| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 13 |
| Topic | Parametric differentiation |
| Type | Tangent/normal meets curve again |
| Difficulty | Standard +0.8 This is a multi-part parametric differentiation question requiring finding dy/dx, determining a normal line equation, and solving a simultaneous system involving trigonometric parametric equations and a linear equation. Part (c) requires substituting the line equation into parametric equations and solving a non-trivial trigonometric equation, which goes beyond routine exercises. The 13 total marks and the need for extended algebraic manipulation place this above average difficulty, though it remains within standard Further Maths techniques. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations1.07s 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 $\frac{\mathrm{d}y}{\mathrm{d}x}$ 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 2023 Q4 [13]}}