| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Solve using sech/tanh identities |
| Difficulty | Standard +0.3 This is a standard Further Maths hyperbolic functions question with routine components: sketching tanh (a well-known curve), proving a standard identity using definitions (straightforward algebra), and solving an equation using the proven identity to get a quadratic in tanh x. All steps are textbook exercises requiring no novel insight, though it's slightly above average A-level difficulty due to being Further Maths content. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07b Hyperbolic graphs: sketch and properties4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 |
**(a) [3 marks]**
Sketch the graph of $y = \tanh x$ and state the equations of its asymptotes.
**(b) [3 marks]**
Use the definitions of $\sinh x$ and $\cosh x$ in terms of $e^x$ and $e^{-x}$ to show that
$$\operatorname{sech}^2 x + \tanh^2 x = 1$$
**(c) [5 marks]**
Solve the equation $6\operatorname{sech}^2 x = 4 + \tanh x$, giving your answers in terms of natural logarithms.2
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $y = \tanh x$ and state the equations of its asymptotes.
\item Use the definitions of $\sinh x$ and $\cosh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$ to show that
$$\operatorname { sech } ^ { 2 } x + \tanh ^ { 2 } x = 1$$
\item Solve the equation $6 \operatorname { sech } ^ { 2 } x = 4 + \tanh x$, giving your answers in terms of natural logarithms.\\[0pt]
[5 marks]
\section*{Answer space for question 2}
(a)\\
\includegraphics[max width=\textwidth, alt={}, center]{bc3aaed2-4aef-4aec-b657-098b1e581e55-04_855_1447_920_324}
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2015 Q2 [11]}}