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
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The curves $C_1$ and $C_2$ have equations

$y = x^4 + 7x^2 - 4 \qquad \text{and} \qquad y = x^2 - 12$

respectively, where $x \in \mathbb{R}$ .

Use algebra to prove, by contradiction, that there is no point that lies on both $C_1$ and $C_2$ .

(4 marks)
Level 3 · 4 marksTry it
2
The curve $C$ has equation

$y = x^5 + 4x - 3\sin x \qquad x \in \mathbb{R}$

Use proof by contradiction to prove that $C$ has no stationary points.

(4 marks)
Level 3 · 4 marksTry it
3
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Given that $180^\circ < x < 270^\circ$ , prove by contradiction that

$\tan x + \cot x \geqslant 2$

(4 marks)
Level 3 · 4 marksTry it
4
Prove by contradiction, using algebra, that if $n$ is an integer and $n^3 - 6n + 7$ is odd, then $n$ is even.

(4 marks)
Level 3 · 4 marksTry it
5
A student is proving by contradiction that

“there are no positive integers $x$ and $y$ for which $xy + 3x + 2y = 32$ ”

The first part of the student's proof is shown below.

Suppose that positive integers $x$ and $y$ exist with $xy + 3x + 2y = 32$

Adding $6$ to both sides gives $(x + 2)(y + 3) = 38$

If $x + 2 = 1$ and $y + 3 = 38$ , then $x = -1$ and $y = 35$ $\quad$ $x$ is not positive

Show the calculations and statements that are needed to complete the proof.

(4 marks)
Level 3 · 4 marksTry it

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