Last updated on Oct 23, 2026

Proof by Contradiction (AQA A-Level Mathematics): Quizzes

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Proof by Contradiction

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What is the first step in proof by contradiction?

Assume the opposite of what you want

What must you explicitly state when doing proof by contradiction?

'a contradiction'

What is the correct negation of 'all numbers are even'?

Not all numbers are even

Negate: 'all multiples of $2$ are multiples of $3$ '.

At least one multiple of $2$ isn't of $3$

To prove no greatest integer exists, what do you assume?

There exists a greatest integer $k$

Proving if $n^2$ is odd then $n$ is odd: assume $n$ is even, so $n = 2a$ . What form is $n^2$ ?

$n^2 = 2(2a^2)$ which is even

Proving if $n^3$ is even then $n$ is even: expand $(2m+1)^3$ to get $8m^3 + 12m^2 + 6m + 1$ . What form?

$2k + 1$ (odd)

A number is rational if it can be written as $rac{a}{b}$ where $a$ and $b$ are integers with what?

No common factors

Proving $sqrt{2}$ irrational: from $2b^2 = a^2$ , why is $a$ even?

$a^2$ is even, so $a$ must be even

Proving $sqrt{2}$ irrational: if both $a$ and $b$ are even, what is contradicted?

They have no common factors

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Proof

Proof by Contradiction

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Proof

Proof by Contradiction

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Proof

Proof by Contradiction

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Proof

Proof by Contradiction

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Proof by Contradiction (AQA A-Level Mathematics): Quizzes

Practise the questions

What is the first step in proof by contradiction?

What must you explicitly state when doing proof by contradiction?

What is the correct negation of 'all numbers are even'?

Negate: 'all multiples of 222 are multiples of 333'.

To prove no greatest integer exists, what do you assume?

Proving if n2n^2n2 is odd then nnn is odd: assume nnn is even, so n=2an = 2an=2a. What form is n2n^2n2?

Proving if n3n^3n3 is even then nnn is even: expand (2m+1)3(2m+1)^3(2m+1)3 to get 8m3+12m2+6m+18m^3 + 12m^2 + 6m + 18m3+12m2+6m+1. What form?

A number is rational if it can be written as rac{a}{b} where aaa and bbb are integers with what?

Proving sqrt2sqrt{2}sqrt2 irrational: from 2b2=a22b^2 = a^22b2=a2, why is aaa even?

Proving sqrt2sqrt{2}sqrt2 irrational: if both aaa and bbb are even, what is contradicted?

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Proof

Proof by Contradiction

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Proof

Proof by Contradiction

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Proof

Proof by Contradiction

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Proof

Proof by Contradiction

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Negate: 'all multiples of $2$ are multiples of $3$ '.

Proving if $n^2$ is odd then $n$ is odd: assume $n$ is even, so $n = 2a$ . What form is $n^2$ ?

Proving if $n^3$ is even then $n$ is even: expand $(2m+1)^3$ to get $8m^3 + 12m^2 + 6m + 1$ . What form?

A number is rational if it can be written as $rac{a}{b}$ where $a$ and $b$ are integers with what?

Proving $sqrt{2}$ irrational: from $2b^2 = a^2$ , why is $a$ even?

Proving $sqrt{2}$ irrational: if both $a$ and $b$ are even, what is contradicted?