| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2016 |
| Session | October |
| Marks | 8 |
| 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 multi-part question combining basic trigonometry (sine rule to find an angle) with standard sector area formulas. Part (a) guides students through finding the angle, and part (b) requires adding triangle and sector areas using bookwork formulas. Slightly easier than average due to the scaffolding and routine application of standard techniques. |
| 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{\sin D}{5} = \frac{\sin 1.1}{6}\) | M1 | Uses sine rule with sides and angles in correct positions |
| \(\sin D = 0.74267\) so \(D = 0.84\) | M1, A1 | Makes \(\sin D\) subject and uses inverse sine; accept awrt 0.84 or in degrees awrt \(48.0°\) |
| \(B = \pi - (1.1 + 0.84) = 1.20^*\) | A1* | Printed answer; expect either \(\pi-(1.1+\text{awrt } 0.84)\) or \(\pi-1.1-\text{awrt } 0.84\) before 1.20 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Uses angle \(DBC = \pi - 1.2 =\) awrt \(1.94\) | B1 | Uses angles on straight line; if degrees accept awrt \(111.2°\) |
| Area of sector \(= \frac{1}{2}r^2\theta = \frac{1}{2} \times 6^2 \times 1.94' = 34.9\) or Area of triangle \(ABD = \frac{1}{2} \times 5 \times 6 \times \sin 1.2 = 14.0\) | M1 | Uses correct area formula for sector or triangle; may follow through on incorrectly found angle \(DBC\) |
| Total area \(= \frac{1}{2} \times 6^2 \times 1.94' + \frac{1}{2} \times 5 \times 6 \times \sin 1.2\) | dM1 | Adds correct area formula for sector and correct area formula for triangle |
| \(= 48.9 \text{ cm}^2\) | A1 | Accept awrt 48.9 (units not required) |
## Question 8:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{\sin D}{5} = \frac{\sin 1.1}{6}$ | M1 | Uses sine rule with sides and angles in correct positions |
| $\sin D = 0.74267$ so $D = 0.84$ | M1, A1 | Makes $\sin D$ subject and uses inverse sine; accept awrt 0.84 or in degrees awrt $48.0°$ |
| $B = \pi - (1.1 + 0.84) = 1.20^*$ | A1* | Printed answer; expect either $\pi-(1.1+\text{awrt } 0.84)$ or $\pi-1.1-\text{awrt } 0.84$ before 1.20 |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Uses angle $DBC = \pi - 1.2 =$ awrt $1.94$ | B1 | Uses angles on straight line; if degrees accept awrt $111.2°$ |
| Area of sector $= \frac{1}{2}r^2\theta = \frac{1}{2} \times 6^2 \times 1.94' = 34.9$ or Area of triangle $ABD = \frac{1}{2} \times 5 \times 6 \times \sin 1.2 = 14.0$ | M1 | Uses correct area formula for sector or triangle; may follow through on incorrectly found angle $DBC$ |
| Total area $= \frac{1}{2} \times 6^2 \times 1.94' + \frac{1}{2} \times 5 \times 6 \times \sin 1.2$ | dM1 | Adds correct area formula for sector and correct area formula for triangle |
| $= 48.9 \text{ cm}^2$ | A1 | Accept awrt 48.9 (units not required) |
---8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{53865e15-3838-4551-b507-fe49549b87db-20_545_1048_212_584}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
The compound shape $A B C D A$, shown in Figure 1, consists of a triangle $A B D$ joined along its edge $B D$ to a sector $D B C$ of a circle with centre $B$ and radius 6 cm . The points $A , B$ and $C$ lie on a straight line with $A B = 5 \mathrm {~cm}$ and $B C = 6 \mathrm {~cm}$. Angle $D A B = 1.1$ radians.
\begin{enumerate}[label=(\alph*)]
\item Show that angle $A B D = 1.20$ radians to 3 significant figures.
\item Find the area of the compound shape, giving your answer to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2016 Q8 [8]}}