| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area between two polar curves |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths polar coordinates question requiring intersection finding, distance calculations, tangency verification, and area between curves. While it involves several steps and FP3 content, each part follows standard techniques: solving r₁=r₂ for intersections, using symmetry for related points, applying the distance formula, and integrating ½r² with appropriate limits. The tangency explanation requires geometric insight but is guided by 'hence'. More routine than typical FP3 proof questions. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
8 The diagram shows a sketch of a curve and a circle.\\
\includegraphics[max width=\textwidth, alt={}, center]{a2bc95fe-5588-4ff7-a8a3-0cd07df412c9-4_460_693_370_680}
The polar equation of the curve is
$$r = 3 + 2 \sin \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$
The circle, whose polar equation is $r = 2$, intersects the curve at the points $P$ and $Q$, as shown in the diagram.
\begin{enumerate}[label=(\alph*)]
\item Find the polar coordinates of $P$ and the polar coordinates of $Q$.
\item A straight line, drawn from the point $P$ through the pole $O$, intersects the curve again at the point $A$.
\begin{enumerate}[label=(\roman*)]
\item Find the polar coordinates of $A$.
\item Find, in surd form, the length of $A Q$.
\item Hence, or otherwise, explain why the line $A Q$ is a tangent to the circle $r = 2$.
\end{enumerate}\item Find the area of the shaded region which lies inside the circle $r = 2$ but outside the curve $r = 3 + 2 \sin \theta$. Give your answer in the form $\frac { 1 } { 6 } ( m \sqrt { 3 } + n \pi )$, where $m$ and $n$ are integers.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2013 Q8 [19]}}