4MA0 Higher June 2016 Paper 3R Q22
22
Diagram NOT accurately drawn
The diagram shows a circle, centre \(C\).
\(PR\) is a chord of the circle.
The area of the shaded region is 100 cm²
Angle \(PCR\) = 30°
Calculate the length of the arc \(PQR\).
Give your answer correct to 3 significant figures.
(6)
| Scheme | Marks |
|---|---|
\(\dfrac{30}{360} \times \pi r^2 - 0.5r^2 \sin 30 = 100\) oe \(r^2\left(\dfrac{30\pi}{360} - 0.5\sin 30\right) = 100\) \(r^2\left(\dfrac{\pi - 3}{12}\right) = 100\) | M2 |
| \(r^2 = \dfrac{1200}{\pi - 3}\) | M1 |
| \(r = \sqrt{\dfrac{1200}{\pi - 3}}\) | M1 |
| \(2\pi \times \sqrt{\dfrac{1200}{\pi - 3}} \times \dfrac{30}{360}\) oe eg \(2\pi \times 92\ldots \times \dfrac{1}{12}\) | M1 |
| 48.2 | A1 |
| (6) |
Notes
M2: For a correct equation involving \(r^2\)
M1 for \(\dfrac{30}{360} \times \pi r^2\) (or \(\dfrac{\pi r^2}{12}\)) or \(0.5r^2 \sin 30\) (or \(0.25r^2\))
M1: For a correct equation with \(r^2\) the subject
M1: For correctly isolating \(r\). Accept 92.(05984992...) rounded or truncated to at least 2SF
M1: A correct expression for the length of are \(PQR\)
A1: Accept answers which round to 48.2