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WMA12/01 Jun 2025, Question 6: Solving Exponential and Log Equations

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 10 marks

The question, in short

Solve a logarithmic equation exactly without a calculator, then find where two exponential curves intersect, giving the point in the form (log₃ a, b).

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

Part (i) · 5 marks

Solve 2log₂(4 − x) = 3 + log₂((x + 11)/2) for x, showing full working without calculator methods.

  1. 12log₂(4 − x) = log₂(4 − x)² and 3 = log₂ 8
  2. 2log₂(4 − x)² − log₂((x + 11)/2) = log₂ 8, so (4 − x)² ÷ ((x + 11)/2) = 8
  3. 3(4 − x)² = 4(x + 11) = 4x + 44
  4. 4x² − 8x + 16 = 4x + 44, so x² − 12x − 28 = 0
  5. 5(x − 14)(x + 2) = 0, so x = 14 or x = −2
  6. 6x = 14 makes 4 − x negative, so reject it: x = −2

Answerx = −2

Part (ii) · 5 marks

Curves y = 3²ˣ⁺¹ and y = 6 × 3ˣ meet at P; find exact coordinates of P as (log₃ a, b), a and b integers.

  1. 13²ˣ⁺¹ = 6 × 3ˣ
  2. 23 × (3ˣ)² = 6 × 3ˣ
  3. 3Divide by 3 × 3ˣ: 3ˣ = 2, so x = log₃ 2
  4. 4y = 6 × 3ˣ = 6 × 2 = 12
  5. 5P = (log₃ 2, 12)

AnswerP = (log₃ 2, 12)

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