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
Use proof by contradiction to prove that $\sqrt{3}$ is irrational. (You may use the result that if $a^2$ is a multiple of $3$ then $a$ is a multiple of $3$ .)
Level 5 · 5 marksTry it
2
Use proof by contradiction to show that there is no smallest positive rational number.
Level 5 · 3 marksTry it
3
Use proof by contradiction to show that $\log_{2} 3$ is irrational. (5 marks)
Level 5 · 5 marksTry it
4
Given that $x$ is a real number such that

$x^3 + x = 12$
(a) use proof by contradiction to prove that $x < 3$

(4 marks)
(b) Show that $2k^2 - 12k + 21$ is positive for all real values of $k$

(2 marks)
Level 5 · 6 marksTry it
5
(a) Prove, using algebra, that $5^{n+1} - 5^n$ is a multiple of 20 for all positive integers $n$

(3 marks)
(b) A student was asked to use proof by contradiction to prove the following statement.

“For all integers $n$ , if $n^2$ is a multiple of 4 then $n$ is even”

The start of the student's proof is shown below.

Assume that $n^2$ is not a multiple of 4 and that $n$ is odd.

$\Rightarrow n = 2k + 1$ for some integer $k$

$\Rightarrow n^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1$

Explain why the student's proof cannot be completed.

(1 mark)
(c) Use proof by contradiction to prove the statement.

(3 marks)
Level 5 · 7 marksTry it