Lemmara logoLemmaraWorked solution

4PM1/02 Jun 2023, Question 4: Simultaneous Equations with a Curve

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 8 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Curve S is y = x²/4 + 2 and line l is 2y − x − 4 = 0, both for x ≥ 0, meeting at A and B. Find B, then the volume when the shaded region is rotated 2π about the y-axis.

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

Part (a) · 4 marks

Show A is (0, 2) and find the coordinates of B where the curve and line meet.

  1. 1Line: y = (x + 4)/2
  2. 2x²/4 + 2 = (x + 4)/2, so x² + 8 = 2x + 8
  3. 3x² − 2x = 0, x(x − 2) = 0
  4. 4x = 0 gives y = 2: A(0, 2)
  5. 5x = 2 gives y = 3: B(2, 3)

Answer(i) A(0, 2) shown (ii) B(2, 3)

Part (b) · 4 marks

Using algebraic integration, find the volume of the solid formed by rotating the region between S and l about the y-axis, in terms of π.

  1. 1About the y-axis: curve x² = 4y − 8, line x = 2y − 4
  2. 2V = π∫ from 2 to 3 of (4y − 8) dy − π∫ from 2 to 3 of (2y − 4)² dy
  3. 3= π∫ from 2 to 3 of (−4y² + 20y − 24) dy
  4. 4= π[−4y³/3 + 10y² − 24y] from 2 to 3
  5. 5= π[(−36 + 90 − 72) − (−32/3 + 40 − 48)] = π(−18 + 56/3)
  6. 6= (2/3)π

Answer(2/3)π

Stuck on a step? Read the free lesson: Simultaneous Equations with a Curve

Practise more simultaneous equations with a curve questions

Real 4PM1 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on simultaneous equations with a curve

Also searched as: 4PM1 2 June 2023 · 4PM1/02 · 4pm1 jun23 2 · question 4.
Lemmara is an independent study tool, not affiliated with or endorsed by Pearson Edexcel. The worked solutions are Lemmara’s own.