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4PM1/01 Jan 2020, Question 9: Binomial Series for Any Rational Power

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 8 marks

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The question, in short

Use a binomial expansion of √(1 − 4x) to estimate √0.76 and then √19, giving answers to stated accuracy.

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

Part (a) · 3 marks

Write the binomial series for √(1 − 4x) in increasing powers of x through the x³ term, with integer coefficients.

  1. 1(1 − 4x)^(1/2) = 1 + (1/2)(−4x) + (1/2)(−1/2)(−4x)²/2! + (1/2)(−1/2)(−3/2)(−4x)³/3!
  2. 2= 1 − 2x − (1/8)(16x²) − (1/16)(64x³)
  3. 3= 1 − 2x − 2x² − 4x³

Answer1 − 2x − 2x² − 4x³

Part (b) · 3 marks

Choose a value of x so the expansion approximates √0.76, and evaluate it to 4 decimal places.

  1. 11 − 4x = 0.76 gives x = 0.06
  2. 21 − 0.12 − 0.0072 − 0.000864
  3. 3= 0.871936 = 0.8719

Answer0.8719

Part (c) · 2 marks

From the previous estimate, deduce a value for √19 accurate to 3 decimal places.

  1. 1√0.76 = √(19/25) = √19/5
  2. 2√19 = 5 × 0.8719…
  3. 3= 4.360

Answer4.360

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