Answer only one of the following two alternatives.

**EITHER**

It is given that $w = \cos y$ and
$$\tan y \frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + 2\tan y \frac{dy}{dx} = 1 + e^{-2x} \sec y.$$

\begin{enumerate}[label=(\roman*)]
\item Show that
$$\frac{d^2 w}{dx^2} + 2\frac{dw}{dx} + w = -e^{-2x}.$$ [4]

\item Find the particular solution for $y$ in terms of $x$, given that when $x = 0$, $y = \frac{1}{4}\pi$ and $\frac{dy}{dx} = \frac{1}{\sqrt{3}}$. [10]
\end{enumerate}

**OR**

The curves $C_1$ and $C_2$ have polar equations, for $0 \leq \theta \leq \frac{1}{2}\pi$, as follows:
\begin{align}
C_1: r &= 2(e^\theta + e^{-\theta}), \\
C_2: r &= e^{2\theta} - e^{-2\theta}.
\end{align}

The curves intersect at the point $P$ where $\theta = \alpha$.

\begin{enumerate}[label=(\roman*)]
\item Show that $e^{2\alpha} - 2e^\alpha - 1 = 0$. Hence find the exact value of $\alpha$ and show that the value of $r$ at $P$ is $4\sqrt{2}$. [6]

\item Sketch $C_1$ and $C_2$ on the same diagram. [3]

\item Find the area of the region enclosed by $C_1$, $C_2$ and the initial line, giving your answer correct to 3 significant figures. [5]
\end{enumerate}

\hfill \mbox{\textit{CAIE FP1 2019 Q11 [28]}}