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WMA12/01 Oct 2024, Question 5: The Binomial Expansion

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 7 marks

The question, in short

Expand a binomial with a symbolic coefficient up to the x² term, then use a given constant term from a related product to determine the constant a.

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

Part (a) · 3 marks

Write the first three terms, in increasing powers of x, of the expansion of (2 + ax)⁶, simplifying each term and leaving answers in terms of a.

  1. 1(2 + ax)⁶ = 2⁶ + 6C1 × 2⁵(ax) + 6C2 × 2⁴(ax)² + …
  2. 2= 64 + 6 × 32ax + 15 × 16a²x²
  3. 3= 64 + 192ax + 240a²x²

Answer64 + 192ax + 240a²x²

Part (b) · 4 marks

Using f(x) = (3 + 1/x)²(2 + ax)⁶ and the fact that its constant term equals 576, work out the value of a.

  1. 1(3 + 1/x)² = 9 + 6/x + 1/x²
  2. 2Constant term: 9 × 64 + (6/x)(192ax) + (1/x²)(240a²x²) = 576 + 1152a + 240a²
  3. 3576 + 1152a + 240a² = 576, so 240a² + 1152a = 0
  4. 4a ≠ 0, so 240a + 1152 = 0
  5. 5a = −24/5

Answera = −24/5

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