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A Level Maths · Pure · Year 12 and 13

A Level Maths Proof revision

Proof questions are short but unforgiving: the mark goes for a complete, clearly-worded argument, not for the right idea. They turn up on every board, often as the first or last question on a pure paper.

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Same content for AQA, Edexcel and OCR
Step-by-step mark schemes
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What you need to know

Proof in Year 12 and Year 13

AS content is usually taught in Year 12. The full A Level adds the Year 13 content. Sitting AS exams or mocks? See the AS Level Maths checklist.

Year 12 (AS content)

Proof by deduction, including simple algebraic proofs about odd, even and consecutive integers
Proof by exhaustion: checking every case
Disproof by counter-example

Year 13 (full A Level only)

Proof by contradiction, including the irrationality of √2 and the infinitude of primes

Worked examplemedium4 marks

Prove algebraically that the difference between the squares of any two consecutive odd integers is always a multiple of $8$.

Mark scheme and worked answer

[M1] for defining two consecutive odd integers, e.g., $2n - 1$ and $2n + 1$, where $n \in \mathbb{Z}$.
[M1] for setting up the difference of their squares: $(2n + 1)^2 - (2n - 1)^2$.
[A1] for expanding and simplifying correctly: $(4n^2 + 4n + 1) - (4n^2 - 4n + 1) = 8n$.
[R1] for concluding that since $n$ is an integer, $8n$ is a multiple of $8$. [AG]

Then practise: skills to practise · open proof practice questions.

Where it's examined

Proof on the AQA, Edexcel and OCR papers

Every board examines the same A Level Maths content, set by the Department for Education. Only the paper it appears on and the section numbering differ.

Board	Specification reference	Examined in
AQA 7357	AQA section A	Papers 1, 2 and 3
Edexcel 9MA0	Edexcel topic Pure 1	Paper 1 and Paper 2
OCR A H240	OCR reference 1.01	Paper 1, and the pure sections of Papers 2 and 3

Practice questions

A Level Maths proof exam-style questions

Three examples from the question bank, one with its full mark scheme open. Every question comes with a step-by-step mark scheme (M, A and B marks) once you start practising.

Question 1easy3 marks

Prove algebraically that the sum of any three consecutive integers is a multiple of $3$.

Mark scheme: in the 7-day free trial.

Question 2medium4 marks

Prove by contradiction that $\sqrt{5}$ is an irrational number.

Full worked solution Free, no sign-in

M1 for assuming the opposite: $\sqrt{5}$ is rational, so it can be written as $\frac{a}{b}$ where $a$ and $b$ are integers with no common factors.
M1 for squaring both sides to get $5 = \frac{a^2}{b^2}$, leading to $a^2 = 5b^2$, and deducing that $a^2$ is a multiple of $5$, so $a$ must also be a multiple of $5$.
M1 for letting $a = 5k$, substituting to get $(5k)^2 = 5b^2 \implies 25k^2 = 5b^2 \implies b^2 = 5k^2$, and deducing $b$ must also be a multiple of $5$.
A1 for stating that both $a$ and $b$ being multiples of $5$ contradicts the assumption that they have no common factors, hence $\sqrt{5}$ must be irrational.

Every question here has a full worked solution. See them all →

Question 3hard6 marks

Prove by contradiction that there are an infinite number of prime numbers.

Mark scheme: in the 7-day free trial.

Questions are original and written in the style of Edexcel 9MA0, and cover content that AQA and OCR examine too. They are not past-paper questions.

Revision notes: Proof: deduction, exhaustion and counter-example · Proof by contradiction

In the practice engine

Skills you can practise

These skills sit in the Pure I unit of the UK A Level practice engine. Pick any mix of them, set the difficulty, and check each answer against its mark scheme.

Proof by deductionProof by exhaustionDisproof by counter-exampleProof by contradiction

Open Pure I practice

Past papers

Past paper questions on proof

Proof can come up on these papers (from each board's paper structure). We haven't mapped individual questions yet, so open a recent paper and look for it.

Board	Recent papers that can test it
Edexcel 9MA0	Paper 1: June 2025, June 2024, June 2023 Paper 2: June 2025, June 2024, June 2023
AQA 7357	Paper 1: June 2025, June 2024, June 2023 Paper 2: June 2025, June 2024, June 2023 Paper 3: June 2025, June 2024, June 2023
OCR H240	Paper 1: June 2025, June 2024, June 2023 Paper 2: June 2025, June 2024, June 2023 Paper 3: June 2025, June 2024, June 2023

Papers are © the exam boards; each page links to the official question paper and mark scheme. All A Level Maths past papers →

Proof: common questions

Is proof on the AQA, Edexcel and OCR A Level Maths exams?

Yes. Proof is part of the A Level Maths content that every board examines: AQA section A, Edexcel Pure 1, OCR 1.01. It is assessed in AQA Papers 1, 2 and 3; Edexcel Paper 1 and Paper 2; OCR A Paper 1, and the pure sections of Papers 2 and 3.

Is proof Year 12 or Year 13 content?

Both. The AS content, usually taught in Year 12, is: Proof by deduction, including simple algebraic proofs about odd, even and consecutive integers; Proof by exhaustion: checking every case; Disproof by counter-example. The full A Level adds: Proof by contradiction, including the irrationality of √2 and the infinitude of primes.

How do I write a proof that gets full marks?

State what you are assuming, show every algebraic step, and finish with a concluding sentence that says what has been proved. Examiners withhold the final mark when the conclusion is missing, even if the algebra is right.

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