| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | February |
| Marks | 10 |
| Topic | Hyperbolic functions |
| Type | Prove inverse hyperbolic logarithmic form |
| Difficulty | Challenging +1.3 Part (a) is a standard Further Maths proof of the inverse hyperbolic tangent formula requiring manipulation of exponentials and logarithms (routine for FM students, k=1). Part (b) requires applying the result to solve a non-standard equation involving composition of hyperbolic and logarithmic functions, demanding careful algebraic manipulation and domain considerations across multiple steps—this elevates it above routine FM exercises but remains within expected FM problem-solving territory. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07f Inverse hyperbolic: logarithmic forms |
\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 $k$.
[5]
\item Hence, or otherwise, solve the equation
$$2x = \tanh\left(\ln \sqrt{2-3x}\right)$$
[5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q7 [10]}}