| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2017 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Compound shape area |
| Difficulty | Moderate -0.3 This is a standard compound shape problem requiring arc length formula (s=rθ), basic trigonometry (tan⁻¹), and perimeter calculation by adding segments. All techniques are routine C2 level with straightforward application, though the multi-part structure and careful tracking of which segments form the perimeter requires moderate attention to detail. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(S = r\theta = 7 \times 0.8 = 5.6\) (cm) | M1A1 | M1: Uses \(S = r\theta\); A1: 5.6 oe e.g. \(28/5\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\angle POC = \frac{\pi}{2} - 0.8 =\) awrt \(0.771\) | M1A1 | M1: Attempts to find \(\frac{\pi}{2} - 0.8\) or \(\pi - \frac{\pi}{2} - 0.8\). Allow attempt to find \(\theta\) from \(\theta + \frac{\pi}{2} + 0.8 = \pi\); A1: awrt 0.771. Answers in degrees only score M1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(4^2 + 5^2 - 2 \times 4 \times 5\cos(0.771)\) or \(\sqrt{4^2 + 5^2 - 2 \times 4 \times 5\cos(0.771)}\) | M1 | Correct use of cosine rule to find \(CP\) or \(CP^2\). NB 0.771 radians is awrt 44 degrees. Ignore LHS; look for e.g. \(4^2 + 5^2 - 2\times4\times5\cos(0.771)\) or 44 |
| \(CP^2 = 4^2 + 5^2 - 2 \times 4 \times 5\cos(0.771)\) | A1 | Correct expression for \(CP\) or \(CP^2\) with LHS consistent with RHS. Allow awrt 0.77 radians or awrt 44 degrees |
| Perimeter \(= 4 + 5 + 2 \times 7 +\) '5.6' \(+\) '3.5' | M1 | \(4+5+2\times7+\) their \(AQ +\) their \(CP\). Need to see all 6 lengths, may be implied by e.g. \(23 +\) '5.6' \(+\) '3.5' |
| \(= 32.11\) (cm) | A1 | Awrt 32.11 (ignore units) |
## Question 3:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $S = r\theta = 7 \times 0.8 = 5.6$ (cm) | M1A1 | M1: Uses $S = r\theta$; A1: 5.6 oe e.g. $28/5$ |
---
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\angle POC = \frac{\pi}{2} - 0.8 =$ awrt $0.771$ | M1A1 | M1: Attempts to find $\frac{\pi}{2} - 0.8$ or $\pi - \frac{\pi}{2} - 0.8$. Allow attempt to find $\theta$ from $\theta + \frac{\pi}{2} + 0.8 = \pi$; A1: awrt 0.771. Answers in degrees only score M1A0 |
---
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4^2 + 5^2 - 2 \times 4 \times 5\cos(0.771)$ or $\sqrt{4^2 + 5^2 - 2 \times 4 \times 5\cos(0.771)}$ | M1 | Correct use of cosine rule to find $CP$ or $CP^2$. NB 0.771 radians is awrt 44 degrees. Ignore LHS; look for e.g. $4^2 + 5^2 - 2\times4\times5\cos(0.771)$ or 44 |
| $CP^2 = 4^2 + 5^2 - 2 \times 4 \times 5\cos(0.771)$ | A1 | Correct expression for $CP$ or $CP^2$ with LHS consistent with RHS. Allow awrt 0.77 radians or awrt 44 degrees |
| Perimeter $= 4 + 5 + 2 \times 7 +$ '5.6' $+$ '3.5' | M1 | $4+5+2\times7+$ their $AQ +$ their $CP$. Need to see all 6 lengths, may be implied by e.g. $23 +$ '5.6' $+$ '3.5' |
| $= 32.11$ (cm) | A1 | Awrt 32.11 (ignore units) |
---3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f39ade34-32e2-4b5c-b80a-9663c6a65c87-04_629_1061_260_555}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
The shape $P O Q A B C P$, as shown in Figure 1, consists of a triangle $P O C$, a sector $O Q A$ of a circle with radius 7 cm and centre $O$, joined to a rectangle $O A B C$.
The points $P , O$ and $Q$ lie on a straight line.\\
$P O = 4 \mathrm {~cm} , C O = 5 \mathrm {~cm}$ and angle $A O Q = 0.8$ radians.
\begin{enumerate}[label=(\alph*)]
\item Find the length of arc $A Q$.
\item Find the size of angle $P O C$ in radians, giving your answer to 3 decimal places.\\
(2)
\item Find the perimeter of the shape $P O Q A B C P$, in cm , giving your answer to 2 decimal places.\\
(4)
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2017 Q3 [8]}}