| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | February |
| Marks | 9 |
| Topic | Parametric differentiation |
| Type | Tangent/normal meets curve again |
| Difficulty | Challenging +1.2 This is a multi-part parametric question requiring finding a parameter value, deriving a normal equation using dy/dx = (dy/dt)/(dx/dt), and finding a second intersection point. While it involves several standard techniques and the double angle formula (cos 2t = 1 - 2sin²t), each step follows routine procedures without requiring novel insight or particularly complex algebra. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
13.
The curve $C$ has parametric equations
$$x = 10 \cos 2 t , \quad y = 6 \sin t , \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$
The point $A$ with coordinates $( 5,3 )$ lies on $C$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $t$ at the point $A$.
\item Show that an equation of the normal to $C$ at $A$ is
$$3 y = 10 x - 41$$
The normal to $C$ at $A$ cuts $C$ again at the point $B$.
\item Find the exact coordinates of $B$.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q13 [9]}}