| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Find stationary points of hyperbolic curves |
| Difficulty | Standard +0.8 This is a Further Maths question requiring differentiation of hyperbolic functions, solving a transcendental equation, and proving uniqueness of a stationary point. While the techniques are standard (knowing d/dx(sinh x) = cosh x and d/dx(cosh x) = sinh x), part (b) requires algebraic manipulation using cosh²x - sinh²x = 1 to show the stationary point exists and has integer y-coordinate. The uniqueness argument connects to part (a)(ii), requiring insight beyond routine calculation. This is moderately challenging for Further Maths but not exceptionally difficult. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07b Hyperbolic graphs: sketch and properties4.07d Differentiate/integrate: hyperbolic functions |
2
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Sketch on the axes below the graphs of $y = \sinh x$ and $y = \cosh x$.
\item Use your graphs to explain why the equation
$$( k + \sinh x ) \cosh x = 0$$
where $k$ is a constant, has exactly one solution.
\end{enumerate}\item A curve $C$ has equation $y = 6 \sinh x + \cosh ^ { 2 } x$. Show that $C$ has only one stationary point and show that its $y$-coordinate is an integer.\\
\includegraphics[max width=\textwidth, alt={}, center]{53d742f4-923b-478c-8ae6-ada6c0bb4a7e-2_560_704_1416_171}\\
\includegraphics[max width=\textwidth, alt={}, center]{53d742f4-923b-478c-8ae6-ada6c0bb4a7e-2_560_711_1416_964}
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2013 Q2 [9]}}