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4PM1/02R Jun 2020, Question 10: Volumes of Revolution

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 14 marks

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The question, in short

Given cubic f(x) = 32x³ − 33x + 1, verify a root, solve it algebraically, then use curves y = √x and y = 1/(8x) with line x = a to find an intersection and a volume condition.

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

Part (a) · 1 mark

Substitute x = 1 into f(x) and show the result equals zero.

  1. 1f(1) = 32 − 33 + 1
  2. 2= 0

Answerf(1) = 0 shown

Part (b) · 4 marks

Use the known factor from part (a) to factorise and solve f(x) = 0 algebraically.

  1. 1(x − 1)(32x² + 32x − 1) = 0
  2. 2x = [−32 ± √(32² + 4 × 32)]/64
  3. 3x = 1, (−4 + 3√2)/8, (−4 − 3√2)/8

Answerx = 1, x = (−4 + 3√2)/8, x = (−4 − 3√2)/8

Part (c) · 2 marks

Find the x-coordinate p where y = √x and y = 1/(8x) intersect.

  1. 1√x = 1/(8x)
  2. 2x^(3/2) = 1/8
  3. 3x = 1/4

Answerp = 1/4

Part (d) · 7 marks

Rotate region R about the x-axis, set the integral equal to 27π/64, and solve for a.

  1. 1V = π ∫ from 1/4 to a of x dx − π ∫ from 1/4 to a of 1/(64x²) dx
  2. 2π[x²/2] from 1/4 to a = π(a²/2 − 1/32)
  3. 3π[−1/(64x)] from 1/4 to a = π(1/16 − 1/(64a))
  4. 4π(a²/2 − 1/32) − π(1/16 − 1/(64a)) = 27π/64
  5. 532a³ − 33a + 1 = 0
  6. 6From (b), a > 1/4 ⇒ a = 1

Answera = 1

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