| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 8 |
| Topic | Hyperbolic functions |
| Type | Solve using substitution u = cosh x or u = sinh x |
| Difficulty | Challenging +1.3 Part (i) is a standard hyperbolic identity proof requiring algebraic manipulation of exponential definitions (routine for Further Maths). Part (ii) requires recognizing the connection to the identity in (i), making the substitution u = sinh x, then solving sinh 3x = 3/16 using logarithms—this demands insight to connect parts and careful algebraic manipulation. The multi-step reasoning and non-obvious substitution elevate this above average difficulty, but it's a recognizable Further Maths technique once the pattern is spotted. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 |
\begin{enumerate}[label=(\roman*)]
\item Using the definition of $\sinh x$ in terms of $e^x$ and $e^{-x}$, show that
$$4\sinh^3 x = \sinh 3x - 3\sinh x.$$ [3]
\item In this question you must show detailed reasoning.
By making a suitable substitution, find the real root of the equation
$$16u^3 + 12u = 3.$$
Give your answer in the form $\frac{(a^{\frac{1}{b}} - a^{-\frac{1}{b}})}{c}$ where $a$, $b$ and $c$ are integers. [5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q8 [8]}}