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WMA12/01 Jan 2025, 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

Question tests irrational products and odd-integer expressions: find a counterexample to a claim about xy, then prove n³ + 3n + 2 is even but not divisible by 4 for odd n.

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

Part (i) · 1 mark

Give a counterexample with two different irrational numbers x and y whose product xy is not irrational.

  1. 1Choose x = √2 and y = √8 (both irrational, x ≠ y)
  2. 2xy = √16 = 4
  3. 34 is rational, so the statement is not always true

AnswerFor example x = √2, y = √8: both irrational, x ≠ y, but √2 × √8 = 4, which is rational

Part (ii) · 4 marks

Using algebra, prove that for every odd integer n, n³ + 3n + 2 is even and is never a multiple of 4.

  1. 1Let n = 2k + 1, where k is an integer
  2. 2n³ + 3n + 2 = (2k + 1)³ + 3(2k + 1) + 2
  3. 3= 8k³ + 12k² + 6k + 1 + 6k + 3 + 2 = 8k³ + 12k² + 12k + 6
  4. 4= 4(2k³ + 3k² + 3k + 1) + 2
  5. 5This is 2 more than a multiple of 4, so it is even but never a multiple of 4

AnswerShown: with n = 2k + 1, n³ + 3n + 2 = 8k³ + 12k² + 12k + 6 = 4(2k³ + 3k² + 3k + 1) + 2, which is even but not a multiple of 4

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