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Date	May 2018	Marks available	3	Reference code	18M.2.SL.TZ1.S_3
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	S_3	Adapted from	N/A

Question

The diagram shows a circle, centre O, with radius 4 cm. Points A and B lie on the circumference of the circle and AÔB = θ , where 0 ≤ θ ≤ $\pi$ $\pi$ .

Find the area of the shaded region, in terms of θ.

[3]

a.

The area of the shaded region is 12 cm². Find the value of θ.

[3]

b.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

valid approach to find area of segment (M1)

eg area of sector – area of triangle, $\frac{1}{2}{r^2}\left( {\theta - {\text{sin}}\theta } \right)$ $\frac{1}{2}{r^2}\left( {\theta - {\text{sin}}\theta } \right)$

correct substitution (A1)

eg $\frac{1}{4}{\left( 4 \right)^2}\theta - \frac{1}{2}{\left( 4 \right)^2}{\text{sin}}\theta ,\,\,\frac{1}{2} \times 16\left[ {\theta - {\text{sin}}\theta } \right]$ $\frac{1}{4}{\left( 4 \right)^2}\theta - \frac{1}{2}{\left( 4 \right)^2}{\text{sin}}\theta ,\,\,\frac{1}{2} \times 16\left[ {\theta - {\text{sin}}\theta } \right]$

area = 80 – 8 sinθ, 8(θ – sinθ) A1 N2

[3 marks]

a.

setting their area expression equal to 12 (M1)

eg 12 = 8(θ – sinθ)

2.26717

θ = 2.27 (do not accept an answer in degrees) A2 N3

[3 marks]

b.

Examiners report

[N/A]

a.

[N/A]

b.

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.4—Circle: radians, arcs, sectors

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19M.1.AHL.TZ1.H_3:
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17M.2.SL.TZ1.S_5:
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