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
(a) Find a counterexample to disprove the statement that $n^2 + n + 11$ is prime for all positive integers $n$ .

(1 mark)
(b) Write down the symbol, $\Rightarrow$ , $\Leftarrow$ or $\Leftrightarrow$ , that should replace $\square$ in each of the following to give a fully correct statement.

(i) $n$ is an integer multiple of $6 \quad \square \quad n$ is an integer multiple of $3$

(ii) $|x| < 2 \quad \square \quad x^2 < 4$

(iii) $\ln x < 0 \quad \square \quad x < 1$

(3 marks)
(c) Prove that, for any odd integers $p$ and $q$ , $p^2 - q^2$ is divisible by $8$ .

(4 marks)
Level 4 · 8 marksTry it
2
(a) Find a counterexample to disprove the statement that $\sqrt{x^2} = x$ for all real numbers $x$ .

(1 mark)
(b) The statement $\cos x° = 0 \Leftrightarrow x = 90$ is not true.

(i) Give a value of $x$ which shows this, explaining your reasoning.

(ii) Replace the symbol $\Leftrightarrow$ with a different symbol so that the statement is true.

(2 marks)
(c) The integers $a$ , $b$ and $c$ are consecutive multiples of $5$ , with $a < b < c$ . Prove that $a + b + c$ is divisible by $15$ .

(3 marks)
Level 4 · 6 marksTry it
3
(a) Show, by means of a counter-example, that the following statement is false:

'For all positive integers $n$ , the number $2^n + 1$ is prime.' (1 mark)
(b) Prove that the square of any odd number is one more than a multiple of $8$ . (4 marks)
Level 4 · 5 marksTry it