Answer only one of the following two alternatives.

\textbf{EITHER}

The curve $C$ is defined parametrically by
$$x = 18t - t^2 \quad \text{and} \quad y = 8t^{\frac{1}{2}},$$
where $0 < t \leqslant 4$.

\begin{enumerate}[label=(\roman*)]
\item Show that at all points of $C$,
$$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = \frac{-3(9 + t)}{2t^2(9 - t)^3}.$$ [4]

\item Show that the mean value of $\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}$ with respect to $x$ over the interval $0 < x \leqslant 56$ is $\frac{3}{70}$. [4]

\item Find the area of the surface generated when $C$ is rotated through $2\pi$ radians about the $x$-axis, showing full working. [6]
\end{enumerate}

\textbf{OR}

Let $I_n = \int_1^{\sqrt{2}} (x^2 - 1)^n \mathrm{d}x$.

\begin{enumerate}[label=(\roman*)]
\item Show that, for $n \geqslant 1$,
$$(2n + 1)I_n = \sqrt{2} - 2nI_{n-1}.$$ [5]

\item Using the substitution $x = \sec \theta$, show that
$$I_n = \int_0^{\frac{1}{4}\pi} \tan^{2n+1} \theta \sec \theta \, \mathrm{d}\theta.$$ [4]

\item Deduce the exact value of
$$\int_0^{\frac{1}{4}\pi} \frac{\sin^7 \theta}{\cos^8 \theta} \, \mathrm{d}\theta.$$ [5]
\end{enumerate}

\hfill \mbox{\textit{CAIE FP1 2018 Q11 [28]}}