Lemmara logoLemmaraWorked solution

4024/22 May/Jun 2021, Question 10: Composite Perimeter, Area and Volume

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 9 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Circle geometry: use a straight line and cyclic triangle to find an angle with reasons; then use tangents with OP = 8 cm and angle POQ = 130° to find PR and a shaded percentage.

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

Part (a) · 3 marks

Find angle ACB using angle ABD = 130° on straight line OBD, giving a reason at each step.

  1. 1Angle ABO = 180 − 130 = 50° (angles on a straight line)
  2. 2OA = OB so angle OAB = 50°, angle AOB = 180 − 100 = 80° (isosceles triangle)
  3. 3Angle ACB = 80 ÷ 2 = 40° (angle at centre is twice angle at circumference)

AnswerAngle ABO = 50° (angles on a straight line) Angle AOB = 80° (angles in an isosceles triangle) Angle ACB = 40° (angle at centre is twice angle at circumference)

Part (bi) · 2 marks

With tangents PR and QR, OP = 8 cm and angle POQ = 130°, find the length PR.

  1. 1Angle POR = 130 ÷ 2 = 65° and angle OPR = 90° (tangent ⟂ radius)
  2. 2PR = 8 tan 65°
  3. 3= 17.15… ≈ 17.2 cm

Answer17.2 or 17.15 to 17.16

Part (bii) · 4 marks

Calculate what percentage of quadrilateral OPRQ is shaded, using the tangent and sector geometry.

  1. 1Area OPRQ = 2 × 1/2 × 8 × 17.156 = 137.25 cm²
  2. 2Sector = 130/360 × π × 8² = 72.61 cm²
  3. 3Shaded = 137.25 − 72.61 = 64.64 cm²
  4. 464.64/137.25 × 100 = 47.1%

Answer47 to 47.4

Stuck on a step? Read the free lesson: Composite Perimeter, Area and Volume

Practise more composite perimeter, area and volume questions

Real 4024 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on composite perimeter, area and volume

Also searched as: 4024/22/M/J/21 · 4024_s21_qp_22 · 4024 s21 22 · question 10.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.