| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| 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 intersection points, area calculation using the polar area formula, and curve sketching. While it involves multiple parts and integration, the techniques are routine for FP2 students: solving r₁=r₂ for intersections, applying ½∫r²dθ with appropriate limits, and recognizing standard polar curves. The 15-mark allocation reflects length rather than exceptional difficulty. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
The diagram above shows the curve $C_1$ which has polar equation $r = a(3 + 2\cos\theta)$, $0 \leq \theta < 2\pi$ and the circle $C_2$ with equation $r = 4a$, $0 \leq \theta < 2\pi$, where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $a$, the polar coordinates of the points where the curve $C_1$ meets the circle $C_2$.(4)
\end{enumerate}
The regions enclosed by the curves $C_1$ and $C_2$ overlap and this common region $R$ is shaded in the figure.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find, in terms of $a$, an exact expression for the area of the region $R$.(8)
\item In a single diagram, copy the two curves in the diagram above and also sketch the curve $C_3$ with polar equation $r = 2a\cos\theta$, $0 \leq \theta < 2\pi$ Show clearly the coordinates of the points of intersection of $C_1$, $C_2$ and $C_3$ with the initial line, $\theta = 0$.(3)(Total 15 marks)
\end{enumerate}
\includegraphics{figure_4}
\hfill \mbox{\textit{Edexcel FP2 2008 Q4}}