| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Standard +0.8 This is a multi-part Further Maths polar coordinates question requiring: (a) standard polar area integration with a cardioid, (b)(i) solving simultaneous polar/Cartesian equations, and (b)(ii) coordinate geometry with surds. While polar area is a standard technique, the combination of skills and the Further Maths context places this moderately above average difficulty. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09c Area enclosed: by polar curve |
4 The diagram shows the curve $C$ with polar equation
$$r = 6 ( 1 - \cos \theta ) , \quad 0 \leqslant \theta < 2 \pi$$
\includegraphics[max width=\textwidth, alt={}, center]{06ae13de-5cf3-421d-ac7a-ee9f74b653be-3_552_903_922_550}
\begin{enumerate}[label=(\alph*)]
\item Find the area of the region bounded by the curve $C$.
\item The circle with cartesian equation $x ^ { 2 } + y ^ { 2 } = 9$ intersects the curve $C$ at the points $A$ and $B$.
\begin{enumerate}[label=(\roman*)]
\item Find the polar coordinates of $A$ and $B$.
\item Find, in surd form, the length of $A B$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2006 Q4 [12]}}