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4PM1/02R Jun 2019, Question 6: Binomial Series for Any Rational Power

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 8 marks

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The question, in short

Rewrite √(9 − x) as p(1 + qx)^(1/2), find p and q, expand up to the x³ term with exact fractional coefficients, then use the expansion with a chosen x to estimate √(31/4) to 5 decimal places.

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

Part (a) · 2 marks

Determine the constants p and q so that √(9 − x) equals p(1 + qx)^(1/2).

  1. 1√(9 − x) = √9 × √(1 − x/9)
  2. 2= 3(1 − x/9)^(1/2)
  3. 3p = 3, q = −1/9

Answerp = 3, q = −1/9

Part (b) · 3 marks

Using your p and q, write the binomial expansion in increasing powers of x through x³, giving each coefficient as a simplified exact fraction.

  1. 13(1 − x/9)^(1/2) = 3[1 + (1/2)(−x/9) + ((1/2)(−1/2)/2!)(−x/9)² + ((1/2)(−1/2)(−3/2)/3!)(−x/9)³ + …]
  2. 2= 3(1 − x/18 − x²/648 − x³/11664)
  3. 3= 3 − x/6 − x²/216 − x³/3888

Answer3 − x/6 − x²/216 − x³/3888

Part (c) · 3 marks

Pick a suitable x, substitute into the part (b) expansion, and obtain a 5 decimal place approximation of √(31/4).

  1. 131/4 = 9 − x ⇒ x = 5/4
  2. 2√(31/4) ≈ 3 − 1.25/6 − 1.25²/216 − 1.25³/3888
  3. 3= 2.783930… ≈ 2.78393

Answer2.78393

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