| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area between two polar curves |
| Difficulty | Standard +0.8 This is a Further Maths polar coordinates question requiring finding intersection points and calculating area between two curves. While the techniques are standard (solving r₁=r₂, using ½∫r²dθ), it requires careful setup with two different integrals for the shaded region and solid understanding of polar geometry. The multi-step nature and Further Maths context place it moderately above average difficulty. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{18620cc5-2377-480b-b815-63bfc6a9760a-15_618_942_255_584}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
The curve $C _ { 1 }$ with equation
$$r = 7 \cos \theta , \quad - \frac { \pi } { 2 } < \theta \leqslant \frac { \pi } { 2 }$$
and the curve $C _ { 2 }$ with equation
$$r = 3 ( 1 + \cos \theta ) , \quad - \pi < \theta \leqslant \pi$$
are shown on Figure 1.\\
The curves $C _ { 1 }$ and $C _ { 2 }$ both pass through the pole and intersect at the point $P$ and the point $Q$.
\begin{enumerate}[label=(\alph*)]
\item Find the polar coordinates of $P$ and the polar coordinates of $Q$.
The regions enclosed by the curve $C _ { 1 }$ and the curve $C _ { 2 }$ overlap, and the common region $R$ is shaded in Figure 1.
\item Find the area of $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2016 Q8 [10]}}