15.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{bc7fb499-9462-40ae-88f4-87fc60f6a005-38_540_741_169_676}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}

Figure 1 shows a sketch of the curve $C$ with parametric equations

$$x = 3 \sin ^ { 3 } \theta \quad y = 1 + \cos 2 \theta \quad - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Show that

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \operatorname { cosec } \theta \quad \theta \neq 0$$

where $k$ is a constant to be found.

The point $P$ lies on $C$ where $\theta = \frac { \pi } { 6 }$
\item Find the equation of the tangent to $C$ at $P$, giving your answer in the form $a x + b y + c = 0$ where $a , b$ and $c$ are integers.
\item Show that $C$ has Cartesian equation

$$8 x ^ { 2 } = 9 ( 2 - y ) ^ { 3 } \quad - q \leq x \leq q$$

where $q$ is a constant to be found.
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Pure 2025 Q15 [9]}}