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WMA11/01 Oct 2023, Question 9: Triangles and Sectors in Radians

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 7 marks

The question, in short

Using a sector diagram with inner circle C, arc PS 0.72 m and angle POS 0.6 radians, work through parts to find OP, derive a quadratic in x, then find the target area's perimeter.

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

Part (a) · 1 mark

Show that the radius OP of circle C equals 1.2 m using the given arc length and angle.

  1. 1Arc length s = rθ
  2. 20.72 = OP × 0.6
  3. 3OP = 0.72/0.6 = 1.2 m

AnswerShown: arc = rθ, so 0.72 = OP × 0.6 and OP = 0.72/0.6 = 1.2 m

Part (b) · 3 marks

Given target area 90 m² and PQ = x, show that 5x² + 12x − 1500 = 0.

  1. 1Large sector radius OQ = x + 1.2, area = (1/2)(x + 1.2)² × 0.6
  2. 2Small sector area = (1/2)(1.2)² × 0.6 = 0.432
  3. 30.3(x² + 2.4x + 1.44) − 0.432 = 90
  4. 40.3x² + 0.72x = 90
  5. 5Multiply by 50/3: 5x² + 12x − 1500 = 0

AnswerShown: (1/2)(x + 1.2)² × 0.6 − (1/2)(1.2)² × 0.6 = 90 ⇒ 0.3x² + 0.72x = 90 ⇒ 5x² + 12x − 1500 = 0

Part (c) · 3 marks

Hence calculate the total perimeter of PQRS, giving the answer to the nearest metre.

  1. 1x = (−12 + √(144 + 30000))/10 = (−6 + 4√471)/5 = 16.16 m (reject the negative root)
  2. 2Arc QR = (x + 1.2) × 0.6 = 17.36 × 0.6 = 10.42 m
  3. 3Perimeter = PQ + RS + arc QR + arc PS = 16.16 + 16.16 + 10.42 + 0.72
  4. 4= 43.46… ≈ 43 m

Answer43 m (perimeter ≈ 43.46 m; 44 m also accepted)

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Also searched as: WMA11 October 2023 · WMA11/01 · wma11 oct23 · IAS Maths Pure Mathematics 1 October 2023 · question 9.
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