Challenging +1.2 This is a structured Further Maths question on Maclaurin series with clear scaffolding. Part (a) involves routine differentiation using logarithm and chain rules, part (b) applies a standard template (Maclaurin's theorem), and parts (c)-(d) require recognizing how to match coefficients to find limits. While it requires multiple techniques and careful algebra across several parts, each step follows predictable patterns for FP3 students, making it moderately above average difficulty but not requiring novel insight.
It is given that \(y = \ln \left( \mathrm { e } ^ { 3 x } \cos x \right)\).
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - \tan x\).
Find \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\).
Hence use Maclaurin's theorem to show that the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \left( \mathrm { e } ^ { 3 x } \cos x \right)\) are \(3 x - \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 12 } x ^ { 4 }\).
(3 marks)
Write down the expansion of \(\ln ( 1 + p x )\), where \(p\) is a constant, in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\).
Find the value of \(p\) for which \(\lim _ { x \rightarrow 0 } \left[ \frac { 1 } { x ^ { 2 } } \ln \left( \frac { \mathrm { e } ^ { 3 x } \cos x } { 1 + p x } \right) \right]\) exists.
Hence find the value of \(\lim _ { x \rightarrow 0 } \left[ \frac { 1 } { x ^ { 2 } } \ln \left( \frac { \mathrm { e } ^ { 3 x } \cos x } { 1 + p x } \right) \right]\) when \(p\) takes the value found in part (d)(i).
6
\begin{enumerate}[label=(\alph*)]
\item It is given that $y = \ln \left( \mathrm { e } ^ { 3 x } \cos x \right)$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - \tan x$.
\item Find $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$.
\end{enumerate}\item Hence use Maclaurin's theorem to show that the first three non-zero terms in the expansion, in ascending powers of $x$, of $\ln \left( \mathrm { e } ^ { 3 x } \cos x \right)$ are $3 x - \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 12 } x ^ { 4 }$.\\
(3 marks)
\item Write down the expansion of $\ln ( 1 + p x )$, where $p$ is a constant, in ascending powers of $x$ up to and including the term in $x ^ { 2 }$.
\item \begin{enumerate}[label=(\roman*)]
\item Find the value of $p$ for which $\lim _ { x \rightarrow 0 } \left[ \frac { 1 } { x ^ { 2 } } \ln \left( \frac { \mathrm { e } ^ { 3 x } \cos x } { 1 + p x } \right) \right]$ exists.
\item Hence find the value of $\lim _ { x \rightarrow 0 } \left[ \frac { 1 } { x ^ { 2 } } \ln \left( \frac { \mathrm { e } ^ { 3 x } \cos x } { 1 + p x } \right) \right]$ when $p$ takes the value found in part (d)(i).
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2013 Q6 [14]}}