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4PM1/01 Jun 2022, Question 5: Definite Integrals and Areas

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 11 marks

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The question, in short

Rewrite (2 + 3x)^(−1) as p(1 + qx)^(−1), expand it up to x³, use this for f(x) = (1 + x)/(2 + 3x), then integrate to estimate the definite integral from 0 to 0.5.

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 in the equivalent form p(1 + qx)^(−1).

  1. 1(2 + 3x)^(−1) = [2(1 + (3/2)x)]^(−1)
  2. 2= (1/2)(1 + (3/2)x)^(−1), so p = 1/2, q = 3/2

Answerp = 1/2, q = 3/2

Part (b) · 3 marks

Using part a, write the binomial expansion up to the x³ term, giving each coefficient as a simplified fraction.

  1. 1(1 + (3/2)x)^(−1) = 1 − (3/2)x + ((−1)(−2)/2!)((3/2)x)² + ((−1)(−2)(−3)/3!)((3/2)x)³
  2. 2= 1 − (3/2)x + (9/4)x² − (27/8)x³
  3. 3Multiply by 1/2: 1/2 − (3/4)x + (9/8)x² − (27/16)x³

Answer1/2 − (3/4)x + (9/8)x² − (27/16)x³

Part (c) · 2 marks

Find the series for f(x) up to x³, with coefficients as exact fractions in lowest terms.

  1. 1(1 + x)(1/2 − (3/4)x + (9/8)x² − (27/16)x³)
  2. 2= 1/2 − (3/4)x + (9/8)x² − (27/16)x³ + (1/2)x − (3/4)x² + (9/8)x³ + …
  3. 3= 1/2 − (1/4)x + (3/8)x² − (9/16)x³

Answer1/2 − (1/4)x + (3/8)x² − (9/16)x³

Part (d) · 4 marks

Integrate the series algebraically to estimate the integral of f(x) from 0 to 0.5, correct to 4 decimal places.

  1. 1∫ (1/2 − (1/4)x + (3/8)x² − (9/16)x³) dx = (1/2)x − (1/8)x² + (1/8)x³ − (9/64)x⁴
  2. 2Substitute x = 0.5: 1/4 − 1/32 + 1/64 − 9/1024
  3. 3= 231/1024 ≈ 0.2256

Answer0.2256

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