| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Logo and design problems |
| Difficulty | Standard +0.3 This is a straightforward C2 sector/triangle problem requiring standard formulas (sector area = ½r²θ, cosine rule) with minimal problem-solving. The symmetry simplifies part (b), and part (c) is just arithmetic combination of areas. Slightly easier than average due to clear setup and routine application of formulas. |
| Spec | 1.05c Area of triangle: using 1/2 ab sin(C)1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{2}r^2\theta = \frac{1}{2} \times 6^2 \times 2.2 = 39.6 \text{ cm}^2\) | M1 A1 | Needs \(\theta\) in radians; answer exactly 39.6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\left(\frac{2\pi - 2.2}{2}\right) = \pi - 1.1 = 2.04\) (rad) | M1 A1 | Needs full method; allow answers rounding to 2.04 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\Delta DAC = \frac{1}{2} \times 6 \times 4 \sin 2.04 \approx 10.7\) | M1 A1ft | Use \(\frac{1}{2} \times 6 \times 4 \sin A\); ft value from (b) |
| Total area \(=\) sector \(+ 2\) triangles \(= 61 \text{ cm}^2\) | M1 A1 | Uses total area \(=\) sector \(+ 2\) triangles |
# Question 7:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{2}r^2\theta = \frac{1}{2} \times 6^2 \times 2.2 = 39.6 \text{ cm}^2$ | M1 A1 | Needs $\theta$ in radians; answer exactly 39.6 |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left(\frac{2\pi - 2.2}{2}\right) = \pi - 1.1 = 2.04$ (rad) | M1 A1 | Needs full method; allow answers rounding to 2.04 |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\Delta DAC = \frac{1}{2} \times 6 \times 4 \sin 2.04 \approx 10.7$ | M1 A1ft | Use $\frac{1}{2} \times 6 \times 4 \sin A$; ft value from (b) |
| Total area $=$ sector $+ 2$ triangles $= 61 \text{ cm}^2$ | M1 A1 | Uses total area $=$ sector $+ 2$ triangles |
---7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{12e54724-64a3-4dc0-b7d5-6ef6cc04124c-09_878_991_233_461}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
The shape $B C D$ shown in Figure 3 is a design for a logo.
The straight lines $D B$ and $D C$ are equal in length. The curve $B C$ is an arc of a circle with centre $A$ and radius 6 cm . The size of $\angle B A C$ is 2.2 radians and $A D = 4 \mathrm {~cm}$.
Find
\begin{enumerate}[label=(\alph*)]
\item the area of the sector $B A C$, in $\mathrm { cm } ^ { 2 }$,
\item the size of $\angle D A C$, in radians to 3 significant figures,
\item the complete area of the logo design, to the nearest $\mathrm { cm } ^ { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2009 Q7 [8]}}