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4PM1/01 Jan 2022, 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 reciprocal square root expression in powers of x up to x³, then use it to estimate a related surd expression.

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

Part (a) · 3 marks

Find the binomial expansion of 1/√(1 − 2x) up to the x³ term, simplifying each coefficient.

  1. 1(1 − 2x)^(−1/2) = 1 + (−1/2)(−2x) + (−1/2)(−3/2)(−2x)²/2! + (−1/2)(−3/2)(−5/2)(−2x)³/3!
  2. 2= 1 + x + (3/8)(4x²) + (5/16)(8x³)
  3. 3= 1 + x + (3/2)x² + (5/2)x³

Answer1 + x + (3/2)x² + (5/2)x³

Part (b) · 2 marks

Demonstrate that 1/√0.96 equals 5√6/12, showing each step.

  1. 1√0.96 = √(96/100) = 4√6/10
  2. 21/√0.96 = 10/(4√6)
  3. 3= 10√6/24 = 5√6/12

Answer1/√0.96 = 5√6/12 shown

Part (c) · 2 marks

Rewrite 1/(5√6 − 12) as a√6/c + b with integer a, b, c, showing working.

  1. 1Multiply top and bottom by (5√6 + 12)
  2. 2(5√6 − 12)(5√6 + 12) = 150 − 144 = 6
  3. 31/(5√6 − 12) = (5√6 + 12)/6 = 5√6/6 + 2

Answer5√6/6 + 2 (a = 5, c = 6, b = 2)

Part (d) · 4 marks

Use the part (a) expansion with a chosen x to estimate 9/(5√6 − 12) to 5 decimal places.

  1. 11 − 2x = 0.96 gives x = 0.02
  2. 29/(5√6 − 12) = 9(5√6/6 + 2) = 9(2 × 5√6/12 + 2) = 9(2/√0.96 + 2)
  3. 31/√0.96 ≈ 1 + 0.02 + (3/2)(0.0004) + (5/2)(0.000008) = 1.02062
  4. 49(2 × 1.02062 + 2) = 36.37116

Answer36.37116

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