| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Arc length with hyperbolic curves |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring differentiation of hyperbolic functions using chain rule and standard identities (part a), followed by arc length integration (part b). While it involves multiple steps and hyperbolic function manipulation, the techniques are standard for FP2 students and the integration likely simplifies nicely to the given form. The difficulty is elevated above average due to the Further Maths content and multi-step nature, but it's a fairly routine application of taught methods rather than requiring novel insight. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07d Differentiate/integrate: hyperbolic functions8.06a Reduction formulae: establish, use, and evaluate recursively |
3 A curve has cartesian equation
$$y = \frac { 1 } { 2 } \ln ( \tanh x )$$
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sinh 2 x }$$
\item The points $A$ and $B$ on the curve have $x$-coordinates $\ln 2$ and $\ln 4$ respectively. Find the arc length $A B$, giving your answer in the form $p \ln q$, where $p$ and $q$ are rational numbers.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2012 Q3 [12]}}