How to Get Full Marks on A-Level Maths Proof Questions (9709)

To get full marks on proof questions in A Level Maths 9709, treat every line as something an examiner can tick. Start from a fact you are allowed to use, write each algebraic step in full, and finish with a short concluding statement. Most marks are lost to missing steps and jumps in logic, not to wrong final answers, especially on "show that" questions where the answer is already printed.

What proof means in 9709

There is no single "proof" chapter to revise and tick off. The 9709 syllabus describes mathematical proof and reasoning as something that runs through the course: your solution should be written so that someone else can follow each line and confirm it is complete and accurate. In the Pure Mathematics papers you meet this as "show that", "hence", "prove" and "deduce" inside algebra, trigonometry, series and coordinate geometry.

Two styles cover most of what you will need:

• Proof by deduction: you build a chain from something known (a definition, an identity, a given equation) to the required result.
• Disproof by counterexample: you show a statement is false by finding one case that breaks it.

Cambridge's own wording for the command word is useful. "Show (that)" means to provide structured evidence that leads to a given result. The word that matters there is structured.

Why marks go missing

The examiner is marking your reasoning. On a "show that" question you are handed the answer, so every mark lives in the working.

If you want to see how marks are labelled and awarded, how Cambridge mark schemes work explains the method and accuracy codes.

Proof by deduction: a reliable structure

1. State your starting point. Pick one side of an identity, usually the more complicated one, and say you will work on it.
2. Change one thing per line. Apply one identity or definition at a time, such as \sin^2\theta + \cos^2\theta = 1 or \tan\theta = \frac{\sin\theta}{\cos\theta}.
3. Keep the target in view. Each line should bring you closer to the printed result.
4. Close it. Write a short line such as "hence the identity is proved".

An example in the style of an exam question: show that \frac{1 - \cos^2\theta}{\cos\theta} = \sin\theta\tan\theta.

Start with the left side. Since 1 - \cos^2\theta = \sin^2\theta, the left side becomes \frac{\sin^2\theta}{\cos\theta}. Write this as \sin\theta \times \frac{\sin\theta}{\cos\theta}, which is \sin\theta\tan\theta, the right side. Then add your closing line. Each of those moves is a point an examiner may be looking for, so none of them should be done in your head.

Proving something is always positive

A common style asks you to show that an expression is always positive, or never zero. Completing the square is the usual route.

Show that x^2 - 6x + 10 > 0 for all real x.

Complete the square: x^2 - 6x + 10 = (x - 3)^2 + 1. A squared real number is never negative, so (x-3)^2 \ge 0, which means (x-3)^2 + 1 \ge 1. Therefore the expression is always greater than 0.

Notice the middle sentence. The logic is "a square is at least zero", and that reason is part of the proof. Writing only the completed square and "so it is positive" leaves the key idea unsaid.

Disproof by counterexample

If a question asks whether a statement is always true, one clean counterexample settles it. You do not need to test many cases.

To disprove "for all real x, x^2 > x", pick x = \tfrac{1}{2}. Then x^2 = \tfrac{1}{4}, which is not greater than \tfrac{1}{2}. So the statement is false.

Name the value, show the calculation, and state the conclusion. Do not leave the examiner to work out why it fails. Also choose your counterexample so the arithmetic is easy to check. A value like x = 0 or x = 1 would also work here, and simple numbers leave less room for slips.

Discriminant proofs

Many questions ask you to show that a quadratic has two distinct real roots, equal roots or no real roots. For ax^2 + bx + c = 0 the discriminant is b^2 - 4ac:

• Two distinct real roots when b^2 - 4ac > 0
• Equal roots when b^2 - 4ac = 0
• No real roots when b^2 - 4ac < 0

Example: show that x^2 + kx - 1 = 0 has two distinct real roots for every real value of k.

Here a = 1, b = k, c = -1, so b^2 - 4ac = k^2 + 4. Since k^2 \ge 0 for every real k, we have k^2 + 4 \ge 4 > 0. The discriminant is positive for all k, so there are always two distinct real roots.

The phrase "for every real k" is the part students forget to cover. Your final inequality must hold for every allowed value, and you should say why.

A checklist for each proof

• Did I decide whether this is deduction or disproof?
• Did I start from something known, not from what I am trying to prove?
• Is every algebraic step on the page?
• For a "show that", does my last line match the printed result exactly?
• Did I write a closing sentence, and does my final inequality or statement cover every case the question allows?

Practising proof properly

Proof is hard to self-check, because gaps in your own logic are invisible to you. The most useful habit is to write a full solution by hand, then compare it line by line with the mark scheme and ask which marks you would really have been given. Command words explained is worth reading alongside this, because the exact verb in the question tells you how much justification is expected.

Try proof questions on a real paper

Pick a paper from the A Level Mathematics past papers, write your solutions by hand and have them marked against the official mark scheme, with each mark traced back to the line that earned it. Reading and downloading papers is free, and a free account adds AI marking. The 2023 and 2024 papers are marked first, with earlier years being added, so check which papers are available before you plan around a particular year.