| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 9 |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Standard +0.3 This is a straightforward polar coordinates question requiring standard techniques: finding maximum by differentiation (or recognizing sin 2θ peaks at π/4), evaluating r, and applying the polar area formula ∫½r²dθ. While polar coordinates is a Further Maths topic, the question involves routine calculus with no novel problem-solving, making it slightly easier than average overall. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
The equation of the curve shown on the graph is, in polar coordinates, $r = 3\sin 2\theta$ for $0 \leqslant \theta \leqslant \frac{1}{2}\pi$.
\includegraphics{figure_2}
\begin{enumerate}[label=(\alph*)]
\item The greatest value of $r$ on the curve occurs at the point $P$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\theta = \frac{1}{4}\pi$ at the point $P$. [2]
\item Find the value of $r$ at the point $P$. [1]
\item Mark the point $P$ on a copy of the graph. [1]
\end{enumerate}
\item In this question you must show detailed reasoning.
Find the exact area of the region enclosed by the curve. [5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q2 [9]}}