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WMA12/01 Jun 2019, Question 3: 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 using algebra, and disprove a claim about 2ⁿ + 1 being prime for all natural numbers n.

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

Part (i) · 3 marks

Use algebra to show (x − 4)² ≥ 2x − 9 holds for every real x.

  1. 1(x − 4)² − (2x − 9) = x² − 8x + 16 − 2x + 9
  2. 2= x² − 10x + 25
  3. 3= (x − 5)²
  4. 4(x − 5)² ≥ 0 for all real x because a square is never negative
  5. 5So (x − 4)² ≥ 2x − 9 for all real x

AnswerShown: (x − 4)² − (2x − 9) = x² − 10x + 25 = (x − 5)² ≥ 0 for all real x, so (x − 4)² ≥ 2x − 9

Part (ii) · 1 mark

Show the statement that 2ⁿ + 1 is prime for all n in ℕ is false.

  1. 1Try n = 3: 2³ + 1 = 8 + 1 = 9
  2. 29 = 3 × 3, so it is not prime, and the statement is untrue

AnswerShown: n = 3 gives 2³ + 1 = 9 = 3 × 3, which is not prime

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Also searched as: WMA12 June 2019 · WMA12/01 · wma12 jun19 · IAS Maths Pure Mathematics 2 June 2019 · question 3.
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