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WMA11/01 Jan 2024, Question 4: Solving Exponential and Log Equations

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

The question, in short

Show full working without calculator aids: use the substitution p = 2ˣ to turn the given exponential equation into a quadratic, then solve the original equation.

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

Part (a) · 3 marks

Substitute p = 2ˣ into 2 × 4ˣ − 2ˣ⁺³ = 17 × 2ˣ⁻¹ − 4 and show it becomes 4p² − 33p + 8 = 0.

  1. 14ˣ = (2ˣ)² = p², 2ˣ⁺³ = 8 × 2ˣ = 8p, 2ˣ⁻¹ = p/2
  2. 22p² − 8p = 17p/2 − 4
  3. 34p² − 16p = 17p − 8
  4. 44p² − 33p + 8 = 0

AnswerShown: 2p² − 8p = 17p/2 − 4, so 4p² − 16p = 17p − 8, giving 4p² − 33p + 8 = 0

Part (b) · 3 marks

Using your quadratic in p, solve 2 × 4ˣ − 2ˣ⁺³ = 17 × 2ˣ⁻¹ − 4 for x.

  1. 1(4p − 1)(p − 8) = 0 ⇒ p = 1/4 or p = 8
  2. 22ˣ = 1/4 = 2⁻² ⇒ x = −2
  3. 32ˣ = 8 = 2³ ⇒ x = 3

Answerx = −2, x = 3

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