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WMA12/01 Oct 2021, Question 1: The Binomial Expansion

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 6 marks

The question, in short

Given the first three terms 1, −4x and px² of the expansion of (1 + kx)¹⁶, determine k and p, then use k to find the x² term of g(x) = (2 + 16/x)(1 + kx)¹⁶.

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Part (ai) · 2 marks

Work out the constant k from the x-coefficient of the first three terms.

  1. 1The x term of (1 + kx)¹⁶ is 16kx
  2. 216k = −4
  3. 3k = −1/4

Answerk = −1/4

Part (aii) · 1 mark

Work out the constant p from the x² coefficient of the first three terms.

  1. 1x² term: (16×15/2)k²x² = 120k²x²
  2. 2p = 120 × (−1/4)² = 120/16
  3. 3p = 15/2

Answerp = 15/2

Part (b) · 3 marks

With the k from part (a), expand g(x) and state its x² term.

  1. 1x² terms come from 2 × (15/2)x² and (16/x) × (x³ term)
  2. 2x³ term of (1 − x/4)¹⁶: (16×15×14/3!)(−1/4)³x³ = −35/4 x³
  3. 32 × (15/2)x² = 15x²
  4. 4(16/x) × (−35/4)x³ = −140x²
  5. 5Term in x² = 15x² − 140x² = −125x²

Answer−125x²

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