| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2011 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Properties of specific curves |
| Difficulty | Challenging +1.2 Part (a) requires standard parametric-to-Cartesian conversion using trigonometric identities (cos²t + sin²t = 1) and completing the square—routine A-level techniques. Parts (b-c) involve straightforward area calculation and single-variable calculus optimization. While the AEA context and multi-step nature elevate it slightly above average, the techniques are all standard Further Maths content with no novel insights required. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07n Stationary points: find maxima, minima using derivatives1.07s Parametric and implicit differentiation |
The curve $C$ has parametric equations
$$x = \cos^2 t$$
$$y = \cos t \sin t$$
where $0 \leq t < \pi$
\begin{enumerate}[label=(\alph*)]
\item Show that $C$ is a circle and find its centre and its radius.
[5]
\end{enumerate}
% Figure 1 shows a sketch of C with point P, rectangle R with diagonal OP
\includegraphics{figure_1}
Figure 1
Figure 1 shows a sketch of $C$. The point $P$, with coordinates $(\cos^2 \alpha, \cos\alpha \sin \alpha)$, $0 < \alpha < \frac{\pi}{2}$, lies on $C$. The rectangle $R$ has one side on the $x$-axis, one side on the $y$-axis and $OP$ as a diagonal, where $O$ is the origin.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that the area of $R$ is $\sin\alpha \cos^3 \alpha$
[1]
\item Find the maximum area of $R$, as $\alpha$ varies.
[7]
\end{enumerate}
[Total 13 marks]
\hfill \mbox{\textit{Edexcel AEA 2011 Q4 [13]}}