| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Segment area calculation |
| Difficulty | Moderate -0.3 This is a straightforward sector/segment question requiring standard formulas (area = ½r²θ, segment = sector - triangle). Part (a) is simple algebra, (b) applies perimeter formula, and (c) requires subtracting triangle area from sector area. All techniques are routine C2 content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{1}{2}r^2\theta = \frac{1}{2}r^2 \times 1.5 = 15\) | M1 A1 | |
| \(r^2 = 20 = \sqrt{(4 \times 5)}\); \(r = 2\sqrt{5}\) | A1 | (*) (3) |
| (b) \(r\theta + 2r = 3\sqrt{5} + 4\sqrt{5} = 7\sqrt{5}\) cm | M1 A1 | (or 15.7, or a.w.r.t 15.65....) (2) |
| (c) \(\triangle OAB: \frac{1}{2}r^2\sin\theta = 10\sin 1.5 (= 9.9749...)\) | M1 | |
| Segment area \(= 15 - \triangle OAB = 5.025\) cm² | M1 A1 | (3) |
**(a)** $\frac{1}{2}r^2\theta = \frac{1}{2}r^2 \times 1.5 = 15$ | M1 A1 |
$r^2 = 20 = \sqrt{(4 \times 5)}$; $r = 2\sqrt{5}$ | A1 | (*) (3)
**(b)** $r\theta + 2r = 3\sqrt{5} + 4\sqrt{5} = 7\sqrt{5}$ cm | M1 A1 | (or 15.7, or a.w.r.t 15.65....) (2)
**(c)** $\triangle OAB: \frac{1}{2}r^2\sin\theta = 10\sin 1.5 (= 9.9749...)$ | M1 |
Segment area $= 15 - \triangle OAB = 5.025$ cm² | M1 A1 | (3)\includegraphics{figure_2}
Fig. 2 shows the sector $OAB$ of a circle of radius $r$ cm. The area of the sector is $15$ cm$^2$ and $\angle AOB = 1.5$ radians.
\begin{enumerate}[label=(\alph*)]
\item Prove that $r = 2\sqrt{5}$. [3]
\item Find, in cm, the perimeter of the sector $OAB$. [2]
\end{enumerate}
The segment $R$, shaded in Fig 1, is enclosed by the arc $AB$ and the straight line $AB$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Calculate, to 3 decimal places, the area of $R$. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q7 [8]}}