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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | GCSE Higher |
State whether each of the following statements is true or false. Give reasons for your answers.
(a) When \(a^2 = 25\) the only value that \(a\) can have is 5.
(b) When \(b\) is a positive integer, the value of \(3b\) is always a factor of the value of \(12b\).
(c) When \(c\) is positive, the value of \(c^2\) is always greater than the value of \(c\).
2. | GCSE Higher |
One is added to the product of two consecutive positive even numbers. Show that the result is a square number.
3. | GCSE Higher |
(a) Give a reason why 0 is an even number.
(b) The lengths of the sides of a right-angled triangle are all integers. Prove that if the lengths of the two shortest sides are even, then the length of the third side must also be even.
4. | GCSE Higher |
Betsy thinks that \((3x)^2\) is always greater than or equal to \(3x\).
Is she is correct?
Show your working to justify your decision
5. | GCSE Higher |
This expression can be used to generate a sequence of numbers.
$$n^2+n + 5$$(a) Work out the first three terms of this sequence.
(b) What is the smallest value of \(n\) that produces a term of the sequence that is not a prime number?
(c) Is it true that odd square numbers have exactly three factors? Explain and justify your answer.
(d) Seymour is thinking of a number.
Find the two possible numbers that Seymour could be thinking of.
6. | GCSE Higher |
Given that \(n\) can be any integer such that \(n \gt 1\), prove that \(n^2 + 3n\) is even.
7. | 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.
8. | 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
$$\frac{p-1}{q-1}-\frac pq\lt 0$$9. | 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.
10. | GCSE Higher |
Prove that the expression below is always positive.
$$ x^2 - 5x + 9 $$11. | IB Analysis and Approaches |
Consider the sum of the squares of any two consecutive odd integers.
(a) Show that \((2n + 1)^2 + (2n + 3)^2 = 8n^2 +16n + 10\) , where \(n \in \mathbb{Z} \)
(b) Hence, or otherwise, prove that the sum of the squares of any two consecutive odd integers is even.
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