| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Arc length with hyperbolic curves |
| Difficulty | Standard +0.8 Part (a) requires arc length integration with hyperbolic functions (5 marks), involving differentiation of cosh, using the identity cosh²-sinh²=1, and integrating sinh. Part (b) requires solving sinh(k)=4a using inverse hyperbolic functions and finding coordinates. This is a standard Further Maths question requiring multiple techniques but follows predictable patterns for catenary problems, making it moderately challenging but not requiring novel insight. |
| Spec | 1.08d Evaluate definite integrals: between limits4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07d Differentiate/integrate: hyperbolic functions |
\includegraphics{figure_13}
A rope is hung from points $P$ and $Q$ on the same horizontal level, as shown in Fig. 2. The curve formed by the rope is modelled by the equation
$$y = a \cosh\left(\frac{x}{a}\right), \quad -ka \leq x \leq ka,$$
where $a$ and $k$ are positive constants.
\begin{enumerate}[label=(\alph*)]
\item Prove that the length of the rope is $2a \sinh k$. [5]
\end{enumerate}
Given that the length of the rope is $8a$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the coordinates of $Q$, leaving your answer in terms of natural logarithms and surds, where appropriate. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q13 [9]}}