Lemmara logoLemmaraWorked solution

WMA12/01 Jan 2021, Question 5: Proof by Exhaustion and Counter-example

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 4 marks

The question, in short

Prove an inequality in x for x ≥ 0 using algebra, and disprove a claim about sums of three consecutive primes.

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

Part (i) · 3 marks

Use algebra to show 3x + 1 ≥ 2√(3x) holds for every x ≥ 0.

  1. 1For x ≥ 0, √(3x) exists and is real
  2. 2Any real number squared is never negative: (√(3x) − 1)² ≥ 0
  3. 3Expand: 3x − 2√(3x) + 1 ≥ 0
  4. 4Rearrange: 3x + 1 ≥ 2√(3x), as required

AnswerShown: (√(3x) − 1)² ≥ 0 gives 3x − 2√(3x) + 1 ≥ 0, so 3x + 1 ≥ 2√(3x)

Part (ii) · 1 mark

Show the statement about three consecutive primes summing to a multiple of 5 is false.

  1. 1Try three consecutive primes: 5, 7, 11
  2. 25 + 7 + 11 = 23
  3. 323 is not a multiple of 5, so the statement is not true

AnswerCounter-example: 5 + 7 + 11 = 23, which is not a multiple of 5

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 2021 · WMA12/01 · wma12 jan21 · IAS Maths Pure Mathematics 2 January 2021 · question 5.
Lemmara is an independent study tool, not affiliated with or endorsed by Pearson Edexcel. The worked solutions are Lemmara’s own.