| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Topic | Parametric curves and Cartesian conversion |
| Type | Convert to Cartesian (sin/cos identities) |
| Difficulty | Standard +0.3 Part (a) is a standard parametric-to-Cartesian conversion using the double angle formula sin(2t) = 2sin(t)cos(t) and the Pythagorean identity, requiring routine algebraic manipulation. Part (b) requires finding where the circle touches the curve by using symmetry or calculus, which is slightly more involved but still a familiar further maths technique. Overall, this is a straightforward further maths question with standard methods. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1 |
8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0e7cab3d-c1e6-4420-93b4-eca5af704432-08_890_919_248_630}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{center}
\end{figure}
Figure 5 shows a sketch of the curve $C _ { 1 }$ with parametric equations
$$x = 2 \sin t , \quad y = 3 \sin 2 t \quad 0 \leq t < 2 \pi$$
\begin{enumerate}[label=(\alph*)]
\item Show that the Cartesian equation of $C _ { 1 }$ can be expressed in the form
$$y ^ { 2 } = k x ^ { 2 } \left( 4 - x ^ { 2 } \right)$$
where $k$ is a constant to be found.
The circle $C _ { 2 }$ with centre $O$ touches $C _ { 1 }$ at four points as shown in Figure 5.
\item Find the radius of this circle.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q8 [9]}}