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ACET (Institute of Actuaries of India Exams) practice questions

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Question 1Mathematics

A car travels 240 km in 3 hours. What is its average speed in km/h?

  1. 60
  2. 80 Correct answer.
  3. 72
  4. 90

Explanation

Average speed equals total distance divided by total time, so 240 km ÷ 3 h = 80 km/h. The other values arise from dividing incorrectly or misreading the time.

Question 2Mathematics

Simplify the expression 3(2x − 4) + 5x.

  1. 6x + 5x − 4
  2. 11x − 12 Correct answer.
  3. 6x − 12
  4. 11x + 4

Explanation

Distribute the 3 to get 6x − 12, then combine like terms 6x + 5x = 11x. The result is 11x − 12.

Question 3Mathematics

What is the value of 2³ × 2² expressed as a single power of 2?

  1. 2¹
  2. 2⁶
  3. 4⁵
  4. 2⁵ Correct answer.

Explanation

When multiplying powers with the same base, add the exponents: 3 + 2 = 5, giving 2⁵. The base stays 2 throughout.

Question 4Mathematics

A triangle has interior angles of 48° and 67°. What is the third angle?

  1. 55°
  2. 65° Correct answer.
  3. 75°
  4. 115°

Explanation

The interior angles of a triangle sum to 180°. Subtracting the known angles gives 180° − 48° − 67° = 65°.

Question 5Mathematics

Find the median of the data set 4, 9, 2, 7, 11, 6.

  1. 7
  2. 6
  3. 7.5
  4. 6.5 Correct answer.

Explanation

Ordering the values gives 2, 4, 6, 7, 9, 11. With an even count, the median is the mean of the two middle values, (6 + 7) ÷ 2 = 6.5.

Question 6Mathematics

Solve for x: 5x − 7 = 3x + 9.

  1. x = 1
  2. x = 2
  3. x = 8 Correct answer.
  4. x = 16

Explanation

Subtract 3x from both sides to get 2x − 7 = 9, then add 7 to get 2x = 16, so x = 8. Checking: 5(8) − 7 = 33 and 3(8) + 9 = 33.

Question 7Mathematics

What is 15% of 240?

  1. 32
  2. 24
  3. 36 Correct answer.
  4. 45

Explanation

15% of 240 is 0.15 × 240 = 36. Alternatively, 10% is 24 and 5% is 12, and 24 + 12 = 36.

Question 8Mathematics

Which of the following numbers is irrational?

  1. −3
  2. √16
  3. √2 Correct answer.
  4. 0.75

Explanation

An irrational number cannot be written as a fraction of integers. √2 has a non-terminating, non-repeating decimal expansion, while √16 = 4, 0.75 = 3/4, and −3 = −3/1 are all rational.

Question 9Mathematics

A rectangle has length 12 cm and width 5 cm. What is the length of its diagonal?

  1. 17 cm
  2. 60 cm
  3. 13 cm Correct answer.
  4. 7 cm

Explanation

The diagonal forms a right triangle with the sides, so by Pythagoras the diagonal is √(12² + 5²) = √(144 + 25) = √169 = 13 cm.

Question 10Mathematics

Factorise x² − 5x + 6.

  1. (x + 2)(x + 3)
  2. (x − 2)(x + 3)
  3. (x − 2)(x − 3) Correct answer.
  4. (x − 1)(x − 6)

Explanation

We need two numbers that multiply to +6 and add to −5, namely −2 and −3. So the factorisation is (x − 2)(x − 3).

Question 11Mathematics

In a right triangle, the angle of elevation to the top of a tower 40 m away is 30°. What is the height of the tower?

  1. 80 m
  2. 20 m
  3. 40√3 m
  4. 40√3/3 m Correct answer.

Explanation

The height is opposite the 30° angle and 40 m is the adjacent side, so tan 30° = h/40. Since tan 30° = 1/√3, h = 40/√3 = 40√3/3 m, about 23.1 m.

Question 12Mathematics

A bag contains 5 red, 3 blue, and 2 green balls. One ball is drawn at random. What is the probability that it is blue?

  1. 3/10 Correct answer.
  2. 3/7
  3. 1/3
  4. 1/2

Explanation

There are 5 + 3 + 2 = 10 balls in total, of which 3 are blue. The probability is therefore 3/10.

Question 13Mathematics

What is the sum of the first 20 positive even integers 2 + 4 + 6 + … + 40?

  1. 440
  2. 400
  3. 210
  4. 420 Correct answer.

Explanation

The sum of the first n even numbers is n(n + 1). With n = 20, the sum is 20 × 21 = 420.

Question 14Mathematics

The line 2x + 3y = 12 crosses the y-axis at which point?

  1. (4, 0)
  2. (0, 4) Correct answer.
  3. (0, 6)
  4. (6, 0)

Explanation

The y-intercept occurs where x = 0. Substituting gives 3y = 12, so y = 4, giving the point (0, 4).

Question 15Mathematics

If f(x) = 2x² − 3x + 1, what is f(−2)?

  1. 3
  2. 11
  3. 15 Correct answer.
  4. −1

Explanation

Substitute x = −2: 2(−2)² − 3(−2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15.

Question 16Mathematics

A circle has circumference 12π cm. What is its area?

  1. 36π cm² Correct answer.
  2. 6π cm²
  3. 144π cm²
  4. 12π cm²

Explanation

From C = 2πr, we get 2πr = 12π, so r = 6. The area is πr² = π(6²) = 36π cm².

Question 17Mathematics

Solve the inequality 3 − 2x < 9.

  1. x < −3
  2. x > 3
  3. x > −3 Correct answer.
  4. x < 3

Explanation

Subtract 3 to get −2x < 6, then divide by −2. Dividing by a negative reverses the inequality, giving x > −3.

Question 18Mathematics

The points A(1, 2) and B(7, 10) are given. What is the midpoint of AB?

  1. (3, 4)
  2. (4, 6) Correct answer.
  3. (6, 8)
  4. (8, 12)

Explanation

The midpoint formula averages the coordinates: ((1 + 7)/2, (2 + 10)/2) = (4, 6).

Question 19Mathematics

A geometric sequence has first term 3 and common ratio 2. What is the 6th term?

  1. 192
  2. 48
  3. 96 Correct answer.
  4. 64

Explanation

The nth term of a geometric sequence is ar^(n−1). Here a = 3, r = 2, n = 6, so the term is 3 × 2⁵ = 3 × 32 = 96.

Question 20Mathematics

Simplify (x³y²)/(x y⁵) for x ≠ 0 and y ≠ 0.

  1. x²y³
  2. 1/(x²y³)
  3. x²/y³ Correct answer.
  4. x⁴/y³

Explanation

Subtract exponents for matching bases: x^(3−1) = x² and y^(2−5) = y⁻³. This gives x²/y³.

Question 21Mathematics

Two similar triangles have corresponding sides in the ratio 2:5. What is the ratio of their areas?

  1. 2:5
  2. 4:25 Correct answer.
  3. 4:10
  4. 8:125

Explanation

For similar figures, areas scale as the square of the linear ratio. Squaring 2:5 gives 4:25.

Question 22Mathematics

The quadratic equation x² + kx + 9 = 0 has equal roots. What are the possible values of k?

  1. k = 3 or k = −3
  2. k = 0 only
  3. k = 9 or k = −9
  4. k = 6 or k = −6 Correct answer.

Explanation

Equal roots require the discriminant b² − 4ac to be zero. Here k² − 4(1)(9) = 0, so k² = 36 and k = ±6.

Question 23Mathematics

A cone has base radius 3 cm and height 4 cm. What is its total surface area, leaving the answer in terms of π?

  1. 21π cm²
  2. 24π cm² Correct answer.
  3. 15π cm²
  4. 12π cm²

Explanation

The slant height is √(3² + 4²) = 5. The curved surface area is πrl = π(3)(5) = 15π and the base area is πr² = 9π, so the total is 24π cm².

Question 24Mathematics

If log₂(x) + log₂(x − 2) = 3, what is the value of x?

  1. x = 4 Correct answer.
  2. x = 2
  3. x = 4 or x = −2
  4. x = 8

Explanation

Combining the logs gives log₂(x(x − 2)) = 3, so x² − 2x = 8, giving x² − 2x − 8 = 0, which factors as (x − 4)(x + 2) = 0. Since logarithms require x > 0 and x − 2 > 0, only x = 4 is valid.

Question 25Mathematics

A sum of money is invested at 10% per annum compound interest. After how many complete years does the amount first exceed 1.5 times the original sum?

  1. 3 years
  2. 4 years
  3. 5 years Correct answer.
  4. 6 years

Explanation

We need 1.1ⁿ > 1.5. Since 1.1⁴ ≈ 1.4641 and 1.1⁵ ≈ 1.6105, the amount first exceeds 1.5 times the original after 5 complete years.

Question 26Mathematics

Evaluate 3⁴ ÷ 3² × 3.

  1. 3³ Correct answer.
  2. 3⁶
  3. 3⁴
  4. 3⁵

Explanation

Using index laws, 3⁴ ÷ 3² = 3², and multiplying by 3 gives 3² × 3¹ = 3³. The exponents add when bases are multiplied and subtract when divided.

Question 27Mathematics

A shirt priced at $80 is reduced by 25%. What is the sale price?

  1. $55
  2. $60 Correct answer.
  3. $20
  4. $65

Explanation

A 25% reduction removes 0.25 × 80 = $20 from the original price, leaving 80 − 20 = $60. Equivalently the sale price is 75% of $80.

Question 28Mathematics

What is the mean of the data set 4, 7, 9, 12, 13?

  1. 10
  2. 9 Correct answer.
  3. 11
  4. 8

Explanation

The mean is the total of the values divided by how many there are: (4 + 7 + 9 + 12 + 13) ÷ 5 = 45 ÷ 5 = 9.

Question 29Mathematics

Which of the following is the equation of a straight line with gradient −2 passing through the point (0, 5)?

  1. y = −2x + 5 Correct answer.
  2. y = 2x + 5
  3. y = −2x − 5
  4. y = 5x − 2

Explanation

The point (0, 5) is the y-intercept, so c = 5, and the gradient is m = −2. Substituting into y = mx + c gives y = −2x + 5.

Question 30Mathematics

Simplify the expression 4(2x − 3) − 2(x − 5).

  1. 6x − 22
  2. 6x + 2
  3. 6x − 2 Correct answer.
  4. 10x − 22

Explanation

Expanding gives 8x − 12 − 2x + 10, because subtracting −10 adds 10. Collecting like terms gives 6x − 2.

Question 31Mathematics

A fair six-sided die is rolled once. What is the probability of obtaining a prime number?

  1. 1/2 Correct answer.
  2. 1/6
  3. 1/3
  4. 2/3

Explanation

The prime outcomes on a die are 2, 3 and 5, so there are 3 favourable outcomes out of 6. The probability is 3/6 = 1/2.

Question 32Mathematics

What is the value of 5! ÷ 3!?

  1. 60
  2. 120
  3. 2
  4. 20 Correct answer.

Explanation

Since 5! = 120 and 3! = 6, the quotient is 120 ÷ 6 = 20. Alternatively, most factors cancel: 5 × 4 = 20.

Question 33Mathematics

The quadratic expression 2x² + 7x + 3 can be written in the form (2x + a)(x + b). What are the values of a and b?

  1. a = 3, b = 7
  2. a = 7, b = 3
  3. a = 1, b = 3 Correct answer.
  4. a = 3, b = 1

Explanation

Expanding (2x + a)(x + b) gives 2x² + (2b + a)x + ab. Matching the constant gives ab = 3 and matching the x-coefficient gives 2b + a = 7, which is satisfied by a = 1 and b = 3.

Question 34Mathematics

The sum of the first n terms of an arithmetic sequence is given by Sₙ = n² + 2n. What is the 10th term of the sequence?

  1. 120
  2. 39
  3. 19
  4. 21 Correct answer.

Explanation

The nth term equals Sₙ − Sₙ₋₁, so the 10th term is S₁₀ − S₉ = (100 + 20) − (81 + 18) = 120 − 99 = 21.

Question 35Mathematics

A sector of a circle of radius 9 cm has area 27π cm². What is the angle of the sector at the centre?

  1. 90°
  2. 135°
  3. 60°
  4. 120° Correct answer.

Explanation

Sector area is (θ/360) × πr², so 27π = (θ/360) × 81π. Solving gives θ/360 = 1/3, hence θ = 120°.

Question 36Mathematics

If sin θ = 3/5 and θ is acute, what is the value of tan θ?

  1. 5/3
  2. 4/3
  3. 4/5
  4. 3/4 Correct answer.

Explanation

Using the identity sin²θ + cos²θ = 1, cos²θ = 1 − 9/25 = 16/25, so cos θ = 4/5 for acute θ. Then tan θ = sin θ ÷ cos θ = (3/5) ÷ (4/5) = 3/4.

Question 37Mathematics

For what values of x is the expression (x − 1)/(x² − 4) undefined?

  1. x = 2 only
  2. x = 1, x = 2 and x = −2
  3. x = 2 and x = −2 Correct answer.
  4. x = 1 only

Explanation

A rational expression is undefined where its denominator equals zero. Solving x² − 4 = 0 gives x = 2 or x = −2, so these are the excluded values.

Question 38Mathematics

In a group of 30 students, 18 study Physics, 15 study Chemistry and 8 study both. How many study neither subject?

  1. 7
  2. 5 Correct answer.
  3. 11
  4. 3

Explanation

By the inclusion–exclusion principle, the number studying at least one subject is 18 + 15 − 8 = 25. Therefore 30 − 25 = 5 students study neither.

Question 39Mathematics

The line y = 3x − 2 is reflected in the x-axis. What is the equation of the image?

  1. y = 3x − 2
  2. y = 3x + 2
  3. y = −3x − 2 Correct answer.
  4. y = −3x + 2

Explanation

Reflecting in the x-axis replaces y with −y, so −y = 3x − 2, which rearranges to y = −3x + 2. Both the gradient and the intercept change sign.

Question 40Mathematics

A cube has a total surface area of 96 cm². What is its volume?

  1. 16 cm³
  2. 48 cm³
  3. 64 cm³ Correct answer.
  4. 512 cm³

Explanation

A cube has six congruent square faces, so each face has area 96 ÷ 6 = 16 cm², giving an edge of 4 cm. The volume is then 4³ = 64 cm³.

Question 41Mathematics

What is the remainder when x³ − 2x² + 4x − 7 is divided by (x − 2)?

  1. 1 Correct answer.
  2. −1
  3. 5
  4. 7

Explanation

By the remainder theorem, the remainder is the value of the polynomial at x = 2. Substituting gives 8 − 8 + 8 − 7 = 1.

Question 42Mathematics

Two events A and B are independent with P(A) = 0.4 and P(B) = 0.5. What is P(A ∪ B)?

  1. 0.2
  2. 0.5
  3. 0.9
  4. 0.7 Correct answer.

Explanation

For independent events P(A ∩ B) = 0.4 × 0.5 = 0.2. Then P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.4 + 0.5 − 0.2 = 0.7.

Question 43Mathematics

The matrix M = [[2, 1], [3, 4]] has determinant 5. What is the determinant of M²?

  1. 10
  2. 25 Correct answer.
  3. 5
  4. 50

Explanation

For square matrices, det(M²) = (det M)², so the determinant is 5² = 25. Determinants multiply when matrices are multiplied.

Question 44Mathematics

A geometric series has first term 8 and common ratio 1/2. What is the sum to infinity?

  1. 16 Correct answer.
  2. 8
  3. 4
  4. 12

Explanation

For a geometric series with |r| < 1, the sum to infinity is a/(1 − r). Here it is 8 ÷ (1 − 1/2) = 8 ÷ 0.5 = 16.

Question 45Mathematics

The function f(x) = x² − 6x + 5 is written in the form (x − h)² + k. What is the minimum value of f(x)?

  1. −3
  2. −4 Correct answer.
  3. 5
  4. 3

Explanation

Completing the square gives f(x) = (x − 3)² − 4, so the vertex is at (3, −4). Since the coefficient of x² is positive, the minimum value is −4.

Question 46Mathematics

How many distinct arrangements of the letters in the word LEVEL are possible?

  1. 60
  2. 120
  3. 20
  4. 30 Correct answer.

Explanation

The word has 5 letters with L repeated twice and E repeated twice. The number of distinct arrangements is 5! ÷ (2! × 2!) = 120 ÷ 4 = 30.

Question 47Mathematics

Given that log₃ 2 = p, which expression equals log₃ 24 in terms of p?

  1. p³ + 1
  2. p + 3
  3. 3p
  4. 3p + 1 Correct answer.

Explanation

Since 24 = 8 × 3 = 2³ × 3, we have log₃ 24 = log₃(2³) + log₃ 3 = 3 log₃ 2 + 1 = 3p + 1. The product rule and power rule for logarithms give this result.

Question 48Mathematics

The equation x² + y² − 4x + 6y − 12 = 0 represents a circle. What is its radius?

  1. 25
  2. 12
  3. √12
  4. 5 Correct answer.

Explanation

Completing the square gives (x − 2)² + (y + 3)² = 12 + 4 + 9 = 25, so the radius is √25 = 5. The centre is (2, −3).

Question 49Mathematics

A curve has equation y = x³ − 3x² + 4. What are the x-coordinates of its stationary points?

  1. x = 2 only
  2. x = 0 and x = 3
  3. x = 1 and x = 2
  4. x = 0 and x = 2 Correct answer.

Explanation

Differentiating gives dy/dx = 3x² − 6x = 3x(x − 2). Setting the derivative to zero gives x = 0 or x = 2, the stationary points.

Question 50Mathematics

The roots of the equation 2x² − 5x + 1 = 0 are α and β. What is the value of α² + β²?

  1. 21/4 Correct answer.
  2. 5/2
  3. 17/4
  4. 1/2

Explanation

Here α + β = 5/2 and αβ = 1/2. Then α² + β² = (α + β)² − 2αβ = 25/4 − 1 = 21/4.