| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | November |
| Marks | 10 |
| Topic | Hyperbolic functions |
| Type | Prove inverse hyperbolic logarithmic form |
| Difficulty | Standard +0.8 This is a standard Further Maths proof of an inverse hyperbolic function's logarithmic form, requiring manipulation of exponentials and logarithms, followed by a non-routine application requiring substitution and algebraic manipulation. The proof itself is bookwork, but part (b) requires insight to apply the result and handle the domain restrictions carefully. |
| Spec | 4.07e Inverse hyperbolic: definitions, domains, ranges4.07f Inverse hyperbolic: logarithmic forms |
2.\\
\begin{enumerate}[label=(\alph*)]
\item Prove that
$$\tanh ^ { - 1 } ( x ) = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) \quad - k < x < k$$
stating the value of the constant $k$.
\item Hence, or otherwise, solve the equation
$$2 x = \tanh ( \ln \sqrt { 2 - 3 x } )$$
\\
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2024 Q2 [10]}}