| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 11 |
| Topic | Parametric curves and Cartesian conversion |
| Type | Show dy/dx equals expression |
| Difficulty | Standard +0.3 This is a standard parametric differentiation question requiring routine techniques: finding dy/dx using the chain rule, locating a stationary point, eliminating the parameter using the double angle formula (cos 2t = 1 - 2sin²t), and sketching. All steps are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
A curve has parametric equations
$$x = 2\sin t, \quad y = \cos 2t + 2\sin t$$
for $-\frac{1}{2}\pi \leqslant t \leqslant \frac{1}{2}\pi$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\frac{dy}{dx} = 1 - 2\sin t$ and hence find the coordinates of the stationary point. [5]
\item Find the cartesian equation of the curve. [3]
\item State the set of values that $x$ can take and hence sketch the curve. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q7 [11]}}