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Exam-Style Questions.

Problems adapted from questions set for previous Mathematics exams.


GCSE Higher

Use algebra to prove that \(0.3\dot1\dot8 \times 0.\dot8\) is equal to \( \frac{28}{99} \).


GCSE Higher

m and n are positive whole numbers with m > n

m2 – n2 = (m + n)(m – n)

If m2 – n2 is a prime number prove that m and n are consecutive numbers.


GCSE Higher

Express as a single fraction and simplify your answer.

$$\frac{p-1}{q-1}-\frac pq$$

Using your answer to part (a), prove that if \(p\) and \(q\) are positive integers and \(p \lt q\), then

$$\frac{p-1}{q-1}-\frac pq\lt 0$$


GCSE Higher

(a) Prove that the product of two consecutive whole numbers is always even.

(b) Prove, by giving a counter example, that the sum of four consecutive integers is not always divisible by 4.


IB Analysis and Approaches

Consider two consecutive positive even numbers, \(2n\) and \(2n + 2\).

Show that the difference of their squares is equal to twice their sum.

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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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