| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| 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 testing standard formulas for arc length and sector area, plus basic trigonometry. Parts (a) and (b) are direct formula application (s=rθ, A=½r²θ). Parts (c) and (d) require recognizing a right triangle and subtracting areas, but involve only routine techniques with no novel insight required. Slightly above average due to the multi-step nature and need to visualize the composite region, but still a standard textbook exercise. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(r\theta = 9 \times 0.7 = 6.3\) | M1 A1 | Also allow 6.30 or awrt 6.30. Use of \(r\theta\) with \(\theta\) in radians |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2}r^2\theta = \frac{1}{2}\times81\times0.7 = 28.35\) | M1 A1 | Also allow 28.3, 28.4, or awrt 28.3 or 28.4. Condone \(28.35^2\) written instead of \(28.35\text{ cm}^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\tan 0.7 = \frac{AC}{9}\) | M1 | Other methods must be fully correct |
| \(AC = 7.58\) (allow awrt) | A1 | NOT 7.59. Premature approximation e.g. taking angle \(C\) as 0.87 radians loses A mark |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Area of triangle \(AOC = \frac{1}{2}(9 \times \text{their } AC)\) | M1 | or other complete method |
| Area \(R =\) "34.11" \(-\) "28.35" (triangle \(-\) sector) | M1 | needs a value for each |
| \(= 5.76\) (allow awrt) | A1 |
## Question 6:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $r\theta = 9 \times 0.7 = 6.3$ | M1 A1 | Also allow 6.30 or awrt 6.30. Use of $r\theta$ with $\theta$ in radians |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2}r^2\theta = \frac{1}{2}\times81\times0.7 = 28.35$ | M1 A1 | Also allow 28.3, 28.4, or awrt 28.3 or 28.4. Condone $28.35^2$ written instead of $28.35\text{ cm}^2$ |
### Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\tan 0.7 = \frac{AC}{9}$ | M1 | Other methods must be fully correct |
| $AC = 7.58$ (allow awrt) | A1 | NOT 7.59. Premature approximation e.g. taking angle $C$ as 0.87 radians loses A mark |
### Part (d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Area of triangle $AOC = \frac{1}{2}(9 \times \text{their } AC)$ | M1 | or other complete method |
| Area $R =$ "34.11" $-$ "28.35" (triangle $-$ sector) | M1 | needs a value for each |
| $= 5.76$ (allow awrt) | A1 | |
---6.\\
\includegraphics[max width=\textwidth, alt={}, center]{571780c2-945b-4636-b7c3-0bd558d28710-07_458_809_258_569}
\section*{Figure 1}
Figure 1 shows the sector $O A B$ of a circle with centre $O$, radius 9 cm and angle 0.7 radians.
\begin{enumerate}[label=(\alph*)]
\item Find the length of the arc $A B$.
\item Find the area of the sector $O A B$.
The line $A C$ shown in Figure 1 is perpendicular to $O A$, and $O B C$ is a straight line.
\item Find the length of $A C$, giving your answer to 2 decimal places.
The region $H$ is bounded by the arc $A B$ and the lines $A C$ and $C B$.
\item Find the area of $H$, giving your answer to 2 decimal places.\\
\section*{LU}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2010 Q6 [9]}}