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4PM1/02 Nov 2021, Question 10: Binomial Series for Any Rational Power

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 13 marks

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The question, in short

Work with the expression 3 over the square root of 9 minus 3x, covering a proof, a binomial expansion, a related function's expansion, and an integral estimate.

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

Part (a) · 2 marks

Demonstrate that 3 divided by √(9 − 3x) can be rewritten as (1 − x/3) raised to the power −1/2.

  1. 1√(9 − 3x) = √9 × √(1 − 3x/9) = 3√(1 − x/3)
  2. 23/√(9 − 3x) = 1/√(1 − x/3) = (1 − x/3)^(−1/2)

Answer3/√(9 − 3x) = (1 − x/3)^(−1/2) shown

Part (b) · 3 marks

Write the binomial expansion of 3/√(9 − 3x) in increasing powers of x through the x³ term, giving each coefficient as a reduced fraction.

  1. 1(1 − x/3)^(−1/2) = 1 + (−1/2)(−x/3) + (−1/2)(−3/2)(−x/3)²/2! + (−1/2)(−3/2)(−5/2)(−x/3)³/3!
  2. 2= 1 + x/6 + x²/24 + 5x³/432

Answer1 + x/6 + x²/24 + 5x³/432

Part (c) · 4 marks

For f(x) = (1 + 2x)/√(9 − 3x), expand 3f(x) in increasing powers of x through x³, with coefficients as exact reduced fractions.

  1. 13f(x) = (1 + 2x)(1 − x/3)^(−1/2)
  2. 2= (1 + 2x)(1 + x/6 + x²/24 + 5x³/432)
  3. 3x: 1/6 + 2 = 13/6; x²: 1/24 + 2/6 = 3/8; x³: 5/432 + 2/24 = 41/432
  4. 43f(x) = 1 + (13/6)x + (3/8)x² + (41/432)x³

Answer3f(x) = 1 + (13/6)x + (3/8)x² + (41/432)x³

Part (d) · 4 marks

Use the earlier expansion and algebraic integration to estimate the integral of (1 + 2x)/√(9 − 3x) from 0.1 to 0.2, accurate to 6 significant figures.

  1. 1∫ f(x) dx ≈ (1/3)∫ from 0.1 to 0.2 of (1 + (13/6)x + (3/8)x² + (41/432)x³) dx
  2. 2= (1/3)[x + (13/12)x² + (1/8)x³ + (41/1728)x⁴] from 0.1 to 0.2
  3. 3= 0.0444702 (6 s.f.)

Answer0.0444702

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