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4PM1/01 Jun 2022, Question 9: Definite Integrals and Areas

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 16 marks

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The question, in short

Expand a quartic in a squared-factor form, then use a sketched cubic curve and equal shaded areas to find a point's coordinates.

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

Part (a) · 4 marks

Compare coefficients to find a, b and c in the squared-factor form of the quartic.

  1. 1Compare x⁴: a = 3
  2. 2Compare constants: 4c = 64 ⇒ c = 16
  3. 3Compare x³: b − 4a = 4 ⇒ b = 16

Answera = 3, b = 16, c = 16

Part (b) · 7 marks

Set up and evaluate integrals for the two shaded areas, equate them, and show n satisfies the given equation.

  1. 1y = x³ + x² − 6x
  2. 2∫ from 0 to 2 of (x³ + x² − 6x) dx = [x⁴/4 + x³/3 − 3x²] from 0 to 2 = 4 + 8/3 − 12 = −16/3
  3. 3Area between n and 0: 0 − (n⁴/4 + n³/3 − 3n²) = 16/3
  4. 4Multiply by −12: 3n⁴ + 4n³ − 36n² + 64 = 0
  5. 5By (a): (n − 2)²(3n² + 16n + 16) = 0

Answer(x − 2)²(3x² + 16x + 16) = 0 shown

Part (c) · 5 marks

Solve the equation from part b to find the exact coordinates of A.

  1. 13x² + 16x + 16 = 0 ⇒ (3x + 4)(x + 4) = 0 ⇒ x = −4/3 or x = −4
  2. 2N lies between M(−3, 0) and O, so n = −4/3
  3. 3y = (−4/3)(−4/3 + 3)(−4/3 − 2) = (−4/3)(5/3)(−10/3) = 200/27
  4. 4A(−4/3, 200/27)

AnswerA(−4/3, 200/27)

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