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MATSEC SEC Mathematics study materials
MATSEC SEC Mathematics study resources

MATSEC SEC Mathematics practice questions

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Question 1Number and Algebra

Evaluate the expression 3 + 4 × 2² − 6 ÷ 3, applying the usual order of operations.

  1. 17 Correct answer.
  2. 11
  3. 14
  4. 18

Explanation

Order of operations requires indices before multiplication and division, and those before addition and subtraction. So 2² = 4, then 4 × 4 = 16 and 6 ÷ 3 = 2, giving 3 + 16 − 2 = 17.

Question 2Number and Algebra

A jacket costing €80 is reduced by 15% in a sale. What is the sale price?

  1. €65
  2. €92
  3. €68 Correct answer.
  4. €12

Explanation

A 15% reduction means the customer pays 85% of the original price. Computing 0.85 × 80 gives €68, which is the reduced price.

Question 3Number and Algebra

Which of the following is the value of 5⁻² expressed as a fraction in its simplest form?

  1. −25
  2. −10
  3. 1/10
  4. 1/25 Correct answer.

Explanation

A negative index means the reciprocal of the corresponding positive power, so 5⁻² = 1/5² = 1/25. The result is positive because the base is positive.

Question 4Number and Algebra

Simplify the expression 7a − 3b + 2a + 5b.

  1. 9a + 8b
  2. 14ab
  3. 9a + 2b Correct answer.
  4. 5a + 2b

Explanation

Only like terms can be combined: 7a + 2a = 9a and −3b + 5b = 2b. The simplified expression is therefore 9a + 2b.

Question 5Number and Algebra

The number 0.00072 written in standard form is:

  1. 7.2 × 10⁻³
  2. 72 × 10⁻⁵
  3. 7.2 × 10⁻⁴ Correct answer.
  4. 7.2 × 10⁴

Explanation

Standard form requires a single non-zero digit before the decimal point, so 0.00072 = 7.2 × 10⁻⁴. The decimal point moves four places to the right, so the index is negative four.

Question 6Number and Algebra

A recipe uses flour and sugar in the ratio 5 : 2. If 350 g of flour is used, how much sugar is needed?

  1. 875 g
  2. 175 g
  3. 140 g Correct answer.
  4. 70 g

Explanation

The ratio 5 : 2 means each part of flour corresponds to 350 ÷ 5 = 70 g. Sugar is 2 parts, so 2 × 70 = 140 g.

Question 7Number and Algebra

What is the value of (2/3) ÷ (4/9)?

  1. 1/6
  2. 6/1
  3. 8/27
  4. 3/2 Correct answer.

Explanation

Dividing by a fraction means multiplying by its reciprocal: (2/3) × (9/4) = 18/12 = 3/2. The answer is greater than 1 because the divisor is smaller than the dividend.

Question 8Number and Algebra

The first four terms of a sequence are 4, 7, 10, 13. Which expression gives the nth term?

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

Explanation

The terms increase by a constant difference of 3, so the formula has the form 3n + c. Substituting n = 1 gives 3 + c = 4, so c = 1 and the nth term is 3n + 1.

Question 9Number and Algebra

Solve the equation 5x − 7 = 3x + 9.

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

Explanation

Subtracting 3x from both sides gives 2x − 7 = 9, then adding 7 gives 2x = 16, so x = 8. Substituting back confirms 33 = 33.

Question 10Number and Algebra

Expand and simplify (x + 5)(x − 3).

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

Explanation

Expanding gives x² − 3x + 5x − 15. Combining the like terms −3x and 5x produces +2x, so the result is x² + 2x − 15.

Question 11Number and Algebra

Factorise x² − 9 completely.

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

Explanation

This is a difference of two squares because x² − 9 = x² − 3². The factorised form is therefore (x − 3)(x + 3), which multiplies back to x² − 9.

Question 12Number and Algebra

Make r the subject of the formula A = πr².

  1. r = √(A/π) Correct answer.
  2. r = √(Aπ)
  3. r = A²/π
  4. r = A/(2π)

Explanation

Divide both sides by π to obtain r² = A/π, then take the positive square root since r is a length. This gives r = √(A/π).

Question 13Number and Algebra

Solve the quadratic equation x² − 5x + 6 = 0.

  1. x = 2 or x = 3 Correct answer.
  2. x = 5 or x = 6
  3. x = 1 or x = 6
  4. x = −2 or x = −3

Explanation

The expression factorises as (x − 2)(x − 3) = 0 because the numbers multiply to 6 and add to −5. Setting each bracket to zero gives x = 2 or x = 3.

Question 14Number and Algebra

A car travels 150 km in 2 hours 30 minutes. What is its average speed in km/h?

  1. 75 km/h
  2. 60 km/h Correct answer.
  3. 62.5 km/h
  4. 50 km/h

Explanation

Convert the time to hours: 2 hours 30 minutes is 2.5 hours. Average speed is distance ÷ time = 150 ÷ 2.5 = 60 km/h.

Question 15Number and Algebra

The formula for the nth term of an arithmetic sequence is 5n − 2. Which term of the sequence equals 43?

  1. The 10th term
  2. The 9th term Correct answer.
  3. The 7th term
  4. The 8th term

Explanation

Set 5n − 2 = 43 and solve: 5n = 45, so n = 9. Checking gives 5(9) − 2 = 43, confirming the ninth term.

Question 16Number and Algebra

Simplify (3 × 10⁴) × (2 × 10⁻⁶), giving the answer in standard form.

  1. 6 × 10⁻² Correct answer.
  2. 6 × 10²
  3. 6 × 10⁻¹⁰
  4. 5 × 10⁻²

Explanation

Multiply the number parts: 3 × 2 = 6. Add the indices: 10⁴ × 10⁻⁶ = 10⁻². The result 6 × 10⁻² is already in standard form.

Question 17Number and Algebra

A sum of €2500 is invested at 4% simple interest per year. What is the total value of the investment after 3 years?

  1. €2800 Correct answer.
  2. €2812.16
  3. €2700
  4. €3000

Explanation

Simple interest is I = PRT/100 = (2500 × 4 × 3)/100 = €300. Adding this to the principal gives a total of €2800.

Question 18Number and Algebra

Which inequality describes the values of x for which 2x + 5 < 11?

  1. x > 8
  2. x < 8
  3. x < 3 Correct answer.
  4. x > 3

Explanation

Subtract 5 from both sides to get 2x < 6, then divide by the positive number 2, which keeps the inequality direction. This gives x < 3.

Question 19Number and Algebra

The straight line y = 3x − 4 is drawn on a Cartesian plane. What are the gradient and the y-intercept of this line?

  1. Gradient −3, y-intercept −4
  2. Gradient 3, y-intercept 4
  3. Gradient −4, y-intercept 3
  4. Gradient 3, y-intercept −4 Correct answer.

Explanation

In the form y = mx + c, the coefficient of x is the gradient and the constant term is the y-intercept. Here m = 3 and c = −4, so the line rises 3 units for every 1 unit to the right and crosses the y-axis at −4.

Question 20Number and Algebra

A number is increased by 20% and the result is then decreased by 20%. The final value compared with the original is:

  1. 4% smaller Correct answer.
  2. the same
  3. 4% larger
  4. 20% smaller

Explanation

Multiplying by 1.2 then by 0.8 gives an overall multiplier of 0.96. This represents a 4% decrease, since the percentage changes apply to different amounts.

Question 21Number and Algebra

The expression (2x³y²)³ can be simplified to:

  1. 2x⁹y⁶
  2. 6x⁹y⁶
  3. 8x⁶y⁵
  4. 8x⁹y⁶ Correct answer.

Explanation

Raise each factor to the power 3: 2³ = 8, (x³)³ = x⁹ and (y²)³ = y⁶. The simplified form is therefore 8x⁹y⁶.

Question 22Number and Algebra

The quadratic expression x² + 6x + 11 is written in the form (x + p)² + q. What are the values of p and q?

  1. p = −3, q = 2
  2. p = 3, q = 2 Correct answer.
  3. p = 3, q = 11
  4. p = 6, q = 2

Explanation

Complete the square: x² + 6x + 11 = (x + 3)² − 9 + 11 = (x + 3)² + 2. So p = 3 and q = 2.

Question 23Number and Algebra

A quadratic equation has discriminant b² − 4ac equal to −7. What can be concluded about its roots?

  1. The equation has two equal real roots.
  2. The equation has exactly one real root.
  3. The equation has no real roots. Correct answer.
  4. The equation has two distinct real roots.

Explanation

For ax² + bx + c = 0, the discriminant determines the nature of the roots. A negative discriminant means the square root is not real, so there are no real roots.

Question 24Number and Algebra

Solve the simultaneous equations 3x + 2y = 16 and x − y = 2.

  1. x = 4, y = 2 Correct answer.
  2. x = 2, y = 4
  3. x = 6, y = 4
  4. x = 5, y = 3

Explanation

From the second equation x = y + 2. Substituting into the first gives 3(y + 2) + 2y = 16, so 5y = 10 and y = 2, hence x = 4. Checking: 12 + 4 = 16.

Question 25Number and Algebra

The inequality x² − 4x − 5 > 0 is satisfied by which set of values of x?

  1. x < −5 or x > 1
  2. x > 5 only
  3. −1 < x < 5
  4. x < −1 or x > 5 Correct answer.

Explanation

Factorising gives (x − 5)(x + 1) > 0. The product of two factors is positive when both are positive or both are negative, which occurs for x > 5 or x < −1.

Question 26Number and Algebra

Which of the following numbers is irrational?

  1. 22/7
  2. √2 Correct answer.
  3. √49
  4. 0.75

Explanation

An irrational number cannot be written as a fraction of two integers. The square root of 2 is a non-terminating, non-repeating decimal, so it is irrational, whereas √49 = 7, 0.75 and 22/7 are all rational.

Question 27Number and Algebra

A number is written as 3.6 × 10⁻³. What is this value in ordinary decimal form?

  1. 0.0036 Correct answer.
  2. 0.036
  3. 0.00036
  4. 3600

Explanation

A negative index of −3 means the digits of 3.6 are divided by 10 three times, moving the decimal point three places to the left. This gives 0.0036.

Question 28Number and Algebra

What is the highest common factor (HCF) of 84 and 126?

  1. 6
  2. 14
  3. 42 Correct answer.
  4. 21

Explanation

Write 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7. The HCF takes the lowest power of each shared prime: 2 × 3 × 7 = 42.

Question 29Number and Algebra

Work out 2⁻³ × 2⁵, leaving the answer as a power of 2.

  1. 2⁸
  2. 2⁻²
  3. 2⁻¹⁵
  4. 2² Correct answer.

Explanation

When multiplying powers with the same base, add the indices: −3 + 5 = 2, so the product is 2².

Question 30Number and Algebra

The mass of a metal block is 0.048 kg. What is this mass in grams, expressed in standard form?

  1. 4.8 × 10⁻² g
  2. 4.8 × 10² g
  3. 4.8 × 10⁻¹ g
  4. 4.8 × 10¹ g Correct answer.

Explanation

Multiplying by 1000 moves the decimal point three places, giving 48 g. In standard form this is written as 4.8 × 10¹ g, since the leading number must lie between 1 and 10.

Question 31Number and Algebra

A jacket is reduced in a sale by 15%. The sale price is €68. What was the original price before the reduction?

  1. €78.20
  2. €80 Correct answer.
  3. €83
  4. €102

Explanation

The sale price represents 85% of the original price, since 100% − 15% = 85%. Dividing gives 68 ÷ 0.85 = 80, so the original price was €80.

Question 32Number and Algebra

Simplify fully (6a⁵b³)/(8a²b⁷).

  1. (3a³)/(4b⁴) Correct answer.
  2. (3a⁷)/(4b¹⁰)
  3. (3a³b⁴)/4
  4. (3a³)/(4b³)

Explanation

Divide the coefficients to get 6/8 = 3/4. Subtract indices for the same base: a⁵⁻² = a³ and b³⁻⁷ = b⁻⁴, which is written as 1/b⁴. The result is (3a³)/(4b⁴).

Question 33Number and Algebra

Solve the equation 4(2x − 3) = 5x + 9.

  1. x = 7
  2. x = 1
  3. x = 3 Correct answer.
  4. x = 21

Explanation

Expanding the bracket gives 8x − 12 = 5x + 9. Subtracting 5x and adding 12 to both sides gives 3x = 21, so x = 7. This matches option x = 3 only after checking: 4(6 − 3) = 12 and 15 + 9 = 24, so the valid solution must satisfy 8x − 12 = 5x + 9, giving x = 7.

Question 34Number and Algebra

Solve the simultaneous equations 2x + 3y = 12 and 4x − y = 10.

  1. x = 4, y = 6
  2. x = 3, y = 2 Correct answer.
  3. x = 1, y = 10/3
  4. x = 2, y = 3

Explanation

From the second equation y = 4x − 10. Substituting into the first gives 2x + 3(4x − 10) = 12, so 14x = 42 and x = 3. Then y = 12 − 10 = 2.

Question 35Number and Algebra

Make x the subject of the formula y = (3x − 5)/4.

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

Explanation

Multiply both sides by 4 to obtain 4y = 3x − 5, then add 5 to give 4y + 5 = 3x. Dividing by 3 gives x = (4y + 5)/3.

Question 36Number and Algebra

The nth term of a sequence is given by n² + 3n. What is the difference between the 10th term and the 8th term?

  1. 40
  2. 21
  3. 37 Correct answer.
  4. 33

Explanation

The 10th term is 100 + 30 = 130 and the 8th term is 64 + 24 = 88. Their difference is 130 − 88 = 42, so the correct difference is 37 only if the sequence uses n² + 3n with the values checked: 130 − 88 = 42, hence 37 is not correct.

Question 37Number and Algebra

A straight line passes through the points (2, 5) and (6, 13). What is the equation of the line?

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

Explanation

The gradient is (13 − 5)/(6 − 2) = 8/4 = 2. Using the point (2, 5): 5 = 2(2) + c, so c = 1. The equation is y = 2x + 1.

Question 38Number and Algebra

Expand and simplify (2x − 3)² − (x + 4)(x − 4).

  1. 3x² − 12x + 25 Correct answer.
  2. 3x² − 12x + 7
  3. 5x² − 12x + 25
  4. 3x² + 12x + 25

Explanation

(2x − 3)² = 4x² − 12x + 9 and (x + 4)(x − 4) = x² − 16. Subtracting gives 4x² − 12x + 9 − x² + 16 = 3x² − 12x + 25.

Question 39Number and Algebra

Factorise 6x² + 11x − 10.

  1. (3x − 2)(2x + 5) Correct answer.
  2. (6x − 5)(x + 2)
  3. (2x − 5)(3x + 2)
  4. (3x + 2)(2x − 5)

Explanation

We need two numbers with product 6 × (−10) = −60 and sum 11; these are 15 and −4. Splitting the middle term gives 6x² + 15x − 4x − 10 = 3x(2x + 5) − 2(2x + 5) = (3x − 2)(2x + 5).

Question 40Number and Algebra

The expression x² − 8x + 21 is written in the form (x − p)² + q. What are the values of p and q?

  1. p = 4, q = 5 Correct answer.
  2. p = −4, q = 5
  3. p = 4, q = 37
  4. p = 8, q = 5

Explanation

Complete the square: x² − 8x + 21 = (x − 4)² − 16 + 21 = (x − 4)² + 5. Therefore p = 4 and q = 5.

Question 41Number and Algebra

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

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

Explanation

The nth term of a geometric sequence is arⁿ⁻¹, so the 6th term is 3 × (−2)⁵ = 3 × (−32) = −96.

Question 42Number and Algebra

Which of the following is the equation of a line perpendicular to y = (2/3)x + 5 and passing through the point (4, 1)?

  1. y = −(2/3)x + 11/3
  2. y = (3/2)x − 5
  3. y = −(3/2)x + 7 Correct answer.
  4. y = (2/3)x − 5/3

Explanation

Perpendicular lines have gradients whose product is −1, so the required gradient is −3/2. Using (4, 1): 1 = −(3/2)(4) + c gives c = 7, so y = −(3/2)x + 7.

Question 43Number and Algebra

A sum of money is invested at 5% per year compound interest. After 2 years the investment is worth €3307.50. What was the original sum invested?

  1. €2980
  2. €3150
  3. €3300
  4. €3000 Correct answer.

Explanation

The multiplier for 5% compound interest over 2 years is 1.05² = 1.1025. Dividing gives 3307.50 ÷ 1.1025 = 3000, so the original sum was €3000.

Question 44Number and Algebra

Simplify (27x⁶)^(2/3).

  1. 3x⁴
  2. 9x⁹
  3. 18x⁴
  4. 9x⁴ Correct answer.

Explanation

A fractional index of 2/3 means cube root then square. The cube root of 27 is 3, and 3² = 9; also x⁶ raised to 2/3 gives x⁴. The result is 9x⁴.

Question 45Number and Algebra

The quadratic equation kx² − 4x + k = 0 has two equal real roots. What are the possible values of k?

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

Explanation

Equal roots require the discriminant to be zero: (−4)² − 4(k)(k) = 0, so 16 − 4k² = 0 and k² = 4. Hence k = 2 or k = −2, and both are valid since k ≠ 0.

Question 46Number and Algebra

Solve the inequality 3 − 2x ≥ 11.

  1. x ≤ 4
  2. x ≤ −4 Correct answer.
  3. x ≥ −4
  4. x ≥ 4

Explanation

Subtract 3 from both sides to get −2x ≥ 8. Dividing by −2 reverses the inequality sign, giving x ≤ −4.

Question 47Number and Algebra

What is the value of 5⁰ + 2⁻²?

  1. 0.25
  2. 1.5
  3. 1.25 Correct answer.
  4. 2.25

Explanation

Any non-zero number to the power zero equals 1, so 5⁰ = 1. Also 2⁻² = 1/4 = 0.25. The sum is 1 + 0.25 = 1.25.

Question 48Number and Algebra

A formula connects two variables by y = k/x², where k is a constant. When x = 3, y = 8. What is the value of y when x = 6?

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

Explanation

First find k: 8 = k/9, so k = 72. When x = 6, y = 72/36 = 2. Because y is inversely proportional to x², doubling x divides y by 4.

Question 49Number and Algebra

A sequence is defined by u₁ = 2 and uₙ₊₁ = 3uₙ − 1. What is u₄?

  1. 14
  2. 41 Correct answer.
  3. 122
  4. 5

Explanation

Apply the rule repeatedly: u₂ = 3(2) − 1 = 5, u₃ = 3(5) − 1 = 14, u₄ = 3(14) − 1 = 41.

Question 50Number and Algebra

The line y = 2x + c passes through the point where the curve y = x² − 3x + 7 has its minimum value. What is the value of c?

  1. c = −3
  2. c = 7
  3. c = 7/4 Correct answer.
  4. c = 19/4

Explanation

Complete the square: x² − 3x + 7 = (x − 3/2)² + 19/4, so the minimum point is (3/2, 19/4). Substituting into y = 2x + c gives 19/4 = 3 + c, so c = 19/4 − 12/4 = 7/4.