Year 13 proof by contradiction test to download as PDF

This Year 13 proof by contradiction test lasts 30 minutes and is marked out of 20. It covers the whole chapter for pupils aged 17-18, and the full answers are on a separate page.

⏱️ Duration : 30 min
🎯 Total : 20 marks
❓ 5 questions

Mark/20

Question 1 – Parity of squares
(a) Write the negation of the statement "\(n\) is an even integer". [1 mark] (b) Prove by contradiction that if \(n^2\) is even, where \(n\) is an integer, then \(n\) is even. [3 marks]
Question 2 – The square root of five
Prove by contradiction that \(\sqrt5\) is irrational. [4 marks]
Question 3 – No integer solutions
Prove by contradiction that there are no integers \(m\) and \(n\) such that \(12m+18n=5\). [4 marks]
Question 4 – Infinitely many primes
Complete Euclid’s proof that there are infinitely many primes. (a) Assume that there are finitely many primes \(p_1,p_2,\dots,p_k\). Define a number \(N\) which gives a contradiction. [1 mark] (b) Show that \(N\) is not divisible by any \(p_i\). [2 marks] (c) State the contradiction. [1 mark]
Question 5 – Irrational products
(a) Prove by contradiction that if \(x\) is a non-zero rational number and \(y\) is irrational, then \(xy\) is irrational. [3 marks] (b) Hence state, with a reason, whether \(\dfrac{\sqrt2}{3}\) is rational. [1 mark]

How to use this Year 13 proof by contradiction test

First, revise with the proof by contradiction lesson and the worksheet. Then sit the test in one go, with a timer set to 30 minutes. After that, mark your work with the proof by contradiction test answers. If you lose marks on a question, redo it the next day without help.

Finally, you can download this Year 13 proof by contradiction test as a PDF and print it like a real exam paper. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday.