GCSE maths practice questions for Higher tier, with worked answers

How to use these practice questions

Cover the answer, attempt the question on paper, then compare your working line by line. Mark schemes give method marks, so a correct method with a slip still scores. For the exam dates, the full formula sheet and grade boundaries, see the GCSE Maths guide. On Higher, Ofqual's rules put 30% of the marks on algebra, the biggest share, so several questions below are algebra.

Grade 5 questions

1. Standard form

Question: work out (3 × 10⁴) × (5 × 10⁻⁷), giving your answer in standard form.

Answer: multiply the numbers and add the powers: 15 × 10⁻³. That isn't standard form, because 15 is bigger than 10, so write it as 1.5 × 10⁻².

2. Expanding double brackets

Question: expand and simplify (2x + 3)(x − 4). Answer: 2x² − 8x + 3x − 12 = 2x² − 5x − 12. Check by substituting x = 1: the left side is 5 × (−3) = −15 and the right side is 2 − 5 − 12 = −15.

3. Solving a quadratic by factorising

Question: solve x² + x − 12 = 0. Answer: find two numbers that multiply to −12 and add to 1: 4 and −3. So (x + 4)(x − 3) = 0, giving x = −4 or x = 3. Now factorise one yourself from our Algebra course and press Mark.

Grade 6 and 7 questions

4. Right-angled trigonometry

Question: in the triangle below, the hypotenuse is 10 cm and the angle at A is 35°. Find the side x, opposite the angle.

Answer: you have the hypotenuse and want the opposite side, so use sine: x = 10 sin 35° = 5.74 cm to 3 significant figures. Check the calculator is in degrees. Practise more in the Trigonometry Calculator.

5. Probability without replacement

Question: a bag has 5 red counters and 3 blue counters. Two are taken out without replacement. Find the probability that both are red. Answer: P(red then red) = 5/8 × 4/7 = 20/56 = 5/14. The second fraction is out of 7 because one counter has gone, and one red has gone with it.

6. Perpendicular lines

Question: find the equation of the line through (6, 1) that is perpendicular to y = 3x − 2. Answer: flip the gradient and change its sign: 3 becomes −⅓. Putting x = 6 and y = 1 into y = −⅓x + c gives 1 = −2 + c, so c = 3, and the line is y = −⅓x + 3.

Grade 8 and 9 questions

7. Proportion

Question: y is directly proportional to x². When x = 3, y = 18. Find y when x = 5. Answer: write y = kx². Then 18 = 9k, so k = 2, and y = 2 × 25 = 50. Always find k first and write the formula out.

8. Composite functions

Question: f(x) = 2x + 1 and g(x) = x². Find fg(3) and gf(3). Answer: fg(3) means do g first: g(3) = 9, then f(9) = 19. gf(3) means do f first: f(3) = 7, then g(7) = 49. The order matters, which is exactly what the question tests.

Mistakes these questions are built to catch

• Standard form left as 15 × 10⁻³. The first number must be at least 1 and less than 10.
• Sign slips when expanding. Check by substituting a number into both sides.
• Calculator in radians for a GCSE trigonometry question.
• With-replacement thinking when the question says without replacement.
• Doing fg in the wrong order. The function nearest the x goes first.

A new Higher paper every day

For more GCSE maths practice questions, sit today's GCSE Maths Higher paper. If your mocks are coming up, the GCSE Maths mocks guide shows how to sit and grade a full paper at home.

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UK · GCSEGCSE Maths Formula Sheet 2027, Dates and Grade BoundariesEvery formula on the 2027 GCSE Maths sheet, what you must still learn, exam dates, real grade boundaries and the hardest questions worked through.