| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Polar curve with exponential function |
| Difficulty | Standard +0.8 This is an FP2 polar coordinates question requiring a sketch of an exponential spiral and derivation of an area formula. Part (i) is straightforward. Part (ii) requires applying the polar area formula, substituting the curve equation, integrating an exponential, then manipulating to express the result in terms of r₁ and r₂ rather than θ₁ and θ₂—this algebraic transformation requires insight but follows standard FP2 techniques. Moderately challenging for Further Maths. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
The equation of a curve, in polar coordinates, is
$$r = e^{-2\theta}, \quad \text{for } 0 \leq \theta \leq \pi.$$
\begin{enumerate}[label=(\roman*)]
\item Sketch the curve, stating the polar coordinates of the point at which $r$ takes its greatest value. [2]
\item The pole is $O$ and points $P$ and $Q$, with polar coordinates $(r_1, \theta_1)$ and $(r_2, \theta_2)$ respectively, lie on the curve. Given that $\theta_2 > \theta_1$, show that the area of the region enclosed by the curve and the lines $OP$ and $OQ$ can be expressed as $k(r_1^2 - r_2^2)$, where $k$ is a constant to be found. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2010 Q4 [7]}}