| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | September |
| Marks | 13 |
| Topic | Parametric curves and Cartesian conversion |
| Type | Find normal equation |
| Difficulty | Standard +0.8 This is a substantial parametric curves question requiring dy/dx calculation, normal line equation, and solving a simultaneous system of parametric and linear equations. Part (c) demands algebraic manipulation with trigonometric identities and careful solution of a non-trivial system, going beyond routine textbook exercises. The multi-step nature and need for exact coordinates elevate this above average difficulty. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
The curve $C$ has parametric equations
$$x = 2\cos t, \quad y = \sqrt{3}\cos 2t, \quad 0 \leq t \leq \pi$$
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\frac{dy}{dx}$ in terms of $t$. [2]
\end{enumerate}
The point $P$ lies on $C$ where $t = \frac{2\pi}{3}$
The line $l$ is the normal to $C$ at $P$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that an equation for $l$ is
$$2x - 2\sqrt{3}y - 1 = 0$$ [5]
\end{enumerate}
The line $l$ intersects the curve $C$ again at the point $Q$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the exact coordinates of $Q$.
You must show clearly how you obtained your answers. [6]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q4 [13]}}