| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2012 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Taylor series |
| Type | Maclaurin series for ln(trigonometric expressions) |
| Difficulty | Challenging +1.2 This is a systematic Further Maths question requiring repeated differentiation of a composite logarithmic-trigonometric function and application of Maclaurin's theorem. While the differentiation chain is moderately lengthy (4th derivative) and requires careful algebra with trigonometric identities, the approach is entirely standard for FP3. Part (c) adds a small twist using logarithm laws, but overall this follows a well-practiced template without requiring novel insight. |
| Spec | 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
\begin{enumerate}[label=(\alph*)]
\item Given that $y = \ln \cos 2 x$, find $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$.
\item Use Maclaurin's theorem to show that the first two non-zero terms in the expansion, in ascending powers of $x$, of $\ln \cos 2 x$ are $- 2 x ^ { 2 } - \frac { 4 } { 3 } x ^ { 4 }$.
\item Hence find the first two non-zero terms in the expansion, in ascending powers of $x$, of $\ln \sec ^ { 2 } 2 x$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2012 Q6 [11]}}