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PMA Entrance Exam study materials
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PMA Entrance Exam practice questions

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Question 1Mathematics

Simplify the expression (3x²y)(−2xy³).

  1. 6x³y⁴
  2. −6x²y³
  3. −5x³y⁴
  4. −6x³y⁴ Correct answer.

Explanation

Multiply the coefficients: 3 × (−2) = −6. For the variables, add exponents of like bases: x² · x = x³ and y · y³ = y⁴. The product is therefore −6x³y⁴.

Question 2Mathematics

Solve for x: 5x − 7 = 2x + 11.

  1. x = 6 Correct answer.
  2. x = 3
  3. x = 4
  4. x = 2

Explanation

Subtract 2x from both sides to get 3x − 7 = 11, then add 7 to obtain 3x = 18. Dividing both sides by 3 gives x = 6. Substituting 6 back checks: 30 − 7 = 23 and 12 + 11 = 23.

Question 3Mathematics

What is the value of the discriminant of the quadratic equation 2x² + 3x − 5 = 0?

  1. 49 Correct answer.
  2. −31
  3. 31
  4. −49

Explanation

For ax² + bx + c = 0 the discriminant is b² − 4ac. Here a = 2, b = 3 and c = −5, so b² − 4ac = 9 − 4(2)(−5) = 9 + 40 = 49. A positive discriminant confirms two distinct real roots.

Question 4Mathematics

A triangle has sides of length 7 cm, 24 cm, and 25 cm. What is the area of the triangle?

  1. 300 cm²
  2. 84 cm² Correct answer.
  3. 175 cm²
  4. 168 cm²

Explanation

Since 7² + 24² = 49 + 576 = 625 = 25², the triangle is right-angled with legs 7 and 24. Its area is ½ × 7 × 24 = 84 cm². The 25 cm side is the hypotenuse and is not used as a base.

Question 5Mathematics

Evaluate the expression 4! ÷ 2!.

  1. 2
  2. 24
  3. 12 Correct answer.
  4. 6

Explanation

4! = 4 × 3 × 2 × 1 = 24 and 2! = 2 × 1 = 2. Dividing gives 24 ÷ 2 = 12, which also equals 4 × 3 directly because the common factors cancel.

Question 6Mathematics

If log₂ 8 = x, what is the value of x?

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

Explanation

The equation log₂ 8 = x means 2ˣ = 8. Since 2³ = 8, the exponent x must equal 3.

Question 7Mathematics

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

  1. 420 Correct answer.
  2. 440
  3. 400
  4. 380

Explanation

This is an arithmetic series with first term 2, last term 40, and 20 terms. The sum equals n(a₁ + aₙ)/2 = 20(2 + 40)/2 = 20 × 21 = 420.

Question 8Mathematics

A bag contains 5 red balls and 7 blue balls. If one ball is drawn at random, what is the probability that it is red?

  1. 5/7
  2. 1/5
  3. 7/12
  4. 5/12 Correct answer.

Explanation

There are 5 red balls out of a total of 5 + 7 = 12 balls. The probability of drawing red is therefore 5/12.

Question 9Mathematics

The mean of five numbers is 12. If four of the numbers are 8, 10, 15, and 14, what is the fifth number?

  1. 17
  2. 11
  3. 13 Correct answer.
  4. 15

Explanation

The total of all five numbers is 5 × 12 = 60. The four known numbers sum to 8 + 10 + 15 + 14 = 47, so the fifth number is 60 − 47 = 13.

Question 10Mathematics

Simplify (x + 3)(x − 5).

  1. x² − 2x − 15 Correct answer.
  2. x² + 2x − 15
  3. x² − 8x − 15
  4. x² − 2x + 15

Explanation

Using the distributive property, multiply each term: x² − 5x + 3x − 15. Combining the middle terms gives x² − 2x − 15.

Question 11Mathematics

Which of the following is the equation of a line parallel to y = 3x − 4 that passes through the point (2, 5)?

  1. y = 3x − 1 Correct answer.
  2. y = −3x + 11
  3. y = 3x + 1
  4. y = 3x + 5

Explanation

Parallel lines share the same slope, so the new line has slope 3. Using point-slope form, y − 5 = 3(x − 2), which simplifies to y = 3x − 1.

Question 12Mathematics

A rectangle has a length that is 4 cm more than its width. If its perimeter is 28 cm, what is its area?

  1. 33 cm²
  2. 40 cm²
  3. 45 cm² Correct answer.
  4. 48 cm²

Explanation

Let the width be w; then the length is w + 4. The perimeter is 2(w + w + 4) = 28, so 2w + 4 = 14 and w = 5. The length is 9, and the area is 5 × 9 = 45 cm².

Question 13Mathematics

What is the value of sin 30° + cos 60°?

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

Explanation

Both sin 30° and cos 60° equal 1/2. Their sum is 1/2 + 1/2 = 1.

Question 14Mathematics

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

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

Explanation

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

Question 15Mathematics

The expression x² − 9 factors as which of the following?

  1. (x + 3)²
  2. (x − 3)²
  3. (x − 3)(x + 3) Correct answer.
  4. (x − 9)(x + 1)

Explanation

This is a difference of squares, a² − b² = (a − b)(a + b). With a = x and b = 3, x² − 9 factors as (x − 3)(x + 3).

Question 16Mathematics

How many distinct ways can the letters of the word LEVEL be arranged?

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

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 17Mathematics

A right circular cylinder has a radius of 3 cm and a height of 10 cm. What is its volume in terms of π?

  1. 60π cm³
  2. 900π cm³
  3. 90π cm³ Correct answer.
  4. 30π cm³

Explanation

The volume of a cylinder is V = πr²h. Substituting r = 3 and h = 10 gives V = π(9)(10) = 90π cm³.

Question 18Mathematics

If 2ˣ = 32, what is the value of 3ˣ?

  1. 27
  2. 125
  3. 243 Correct answer.
  4. 81

Explanation

Since 32 = 2⁵, we have x = 5. Then 3ˣ = 3⁵ = 243.

Question 19Mathematics

The sum of two numbers is 20 and their product is 96. What is the positive difference between the numbers?

  1. 6
  2. 2
  3. 8
  4. 4 Correct answer.

Explanation

The numbers satisfy t² − 20t + 96 = 0, which factors as (t − 8)(t − 12) = 0, giving 8 and 12. Their difference is 12 − 8 = 4.

Question 20Mathematics

What is the slope of the line passing through the points (−2, 3) and (4, −9)?

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

Explanation

The slope is (y₂ − y₁)/(x₂ − x₁) = (−9 − 3)/(4 − (−2)) = −12/6 = −2.

Question 21Mathematics

In a geometric sequence, the first term is 3 and the common ratio is 2. What is the 6th term?

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

Explanation

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

Question 22Mathematics

A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls is 7?

  1. 5/36
  2. 7/36
  3. 1/6 Correct answer.
  4. 1/12

Explanation

There are 36 equally likely ordered outcomes. The sums equal to 7 occur in 6 cases: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Thus the probability is 6/36 = 1/6.

Question 23Mathematics

Simplify the expression (x³)⁴ ÷ x⁵.

  1. x⁶
  2. x¹²
  3. x²
  4. x⁷ Correct answer.

Explanation

First apply the power rule: (x³)⁴ = x¹². Then divide by x⁵ by subtracting exponents: x¹² ÷ x⁵ = x⁷.

Question 24Mathematics

The ratio of boys to girls in a class is 3:5. If there are 40 students in total, how many girls are there?

  1. 25 Correct answer.
  2. 15
  3. 20
  4. 24

Explanation

The ratio 3:5 means there are 8 parts in total. Each part is 40 ÷ 8 = 5 students, so the number of girls is 5 parts × 5 = 25.

Question 25Mathematics

For what values of k does the equation x² + kx + 9 = 0 have exactly one real solution?

  1. k = ±6 Correct answer.
  2. k = 9 only
  3. k = 6 only
  4. k = ±3

Explanation

A quadratic has exactly one real root when its discriminant equals zero: k² − 4(1)(9) = 0, so k² = 36 and k = ±6. Both values give a perfect-square trinomial.

Question 26Mathematics

A cadet's score on a physical fitness test is recorded as 78, 85, 91, 85, and 66. What is the median score of this data set?

  1. 81
  2. 85 Correct answer.
  3. 91
  4. 78

Explanation

Arranging the five scores in increasing order gives 66, 78, 85, 85, 91. The median is the middle value of an odd-sized ordered set, which is the third score, 85. The repeated value does not change the position of the middle term.

Question 27Mathematics

A square parade ground has an area of 2,025 square metres. What is the length of one side of the ground?

  1. 35 m
  2. 25 m
  3. 45 m Correct answer.
  4. 55 m

Explanation

For a square, area equals side squared, so the side is the positive square root of the area. Since 45 × 45 = 2,025, the side length is 45 metres. The negative root is discarded because a length must be positive.

Question 28Mathematics

Convert the binary number 1101₂ to base ten.

  1. 11
  2. 26
  3. 14
  4. 13 Correct answer.

Explanation

Each binary digit is multiplied by the corresponding power of two: 1×8 + 1×4 + 0×2 + 1×1. Adding these place values gives 8 + 4 + 0 + 1 = 13.

Question 29Mathematics

A right triangle has legs measuring 9 cm and 12 cm. What is the length of its hypotenuse?

  1. 13 cm
  2. 15 cm Correct answer.
  3. 21 cm
  4. 18 cm

Explanation

By the Pythagorean theorem the hypotenuse satisfies c² = 9² + 12² = 81 + 144 = 225. Taking the positive square root gives c = 15 cm. The 9-12-15 triple is a scaled version of the 3-4-5 triple.

Question 30Mathematics

Evaluate the expression 5 + 3 × (8 − 2)² ÷ 4.

  1. 54
  2. 72
  3. 17
  4. 32 Correct answer.

Explanation

Following the order of operations, first evaluate the bracket to get 6, then square it to get 36. Next multiply 3 × 36 = 108, divide by 4 to get 27, and finally add 5, giving 32.

Question 31Mathematics

A troop of 24 scouts is to be divided into patrols of equal size, with at least 2 patrols and no patrol smaller than 3 scouts. Which patrol size is impossible?

  1. 5 Correct answer.
  2. 4
  3. 8
  4. 6

Explanation

A patrol size is possible only if it divides 24 exactly. The divisors of 24 that are at least 3 are 3, 4, 6, 8, 12 and 24, so sizes 4, 6 and 8 all work. Since 5 does not divide 24, splitting the troop into patrols of 5 would leave a remainder, making it impossible.

Question 32Mathematics

A vehicle travels 150 kilometres in 2.5 hours. At the same average speed, how far will it travel in 4 hours?

  1. 375 km
  2. 200 km
  3. 300 km
  4. 240 km Correct answer.

Explanation

The average speed is 150 ÷ 2.5 = 60 kilometres per hour. Multiplying by 4 hours gives a distance of 240 kilometres. Setting up the proportion 150/2.5 = d/4 yields the same result.

Question 33Mathematics

Which of the following numbers is irrational?

  1. 0.25
  2. √7 Correct answer.
  3. √49
  4. −3/8

Explanation

An irrational number cannot be written as a ratio of two integers. Since 7 is not a perfect square, √7 has a non-terminating, non-repeating decimal expansion and is irrational. The other options are rational: a terminating decimal, a fraction, and a perfect square root.

Question 34Mathematics

A rectangular field measures 60 m by 40 m. A path of uniform width 2 m runs around the outside of the field. What is the area of the path?

  1. 336 m²
  2. 400 m²
  3. 416 m² Correct answer.
  4. 240 m²

Explanation

The outer rectangle measures 64 m by 44 m because the path adds 2 m on each side. Its area is 64 × 44 = 2,816 m², while the field is 60 × 40 = 2,400 m². Subtracting gives the path area of 416 m².

Question 35Mathematics

If 3x − 2y = 12 and x + y = 4, what is the value of x?

  1. 6
  2. 2
  3. 4 Correct answer.
  4. 0

Explanation

From x + y = 4 we get y = 4 − x. Substituting into 3x − 2y = 12 gives 3x − 8 + 2x = 12, so 5x = 20 and x = 4. Substituting back gives y = 0, which satisfies both equations.

Question 36Mathematics

The 3rd term of an arithmetic sequence is 14 and the 7th term is 30. What is the first term of the sequence?

  1. 6 Correct answer.
  2. 10
  3. 4
  4. 8

Explanation

The difference between the 7th and 3rd terms spans four common differences, so 4d = 30 − 14 = 16 and d = 4. Working back from the 3rd term, the first term is 14 − 2(4) = 6.

Question 37Mathematics

Two similar triangles have corresponding sides in the ratio 2 : 5. If the area of the smaller triangle is 12 cm², what is the area of the larger triangle?

  1. 30 cm²
  2. 75 cm² Correct answer.
  3. 60 cm²
  4. 48 cm²

Explanation

For similar figures, areas scale as the square of the linear ratio. The area ratio is therefore (5/2)² = 25/4, so the larger area is 12 × 25/4 = 75 cm².

Question 38Mathematics

A circle has the equation x² + y² − 6x + 8y − 11 = 0. What is its radius?

  1. 4
  2. 8
  3. 6 Correct answer.
  4. √11

Explanation

Completing the square gives (x − 3)² + (y + 4)² = 11 + 9 + 16 = 36. The right side equals r², so the radius is 6. The centre is (3, −4).

Question 39Mathematics

A committee of 2 students is to be chosen from 6 candidates. In how many ways can this be done?

  1. 30
  2. 15 Correct answer.
  3. 12
  4. 36

Explanation

Because the order of selection does not matter, this is a combination: C(6,2) = 6!/(2!·4!) = 15. Choosing students A then B is the same committee as B then A.

Question 40Mathematics

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

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

Explanation

By the remainder theorem, the remainder equals the polynomial evaluated at x = 2. Substituting gives 8 − 8 + 8 − 7 = 1, so the remainder is 1.

Question 41Mathematics

Solve the inequality 2x − 5 < 3x + 1.

  1. x < −6
  2. x < 6
  3. x > −6 Correct answer.
  4. x > 6

Explanation

Subtracting 2x from both sides gives −5 < x + 1, and subtracting 1 gives −6 < x, that is x > −6. No multiplication by a negative number occurs, so the inequality sign stays the same.

Question 42Mathematics

If tan θ = 3/4 and θ lies in the first quadrant, what is the value of sin θ?

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

Explanation

Tangent equals opposite over adjacent, so take the opposite side as 3 and the adjacent side as 4. The hypotenuse is √(9 + 16) = 5, and sine is opposite over hypotenuse, giving 3/5.

Question 43Mathematics

A geometric sequence has first term 8 and sum to infinity 12. What is its common ratio?

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

Explanation

The sum to infinity of a convergent geometric series is a/(1 − r). Setting 8/(1 − r) = 12 gives 1 − r = 2/3, so r = 1/3, which lies between −1 and 1 as required.

Question 44Mathematics

A fair coin is tossed three times. What is the probability of getting at least one head?

  1. 1/8
  2. 1/2
  3. 3/8
  4. 7/8 Correct answer.

Explanation

The complement of at least one head is no heads, meaning three tails, which has probability (1/2)³ = 1/8. Subtracting from 1 gives 1 − 1/8 = 7/8.

Question 45Mathematics

The polynomial P(x) = x³ + ax² + bx + 6 has x − 1 as a factor and leaves a remainder of 12 when divided by x − 2. What is the value of a?

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

Explanation

From the factor theorem, P(1) = 1 + a + b + 6 = 0, so a + b = −7. From the remainder theorem, P(2) = 8 + 4a + 2b + 6 = 12, so 2a + b = −1. Subtracting the first equation from the second gives a = 6, which conflicts with the sign pattern, so solving directly gives a = −3 and b = −4.

Question 46Mathematics

What is the sum of the solutions of the equation 2^(x² − 3x) = 16?

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

Explanation

Rewrite 16 as 2⁴, so x² − 3x = 4, giving x² − 3x − 4 = 0. This factors as (x − 4)(x + 1) = 0, with solutions 4 and −1. Their sum is 3, which also equals the negated coefficient of x by Vieta's formula.

Question 47Mathematics

How many real solutions does the equation |x − 2| + |x + 3| = 4 have?

  1. 0 Correct answer.
  2. 2
  3. Infinitely many
  4. 1

Explanation

The expression |x − 2| + |x + 3| equals the distance from x to 2 plus the distance from x to −3. The smallest possible value is the distance between the points 2 and −3, which is 5. Since 4 is less than 5, no real value of x can satisfy the equation.

Question 48Mathematics

In how many ways can 5 distinct books be arranged on a shelf so that two particular books are never adjacent?

  1. 24
  2. 72 Correct answer.
  3. 48
  4. 120

Explanation

First arrange the other three books, which can be done in 3! = 6 ways. These create four gaps where the two special books may be placed, giving P(4,2) = 12 ordered placements. The total is 6 × 12 = 72.

Question 49Mathematics

A geometric sequence has terms with positive common ratio r. If the sum of the first three terms is 26 and the sum of the next three terms is 702, what is r?

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

Explanation

The sum of terms 4 to 6 equals r³ times the sum of terms 1 to 3, because each later term is r³ times the corresponding earlier term. Thus r³ = 702/26 = 27, giving r = 3 since the ratio is positive.

Question 50Mathematics

A cone and a cylinder have equal radii and equal heights. If the cylinder holds 90 cm³ of water, how much water can the cone hold?

  1. 30 cm³ Correct answer.
  2. 45 cm³
  3. 60 cm³
  4. 90 cm³

Explanation

The volume of a cone is one third of the volume of a cylinder with the same base radius and height. Since the cylinder holds 90 cm³, the cone holds 90 ÷ 3 = 30 cm³.