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4PM1/01 Nov 2023, Question 9: Binomial Series for Any Rational Power

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 11 marks

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The question, in short

Expand a binomial with a negative fractional power up to the x⁶ term, then use a related series with unknown prime constants to find them and estimate a definite integral.

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

Part (a) · 3 marks

Write the binomial expansion of (1 − 8x²)^(−1/2) in ascending powers of x through x⁶, with integer coefficients.

  1. 1(1 − 8x²)^(−1/2) = 1 + (−1/2)(−8x²) + (−1/2)(−3/2)(−8x²)²/2! + (−1/2)(−3/2)(−5/2)(−8x²)³/3!
  2. 2= 1 + 4x² + 24x⁴ + 160x⁶

Answer1 + 4x² + 24x⁴ + 160x⁶

Part (b) · 4 marks

Given the x³ and x⁴ terms of g(x) are 20x³ and 48x⁴, determine the prime values of a and b.

  1. 1g(x) = (a + bx)(1 + 4x² + 24x⁴ + …)
  2. 2= a + bx + 4ax² + 4bx³ + 24ax⁴ + …
  3. 34b = 20, so b = 5
  4. 424a = 48, so a = 2

Answera = 2, b = 5

Part (c) · 4 marks

Using the first five terms of g(x)'s expansion, estimate the integral of g(x) from 0 to 0.2 to 4 significant figures.

  1. 1g(x) ≈ 2 + 5x + 8x² + 20x³ + 48x⁴
  2. 2∫ from 0 to 0.2 = [2x + 5x²/2 + 8x³/3 + 5x⁴ + 48x⁵/5] from 0 to 0.2
  3. 3= 0.4 + 0.1 + 0.021333… + 0.008 + 0.003072
  4. 4= 0.5324 (4 s.f.)

Answer0.5324

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