Proof opens the Pure specification, but it rarely stays confined to one question. It comprises four argument styles — deduction, exhaustion, disproof by counterexample, and (from year two) contradiction — that examiners expect wherever "prove" or "show that" appears, in algebra, trigonometry or sequences questions alike. As a standalone topic it's typically worth only around 6 of the 300 Pure marks, but that understates its reach: shaky reasoning costs marks elsewhere too.
Deduction — a logical chain from what's given to what's required — is the default, used from year one in work like completing the square to show an expression stays positive. Exhaustion checks every case in a finite set rather than arguing generally; one counterexample disproves a universal claim outright. Contradiction, from year two, means assuming the opposite of what you want and showing that assumption collapses, as with √2's irrationality and the infinitude of primes.
The specification statements this topic covers. AS = Year-1 content, also assessed in the standalone AS course (8MA0); A2 = full A level only. Typical share of a 300-mark series: ≈6 marks — our estimate from the 2018–2025 papers, not an official weighting.
| Ref | Spec statement | Level |
|---|---|---|
| 1.1 | Proof by deduction | AS |
| 1.1 | Proof by exhaustion | AS |
| 1.1 | Disproof by counterexample | AS |
| 1.1 | Proof by contradiction | A2 |
Because the target is printed, examiners can see exactly where an argument turns circular or a step doesn't follow from the last. These answers are often marked cso — correct solution only — so one unjustified jump loses the line's mark even if the final expression matches. Write out every algebraic move, including the "obvious" one.
Examiners repeatedly report candidates doing correct algebra then stopping short — reaching k − 12 = 0 and writing k = 12 without the step, or leaving an expression unequated to zero when the argument needs it. The mark sits on the last line reading exactly what was asked, so finish by stating the conclusion.
A recurring weakness is substituting a few values — n = 1, 2, 3 — and treating that as proof for all n, when it only proves what it tests. Deduction needs an argument holding for an arbitrary case; exhaustion needs every case of a genuinely finite set, not a sample.
The method only works if the negation is stated precisely at the start — for √2, that means a/b already in lowest terms, not just "a fraction". Marks are also lost when the argument reaches a contradiction but never says which assumption it contradicts.
Manipulating the printed result on both sides at once effectively assumes what you're trying to show. Work from one side, or the given information, so each line follows genuinely from the last rather than rearranging the target.
Stopping at the final piece of algebra without stating the required equation or conclusion in words hasn't answered the question, however correct the working. Check your last line could stand alone as a direct answer to "prove that…".
Trying three or four values of n and concluding a statement holds "for all n" isn't valid deduction, and isn't exhaustion unless those really are every case of a stated finite set. If the set is small and finite, check it in full; otherwise argue with a general variable throughout.
Assuming only part of the opposite statement, or the original statement by accident, undermines the argument before it starts. Write the exact negation as your opening line, including any condition — lowest terms, smallest counterexample — the statement implies.
Some candidates build a long general argument, or offer several examples, when one value meeting the statement's conditions but breaking its conclusion is already a complete disproof. Check the value meets the conditions, then stop.
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There's little to memorise for proof itself — it's a way of arguing, not a formula — so revision time is better spent on the standard set pieces you're expected to reproduce: a completing-the-square deduction, an exhaustion argument over a small finite set, a counterexample search, and the two classic contradiction proofs for √2 and the infinitude of primes.
Because proof rarely appears in isolation, the most useful practice is spotting "prove", "show that" and "hence" instructions inside ordinary algebra and trigonometry questions and treating each as a mini proof-writing exercise: state what's given, show every step, finish with the literal conclusion asked for. There's no calculator role here — mark your own attempts strictly for how marks vanish: an unjustified jump, or a final line that doesn't quite say what was asked.
All 19 topics: Edexcel A level Maths topic guides. Reference: formula booklet vs memorise and grade boundaries.
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Original questions written for the Pearson Edexcel A Level Mathematics (9MA0) specification. Not affiliated with or endorsed by Pearson.