| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Taylor series |
| Type | Maclaurin series for ln(trigonometric expressions) |
| Difficulty | Challenging +1.2 This is a standard Further Maths FP3 question testing routine Maclaurin series techniques. Part (a) requires straightforward differentiation using chain rule, part (b) applies the standard Maclaurin formula with given derivatives, and part (c) uses series expansions to evaluate a limit—all textbook procedures. While it requires multiple steps and FP3 content, it involves no novel insight or problem-solving beyond applying learned techniques systematically. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07l 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 |
5
\begin{enumerate}[label=(\alph*)]
\item Given that $y = \ln ( 1 + 2 \tan x )$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.\\
(You may leave your expression for $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ unsimplified.)
\item Hence, using Maclaurin's theorem, find the first two non-zero terms in the expansion, in ascending powers of $x$, of $\ln ( 1 + 2 \tan x )$.\\
(2 marks)
\item Find
$$\lim _ { x \rightarrow 0 } \left[ \frac { \ln ( 1 + 2 \tan x ) } { \ln ( 1 - x ) } \right]$$
(4 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2011 Q5 [10]}}