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Exam-Style Questions.

Problems adapted from questions set for previous Mathematics exams.


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 \(c\).


GCSE Higher

One is added to the product of two consecutive positive even numbers. Show that the result is a square number.


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.

Right-Angled Triangle


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


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.


GCSE Higher

Given that \(n\) can be any integer such that \(n \gt 1\), prove that \(n^2 + 3n\) is even.


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.


GCSE Higher

The number \(T\) can be expressed as \(T = 4k + 3\) where \(k\) is a positive integer.

(a) Show that \(T^2\) is always an odd number.

\(T\) and \(U\) are consecutive odd numbers where \(U > T\).

(b) Write down an expression for \(U\), in terms of \(k\).

(c) Show that \(U^2 - T^2\) is always a multiple of 16.


GCSE Higher

Prove that the expression below is always positive.

$$ x^2 - 5x + 9 $$


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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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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