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WMA11/01 Jan 2023, Question 5: 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 = 3ˣ to turn the given exponential equation into 9p² + 26p − 3 = 0, 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 = 3ˣ into 3 × 9ˣ + 3^(x+2) = 1 + 3^(x−1) and show it rearranges to 9p² + 26p − 3 = 0.

  1. 19ˣ = (3ˣ)² = p², 3^(x+2) = 9 × 3ˣ = 9p, 3^(x−1) = 3ˣ/3 = p/3
  2. 2So 3p² + 9p = 1 + p/3
  3. 3Multiply by 3: 9p² + 27p = 3 + p
  4. 49p² + 26p − 3 = 0

AnswerShown: 3p² + 9p = 1 + p/3, so 9p² + 26p − 3 = 0

Part (b) · 3 marks

Using the result from part a, solve 3 × 9ˣ + 3^(x+2) = 1 + 3^(x−1) for x.

  1. 19p² + 26p − 3 = 0, so (9p − 1)(p + 3) = 0
  2. 2p = 1/9 or p = −3; 3ˣ cannot be negative
  3. 33ˣ = 1/9 = 3^(−2), so x = −2

Answerx = −2

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