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4PM1/02 Jun 2023, Question 7: Binomial Series for Any Rational Power

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 11 marks

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The question, in short

Use binomial expansion to find series, state validity, rewrite a rational expression, expand f(x), then integrate to estimate a definite integral numerically.

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

Part (a) · 3 marks

Expand (1 + x/3)^(−3) up to x³, giving each coefficient as an exact fraction in lowest terms.

  1. 1(1 + x/3)^(−3) = 1 + (−3)(x/3) + (−3)(−4)/2! (x/3)² + (−3)(−4)(−5)/3! (x/3)³
  2. 2= 1 − x + 6(x²/9) − 10(x³/27)
  3. 3= 1 − x + (2/3)x² − (10/27)x³

Answer1 − x + (2/3)x² − (10/27)x³

Part (b) · 1 mark

State the interval of x values for which the expansion converges.

  1. 1Valid when |x/3| < 1
  2. 2So −3 < x < 3

Answer−3 < x < 3

Part (c) · 2 marks

Rewrite (3 + x)^(−3) as P(1 + Qx)^(−3), giving rational P and Q.

  1. 1(3 + x)^(−3) = [3(1 + x/3)]^(−3)
  2. 2= 3^(−3)(1 + x/3)^(−3) = (1/27)(1 + x/3)^(−3)
  3. 3P = 1/27, Q = 1/3

AnswerP = 1/27, Q = 1/3

Part (d) · 2 marks

Expand f(x) = (1 + 4x)/(3 + x)³ up to the x² term.

  1. 1f(x) = (1 + 4x) × (1/27)(1 − x + (2/3)x² + …)
  2. 2= (1/27)(1 − x + (2/3)x² + 4x − 4x² + …)
  3. 3= (1/27)(1 + 3x − (10/3)x²)
  4. 4= 1/27 + x/9 − (10/81)x²

Answer1/27 + x/9 − (10/81)x²

Part (e) · 3 marks

Integrate the series term by term from 0 to 0.2 to estimate the integral to 5 significant figures.

  1. 1∫ from 0 to 0.2 of (1/27 + x/9 − 10x²/81) dx = [x/27 + x²/18 − 10x³/243] from 0 to 0.2
  2. 2= 0.2/27 + 0.04/18 − 10(0.008)/243
  3. 3= 0.0074074 + 0.0022222 − 0.0003292
  4. 4= 0.0093004

Answer0.0093004

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