4MA0 Higher June 2015 Paper 4R Q20
20
Diagram NOT accurately drawn
\(ABC\) is an arc of a circle with centre \(O\) and radius 8 cm.
\(AC\) is a chord of the circle.
Angle \(AOC = 120^\circ\)
Calculate the perimeter of the shaded segment.
Give your answer correct to 3 significant figures.
(5)
| Scheme | Marks |
|---|---|
\((AC^2 =)\;8^2 + 8^2 - 2 \times 8 \times 8\cos 120^\circ\) or \((AC^2 =)\) 192 oe or \(\dfrac{AC}{\sin 120^\circ} = \dfrac{8}{\sin 30}\) oe or \(8\cos(30^\circ)\) or \(8\sin(60^\circ)\) or \(4\sqrt{3}\) or 6.92(820...) oe | M1 |
\((AC =)\;2 \times 8\sin(60^\circ)\) oe or \((AC =)\;2 \times 8\cos(30^\circ)\) oe or \((AC =)\;\dfrac{8}{\sin 30} \times \sin 120\) oe or \((AC =)\;\sqrt{8^2 + 8^2 - 2 \times 8 \times 8\cos 120^\circ}\) or \(\sqrt{128 + 64}\) or \(\sqrt{192}\) or \(8\sqrt{3}\) or 13.8(56…) oe | M1 |
(arc \(ABC\)) = \(\dfrac{120}{360} \times 2\pi \times 8\) or (arc \(ABC\)) = \(\dfrac{120}{360} \times \pi \times 16\) or (arc \(ABC\)) = \(\dfrac{2\pi \times 8}{3}\) or \(\dfrac{\pi \times 16}{3}\) oe or 16.7(55…) | M1 |
| “13.8(56...)” + “16.7(55...)” | M1 |
| 30.6 | A1 |
| (5) |
Notes
M1: Allow 6.92(820...) rounded or truncated to at least 3 significant figures.
M1: Allow 13.8(5640646) rounded or truncated to at least 3 significant figures.
Implies first M1
M1: For attempting to find the length of arc \(ABC\)
Allow 16.7(5516082) rounded or truncated to at least 3 significant figures.
M1: dep on first and third method marks
A1: Accept 30.6 - 30.7 inclusive