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4PM1/02 Nov 2023, Question 9: Radians, Arcs and Sectors

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 13 marks

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The question, in short

Logo AEBCD shaded in Figure 2 with straight line ABC as diameter of semicircle ADC and arc AEB centred at O. Given BC = 2x, OA = OB = x, arc AEB = 1.8x, find constants and area.

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

Part (a) · 7 marks

Show perimeter P = ax(π + π sin 0.9 + b), finding constants a and b.

  1. 1Angle AOB = 1.8x/x = 1.8 rad
  2. 2AB = 2x sin 0.9
  3. 3Diameter AC = 2x sin 0.9 + 2x
  4. 4Arc ADC = πAC/2 = πx(1 + sin 0.9)
  5. 5P = πx(1 + sin 0.9) + 2x + 1.8x = x(π + π sin 0.9 + 3.8)

AnswerP = x(π + π sin 0.9 + 3.8), so a = 1, b = 3.8

Part (b) · 6 marks

Given x = 10 cm, find the area of the logo in cm² to 3 significant figures.

  1. 1D = 2(10)sin 0.9 + 20 = 35.6665…
  2. 2Semicircle area = (1/2)π(35.6665…/2)² = 499.553…
  3. 3Sector area = (1/2)(10²)(1.8) = 90
  4. 4Segment area = 90 − (1/2)(10²)sin 1.8 = 41.307…
  5. 5Logo area = 499.553… − 41.307… = 458 cm²

Answer458 cm²

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