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WMA12/01 Oct 2022, Question 2: The Binomial Expansion

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

The question, in short

Show a constant equals 2/5 using a remainder, expand a binomial to three terms, then find a gradient at x = 0.

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

Part (a) · 2 marks

Show k = 2/5 using the remainder 243/32 when f(x) = (2 − kx)⁵ is divided by 4x − 5.

  1. 1Remainder theorem: remainder = f(5/4)
  2. 2(2 − 5k/4)⁵ = 243/32
  3. 3Take the fifth root: 2 − 5k/4 = 3/2
  4. 45k/4 = 1/2, so k = 2/5

AnswerShown: (2 − 5k/4)⁵ = 243/32 ⇒ 2 − 5k/4 = 3/2 ⇒ k = 2/5

Part (b) · 3 marks

Expand (2 − (2/5)x)⁵ in ascending powers of x, giving the first three terms in simplest form.

  1. 1(2 − (2/5)x)⁵ = 2⁵ + 5C1 × 2⁴ × (−(2/5)x) + 5C2 × 2³ × (−(2/5)x)² + …
  2. 2First term: 32
  3. 3x term: 5 × 16 × (−2/5)x = −32x
  4. 4x² term: 10 × 8 × (4/25)x² = (64/5)x²
  5. 5So 32 − 32x + (64/5)x²

Answer32 − 32x + (64/5)x²

Part (c) · 2 marks

Use the expansion from part (b) to find the gradient of C at x = 0.

  1. 1f(x) ≈ 32 − 32x + (64/5)x²
  2. 2f′(x) ≈ −32 + (128/5)x
  3. 3At x = 0: gradient = −32

Answer−32

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