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
3
(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
4
(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
5
(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

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