| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2012 |
| Session | January |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Standard +0.8 This is a multi-part polar coordinates question requiring (a) standard area integration with cos²θ, (b)(i) converting between polar and Cartesian to find intersection points and chord length, and (b)(ii) finding arc length. While polar area is routine for FP3, the intersection problem requires coordinate conversion and geometric reasoning across multiple steps, making it moderately challenging but still within standard Further Maths scope. |
| Spec | 4.09c Area enclosed: by polar curve |
8 The diagram shows a sketch of the curve $C$ with polar equation
$$r = 3 + 2 \cos \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$
\includegraphics[max width=\textwidth, alt={}, center]{80c4336c-b0ca-46de-a871-812e6923f7f2-4_463_668_1468_699}
\begin{enumerate}[label=(\alph*)]
\item Find the area of the region bounded by the curve $C$.
\item A circle, whose cartesian equation is $( x - 4 ) ^ { 2 } + y ^ { 2 } = 16$, intersects the curve $C$ at the points $A$ and $B$.
\begin{enumerate}[label=(\roman*)]
\item Find, in surd form, the length of $A B$.
\item Find the perimeter of the segment $A O B$ of the circle, where $O$ is the pole.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2012 Q8 [15]}}