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4PM1/01R Jun 2022, Question 3: Radians, Arcs and Sectors

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 6 marks

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The question, in short

Sector ORS has centre O, radius 4 cm, angle θ radians and area 2π cm². Find θ, the sector perimeter to 3 s.f., and the exact shaded area bounded by TR, TS and arc RS, where T is on OR with OT:TR = 1:3.

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

Part (a) · 2 marks

Use the sector area formula with radius 4 cm and area 2π cm² to find the exact value of θ in radians.

  1. 1(1/2) × 4² × θ = 2π
  2. 28θ = 2π ⇒ θ = π/4

Answerθ = π/4

Part (b) · 2 marks

Calculate the perimeter of sector ORS in cm, giving the answer to 3 significant figures.

  1. 1Arc RS = 4 × π/4 = π
  2. 2Perimeter = 4 + 4 + π = 11.1 cm

Answer11.1 cm

Part (c) · 2 marks

With OT:TR = 1:3 on OR, find the exact area A of the shaded region bounded by TR, TS and arc RS.

  1. 1OT = 1 cm (OT : TR = 1 : 3 with OR = 4)
  2. 2Area of triangle OTS = (1/2) × 1 × 4 × sin(π/4) = √2
  3. 3A = sector − triangle = 2π − √2

AnswerA = 2π − √2

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