A-level · Proof

Proof by contradictionExam-style questions

Assuming the opposite of what you want to prove, deriving a logical contradiction, and concluding the original must be true. Negating a statement correctly, the classic results (√2 is irrational), and why reaching an actual contradiction is the whole point. (A-level / Year 2.)

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

DifficultyAll Level 3 Level 4 Level 5

1
(a) Prove that the square of any odd integer is $1$ more than a multiple of $4$ .

(2 marks)
(b) Hence, or otherwise, use proof by contradiction to show that the equation $a^2 = 4b + 7$ has no solutions in which $a$ and $b$ are both integers.

(3 marks)
Level 4 · 5 marksTry it
2
A student is using proof by contradiction to show that, for all integers $n$ ,

“if $n^3$ is a multiple of $5$ , then $n$ is a multiple of $5$ ”

The student's working so far is shown below.

Suppose that $n$ is an integer for which $n^3$ is a multiple of $5$ but $n$ is not a multiple of $5$

When $n = 5m \pm 1$ , where $m \in \mathbb{Z}$ ,

$n^3 = 125m^3 \pm 75m^2 + 15m \pm 1 = 5(25m^3 \pm 15m^2 + 3m) \pm 1$ , which is $1$ more or $1$ less than a multiple of $5$
(a) Show the calculations and statements that are needed to complete the proof.

(3 marks)
(b) Hence use proof by contradiction to show that $\sqrt[3]{5}$ is irrational.

(5 marks)
Level 4 · 8 marksTry it
3
Use proof by contradiction to show that there is no greatest even integer.
Level 4 · 3 marksTry it
4
Use proof by contradiction to prove that $\sqrt{2}$ is irrational.
Level 4 · 5 marksTry it
5
Use proof by contradiction to show that the sum of a rational number and an irrational number is irrational.
Level 4 · 3 marksTry it

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