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4PM1/01R Jun 2020, Question 5: Binomial Series for Any Rational Power

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

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The question, in short

Use binomial expansion of a square root expression, then apply it to estimate two related square roots numerically.

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 − x) through the x³ term, giving each coefficient as a simplified exact fraction.

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

Answer1 − (1/2)x − (1/8)x² − (1/16)x³

Part (b) · 3 marks

Choose a suitable x value in the series to estimate √0.92, correct to 6 decimal places.

  1. 11 − x = 0.92 ⇒ x = 0.08
  2. 2√0.92 ≈ 1 − 0.04 − 0.0008 − 0.000032
  3. 3= 0.959168

Answer0.959168

Part (c) · 2 marks

Use the earlier result to estimate √23, correct to 5 decimal places.

  1. 1√0.92 = √(23/25) = √23/5
  2. 2√23 ≈ 5 × 0.959168 = 4.79584

Answer4.79584

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