ACET Aug 2025 Question Paper

Institute of Actuaries of India
ACET August 2025

Mathematics
Q1. If 𝑓(𝑥) = 𝑥 3 − 𝑥 2 + 2𝑥, 𝑡ℎ𝑒𝑛𝑡ℎ𝑒𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑓(𝑥) = 4 has:

A. No real roots
B. Exactly 1 real root
C. Exactly 2 real roots
D. Exactly 3 real roots

[1]
Q2. Evaluate the limit:

𝑥 𝑥
𝑡𝑎𝑛 ( ) − 𝑡𝑎𝑛 ( )
𝑙𝑖𝑚𝑥→0 5 10
𝑥
𝑠𝑖𝑛 ( )
20

A. −2
B. 2
C. 0
D. Undefined

[1]
Q3. If 𝑝⃗ = 3𝑖^ − 2𝑗^ ∧ 𝑞⃗ = 4𝑖^ + 5𝑗^, then the scalar product 𝑝⃗ ⋅ 𝑞⃗ is:

A. 2
B. 1
C. 0
D. −2

[1]

Q4. Which of the following matrix operations is undefined, given that matrix A has dimensions 3 × 2,
matrix B has dimensions 2 × 4, and matrix C has dimensions 3 × 4?

A. A · B + C
B. AT · C
C. C · A
D. BT · AT

[3]

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Q5. Use the Newton-Raphson method to estimate the root of the equation

𝑓(𝑥) = 𝑥 3 − 2𝑥 − 5 = 0

Start with an initial guess of x0 = 2. Calculate x1 and x2, rounding your answers to four decimal places.

Option x1 x2
A 2.0009 2.0999
B 2.0999 2.0946
C 2.0009 2.0999
D 2.1000 2.0946

[3]

Q6. Using the Newton-Raphson method, find the first approximation x1 for the root of the equation

𝑓(𝑥) = 𝑙𝑛 (√𝑥 2 + 𝑠𝑖𝑛(𝑥)) − 2 = 0

starting with an initial guess of 𝑥0 = 1. Round your answer to five decimal places.

A. 3.45679
B. 4.45679
C. −3.45679
D. UNDEFINED

[3]

Q7. What does the modulus of a complex number z = a + bi (denoted as |z|) geometrically represent in the
complex plane?

A. The angle the complex number forms with the positive real axis.
B. The distance of the complex number from the origin (0, 0).
C. The real part of the complex number.
D. The complex conjugate of the complex number.

[1]

Page 2 of 17

Q8. The value of the integral
5
∫ |𝑥 − 1| 𝑑𝑥
−2

A. 12.5
B. 10
C. 15
D. 7.5
[1]
Q9. If M is a square matrix such that 𝑀2 = 𝐼 (where 𝐼 is the identity matrix of the same dimensions),
then which of the following statements must be true about M?
A. M may have determinant −1 or 1
B. M is a zero matrix
C. M is a diagonal matrix
D. M has a determinant 0

[1]
Q10. If 𝜔 is a non-real cube root of unity, then what is the value of (1 − 𝜔 − 𝜔2 )5 ?
A. 32

B. 𝑖
C. 0
D. -32

[1]
−√3
Q11. The principal value of 𝑎𝑟𝑐𝑠𝑖𝑛 ( ) is:
2

𝜋
A. 0 +
3
𝜋
B. 0 −
3
𝜋
C. 0 −
6
2𝜋
D. 0 +
6

[1]

1 7
Q12. Find the coefficient of 𝑥 3 in the binomial expansion of (2𝑥 − )
𝑥

A. 572
B. 670
C. 762
D. 672
[2]
𝑦 𝜕𝑧 𝜕𝑧
Q13. If 𝑧 = 𝑡𝑎𝑛 ( ) evaluate 𝑥 + 𝑦 𝜕𝑦
𝑥 𝜕𝑥

A. 0
B. 1
Page 3 of 17

C. -1
D. Can’t be determined
[3]

Q14. The interval in which the 𝑓(𝑥) = 𝑥 4 − 4𝑥 3 + 5 is strictly increasing in x is:

A. (−∞, 0)
B. (0, 3)
C. (3, ∞)
D. (−∞, 3)

[1]

Q15. The inverse of
1 0
( )
2 −1
−1 0
A. ( )
2 1
1 0
B. ( )
−2 −1
1 0
C. ( )
2 −1
0 1
D. ( )
−1 2
[1]

Q16. Which of the following is True regarding the Cross Vector multiplication?

A. The resultant vector has no direction

B. Vector multiplication is associative

C. Vector multiplication is commutative
D. The magnitude of the Vector multiplication represents area of a parallelogram

[1]

Q17. What is the value of 𝑥 2 + 𝑦 2 if 𝑥 + 𝑦 = 5and 𝑥𝑦 = 6?

A. 3
B. 10
C. 13
D. 0

[1]

Q18. The number of terms in the expansion of (1 + 𝑥 10 )(1 + 𝑥)8 is:
A. 19
B. 23

C. 10
D. 18

[1]

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𝑑𝑦 𝑒𝑥
Q19. Find the derivative of the function 𝑦 =
𝑑𝑥 𝑐𝑜𝑠𝑥

𝑒 𝑥 (𝑐𝑜𝑠𝑥+𝑠𝑖𝑛𝑥)
A.
𝑐𝑜𝑠 2 𝑥
𝑒 𝑥 (𝑐𝑜𝑠𝑥−𝑠𝑖𝑛𝑥)
B.
𝑐𝑜𝑠 2 𝑥
𝑒 𝑥 𝑠𝑖𝑛𝑥
C.
𝑐𝑜𝑠 2 𝑥
𝑒𝑥
D.
𝑐𝑜𝑠𝑥+𝑠𝑖𝑛𝑥

[2]
Q20. Evaluate the definite integral
𝜋
2 𝑠𝑖𝑛𝑥
∫ 𝑑𝑥
0 𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥

A. 𝜋/2

B. π/3

C. π/4

D. π/6
[1]

Statistics

Q21. Let A, B, and C be three events in a sample space such that:

• 𝑃(𝐴) = 0.5

• 𝑃(𝐵) = 0.6

• 𝑃(𝐶) = 0.4
• 𝑃(𝐴 ∩ 𝐵) = 0.2
• 𝑃(𝐴 ∩ 𝐶) = 0.15
• 𝑃(𝐵 ∩ 𝐶) = 0.1
• 𝑃(𝐴 ∩ 𝐵 ∩ 𝐶) = 0.05
Find 𝑃((𝐴 ∪ 𝐵) ∩ 𝐶 𝑐 )
A. 5/6
B. 3/4

C. 11/12
D. 7/10

[2]

Q22. A factory has two machines, Machine A and Machine B, producing light bulbs. Machine A produces

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60% of the bulbs, and Machine B produces 40%. The defect rate for bulbs from Machine A is 3%, and for
Machine B it is 5%. A bulb is chosen at random from the total production, and it is found to be defective.
What is the probability that this defective bulb was produced by Machine B?
A. 0.33
B. 0.4

C. 0.45
D. 0.53

[2]

Q23. A gamer is testing a new online game where there are two types of treasure chests that appear at random:
rare and common. The game developers claim that one-quarter of the treasure chests are rare, while the
remaining three-quarters are common. The gamer decides to test this claim by opening chests one by one and
tracking their types.
He assumes that each chest has a 1/4 chance of being rare and a 3/4 chance of being common. He can clearly
distinguish between the two types.
If the assumption is correct, what is the probability that the gamer will open at least 2 rare chests before he
opens 3 common chests?

A. 0.2222

B. 0.2513
C. 0.2032

D. 0.2617

[3]

Q24. In a survey of 100 people, 60 people like tea, 50 like coffee, and 30 like both tea and coffee. If one
person is selected at random, then the probability that the person likes neither tea nor coffee is:
A. 1/10
B. 1/5

C. 2/25

D. 4/25

[1]

Q25. For an exponential distribution, which of the following is the correct relationship among the mean,
median, and mode?

A. Mean > Median > Mode
B. Mode > Median > Mean

C. Mean = Median = Mode
D. Mean < Median < Mode

[1]

Q26. Let 𝑋, 𝑌, and 𝑍 be random variables such that 𝑉𝑎𝑟(𝑋) = 4, 𝑉𝑎𝑟(𝑌) = 9, 𝑉𝑎𝑟(𝑍) = 1,∧ 𝐶𝑜𝑣(𝑋, 𝑌) = 3.
Assume that 𝑋 and 𝑍 are uncorrelated, and 𝑌 and 𝑍 are uncorrelated. Find 𝑉𝑎𝑟(2𝑋 + 3𝑌 + 𝑍).
Page 6 of 17

A. 58
B. 55

C. 52
D. 134

[2]

Q27. A multiple-choice quiz has 8 questions, each with 4 options and only one correct answer. A student
guesses all questions at random. What is the probability that the student gets exactly 3 answers correct?

A. 0.278
B. 0.293
C. 0.312
D. 0.207

[1]

Q28. The random variable X has probability density function:

𝑘(3𝑥 − 𝑥 2 ),∧ 0 ≤ 𝑥 ≤ 3
𝑓(𝑥) = {
0,∧ otherwise

Evaluate the constant k and hence calculate Var(X). What is the value of Var(X)?
A. 27/80

B. 9/20
C. 81/160

D. 21/50

[1]

Q29. The time taken by employees to complete a routine task in a company is normally distributed with
a mean of 50 minutes and a standard deviation of 10 minutes. What is the probability that a randomly
selected employee completes the task in between 40 and 65 minutes? Use the following values from the
standard normal distribution:
Φ (1.5) = 0.9332, Φ(1) = 0.8413
A. 0.6826

B. 0.7745

C. 0.8186
D. 0.9332
[2]

Q30. Let 𝑋 ∼ 𝑈𝑛𝑖𝑓𝑜𝑟𝑚[𝑎, 𝑏], where 𝑎 < 𝑏, and let 𝑌 = 𝑐𝑋, where 𝑐 is a non-zero constant. A student is
attempting to derive 𝑉𝑎𝑟(𝑌) using properties of variance for transformed variables. Which of the following
correctly represents 𝑉𝑎𝑟(𝑌) in terms of a, b, and c

Page 7 of 17

𝑐 2 (𝑏−𝑎)2
A. 12
𝑐(𝑏−𝑎)2
B.
12
(𝑏−𝑎+𝑐)2
C.
12
𝑐 2 (𝑏2 −𝑎2 )
D.
12

[1]

Q31. In a simple linear regression model 𝑌 = 𝑎 + 𝑏𝑋 + 𝜀, the least squares estimator 𝑏 is:
A. The average of 𝑋
∑(𝑋𝑖 −𝑋)(𝑌𝑖 −𝑌)
B. 2
∑(𝑋𝑖 −𝑋)

∑ 𝑋𝑖
C. ∑ 𝑌𝑖

D. Always equal to 1 if correlation is perfect

[1]

Q32. Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be i.i.d. with mean µ and variance 𝜎 2 . Consider the sample variance

1 2
𝑆 2 = ∑𝑛𝑖=1(𝑋𝑖 − 𝑋)
𝑛

Which of the following is true about E[S2]?
𝑛−1
A. 𝜎2
𝑛

B. 𝜎 2
𝑛+1 2
C. 𝜎
𝑛
1 2
D. 𝜎
𝑛

[1]
Q33. The marginal distributions of X and Y are given in the following table:
y \x 1 3 Total
1 q 0.4 − q 0.4
2 0.2 0.4 0.6
The value of q for which the correlation between X and Y is 0.25 is:
A. 0.18
B. 0.22
C. 0.5
D. None of the above

[2]
Page 8 of 17

Q34. A joint pdf is defined by
1
𝑓(𝑥, 𝑦) = 𝑘𝑥𝑦 2 + , 0 ≤ 𝑥 ≤ 2,0 ≤ 𝑦 ≤ 1.
5

Which of the following statements is correct?

A. 𝑋 and 𝑌 are independent.

B. 𝐶𝑜𝑣(𝑋, 𝑌) = 0.

C. Marginal of 𝑋 is uniform on [0,2].
D. None of the above.

[2]
Q35. In a simple linear regression with intercept, which of the following is always true?
A. The regression line passes through the origin.
B. The residuals are uncorrelated with the fitted values.
C. The sum of residuals is aways negative.
D. The slope is always less than 1.

[1]

Q36. You are estimating a population mean using sample mean 𝑋 , where 𝑋1 , … , 𝑋𝑛 ∼ 𝑁(𝜇, 𝜎 2 ). What
happens to the standard error if sample size is multiplied by 4?
A. It doubles
B. It is halved
C. It becomes one-fourth
D. It remains the same

[1]

Q37. A manager needs to select a 4-member team from a group of 6 accountants and 4 analysts. At least
one analyst must be included in the team. How many such teams can be formed?
A. 185
B. 195
C. 205
D. 215

[1]
2
Q38. Given joint pdf 𝑓(𝑥, 𝑦) = < 𝑦 < 𝑥 < 1 find marginal pdf of 𝑋.
0

A. 𝑓𝑋 (𝑥) = 2𝑥, 𝑥 ∈ [0,1]
B. 𝑓𝑋 (𝑥) = 2(1 − 𝑥)
C. 𝑓𝑋 (𝑥) = 2(1 + 𝑥)

D. 𝑓𝑋 (𝑥) = 𝑥
Page 9 of 17

[2]
Q39. Let 𝑓(𝑥) = 6𝑥(1 − 𝑥), 0 < 𝑥 < 1. Find the median of the distribution.
A. 0.5
1
B. 1 −
√2
1
C.
√2
D. Cannot be determined

[2]

Q40. In a city’s central station, trains arrive from two independent railway lines. Line 1 follows a Poisson
process with a rate of 4 trains per hour, and Line 2 follows a Poisson process with a rate of 6 trains per
hour. All arrivals are independent. What is the probability that exactly 7 trains arrive at the station in a
1-hour period?

(10)7 𝑒 −10
A.
7!
(6)7 𝑒 −10
B.
7!
(4)7 𝑒 −6
C.
7!
(10)7 𝑒 −6
D.
7!
[1]

Data Interpretation

A logistics company tracks the number of packages delivered by its five regional hubs over five years.

The data below shows the number (in thousands) of packages delivered by each hub.
Page 10 of 17

Year North South East West Central
2020 120 150 130 100 90
2021 135 165 145 110 100
2022 150 180 155 130 120
2023 140 200 160 140 130
2024 160 220 170 150 150

Q41. In which year did the Central hub experience the highest percentage increase in deliveries over the previous year?

A. 2021
B. 2022
C. 2023
D. 2024
[1]

Q42. What is the average annual growth rate (in %) of total packages delivered across all hubs from 2020 to 2024?

A. 8.7%
B. 9.56%
C. 10.1%

D. 10.5%
[2]

Q43. If each delivery generates a profit of ₹15,000 per 1,000 packages, what was the total profit from the South and
East hubs combined in 2022?
A. ₹5.03 million

B. ₹4.95 million
C. ₹5.05 million

D. ₹4.65 million
[1]

Q44. Which region showed the most consistent year-over-year growth in terms of the lowest standard deviation of
deliveries?
A. North
B. South
C. Central

D. West
[2]

Q45. Assume the West hub can only operate at a maximum of 150,000 packages per year. In which year(s) did the
West hub reach or exceed 90% of its operational capacity?
A. 2022, 2023, 2024
B. 2023 and 2024
C. 2022 only
D. 2024 only

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[1]

An investor’s portfolio is divided across four asset classes as follows: Equity (40%), Debt (35%), Real Estate (15%),
and Others (10%).

Q46. If the total investment is ₹50 lakhs, how much is invested in Real Estate?

A. ₹6,50,000
B. ₹7,00,000
C. ₹7,50,000
D. ₹8,00,000
[1]

Q47. What is the difference in investment amount between Equity and Debt?
A. ₹1,50,000

B. ₹2,50,000
C. ₹3,00,000
D. ₹3,50,000
[1]

Q48. Equity is expected to grow by 10% next year. What will be the new amount invested in Equity?
A. ₹21,00,000
B. ₹21,50,000
C. ₹22,00,000
D. ₹22,50,000
[1]
Page 12 of 17

Q49. If the investor wants Debt and Others combined to represent no more than 40% of the portfolio, by what
percentage must the Debt investment be reduced (assuming Others remain constant)?

A. 12.5%
B. 13.5%
C. 14.29%
D. 15%
[2]

Q50. What percentage of the portfolio is not invested in Equity and Debt combined?
A. 15%
B. 20%

C. 30%

D. 25%
[1]

Q51. Assuming all else remained equal and only Equity increased by 10% next year, calculate the percentage of
portfolio that Real estate comprises of

A. 14.42%
B. 15%
C. 20%

D. 25%

[2]

English

Q52. Choose the word that best completes the sentence:
The manager was praised for her ______ handling of the crisis, preventing any serious damage.
A. Careless

B. Meticulous
C. Reckless
D. Negligent
[1]

Q53. Choose the word closest in meaning to “vigilant.”

A. Careless
B. Attentive
C. Indifferent
D. Timid
[1]

Page 13 of 17

Q54. Choose the word that is the opposite of “transient.”

A. Temporary

B. Permanent

C. Fleeting
D. Momentary
[1]

Q55. Find the part of the sentence with a grammatical error:
Neither the employees nor the manager are willing to accept the new schedule.
A. Neither the employees
B. Nor the manager

C. Are willing

D. To accept the new schedule
[2]

Q56. He succeeded ______ convincing the board to approve his project.

A. At
B. In

C. On
D. For
[1]

Q57. What does the phrase “break the ice” mean?
A. To shatter a conversation in an awkward situation
B. To shatter frozen water

C. To create a conflict
D. To freeze up emotionally
[2]

Q58. Choose the correct order of the following parts to form a meaningful sentence:

(1) the committee approved the proposal
(2) which had been under review
(3) after a series of heated discussions

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(4) for several weeks
A. 1,2,3,4

B. 1,3,4,2
C. 4,3,1,2

D. 3,1,2,4

[2]

Q59. Choose the correct word:
A person who writes poems is called a ______.
A. Novelist
B. Poet

C. Playwright

D. Narrator
[1]

Q60.The CEO’s decision was considered __________ since it brought a fresh perspective to the company’s
culture.

A. Conventional
B. Innovative

C. Obsolete
D. Insignificant
[1]

Q61. Choose the sentence that best expresses the meaning of the original sentence:
“Despite the heavy rain, the event continued as planned.”
A. The event was canceled because of the heavy rain.
B. The event was postponed due to heavy rain.

C. The event took place despite the heavy rain.
D. The event did not happen because of the rain.
[2]

Q62. Change the sentence into indirect speech:
She said, “I will finish the task tomorrow.”
A. She said she will finish the task tomorrow.
B. She said she would finish the task tomorrow.

C. She said she would finish the task the next day.
D. She said she will finish the task the next day.

[1]

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Logical Reasoning

Q63.Rahul introduces Priya saying, "She is the daughter of my grandfather's only child." How is Priya related to
Rahul?
A. Cousin
B. Sister
C. Aunt
D. Niece
[2]

Q64. If September 15, 2029, is a Saturday, what day of the week will December 31, 2029, be?
A. Monday
B. Tuesday

C. Thursday

D. Sunday
[2]

Q65. The pattern of figures is as follows: black triangle, white square, black pentagon, white hexagon, black
heptagon, ... What color and shape come next?

A. White octagon
B. Black hexagon

C. White nonagon
D. Black octagon
[1]

Q66. Five colleagues – R, S, T, U, V – are sitting in a straight line, all facing north.

S does not sit at any end.
U sits second to the left of V.
R sits to the immediate right of T.
V is not at an end.
Who is in the middle?

A. R

B. V
C. T
D. U
[1]

Q67. Statements:
1. Some engineers are musicians.
2. All musicians are dancers.
3. No dancer is a scientist.

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Choose the correct option:

A. Some engineers are definitely not scientists.
B. All musicians are not engineers.

C. Some dancers are engineers.
D. All of the above
[1]

Q68. If ACTION is coded as BDUJPO and REACT as SFBDU, how is TEACH coded?
A. UFBDI

B. SDBDI
C. UFDBI

D. SDBDJ
[1]

Q69. Find the next number in the sequence: 10, 18, 33, 55, 84, . . .

A. 187

B. 205
C. 195
D. 120
[1]

Q70. Six people—Nina, Omar, Priya, Raj, Surya, Tia—sit around a circular table, facing the center.

Priya sits to the right of Tia.
Raj is not adjacent to Priya or Surya.
Nina is between Omar and Surya.
Who sits to the immediate right of Surya?

A. Priya

B. Omar
C. Tia
D. Nina
[1]

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

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