| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Paper | Download PDF ↗ |
| Topic | Taylor series |
| Type | Maclaurin series for ln(exponential expressions) |
| Difficulty | Challenging +1.2 This is a structured Further Maths question on Taylor/Maclaurin series with clear scaffolding through parts (a)-(e). While it requires successive differentiation of a composite function and manipulation of series expansions, each step is guided and uses standard FP1 techniques. The final limit problem in part (e) requires insight about cancelling leading terms but follows naturally from the previous parts. Moderately above average difficulty due to the multi-step nature and being Further Maths content, but not exceptionally challenging. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07l Derivative of ln(x): and related functions4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
6.
$$y = \ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right) \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 3 \tan 3 x$$
\item Determine $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$
\item 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.
\item 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.
\item 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.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2023 Q6}}