Proof — Edexcel A-Level Mathematics
Test yourself on Proof with PEARSON EDEXCEL A-Level practice questions.
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Proof explained
A mathematical proof starts from given assumptions and moves by logical steps to a conclusion.
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Deduction uses algebra or known results to show a statement is always true, for example completing the square to prove x² + 4x + 7 > 0 for all real x. Exhaustion checks every possible case, such as testing all residue classes modulo 3. Disproof by counter example gives one valid case where the statement fails. Contradiction assumes the opposite and derives an impossibility; classic examples prove that √2 is irrational and that there are infinitely many primes. You must also apply these methods to unfamiliar statements, choosing a suitable structure and writing each step so the logic is clear. Assessment rewards correct assumptions, valid implications, and a conclusion that explicitly addresses the original statement.
Your focus
- Construct a deductive proof from given assumptions to a conclusion.
- Use exhaustion to prove a statement by checking all possible cases.
- Disprove a statement by providing a valid counter example.
Show all 4 objectives
- Apply proof by contradiction, including the irrationality of √2 and the infinity of primes, to unfamiliar statements.
Proof exam tips
Marking Points
- State the given assumptions or definitions that the proof will use.
- Choose an appropriate method: deduction, exhaustion, counter example, or contradiction.
- Write a chain of logical steps where each step follows from the previous one or from a stated assumption.
- For exhaustion, confirm that every possible case has been considered.
- For contradiction, assume the negation, derive a contradiction, and conclude the original statement is true.
- For disproof, provide a specific counter example that satisfies the conditions but makes the statement false.
- End with a clear conclusion that directly answers the statement to be proved or disproved.
Examiner Tips
- 💡Plan the proof structure before writing: identify whether the statement is general or existential, and whether a counter example is enough.
- 💡Show every algebraic step and use words such as 'assume', 'therefore' and 'this contradicts' to make the logic explicit.
- 💡For irrationality of √2, set √2 = a/b in lowest terms, square, and show both a and b must be even, contradicting lowest terms.
Common Mistakes
- Assuming what you are trying to prove; correct by starting only from given assumptions and previously established results.
- Using a single example as proof of a general statement; correct by using deduction or exhaustion, or by labelling the example as a counter example only when disproving.
- In proof by contradiction, failing to state the negation clearly; correct by writing the exact opposite of the conclusion before deriving the contradiction.