| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | February |
| Marks | 8 |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Challenging +1.3 This is a Further Maths polar coordinates question requiring a sketch and area calculation using the standard formula A = ½∫r²dθ. While the integration involves sec and tan terms requiring trigonometric identities (sec²θ = 1 + tan²θ), the setup is straightforward and the techniques are standard for FM students. The restricted domain and exact answer requirement add moderate complexity, but this remains a typical textbook-style polar area question without requiring novel insight. |
| 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
$$r = \sec \theta + \tan \theta, \quad \text{for } 0 \leq \theta \leq \frac{1}{4}\pi.$$
\begin{enumerate}[label=(\roman*)]
\item Sketch the curve. [2]
\item Find the exact area of the region bounded by the curve and the lines $\theta = 0$ and $\theta = \frac{1}{4}\pi$. [6]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q7 [8]}}