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4PM1/02 Jun 2024, Question 3: Radians, Arcs and Sectors

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 6 marks

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The question, in short

Sector AOB has radius 3r cm and angle θ rad; a circle of radius r cm touches OA, OB and arc AB. Find θ exactly, then find r given the shaded area is 8π cm².

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

Part (a) · 2 marks

Use the geometry of the small circle touching both radii and the arc to find the exact value of θ.

  1. 1The centre of the small circle is 3r − r = 2r from O
  2. 2sin(θ/2) = r/(2r) = 1/2
  3. 3θ/2 = π/6, so θ = π/3

Answerθ = π/3

Part (b) · 4 marks

Given the shaded area between sector and small circle equals 8π cm², calculate r.

  1. 1Area of circle = πr²
  2. 2Area of sector = ½ × (π/3) × (3r)² = (3/2)πr²
  3. 3Shaded area = (3/2)πr² − πr² = πr²/2
  4. 4πr²/2 = 8π, so r² = 16, r = 4

Answerr = 4

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