| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Topic | Parametric differentiation |
| Type | Find normal equation at parameter |
| Difficulty | Standard +0.8 This is a multi-part parametric curves question requiring: (a) finding the parameter value from coordinates, computing dy/dx via the chain rule, finding the normal equation—all standard but requiring careful algebra; (b) eliminating the parameter using double-angle identities and algebraic manipulation to reach a specific Cartesian form. The techniques are A-level standard but the execution requires precision across multiple steps, placing it moderately above average difficulty. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
The curve defined by the parametric equations
$$x = 2\cos\theta, \quad y = 3\sin(2\theta) \quad \text{and} \quad \theta \in [0, 2\pi]$$
is shown below.
The point $P\left(\sqrt{3}, \frac{3\sqrt{3}}{2}\right)$ is marked on the curve.
\includegraphics{figure_curve}
\begin{enumerate}[label=(\alph*)]
\item Show that the equation of the normal to the curve at $P$ can be written as $3y - x = \frac{7\sqrt{3}}{2}$ [5]
\item Show that the Cartesian equation of the curve may be written as $ay^2 + bx^4 + cx^2 = 0$ where $a$, $b$ and $c$ are integers to be found. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2022 Q10 [8]}}