18M.1.SL.TZ2.S_4

pestleMathematicsAASLPaper 118M· sl-3-4-circle-radians-arcs-sectorssource ↗

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

The points A and B lie on the circumference of the circle, and ${\text{A}}\mathop {\text{O}}\limits^ \wedge {\text{B}}$ = θ. The area of the shaded sector AOB is 12 cm² and the length of arc AB is 6 cm.

Find the value of r.

Markscheme / solution

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

evidence of correctly substituting into circle formula (may be seen later) A1A1
eg $\frac{1}{2}\theta {r^2} = 12,\,\,r\theta = 6$

attempt to eliminate one variable (M1)
eg $r = \frac{6}{\theta },\,\,\theta = \frac{1}{r},\,\,\frac{{\frac{1}{2}\theta {r^2}}}{{r\theta }} = \frac{{12}}{6}$

correct elimination (A1)
eg $\frac{1}{2} \times \frac{6}{r} \times {r^2} = 12,\,\,\frac{1}{2}\theta \times {\left( {\frac{6}{\theta }} \right)^2} = 12,\,\,A = \frac{1}{2} \times {r^2} \times \frac{l}{r},\,\,\frac{{{r^2}}}{{2r}} = 2$

correct equation (A1)
eg $\frac{1}{2} \times 6r = 12,\,\,\frac{1}{2} \times \frac{{36}}{\theta } = 12,\,\,12 = \frac{1}{2} \times {r^2} \times \frac{6}{r}$

correct working (A1)
eg $3r = 12,\,\,\frac{{18}}{\theta } = 12,\,\,\frac{r}{2} = 2,\,\,24 = 6r$

r = 4 (cm) A1 N2

[7 marks]

Examiners’ report

[N/A]