| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Prove hyperbolic identity from exponentials |
| Difficulty | Standard +0.3 This is a standard FP2 hyperbolic functions question requiring routine manipulation of definitions and identities. Part (a) is direct substitution of exponential definitions (textbook exercise), part (b)(i) involves applying the result from (a) and simplifying, and part (b)(ii) is straightforward application of the inverse tanh formula. While it requires multiple steps, all techniques are standard for FP2 with no novel insight needed. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 |
\begin{enumerate}[label=(\alph*)]
\item Use the definitions of $\cosh \theta$ and $\sinh \theta$ in terms of $e^\theta$ to show that
$$\cosh x \cosh y - \sinh x \sinh y = \cosh(x - y)$$ [4 marks]
\item It is given that $x$ satisfies the equation
$$\cosh(x - \ln 2) = \sinh x$$
\begin{enumerate}[label=(\roman*)]
\item Show that $\tanh x = \frac{5}{4}$. [4 marks]
\item Express $x$ in the form $\frac{1}{2} \ln a$. [2 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2011 Q2 [10]}}