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WMA12/01 Jun 2019, Question 4: The Binomial Expansion

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

The question, in short

Expand (2 − ¼x)⁶ up to the x³ term, then use the small-x approximation to combine it with (2 + ¼x)⁶ and determine the constants a and b.

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

Part (a) · 4 marks

Write out the first four terms of the expansion of (2 − ¼x)⁶, giving powers of x in increasing order.

  1. 1(2 − ¼x)⁶ = 2⁶ + ⁶C₁ 2⁵(−¼x) + ⁶C₂ 2⁴(−¼x)² + ⁶C₃ 2³(−¼x)³ + ...
  2. 2= 64 + 6 × 32 × (−¼x) + 15 × 16 × (1/16)x² + 20 × 8 × (−1/64)x³
  3. 3= 64 − 48x + 15x² − 2.5x³

Answer64 − 48x + 15x² − 2.5x³

Part (b) · 3 marks

Ignoring x⁴ and higher terms, add the two expansions and find the constants a and b in a + bx².

  1. 1(2 + ¼x)⁶ = 64 + 48x + 15x² + 2.5x³ + ... (replace x by −x)
  2. 2Add: (64 − 48x + 15x² − 2.5x³) + (64 + 48x + 15x² + 2.5x³)
  3. 3The x and x³ terms cancel
  4. 4= 128 + 30x², so a = 128 and b = 30

AnswerShown: a = 128, b = 30, so the sum ≈ 128 + 30x²

Stuck on a step? Read the free lesson: The Binomial Expansion

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