| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Find intersection points |
| Difficulty | Challenging +1.8 This AEA question requires finding intersection points by solving a system involving parametric equations and a circle, then computing an area using integration. It demands careful algebraic manipulation (substituting parametric forms into x² + y² = r², solving a trigonometric equation), understanding of parametric integration, and likely polar coordinates or sector area formulas. The multi-step nature, combination of techniques, and AEA context place it well above average difficulty, though it follows a relatively standard framework for this type of problem. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03g Parametric equations: of curves and conversion to cartesian1.08e Area between curve and x-axis: using definite integrals |
3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{78ba3acc-4cca-4d15-8362-a27e425c5859-08_752_586_251_742}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the curve $C$ given by the parametric equations
$$x = \frac { 5 } { \sqrt { 3 } } \sin t \quad y = 5 ( 1 - \cos t ) \quad 0 \leqslant t \leqslant 2 \pi$$
The circle with centre at the origin $O$ and with radius $\frac { 5 \sqrt { 2 } } { 2 }$ meets the curve $C$ at the points $A$ and $B$ as shown in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Determine the value of $t$ at the point $B$ .
The region $R$ ,shown shaded in Figure 1,is bounded by the curve $C$ and the circle.
\item Determine the area of the region $R$ .
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2023 Q3 [10]}}