| Exam Board | Edexcel |
|---|---|
| Module | F2 (Further Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area between two polar curves |
| Difficulty | Challenging +1.2 This is a standard Further Maths polar coordinates question requiring finding intersection points by equating curves (straightforward algebra) and computing area between curves using the standard formula ∫½r²dθ. While it involves multiple steps and integration of trigonometric functions, the techniques are routine for F2 students with no novel problem-solving required. The 'exact answer' requirement adds minor complexity but is standard at this level. |
| Spec | 4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{3}{2}\cos\theta = 3\sqrt{3}-\frac{9}{2}\cos\theta\) or \(\cos\theta=\frac{2r}{3}\Rightarrow r=3\sqrt{3}-3r\) | M1 | Puts \(C_1=C_2\) and attempts to solve for \(\theta\), or eliminates \(\cos\theta\) and solves for \(r\) |
| \(\theta=\frac{\pi}{6}\) or \(r=\frac{3\sqrt{3}}{4}\) | A1 | Correct \(\theta\) or correct \(r\); allow \(\theta\) awrt 0.524, \(r\) awrt 1.3 |
| \(r=\frac{3\sqrt{3}}{4}\) and \(\theta=\frac{\pi}{6}\) | A1 | Correct \(r\) and \(\theta\); allow e.g. \(\left(\frac{\pi}{6},\frac{3\sqrt{3}}{4}\right)\); allow awrt 0.524, awrt 1.3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{2}\int\left(3\sqrt{3}-\frac{9}{2}\cos\theta\right)^2\,d\theta\) or \(\frac{1}{2}\int\left(\frac{3}{2}\cos\theta\right)^2\,d\theta\) | M1 | Attempts to use correct formula on either curve; \(\frac{1}{2}\) may be implied by later work |
| \(\left(3\sqrt{3}-\frac{9}{2}\cos\theta\right)^2 = 27-27\sqrt{3}\cos\theta+\frac{81}{4}\cos^2\theta = 27-27\sqrt{3}\cos\theta+\frac{81}{4}\cdot\frac{\cos 2\theta+1}{2}\) | M1 | Expands to form \(a+b\cos\theta+c\cos^2\theta\) and uses \(\cos^2\theta=\pm\frac{1}{2}\pm\frac{\cos 2\theta}{2}\) |
| \(\frac{1}{2}\int\left(3\sqrt{3}-\frac{9}{2}\cos\theta\right)^2\,d\theta = \frac{1}{2}\left[\frac{297}{8}\theta - 27\sqrt{3}\sin\theta+\frac{81}{16}\sin 2\theta\right]\) | M1A1 | M1: Integrates to get at least two terms from \(\alpha\theta\), \(\beta\sin\theta\), \(\gamma\sin 2\theta\). A1: Correct integration with or without the \(\frac{1}{2}\) |
| Applies limits \(0\) to \(\frac{\pi}{6}\) | M1 | Uses limits 0 and \(\frac{\pi}{6}\); if \(\theta=0\) evaluates to 0 it need not be shown |
| \(\frac{1}{2}\int\left(\frac{3}{2}\cos\theta\right)^2\,d\theta = \frac{9}{16}\int(\cos 2\theta+1)\,d\theta = \frac{9}{16}\left[\frac{1}{2}\sin 2\theta+\theta\right]_{\frac{\pi}{6}}^{\frac{\pi}{2}} = \frac{9}{16}\left(\frac{\pi}{3}-\frac{\sqrt{3}}{4}\right)\) | M1 | Uses \(\cos^2\theta=\pm\frac{1}{2}\pm\frac{\cos 2\theta}{2}\), integrates to get at least \(k\sin 2\theta\), uses limits \(\frac{\pi}{6}\) to \(\frac{\pi}{2}\) |
| \(\frac{297}{96}\pi - \frac{351\sqrt{3}}{64}+\frac{9}{16}\left(\frac{\pi}{3}-\frac{\sqrt{3}}{4}\right) = \frac{105}{32}\pi - \frac{45}{8}\sqrt{3}\) | M1A1 | M1: Adds two areas both of form \(a\pi+b\sqrt{3}\). A1: Correct answer; allow equivalent fractions for \(\frac{105}{32}\) and \(\frac{45}{8}\) |
# Question 7:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{3}{2}\cos\theta = 3\sqrt{3}-\frac{9}{2}\cos\theta$ or $\cos\theta=\frac{2r}{3}\Rightarrow r=3\sqrt{3}-3r$ | M1 | Puts $C_1=C_2$ and attempts to solve for $\theta$, or eliminates $\cos\theta$ and solves for $r$ |
| $\theta=\frac{\pi}{6}$ or $r=\frac{3\sqrt{3}}{4}$ | A1 | Correct $\theta$ or correct $r$; allow $\theta$ awrt 0.524, $r$ awrt 1.3 |
| $r=\frac{3\sqrt{3}}{4}$ and $\theta=\frac{\pi}{6}$ | A1 | Correct $r$ and $\theta$; allow e.g. $\left(\frac{\pi}{6},\frac{3\sqrt{3}}{4}\right)$; allow awrt 0.524, awrt 1.3 |
**Total: (3)**
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{2}\int\left(3\sqrt{3}-\frac{9}{2}\cos\theta\right)^2\,d\theta$ or $\frac{1}{2}\int\left(\frac{3}{2}\cos\theta\right)^2\,d\theta$ | M1 | Attempts to use correct formula on either curve; $\frac{1}{2}$ may be implied by later work |
| $\left(3\sqrt{3}-\frac{9}{2}\cos\theta\right)^2 = 27-27\sqrt{3}\cos\theta+\frac{81}{4}\cos^2\theta = 27-27\sqrt{3}\cos\theta+\frac{81}{4}\cdot\frac{\cos 2\theta+1}{2}$ | M1 | Expands to form $a+b\cos\theta+c\cos^2\theta$ and uses $\cos^2\theta=\pm\frac{1}{2}\pm\frac{\cos 2\theta}{2}$ |
| $\frac{1}{2}\int\left(3\sqrt{3}-\frac{9}{2}\cos\theta\right)^2\,d\theta = \frac{1}{2}\left[\frac{297}{8}\theta - 27\sqrt{3}\sin\theta+\frac{81}{16}\sin 2\theta\right]$ | M1A1 | M1: Integrates to get at least two terms from $\alpha\theta$, $\beta\sin\theta$, $\gamma\sin 2\theta$. A1: Correct integration with or without the $\frac{1}{2}$ |
| Applies limits $0$ to $\frac{\pi}{6}$ | M1 | Uses limits 0 and $\frac{\pi}{6}$; if $\theta=0$ evaluates to 0 it need not be shown |
| $\frac{1}{2}\int\left(\frac{3}{2}\cos\theta\right)^2\,d\theta = \frac{9}{16}\int(\cos 2\theta+1)\,d\theta = \frac{9}{16}\left[\frac{1}{2}\sin 2\theta+\theta\right]_{\frac{\pi}{6}}^{\frac{\pi}{2}} = \frac{9}{16}\left(\frac{\pi}{3}-\frac{\sqrt{3}}{4}\right)$ | M1 | Uses $\cos^2\theta=\pm\frac{1}{2}\pm\frac{\cos 2\theta}{2}$, integrates to get at least $k\sin 2\theta$, uses limits $\frac{\pi}{6}$ to $\frac{\pi}{2}$ |
| $\frac{297}{96}\pi - \frac{351\sqrt{3}}{64}+\frac{9}{16}\left(\frac{\pi}{3}-\frac{\sqrt{3}}{4}\right) = \frac{105}{32}\pi - \frac{45}{8}\sqrt{3}$ | M1A1 | M1: Adds two areas both of form $a\pi+b\sqrt{3}$. A1: Correct answer; allow equivalent fractions for $\frac{105}{32}$ and $\frac{45}{8}$ |
**Total: (8)**7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{84aadcb2-399f-4168-94c6-4e6ed0450d6d-12_866_1026_274_468}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of the curves $C _ { 1 }$ and $C _ { 2 }$ with polar equations
$$\begin{array} { l l }
C _ { 1 } : r = \frac { 3 } { 2 } \cos \theta , & 0 \leqslant \theta \leqslant \frac { \pi } { 2 } \\
C _ { 2 } : r = 3 \sqrt { 3 } - \frac { 9 } { 2 } \cos \theta , & 0 \leqslant \theta \leqslant \frac { \pi } { 2 }
\end{array}$$
The curves intersect at the point $P$.
\begin{enumerate}[label=(\alph*)]
\item Find the polar coordinates of $P$.
The region $R$, shown shaded in Figure 1, is enclosed by the curves $C _ { 1 }$ and $C _ { 2 }$ and the initial line.
\item Find the exact area of $R$, giving your answer in the form $p \pi + q \sqrt { 3 }$ where $p$ and $q$ are rational numbers to be found.
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel F2 2016 Q7 [11]}}