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4PM1/02 Jun 2021, Question 3: Radians, Arcs and Sectors

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 9 marks

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The question, in short

Sector OPQ has radius 18 cm and angle 2π/3 rad. Find the arc length as a multiple of π, then use tangents at P and Q meeting at T to find angle α and the shaded area between arc and tangents.

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

Part (a) · 2 marks

Calculate the length of arc PQ for radius 18 cm and angle 2π/3, giving the result as a multiple of π.

  1. 1Arc length = rθ
  2. 2= 18 × 2π/3 = 12π cm

Answer12π cm

Part (bi) · 1 mark

Using tangents PT and QT with right angles at P and Q, find angle PTQ = α in terms of π.

  1. 1Angles of kite OPTQ sum to 2π, with right angles at P and Q
  2. 2α = 2π − π/2 − π/2 − 2π/3 = π/3

Answerα = π/3

Part (bii) · 6 marks

Find, to 3 significant figures, the shaded area bounded by arc PQ and tangents PT and QT.

  1. 1Angle POT = π/3, so PT = 18 tan(π/3) = 18√3
  2. 2Area of kite OPTQ = 2 × (1/2) × 18 × 18√3 = 324√3 (≈ 561.18)
  3. 3Area of sector OPQ = (1/2) × 18² × (2π/3) = 108π (≈ 339.29)
  4. 4Shaded area = 324√3 − 108π ≈ 222 cm²

Answer222 cm²

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