| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Topic | Taylor series |
| Type | Limit using series expansion |
| Difficulty | Standard +0.8 This is a Further Maths question requiring Taylor series manipulation and limit evaluation. Part (a) is routine recall, but part (b) requires recognizing that both numerator and denominator must vanish to order x² (requiring cancellation of the x³ term), then carefully expanding ln(1+kx) and matching coefficients to find k. This involves more sophisticated series manipulation than typical A-level work, though it's a standard FP3 technique. |
| Spec | 4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
2
\begin{enumerate}[label=(\alph*)]
\item Write down the expansion of $\sin 2 x$ in ascending powers of $x$ up to and including the term in $x ^ { 5 }$.
\item Show that, for some value of $k$,
$$\lim _ { x \rightarrow 0 } \left[ \frac { 2 x - \sin 2 x } { x ^ { 2 } \ln ( 1 + k x ) } \right] = 16$$
and state this value of $k$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2012 Q2 [5]}}