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

Short answer: To get full marks on 9709 proof questions, treat every line as a marked step. Start from a definition or known result, justify each move with algebra you actually write down, and finish with a clear concluding statement. Most lost marks come from missing steps or a jump in logic, not from a wrong final answer, especially on "show that" questions where the answer is already printed.

What proof means in 9709

In Cambridge A-Level Mathematics (9709), proof is not a separate topic you can revise once and forget. It runs through the whole Pure syllabus. You will meet it as "show that", "prove that", "hence prove", and "deduce" questions inside trigonometry, algebra, series, and coordinate geometry.

The two styles you must be fluent in are:

• Proof by deduction: you build a logical 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 a single case that breaks it.

Knowing which one the question wants is half the battle.

Why students lose marks

The examiner is marking your reasoning, not just your final line. On a "show that" question the answer is handed to you, so every mark lives in the working. Common ways to drop marks:

Mistake	Why it costs marks	Fix
Skipping algebra steps	Method marks reward shown working	Write each rearrangement in full
Working backwards from the answer	Circular logic proves nothing	Start from a known fact, move towards the result
No statement of assumptions	Examiner cannot follow your logic	Say what you start from
No conclusion	The proof feels unfinished	End with "as required" or similar
Using the thing you are proving	You cannot assume what you must prove	Only use established results

Proof by deduction: a worked structure

Suppose you are asked to show that an expression simplifies to a given identity. The reliable structure is:

1. State your starting point. Pick one side of the identity, usually the more complicated one, and say you will work on it.
2. Transform one step at a time. Apply one definition or identity per line, such as $\sin^2\theta + \cos^2\theta = 1$ or $\tan\theta = \frac{\sin\theta}{\cos\theta}$.
3. Keep the target in view. Each step should move you closer to the printed answer.
4. Conclude. Write a short line like "hence the identity is proved" so the examiner sees you have finished.

A generic trigonometric example: to 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 $\frac{\sin^2\theta}{\cos\theta}$. Splitting this as $\sin\theta \times \frac{\sin\theta}{\cos\theta}$ gives $\sin\theta\tan\theta$, which equals the right side. Then write your closing line. Every one of those moves is a potential mark, so none should be skipped.

Disproof by counterexample

If a question says "prove or disprove" or asks whether a statement is always true, one clean counterexample is worth full marks. You do not need to test many cases. You need one that clearly fails.

For example, to disprove the claim "for all real $x$, $x^2 > x$", pick $x = \tfrac{1}{2}$. Then $x^2 = \tfrac{1}{4}$, which is not greater than $\tfrac{1}{2}$. State the value, show the calculation, and write "so the statement is false". That is the whole answer.

The discipline here is: name your counterexample, show the numbers, and state the conclusion. Do not leave the marker to infer why it works.

Discriminant proofs

A very common 9709 proof asks you to show an equation has two distinct real roots, equal roots, or no real roots. This is a deduction proof using the discriminant $b^2 - 4ac$ of a quadratic $ax^2 + bx + c = 0$:

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

To score full marks, always write which case you are proving, substitute $a$, $b$, and $c$ correctly, simplify fully, and finish by stating the inequality holds and therefore the conclusion follows. If the result depends on a condition such as "for all $k$", make sure your final inequality genuinely covers every allowed value.

A checklist for every proof question

Before you move on, run through this:

• Did I identify deduction versus disproof correctly?
• Did I start from something known, not the thing being proved?
• Is every algebraic step written out, not done in my head?
• For a "show that", does my last line match the printed answer exactly?
• Did I write a clear concluding sentence?

If you can tick all five, you have earned the marks that reasoning-heavy questions carry.

Practising with feedback

Proof improves fastest when someone checks your logic line by line, because the gaps are usually invisible to the person who wrote them. ExamPal is an AI tutor that already knows the 9709 syllabus and marks your written proof against the Cambridge mark scheme, so it can point to the exact step where a mark would be lost and explain why. It is a low-pressure way to spot the jumps in your reasoning before an examiner does.