Proof

Section 1

Both years

1 specification point

Notes and three levels of exam-style practice for each registered specification point in this section.

Checked against Edexcel 9MA0 section 1

Checked against Edexcel 9MA0 section 1. Review basis: the qualification registry sourced from the Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0) specification; registry verification recorded 11 July 2026.

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1.1

Understand and use the structure of mathematical proof, from assumptions through logical steps to a conclusion; use proof by deduction, exhaustion, disproof by counter example, and contradiction (irrationality of √2, infinity of primes).

Notes

Worked answers & exam appearances →

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Explanation

A proof begins with stated assumptions, uses valid implications at every step and ends with a conclusion that matches the claim.
Choose the proof form deliberately: deduction for a general algebraic chain, exhaustion for finitely many cases, a counterexample to disprove a universal claim, or contradiction by assuming the opposite.
In the classic contradiction for $\sqrt2$ , writing $\sqrt2=a/b$ in lowest terms leads to both $a$ and $b$ being even, contradicting the lowest-terms assumption.
Checking several examples is not a proof of a universal statement; in contradiction proofs, also make clear exactly why the derived result conflicts with the assumption.
To prove there are infinitely many primes, assume a finite list, form their product plus one, and note that its prime divisor is not on the list, giving a contradiction.

Worked example

Prove by exhaustion that $n^2+n+2$ is even whenever $n$ is an integer.

1.Every integer is even or odd.
2.If $n=2k$ , then $n^2+n+2=4k^2+2k+2=2(2k^2+k+1)$ .
3.If $n=2k+1$ , then $n^2+n+2=4k^2+6k+4=2(2k^2+3k+2)$ .
4.Both forms are divisible by $2$ , so the result holds for every integer $n$ .

Answer: The expression is even in both the even and odd cases for $n$ .

Common mistakes

Don't attempt proof by exhaustion over a pattern with infinitely many possible cases.
Don't assume the desired conclusion inside the argument instead of deriving it from stated assumptions.

Exam tip

For an exhaustion proof, identify the finite cases, show why each case is accepted or rejected, and finish with one conclusion. A cso final mark requires the complete proof to be correct, although earlier method marks depend on the question-specific scheme.

Tier 1 · Easy

ORIGINAL

Disprove the claim that

n^2+n+41

is prime for every non-negative integer

n

(2)

(Total for Question 1 is 2 marks)

Tier 2 · Standard

ORIGINAL

Prove by contradiction that

\sqrt3

is irrational.

(5)

(Total for Question 1 is 5 marks)

Tier 3 · Hard

ORIGINAL

Suppose someone lists all primes as

p_1,p_2,\ldots,p_k

. Construct an integer from this list and use contradiction to prove that the list cannot be complete.

(5)

(Total for Question 1 is 5 marks)

Your progress and exam materials

This section: Evidence from your answers: 0/1 secureYour confidence: 0 self-rated secureTracker status: 0/1 secure, 0 shaky, 1 unseen

Overall: Evidence from your answers: 0/89 secureYour confidence: 0 self-rated secureTracker status: 0/89 secure, 0 shaky, 89 unseen

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Answer conventions

Follow the wording on the question and its mark scheme. awrt means an appropriately rounded value is accepted; an exact answer must stay as a fraction, surd, logarithm or multiple of π when required, and a rounded decimal may be disallowed. Include requested units and forms. A cso tag protects that accuracy mark, while earlier method marks follow the question-specific dependencies.

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