| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Sector and arc length |
| Difficulty | Moderate -0.3 This is a standard C2 circle geometry question testing arc length, chord length using cosine rule, and segment area. Part (a) is routine arc length formula application. Part (b) requires finding chord length via triangle geometry (standard technique) and algebraic manipulation to match the given form. Part (c) combines sector and triangle area formulas. All techniques are core C2 curriculum with no novel problem-solving required, making it slightly easier than average but not trivial due to the multi-step nature and exact form manipulation. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(= 12 \times (2\pi - \frac{2\pi}{3}) = 16\pi\) cm | M1 A1 | |
| (b) chord \(= 2 \times 12 \sin \frac{\pi}{3} = 24 \times \frac{\sqrt{3}}{2} = 12\sqrt{3}\) | M1 A1 | |
| \(P = (12 \times \frac{2\pi}{3}) + 12\sqrt{3}\) | M1 | |
| \(= 8\pi + 12\sqrt{3} = 4(2\pi + 3\sqrt{3})\) cm | A1 | [\(k = 4\)] |
| (c) area of segment \(= (\frac{1}{2} \times 12^2 \times \frac{2\pi}{3}) - (\frac{1}{2} \times 12^2 \times \sin \frac{2\pi}{3})\) | M2 | |
| \(= 72(\frac{2\pi}{3} - \frac{\sqrt{3}}{2}) = 88.443\) | ||
| as % of area of circle \(= \frac{88.443}{\pi \times 12^2} \times 100\% = 19.6\%\) (1dp) | M1 A1 |
**(a)** $= 12 \times (2\pi - \frac{2\pi}{3}) = 16\pi$ cm | M1 A1 |
**(b)** chord $= 2 \times 12 \sin \frac{\pi}{3} = 24 \times \frac{\sqrt{3}}{2} = 12\sqrt{3}$ | M1 A1 |
$P = (12 \times \frac{2\pi}{3}) + 12\sqrt{3}$ | M1 |
$= 8\pi + 12\sqrt{3} = 4(2\pi + 3\sqrt{3})$ cm | A1 | [$k = 4$] |
**(c)** area of segment $= (\frac{1}{2} \times 12^2 \times \frac{2\pi}{3}) - (\frac{1}{2} \times 12^2 \times \sin \frac{2\pi}{3})$ | M2 |
$= 72(\frac{2\pi}{3} - \frac{\sqrt{3}}{2}) = 88.443$ | |
as % of area of circle $= \frac{88.443}{\pi \times 12^2} \times 100\% = 19.6\%$ (1dp) | M1 A1 | | (10 marks)\includegraphics{figure_2}
Figure 2 shows a circle of radius 12 cm which passes through the points $P$ and $Q$. The chord $PQ$ subtends an angle of $120°$ at the centre of the circle.
\begin{enumerate}[label=(\alph*)]
\item Find the exact length of the major arc $PQ$. [2]
\item Show that the perimeter of the shaded minor segment is given by $k(2\pi + 3\sqrt{3})$ cm, where $k$ is an integer to be found. [4]
\item Find, to 1 decimal place, the area of the shaded minor segment as a percentage of the area of the circle. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q8 [10]}}