| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795 (Pre-U Further Mathematics) |
| Session | Specimen |
| Marks | 8 |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Challenging +1.2 This is a standard Further Maths polar coordinates question requiring knowledge of symmetry properties, double-angle identities, and the polar area formula. Part (i) tests conceptual understanding of polar equations (2 marks of straightforward explanation). Part (ii) involves routine algebraic manipulation (squaring and using cos²(2θ) identity) followed by standard integration—all textbook techniques with no novel insight required. The 8-mark total and multi-step nature elevate it slightly above average difficulty, but it remains a conventional Further Maths exercise. |
| Spec | 1.05l Double angle formulae: and compound angle formulae4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
\includegraphics{figure_6}
The diagram shows a sketch of the curve $C$ with polar equation $r = a \cos^2 \theta$, where $a$ is a positive constant and $-\frac{1}{2}\pi \leqslant \theta \leqslant \frac{1}{2}\pi$.
\begin{enumerate}[label=(\roman*)]
\item Explain briefly how you can tell from this form of the equation that $C$ is symmetrical about the line $\theta = 0$ and that the tangent to $C$ at the pole $O$ is perpendicular to the line $\theta = 0$. [2]
\item The equation of $C$ may be expressed in the form $r = \frac{1}{2}a(1 + \cos 2\theta)$. Using this form, show that the area of the region enclosed by $C$ is given by
$$\frac{1}{16}a^2 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (3 + 4 \cos 2\theta + \cos 4\theta) \, \mathrm{d}\theta,$$
and find this area. [6]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795 Q6 [8]}}