The positive variables $y$ and $t$ are related by

$$y = a ^ { t }$$

where $a$ is a positive constant.
\begin{enumerate}[label=(\roman*)]
\item (a) By differentiating $\ln y$ with respect to $t$, show that $\frac { \mathrm { d } y } { \mathrm {~d} t } = a ^ { t } \ln a$.\\

(b) Write down $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }$.
\item Determine the set of values of $a$ for which the infinite series

$$y + \frac { \mathrm { d } y } { \mathrm {~d} t } + \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} t ^ { 3 } } + \ldots$$

is convergent.\\

A curve has parametric equations

$$x = t ^ { a } , \quad y = a ^ { t }$$
\item Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ in terms of $a$ and $t$, and show that, when $t = 2$,

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 ^ { 1 - 2 a } ( 1 - a + 2 \ln a ) \ln a$$

If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}

\hfill \mbox{\textit{CAIE FP1 2019 Q11 OR}}