| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2010 |
| 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 conversion to Cartesian form, area calculations using standard polar area formula, and proving non-intersection. While it involves several techniques and is from FP3, each part follows standard procedures: (a) uses routine polar-to-Cartesian conversion and completing the square, (b)(i) is a direct application of the polar area formula, (b)(ii) requires algebraic manipulation to show no solutions exist, and (b)(iii) combines previous results. The question is methodical rather than requiring novel insight, making it moderately above average difficulty but not exceptionally challenging for Further Maths students. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
6 The polar equation of a curve $C _ { 1 }$ is
$$r = 2 ( \cos \theta - \sin \theta ) , \quad 0 \leqslant \theta \leqslant 2 \pi$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the cartesian equation of $C _ { 1 }$.
\item Deduce that $C _ { 1 }$ is a circle and find its radius and the cartesian coordinates of its centre.
\end{enumerate}\item The diagram shows the curve $C _ { 2 }$ with polar equation
$$r = 4 + \sin \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$
\includegraphics[max width=\textwidth, alt={}, center]{90a59b47-3799-46a2-b76b-ced5cc3e1aac-4_519_847_443_593}
\begin{enumerate}[label=(\roman*)]
\item Find the area of the region that is bounded by $C _ { 2 }$.
\item Prove that the curves $C _ { 1 }$ and $C _ { 2 }$ do not intersect.
\item Find the area of the region that is outside $C _ { 1 }$ but inside $C _ { 2 }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2010 Q6 [19]}}