| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Find stationary points of hyperbolic curves |
| Difficulty | Standard +0.3 Part (a) is straightforward substitution and manipulation using the definition cosh(2x) = (e^(4x) + 1)/(2e^(2x)), which with x = ln k becomes routine algebra. Part (b) requires differentiating tanh(2x) using sech²(2x), setting f'(ln 2) = 0, and solving for p—standard calculus with hyperbolic functions. This is typical FP3 bookwork with no novel insight required, making it slightly easier than average. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07d Differentiate/integrate: hyperbolic functions |
\begin{enumerate}[label=(\alph*)]
\item Show that, for $x = \ln k$, where $k$ is a positive constant,
$$\cosh 2x = \frac{k^4 + 1}{2k^2}.$$ [3]
\end{enumerate}
Given that $f(x) = px - \tanh 2x$, where $p$ is a constant,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the value of $p$ for which $f(x)$ has a stationary value at $x = \ln 2$, giving your answer as an exact fraction. [4]
\end{enumerate}
(Total 7 marks)
\hfill \mbox{\textit{Edexcel FP3 Q30 [7]}}