A-level · Proof

Disproof by counter-exampleExam-style questions

Disproving a universal ("for all") claim with a single example where it fails — shown with the arithmetic worked out. Why one counter-example is enough to disprove, why no number of supporting examples can prove, and where to hunt for the case that breaks a statement.

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11 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4

1
SECTION A: PURE MATHEMATICS
(a) Disprove the following statement:

" $2^n + 1$ is prime for all positive integers $n$ ." (2 marks)
(b) Prove that the sum of the squares of any two consecutive odd integers is an even number that is not divisible by $4$ . (3 marks)
Level 2 · 5 marksTry it
2
A student writes the following two statements.
(a) “For all positive integers $n$ , $2^n > n^2$ ”

Prove by counter example that this statement is not true.

(2 marks)
(b) “For all real numbers $x$ , if $x^3 > x$ then $x > 1$ ”

Prove by counter example that this statement is not true.

(2 marks)
Level 2 · 4 marksTry it