Year 12 proof test to download as PDF

This Year 12 proof test lasts 60 minutes and is marked out of 20. It covers the whole chapter for pupils aged 16-17, and the full answers are on a separate page.

⏱️ Duration : 60 min
🎯 Total : 20 marks
❓ 5 questions

Mark/20

Question 1 – Proving with algebra
Prove that the sum of the squares of any two consecutive odd numbers is always 2 more than a multiple of 8.
1. Write down two consecutive odd numbers in terms of \(n\). (1 mark)
2. Prove that the sum of their squares is always 2 more than a multiple of 8. (3 marks)
Question 2 – Counter-example
Show by a counter-example that each statement is false.
1. "If \(n\) is prime then \(n+2\) is prime." (2 marks)
2. "\((a+b)^2=a^2+b^2\) for all \(a,b\)." (2 marks)
Question 3 – Exhaustion
Prove that for any integer \(n\), \(n^2+n\) is even, by considering the cases \(n\) even and \(n\) odd.
Question 4 – Proof by contradiction
Prove that there is no largest even number.
Question 5 – Problem: a table of seats
A hall has \(n\) rows. Row 1 has 10 seats and each row has 2 more seats than the row before.
1. Show that row \(n\) has \(2n+8\) seats. (1 mark)
2. Prove that the number of seats in every row is even. (1 mark)
3. Find the total number of seats in a hall with 15 rows. (2 marks)

How to use this Year 12 proof test

First, revise with the proof lesson and the worksheet. Then sit the test in one go, with a timer set to 60 minutes. After that, mark your work with the proof test answers. If you lose marks on a question, redo it the next day without help.

Finally, you can download this Year 12 proof 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.