| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Challenging +1.3 This is a Further Pure 2 polar coordinates question requiring understanding of symmetry properties and polar area integration. Part (i) involves recognizing that perpendicular lines correspond to angles differing by π/2 and using the cosine addition formula—straightforward for FP2 students. Part (ii) is a standard polar area integral with substitution. While this requires FP2-specific knowledge (making it harder than typical A-level), it's a routine application of polar techniques without requiring novel insight or complex multi-step reasoning. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(OP = 3 + 2\cos \alpha\). \(OQ = 3 + 2\cos(\frac{1}{2}\pi+\alpha) = 3 - 2\sin \alpha\). Similarly \(OR = 3 - 2\cos \alpha\). \(OS = 3 + 2\sin \alpha\). Sum = 12 | M1, M1, M1, A1 | Any other unsimplified value; Attempt at simplification of at least two correct expressions; cao |
| (ii) Correct formula with attempt at \(r^2\). Square \(r\) correctly. Attempt to replace \(\cos^2\theta\) with \(a(\cos 2\theta \pm 1)\). Integrate their expression. Get \(\frac{1}{3}r^2\theta - 1\) | M1, A1, M1, A1V, A1 | Need not be expanded, but three terms if it is; Need three terms; cao |
(i) $OP = 3 + 2\cos \alpha$. $OQ = 3 + 2\cos(\frac{1}{2}\pi+\alpha) = 3 - 2\sin \alpha$. Similarly $OR = 3 - 2\cos \alpha$. $OS = 3 + 2\sin \alpha$. Sum = 12 | M1, M1, M1, A1 | Any other unsimplified value; Attempt at simplification of at least two correct expressions; cao
(ii) Correct formula with attempt at $r^2$. Square $r$ correctly. Attempt to replace $\cos^2\theta$ with $a(\cos 2\theta \pm 1)$. Integrate their expression. Get $\frac{1}{3}r^2\theta - 1$ | M1, A1, M1, A1V, A1 | Need not be expanded, but three terms if it is; Need three terms; cao\includegraphics{figure_7}
The diagram shows the curve with equation, in polar coordinates,
$$r = 3 + 2\cos \theta, \quad \text{for } 0 \leq \theta < 2\pi.$$
The points $P$, $Q$, $R$ and $S$ on the curve are such that the straight lines $POR$ and $QOS$ are perpendicular, where $O$ is the pole. The point $P$ has polar coordinates $(r, \alpha)$.
\begin{enumerate}[label=(\roman*)]
\item Show that $OP + OQ + OR + OS = k$, where $k$ is a constant to be found. [3]
\item Given that $\alpha = \frac{1}{4}\pi$, find the exact area bounded by the curve and the lines $OP$ and $OQ$ (shaded in the diagram). [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2009 Q7 [8]}}