| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | November |
| Marks | 4 |
| Topic | Hyperbolic functions |
| Type | Solve using substitution u = cosh x or u = sinh x |
| Difficulty | Standard +0.8 This question requires recognizing the identity cosh²x - sinh²x = 1 to simplify the quartic equation, then making a substitution u = cosh x to reduce it to a quadratic. The algebraic manipulation is non-trivial, and expressing the final answer in the form q ln 2 requires careful work with inverse hyperbolic functions. While systematic, it demands more insight and technical facility than a standard A-level question. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 14.07f Inverse hyperbolic: logarithmic forms |
\begin{enumerate}
\item In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
\end{enumerate}
Determine the values of $x$ for which
$$64 \cosh ^ { 4 } x - 64 \sinh ^ { 2 } x - 73 = 0$$
Give your answer in the form $q \ln 2$ where $q$ is rational and in simplest form.\\
\hfill \mbox{\textit{SPS SPS FM 2024 Q1 [4]}}