| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | February |
| Marks | 11 |
| Topic | Polar coordinates |
| Type | Polar curve with exponential function |
| Difficulty | Challenging +1.8 This is a challenging Further Maths polar coordinates question requiring (a) integration of a non-standard polar area formula with substitution, and (b) optimization using calculus on a composite function involving both trigonometric and exponential terms. Part (b) particularly demands careful differentiation, algebraic manipulation, and verification of the maximum. The combination of techniques and the non-routine nature of the functions involved places this well above average difficulty, though it follows standard FM polar curve methodology. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)4.09c Area enclosed: by polar curve |
In this question you must show detailed reasoning.
The diagram below shows the curve $r = \sqrt{\sin \theta} e^{\frac{1}{2}\cos \theta}$ for $0 \leqslant \theta \leqslant \pi$.
\includegraphics{figure_13}
\begin{enumerate}[label=(\alph*)]
\item Find the exact area enclosed by the curve.
[4]
\item Show that the greatest value of $r$ on the curve is $\sqrt{\frac{3}{2}} e^{\frac{1}{6}}$.
[7]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q13 [11]}}