| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Simultaneous equations with arc/area |
| Difficulty | Standard +0.3 This is a straightforward application of standard arc length and sector area formulas to form two simultaneous equations in r and θ. The setup requires recognizing that perimeter = 2r + rθ = 4(rθ), which gives θ = 2, then substituting into the area formula ½r²θ = 12 to find r. While it involves multiple steps, each is routine and the problem-solving required is minimal—slightly easier than average for A-level. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Arc length \(AB = r\theta\) | B1 | Correct formula stated or used |
| Perimeter \(= 2r + r\theta\) | M1 | Correct expression for perimeter |
| \(2r + r\theta = 4r\theta\) | M1 | Setting perimeter \(= 4 \times\) arc |
| \(2r = 3r\theta \Rightarrow \theta = \frac{2}{3}\) | A1 | Correct value of \(\theta\) |
| Area \(= \frac{1}{2}r^2\theta = 12\) | M1 | Using area formula with their \(\theta\) |
| \(\frac{1}{2}r^2 \times \frac{2}{3} = 12 \Rightarrow r^2 = 36 \Rightarrow r = 6\) | A1 | \(r = 6\) |
## Question 5:
| Arc length $AB = r\theta$ | B1 | Correct formula stated or used |
| Perimeter $= 2r + r\theta$ | M1 | Correct expression for perimeter |
| $2r + r\theta = 4r\theta$ | M1 | Setting perimeter $= 4 \times$ arc |
| $2r = 3r\theta \Rightarrow \theta = \frac{2}{3}$ | A1 | Correct value of $\theta$ |
| Area $= \frac{1}{2}r^2\theta = 12$ | M1 | Using area formula with their $\theta$ |
| $\frac{1}{2}r^2 \times \frac{2}{3} = 12 \Rightarrow r^2 = 36 \Rightarrow r = 6$ | A1 | $r = 6$ |5 The diagram shows a sector $O A B$ of a circle with centre $O$ and radius $r \mathrm {~cm}$.\\
\includegraphics[max width=\textwidth, alt={}, center]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-10_346_360_360_824}
The angle $A O B$ is $\theta$ radians.\\
The area of the sector is $12 \mathrm {~cm} ^ { 2 }$.\\
The perimeter of the sector is four times the length of the $\operatorname { arc } A B$.\\
Find the value of $r$.\\[0pt]
[6 marks]
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-10_1533_1712_1174_150}
\end{center}
\hfill \mbox{\textit{AQA C2 2014 Q5 [6]}}