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WMA11/01 Jun 2024, Question 2: Solving Equations with Surds

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 6 marks

The question, in short

Given m = 2ⁿ, rewrite two expressions in terms of m, then solve x√3 − 3 = x + √3 exactly, giving the answer as p + q√3 with integers p and q.

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

Part (ia) · 1 mark

Express 2ⁿ⁺³ in simplest form using m.

  1. 12ⁿ⁺³ = 2ⁿ × 2³
  2. 2= 8m

Answer8m

Part (ib) · 2 marks

Express 16³ⁿ in simplest form using m.

  1. 116 = 2⁴, so 16³ⁿ = (2⁴)³ⁿ = 2¹²ⁿ
  2. 22¹²ⁿ = (2ⁿ)¹² = m¹²

Answerm¹²

Part (ii) · 3 marks

Solve x√3 − 3 = x + √3 showing all working, answer as p + q√3 with integers p, q.

  1. 1x√3 − x = 3 + √3
  2. 2x(√3 − 1) = 3 + √3, so x = (3 + √3)/(√3 − 1)
  3. 3Multiply top and bottom by (√3 + 1): x = (3 + √3)(√3 + 1)/((√3 − 1)(√3 + 1))
  4. 4= (3√3 + 3 + 3 + √3)/2 = (6 + 4√3)/2
  5. 5x = 3 + 2√3

Answerx = 3 + 2√3

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