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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | GCSE Higher |
Use algebra to prove that \(0.3\dot1\dot8 \times 0.\dot8\) is equal to \( \frac{28}{99} \).
2. | 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.
3. | 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
$$\frac{p-1}{q-1}-\frac pq\lt 0$$4. | 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.
5. | 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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