19M.2.SL.TZ2.S_4

pestleMathematicsAISLPaper 219M· ahl-3-7-radianssource ↗

OAB is a sector of the circle with centre O and radius $r$ , as shown in the following diagram.

The angle AOB is $\theta$ radians, where $0 < \theta < \frac{\pi }{2}$ .

The point C lies on OA and OA is perpendicular to BC.

Find the area of triangle OBC in terms of $r$ and θ.

Markscheme / solution

valid approach (M1)

eg $\frac{1}{2}{\text{OC}} \times {\text{OB}}\,{\text{sin}}\,\theta$ , ${\text{BC}} = r\,{\text{sin}}\,\theta$ , $\frac{1}{2}r\,{\text{cos}}\,\theta \times {\text{BC}}$ , $\frac{1}{2}r\,{\text{sin}}\,\theta \times {\text{OC}}$

area $= \frac{1}{2}{r^2}\,{\text{sin}}\,\theta \,{\text{cos}}\,\theta$ $\left( { = \frac{1}{4}{r^2}\,{\text{sin}}\left( {2\theta } \right)} \right)$ (must be in terms of $r$ and θ) A1 N2

[2 marks]

Examiners’ report

[N/A]