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WMA12/01 Jan 2022, Question 3: 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 constant k to four ascending terms, then use a constant-term condition in a related product to determine k.

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

Part (a) · 4 marks

Write the first four terms in ascending powers of x for the expansion of (2 − kx/4)⁸, simplifying each term.

  1. 1(2 − kx/4)⁸ = 2⁸ + ⁸C₁ 2⁷(−kx/4) + ⁸C₂ 2⁶(−kx/4)² + ⁸C₃ 2⁵(−kx/4)³ + …
  2. 2= 256 + 8 × 128 × (−kx/4) + 28 × 64 × (k²x²/16) + 56 × 32 × (−k³x³/64) + …
  3. 3= 256 − 256kx + 112k²x² − 28k³x³ + …

Answer256 − 256kx + 112k²x² − 28k³x³

Part (b) · 3 marks

Given f(x) = (5 − 3x)(2 − kx/4)⁸ and that the constant term is 3 times the x coefficient, work out k.

  1. 1Constant term of f(x) = 5 × 256 = 1280
  2. 2Coefficient of x = 5 × (−256k) − 3 × 256 = −1280k − 768
  3. 31280 = 3(−1280k − 768)
  4. 41280 = −3840k − 2304, so 3840k = −3584
  5. 5k = −14/15

Answerk = −14/15

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