Lemmara logoLemmaraWorked solution

WMA12/01 Jan 2022, Question 10: Proof by Exhaustion and Counter-example

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 5 marks

The question, in short

Show a prime p where 2p + 1 is not prime, then prove by exhaustion that 5n² + n + 12 is even for integer n.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (i) · 1 mark

Give one prime p for which 2p + 1 is not prime, showing the value fails.

  1. 1Try p = 7 (prime): 2p + 1 = 15
  2. 215 = 3 × 5 is not prime, so the statement is false

AnswerShown: e.g. p = 7 is prime but 2p + 1 = 15 = 3 × 5 is not prime

Part (ii) · 4 marks

Check all cases of n to show 5n² + n + 12 is always even.

  1. 1Every integer is even (n = 2k) or odd (n = 2k + 1)
  2. 2n = 2k: 5(2k)² + 2k + 12 = 20k² + 2k + 12 = 2(10k² + k + 6), which is even
  3. 3n = 2k + 1: 5(2k + 1)² + 2k + 1 + 12 = 20k² + 22k + 18 = 2(10k² + 11k + 9), which is even
  4. 4Both cases are even, so 5n² + n + 12 is even for all integers n

AnswerShown: n = 2k gives 2(10k² + k + 6) and n = 2k + 1 gives 2(10k² + 11k + 9), both even

Stuck on a step? Read the free lesson: Proof by Exhaustion and Counter-example

Practise more proof by exhaustion and counter-example questions

Real XMA01 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on proof by exhaustion and counter-example

Also searched as: WMA12 January 2022 · WMA12/01 · wma12 jan22 · IAS Maths Pure Mathematics 2 January 2022 · question 10.
Lemmara is an independent study tool, not affiliated with or endorsed by Pearson Edexcel. The worked solutions are Lemmara’s own.