A-level · Proof

Proof by deductionExam-style questions

Direct proof — starting from things known to be true and reasoning in valid algebraic steps to the conclusion. Representing the general case (2n, 2n+1, consecutive integers), "show that" manipulations, completing the square for positivity, and always finishing with a clear conclusion.

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

DifficultyAll Level 3 Level 4 Level 5

1
Aled states that, whenever $n$ is a positive multiple of $3$ , the number $2^n + 1$ is not prime.
(a) Prove that Aled's statement is correct.

(4 marks)

Bethan goes further, stating that $2^n + 1$ is not prime for any positive integer $n$ .
(b) Show that Bethan's statement is wrong.

(1 mark)
Level 4 · 5 marksTry it
2
In this question $n$ is an integer greater than $1$ .
(a) Expand and simplify $(n^2 + n + 1)(n^2 - n + 1)$ .

(1 mark)
(b) Hence prove that $n^4 + n^2 + 1$ is the product of two different integers, each of which is greater than $1$ .

(5 marks)
Level 4 · 6 marksTry it
3
In this question you must show detailed reasoning.
(a) Prove that, for all positive numbers $x$ and $y$ with $x \neq y$ ,

$x^3 + y^3 > xy(x + y)$

(4 marks)
(b) Show, by means of a counter example, that the inequality in part (a) does not hold for all real values of $x$ and $y$ with $x \neq y$ .

(2 marks)
Level 4 · 6 marksTry it
4
Prove, using algebra, that

$n^4 - n^2$

is a multiple of 12 for all $n \in \mathbb{N}$

(4 marks)
Level 4 · 4 marksTry it
5
A student is asked to prove the following statement.

“For all real values of $a$ and $b$ , $a^2 + b^2 + 2 \geq 2(a + b)$ ”

The student's proof is shown below.

$a^2 + b^2 + 2 \geq 2a + 2b$

$\Rightarrow a^2 - 2a + 1 + b^2 - 2b + 1 \geq 0$

$\Rightarrow (a - 1)^2 + (b - 1)^2 \geq 0$

A square is never negative, so the statement is true.
(a) Explain the error in the student's proof.

(1 mark)
(b) Prove the statement.

(3 marks)
(c) Deduce the least possible value of $a^2 + b^2 - 2a - 2b$

(1 mark)
Level 4 · 5 marksTry it