| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Segment area calculation |
| Difficulty | Standard +0.3 This is a straightforward C2 segment problem requiring standard formulas: arc length = rθ, isosceles triangle angle sum, and perimeter calculation. All steps are routine applications of basic radian geometry with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Arc \(PQ = r\theta = 6\pi\) (cm) | M1, A1 (2 marks) | \(r\theta\); Condone missing units throughout the paper |
| (b) \(\alpha + \alpha + \frac{3\pi}{7} = \pi\); \(\alpha = \frac{2\pi}{7}\) | M1, A1 (2 marks) | OE; Accept equivalent fractions eg \(\frac{4\pi}{14}\) and condone \(0.286\pi\) or better |
| (c) Chord \(PQ = 2 \times 14 \times \cos\alpha\) | M1 | OE eg \(2 \times 14 \times \sin\frac{3\pi}{14}\) or \(17.45-17.5\) inclusive or \(\sqrt{14^2 + 14^2 - 2 \times 14^2 \times \cos\frac{3\pi}{7}}\) |
| Perimeter \(= 17.45\ldots + 6\pi = 36.307\ldots = 36.3\) (cm) | A1 (2 marks) | Condone > 3sf |
**(a)** Arc $PQ = r\theta = 6\pi$ (cm) | M1, A1 (2 marks) | $r\theta$; Condone missing units throughout the paper
**(b)** $\alpha + \alpha + \frac{3\pi}{7} = \pi$; $\alpha = \frac{2\pi}{7}$ | M1, A1 (2 marks) | OE; Accept equivalent fractions eg $\frac{4\pi}{14}$ and condone $0.286\pi$ or better
**(c)** Chord $PQ = 2 \times 14 \times \cos\alpha$ | M1 | OE eg $2 \times 14 \times \sin\frac{3\pi}{14}$ or $17.45-17.5$ inclusive or $\sqrt{14^2 + 14^2 - 2 \times 14^2 \times \cos\frac{3\pi}{7}}$
Perimeter $= 17.45\ldots + 6\pi = 36.307\ldots = 36.3$ (cm) | A1 (2 marks) | Condone > 3sf
**Total for Q2: 6 marks**
---2 The diagram shows a shaded segment of a circle with centre $O$ and radius 14 cm , where $P Q$ is a chord of the circle.\\
\includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-2_423_551_1270_740}
In triangle $O P Q$, angle $P O Q = \frac { 3 \pi } { 7 }$ radians and angle $O P Q = \alpha$ radians.
\begin{enumerate}[label=(\alph*)]
\item Find the length of the arc $P Q$, giving your answer as a multiple of $\pi$.
\item Find $\alpha$ in terms of $\pi$.
\item Find the perimeter of the shaded segment, giving your answer to three significant figures.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2008 Q2 [6]}}