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WMA12/01 Jun 2023, Question 8: Proof by Exhaustion and Counter-example

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

The question, in short

Prove a claim about consecutive primes and a² + b² false using a counterexample, then prove an inequality for even x, y between 0 and 6 by exhaustion.

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

Part (i) · 2 marks

Find consecutive primes a and b where a² + b² is a multiple of 10, disproving the statement.

  1. 17 and 11 are consecutive prime numbers
  2. 27² + 11² = 49 + 121 = 170
  3. 3170 = 10 × 17, a multiple of 10
  4. 4So the statement is not true

AnswerShown: 7 and 11 are consecutive primes and 7² + 11² = 170 = 10 × 17, a multiple of 10, so the statement is not true

Part (ii) · 3 marks

For even x and y with 0 < x, y < 6, check all cases to show 1 < x² − xy/4 < 15.

  1. 1x and y can only be 2 or 4
  2. 2x = 2, y = 2: 4 − 1 = 3; x = 2, y = 4: 4 − 2 = 2
  3. 3x = 4, y = 2: 16 − 2 = 14; x = 4, y = 4: 16 − 4 = 12
  4. 4All four values lie between 1 and 15, so 1 < x² − xy/4 < 15 is true

AnswerShown: x, y ∈ {2, 4}; (2,2) gives 3, (2,4) gives 2, (4,2) gives 14, (4,4) gives 12, all between 1 and 15

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Also searched as: WMA12 June 2023 · WMA12/01 · wma12 jun23 · IAS Maths Pure Mathematics 2 June 2023 · question 8.
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