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Date	May 2021	Marks available	3	Reference code	21M.1.SL.TZ2.1
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 2
Command term	Hence and Find	Question number	1	Adapted from	N/A

Question

The following diagram shows a circle with centre $O$ $O$ and radius $r$ $r$ .

Points $A$ $A$ and $B$ $B$ lie on the circumference of the circle, and $A ˆ O B = 1 radian$ $A \hat{O} B = 1 radian$ .

The perimeter of the shaded region is $12$ $12$ .

Find the value of $r$ $r$ .

[3]

Hence, find the exact area of the non-shaded region.

[3]

Markscheme

minor arc $AB$ $AB$ has length $r$ $r$ (A1)

recognition that perimeter of shaded sector is $3 r$ $3 r$ (A1)

$3 r = 12$ $3 r = 12$

$r = 4$ $r = 4$ A1

[3 marks]

EITHER
$θ = 2 π - A ˆ O B (= 2 π - 1)$ $θ = 2 π - A \hat{O} B (= 2 π - 1)$ (M1)

Area of non-shaded region $= \frac{1}{2} (2 π - 1) (4^{2})$ $= \frac{1}{2} (2 π - 1) (4^{2})$ (A1)

area of circle $-$ $-$ area of shaded sector (M1)

$16 π - (\frac{1}{2} \times 1 \times 4^{2})$ $16 π - (\frac{1}{2} \times 1 \times 4^{2})$ (A1)

THEN

area $= 16 π - 8 (= 8 (2 π - 1))$ $= 16 π - 8 (= 8 (2 π - 1))$ A1

[3 marks]

Examiners report

[N/A]

Syllabus sections

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

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