Higher November 2024 Paper 2 Q21
21. The circle below is split into sectors, segments and triangles.
\(A\), \(B\), \(C\) and \(D\) are points on the circumference of the circle with centre \(O\).
The lines \(AC\) and \(BD\) are diameters of the circle.
Diagram not drawn to scale
The area of sector \(AOB\) is 12 cm2.
Find the perimeter of the shaded segment.
You must show all your working. [9]
| Scheme | Marks | Comments |
|---|---|---|
| \((r =)\ \sqrt{\dfrac{12 \times 360}{55\pi}}\) or \(\sqrt{\dfrac{12}{\frac{55}{360} \times \pi}}\) oe | M2 | M1 for \(12 = \dfrac{55}{360} \times \pi \times r^2\) or better |
| r = 5(cm) oe | A1 | \(r = \sqrt{\dfrac{864}{11\pi}}\) |
| (Perimeter =) \(\sqrt{5^2 + 5^2 - 2 \times 5 \times 5\cos(180 - 55)}\) \(+\ \dfrac{180 - 55}{360} \times \pi \times 2 \times 5\) OR \(2 \times 5\sin\left(\dfrac{180 - 55}{2}\right) + \dfrac{180 - 55}{360} \times \pi \times 2 \times 5\) OR \(\dfrac{5\sin125}{\sin((180 - 125) \div 2)} + \dfrac{180 - 55}{360} \times \pi \times 2 \times 5\) | M3 | FT ‘their derived r’ M2 for one of the following:
|
| 19.7(7…) (cm) or 19.8(cm) | A3 | Allow answers in the range 19.7 to 20 inclusive from correct working. A1 for each of the following:
|
| (9) |