counts of eight countries in Asian Games 2023
250

Number of Medals 200

150

100

50

0
China Japan South India Uzbekistan Kazakhstan Chinese Thailand
Korea Taipei
Country

Gold Silver Bronze

44. Which of the eight countries won the maximum number of bronze medals?
A. China.
B. Japan.
C. South Korea.
D. Kazakhstan.

45. Among Uzbekistan, Kazakhstan, Chinese Taipei and Thailand, which country won the
maximum number of medals?
A. Uzbekistan.
B. Kazakhstan.
C. Chinese Taipei.
D. Thailand.
Answer questions 46-47 based on the following box and whisker plot of lifetime (in hours) of
light bulbs of two manufacturers M1 and M2.

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Box and Whisker Plot of Lifetimes (in hrs) of Light Bulbs
M1 M2
350

300

250
Lifetime (in hrs)

200

150

100

50

0

46. The lifetime distribution of lifetime of light bulbs of M1 is
A. positively skewed.
B. negatively Skewed.
C. symmetric but not uniform.
D. uniform.

47. Which of the following statements is true?
A. The median lifetime of light bulbs of M1 is greater than that of M2.
B. The median lifetime of M2 is greater than 100 hrs.
C. The lifetime distribution of light bulbs of M2 is negatively skewed.
D. The mean lifetime of lightbulbs of M1 is less than their median lifetime.

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Answer questions 48-49 based on the following scatter plots.

Plot 1 Plot 2
40 10

30
9.5
20
y

y
9
10

0 8.5
0 5 10 15 0 5 10 15
x x

Plot 3 Plot 4
40.0 1000

30.0 800
600
20.0
y

y
400
10.0 200
0.0 0
0 5 10 15 0 10 20 30 40 50
x x

48. The correlation coefficient between 𝑥 and 𝑦 in Plot 3 is
A. 0.75.
B. −1.
C. 1.
D. nearly zero.

49. Which of the following statements is true?
A. The correlation coefficient between 𝑥 and 𝑦 is negative in Plot 1.
B. The correlation coefficient between 𝑥 and 𝑦 is positive in Plot 4.
C. The correlation coefficient between 𝑥 and 𝑦 is nearly 1 in Plot 2.
D. The correlation coefficient between 𝑥 and 𝑦 is 0 in Plot 2.

The following table gives the number of vehicles arriving in a fueling station in 200 five-
minute intervals. Answer questions 50-51 based on the data in the table.

No. of vehicles 0 1 2 3 4 5 6 7 8 9 10 Total

Frequency 2 6 8 17 38 65 30 20 8 4 2 200

50. The percentage of time intervals with at least 5 arriving vehicles is

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A. 64.5.
B. 70.
C. 72.7.
D. 81.8.

51. The median of the distribution of number of vehicles is
A. 4.
B. 4.5.
C. 5.
D. 5.5.

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English
52. Which of the following is a synonym for the word "Benevolent"?

A. Malevolent.
B. Kindhearted.
C. Cruel.
D. Sinister.

53. What is a synonym for the word "Eloquent"?
A. Fluent.
B. Inarticulate.
C. Reserved.
D. Mute.

54. What is the antonym of the word "Humble"?
A. Arrogant.
B. Modest.
C. Meek.
D. Timid.

55. What is the antonym of the word "Voracious"?
A. Gluttonous.
B. Ravenous.
C. Undesirous.
D. Insatiable.

56. Which phrase means "to take responsibility for one's actions"?
A. Take up.
B. Cut down.
C. Bring up.
D. Own up.

57. Which phrase means "to reveal a secret"?
A. Keep a stiff upper lip.

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B. Break the ice.
C. Let the cat out of the bag.
D. Hit the sack.

58. What is the term for the act of speaking or writing in an evasive or indirect manner?
A. Eloquence.
B. Verbose.
C. Concise.
D. Circumlocution.

59. What is the term for the quality of being easily broken or damaged?
A. Resilience.
B. Fragility.
C. Robustness.
D. Tenacity.

60. Select the correct sentence.
A. Her dedication to work and the quality of her performance is truly remarkable.
B. Her dedications to work and the quality of her performances are truly remarkable.
C. Her dedication to work and the quality of her performance are truly remarkable.
D. Her dedications to work and the quality of her performance is truly remarkable.

61. Select the correct sentence.
A. His acumen in deciphering complex algorithms and applying it in practical scenarios
have earned him numerous accolades.
B. His acumen in deciphering complex algorithms and applying them in practical
scenarios has earned him numerous accolades.
C. His acumen in deciphering complex algorithms and applying them in practical
scenarios have earned his numerous accolades.
D. His acumen in deciphering complex algorithms and applying it in practical scenarios
has earned his numerous accolades.

Read the passage below and answer Question No. 62.

In the heart of the city, nestled between towering skyscrapers, lies Central Park—a verdant oasis
in the midst of urban chaos. This sprawling 843-acre park is a sanctuary for both New Yorkers
and visitors alike. Lush meadows, tranquil lakes, and winding paths offer respite from the hustle
and bustle of city life.

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Central Park is not only a haven for nature enthusiasts, but also a cultural hub. Its iconic
landmarks, such as the Bethesda Terrace and Bow Bridge, have been featured in countless films
and novels. The park's extensive collection of sculptures and public art installations enriches the
visitor's experience, providing a glimpse into the vibrant art scene of the city.
Throughout the year, Central Park plays host to a variety of events, from open-air concerts to
Shakespearean performances. Families gather for picnics, children ride carousels, and joggers
traverse the network of trails. It's a place where diversity thrives, where people from all walks of
life come together to share in the simple joy of being surrounded by natural beauty.

I. What makes Central Park a unique place in the city?
i. Its proximity to skyscrapers.
ii. Its extensive collection of sculptures.
iii. Its location in the heart of the city.

II. Which of the following is NOT mentioned as an activity in Central Park?
i. Attending open-air concerts.
ii. Visiting art galleries.
iii. Jogging on trails.

III. What role does Central Park play in the cultural life of the city?
i. Its collection of sculptures and public art installations.
ii. It is a popular spot for picnics and gatherings.
iii. It is primarily a sanctuary for nature enthusiasts.
62. The correct answers to I, II and III are
A. ii, iii, iii, respectively.
B. i, ii, iii, respectively.
C. iii, ii, i, respectively.
D. iii, ii, ii, respectively.

3 marks

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Logical Reasoning
63. How many days will be there between January 15, 2024 and March 05, 2024 (both the days
are inclusive)?
A. 48.
B. 49.
C. 50.
D. 51.

64. A cube of side 𝑛 is painted red on all its sides. It is then cut into 𝑛3 identical cubes. How
many of the smaller cubes will have three sides painted?

A. 4.
B. 8.
C. 6.
D. 6𝑛2 .

65. Given below are two statements numbered 1 and 2:
Statement 1: The depositors in the bank lost their money.

Statement 2: The bank was declared bankrupt.
A. Statement 1 is the cause and Statement 2 is the effect.
B. Statement 2 is the cause and Statement 1 is the effect.
C. Either of the two can cause the other.
D. Neither of the two can cause the other.

66. Hardik introduces Axar as the son of the only brother of his father’s wife. How is Axar
related to Hardik?
A. Cousin.
B. Son.
C. Uncle.
D. Son-in-law.

67. A watch gains 5 seconds in 3 minutes and was set right at 8 AM. What time will it show
at 10 PM on the same day?
A. 10 hours 23 minutes 20 seconds.
B. 10 hours 22 minutes 20 seconds.
C. 10 hours 25 minutes 20 seconds.

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D. 10 hours 26 minutes 20 seconds.

68. Choose the set of three statements where the third statement can be logically derived from
the preceding two:
1. All Turns are Good.
2. All Cans are Good.
3. Some Turns are Cans.
4. No Vans are Cans.
5. Some Vans are not Good.
6. Some Cans are Good.
A. 2, 1, 3.
B. 1, 3, 6.
C. 3, 4, 2.
D. 4, 2, 5.

69. Nine persons A, B, C, D, E, F, G, H, I are watching a movie in a theatre sitting in a row
which has nine seats. B is at one end of the row. H is seated adjacent to both F and G. C
is to the immediate right of D and at the third place to the right of E. A is immediate left
of F. F is at the third place to the left of B.
What is the position of I from the left?
A. 1.
B. 3.
C. 8.
D. 2.

70. In the parlance of reliability theory, two components of a system are said to be in series if
the system fails whenever any of the components fails, and are said to be in parallel if both
of them have to fail for the system to fail. Going by these definitions, identify the correct
statement.

A. A motorbike is a system with two tires in parallel.
B. The renal system of a human being is a system with two kidneys in series.
Entry into an email account is a system in which the username and the password
C.
are two components in series.
A committee that is supposed to keep its report confidential till submission is a
D.
system with all its members in parallel.

*********************

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