21M.1.SL.TZ2.1

pestleMathematicsAASLPaper 121M· sl-3-4-circle-radians-arcs-sectorssource ↗

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

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

The perimeter of the shaded region is $12$ .

Find the value of $r$ .

[3]

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

[3]

Markscheme / solution

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

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

$3 r = 12$

$r = 4$ A1

[3 marks]

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

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

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

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

THEN

area $= 16 π - 8 (= 8 (2 π - 1))$ A1

[3 marks]

Examiners’ report

[N/A]