6.

$$y = \ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right) \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$

(a) Show that

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 3 \tan 3 x$$

(b) Determine $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$\\
(c) Hence determine the first 3 non-zero terms in ascending powers of $x$ of the Maclaurin series expansion of $\ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right)$, giving each coefficient in simplest form.\\
(d) Use the Maclaurin series expansion for $\ln ( 1 + x )$ to write down the first 4 non-zero terms in ascending powers of $x$ of the Maclaurin series expansion of $\ln ( 1 + k x )$, where $k$ is a constant.\\
(e) Hence determine the value of $k$ for which

$$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } \ln \frac { \mathrm { e } ^ { 2 x } \cos 3 x } { 1 + k x } \right)$$

exists.

\hfill \mbox{\textit{Edexcel FP1 2023 Q6}}