June 2024 Paper 3 Q5
5 The diagram below shows a sector of a circle \(OAB\).
The chord \(AB\) divides the sector into a triangle and a shaded segment.
Angle \(AOB\) is \(\dfrac{\pi}{6}\) radians.
The radius of the sector is 18 cm.
Show that the area of the shaded segment is
\[k(\pi - 3)\text{ cm}^2\]where \(k\) is an integer to be found. [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Obtains \(\frac{1}{2} \times 18^2 \times \frac{\pi}{6}\) or \(\frac{1}{2} \times 18^2 \sin\frac{\pi}{6}\) Allow 0.5 instead of \(\sin\frac{\pi}{6}\) Accept use of degrees eg \(\frac{30}{360} \times \pi \times 18^2\) or \(\frac{1}{2} \times 18^2 \sin 30\) | M1 | 3.1a |
| Obtains \(\frac{1}{2} \times 18^2 \times \frac{\pi}{6}\) and \(\frac{1}{2} \times 18^2 \sin\frac{\pi}{6}\) Allow 0.5 instead of \(\sin\frac{\pi}{6}\) Accept use of degrees as above | A1 | 1.1b |
| Completes reasoned argument to obtain \(27(\pi - 3)\) cm2 Must see \(27\pi - 81\) or \(162\left(\frac{\pi}{6} - \frac{1}{2}\right)\) Accept use of degrees as above ISW Condone missing units | R1 | 2.1 |
| (3 marks) |