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, for all non-zero real numbers $a$ and $b$ , if $a < b$ then $\dfrac{1}{a} > \dfrac{1}{b}$ .

(1 mark)
(b) Replace $\square$ in each of the following with $\Rightarrow$ , $\Leftarrow$ or $\Leftrightarrow$ so that each statement is fully correct.

(i) $(x - 3)(x + 5) = 0 \quad \square \quad x = -5$

(ii) $x^3 < -27 \quad \square \quad x < -3$

(iii) $\cos x° = -\tfrac{1}{2} \quad \square \quad x = 120$

(3 marks)
Level 3 · 4 marksTry it
2
(a) Show, by means of a counter example, that the following statement is not true.

"For every $n \in \mathbb{N}$ , the number $n^2 - n + 17$ is prime."

(2 marks)
(b) Using algebra, prove by exhaustion that there is no $n \in \mathbb{N}$ for which $n^2 + 2n + 3$ is a multiple of 4

(4 marks)
Level 3 · 6 marksTry it
3
(a) Consider the statement

"Whenever $n^2$ is a multiple of 8, $n$ must also be a multiple of 8"

Give a counter example to show that this statement is false.

(1 mark)
(b) Given that $n$ is an odd integer, use algebra to prove that

$n^2 + 4n + 7$

is a multiple of 4 but is not a multiple of 8

(4 marks)
Level 3 · 5 marksTry it
4
Disprove, by counter-example: "for all real numbers $a$ and $b$ , if $a^2 = b^2$ then $a = b$ ".
Level 3 · 2 marksTry it
5
Disprove, by counter-example: "the product of two irrational numbers is always irrational".
Level 3 · 2 marksTry it

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