| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area between two polar curves |
| Difficulty | Standard +0.8 This is a Further Maths FP2 polar coordinates question requiring intersection finding and area calculation between curves. Part (a) involves solving a trigonometric equation (2 = 1.5 + sin 3θ), which is straightforward. Part (b) requires setting up and evaluating the polar area integral ½∫(r₂² - r₁²)dθ with correct limits, integrating sin²(3θ) terms, and simplifying to the exact form requested. While methodical, it demands careful setup, trigonometric integration techniques, and algebraic manipulation—typical of standard FP2 polar area problems but not requiring novel insight. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09c Area enclosed: by polar curve |
\includegraphics{figure_1}
Figure 1
Figure 1 shows the curves given by the polar equations
$$r = 2, \quad 0 \leq \theta \leq \frac{\pi}{2},$$
and
$$r = 1.5 + \sin 3\theta, \quad 0 \leq \theta \leq \frac{\pi}{2}.$$
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the points where the curves intersect. [3]
\end{enumerate}
The region $S$, between the curves, for which $r > 2$ and for which $r < (1.5 + \sin 3\theta)$, is shown shaded in Figure 1.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find, by integration, the area of the shaded region $S$, giving your answer in the form $a\pi + b\sqrt{3}$, where $a$ and $b$ are simplified fractions. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q5 [10]}}