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

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 8 marks

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The question, in short

Sector OAB has radius r cm and angle π/3; D divides OA in ratio 3:1; shaded region R has area 79.5 cm². Find r and the perimeter of R, each to 2 significant figures.

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Part (a) · 4 marks

Use the shaded area 79.5 cm² and the ratio 3:1 on OA to calculate r, giving the value to 2 significant figures.

  1. 1Sector area = ½r²(π/3)
  2. 2Triangle ODB = ½ × r × (3r/4) × sin(π/3)
  3. 3½r²(π/3) − ½ × r × (3r/4) × sin(π/3) = 79.5
  4. 4r²(π/6 − 3√3/16) = 79.5
  5. 5r = 19.99… = 20

Answerr = 20

Part (b) · 4 marks

Using your value of r, calculate the perimeter of the shaded region R in cm to 2 significant figures.

  1. 1BD² = 20² + 15² − 2 × 20 × 15 × cos(π/3) = 325
  2. 2BD = 5√13 ≈ 18.03
  3. 3Arc AB = 20 × π/3 ≈ 20.94
  4. 4DA = 20 − 15 = 5
  5. 5Perimeter = 5 + 20.94 + 18.03 ≈ 44 cm

Answer44 cm

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