| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Triangle and sector combined - area/perimeter with given values |
| Difficulty | Standard +0.3 This is a straightforward C2 question combining standard arc length formula (s=rθ), cosine rule for finding BC, and area calculations using sector and triangle formulas. All techniques are routine applications with no novel insight required, making it slightly easier than average. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05c 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 |
|---|---|---|
| (a) \(r\theta = 8 \times 0.7 = 5.6\) (cm) | M1, A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Rightarrow BC = 7.098\) or \(7.10\) (Awrt) or \(\sqrt{(50.4)}\) or better | M1, A1, M1, A1 cao | (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Area of \(R = 28.345 \ldots - 22.4 = 5.9455 = 5.95\) (cm²) | M1, A1, M1, A1, A1 | (5 marks) |
**(a)** $r\theta = 8 \times 0.7 = 5.6$ (cm) | M1, A1 | (2 marks)
**(b)** $BC^2 = 8^2 + 11^2 - 2 \times 8 \times 11 \times \cos 0.7$
$\Rightarrow BC = 7.098$ or $7.10$ (Awrt) or $\sqrt{(50.4)}$ or better | M1, A1, M1, A1 cao | (4 marks)
Perimeter $= (a) + (11-8) + BC_c = 15.7$ (cm)
**(c)** $\triangle = \frac{1}{2}ab\sin c = \frac{1}{2} \times 11 \times 8 \times \sin 0.7$
Sector $= \frac{1}{2}r^2\theta = \frac{1}{2} \times 8^2 \times 0.7$
Area of $R = 28.345 \ldots - 22.4 = 5.9455 = 5.95$ (cm²) | M1, A1, M1, A1, A1 | (5 marks)
**(Subtotal: 11 marks)**
Guidance: (c) Final A1 accept 3sf or better. (a) and (c) M1 for quoting and attempting to use correct formula. (b) 1st M1 for attempting to use cosine rule (formula given).7.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{13bca882-27da-40f2-99d8-4fdeb6629c4e-12_707_1072_301_434}
\end{center}
\end{figure}
Figure 1 shows the triangle $A B C$, with $A B = 8 \mathrm {~cm} , A C = 11 \mathrm {~cm}$ and $\angle B A C = 0.7$ radians. The $\operatorname { arc } B D$, where $D$ lies on $A C$, is an arc of a circle with centre $A$ and radius 8 cm . The region $R$, shown shaded in Figure 1, is bounded by the straight lines $B C$ and $C D$ and the $\operatorname { arc } B D$.
Find
\begin{enumerate}[label=(\alph*)]
\item the length of the $\operatorname { arc } B D$,
\item the perimeter of $R$, giving your answer to 3 significant figures,
\item the area of $R$, giving your answer to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2005 Q7 [11]}}