| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2026 |
| Session | November |
| Marks | 9 |
| Topic | Hyperbolic functions |
| Type | Intersection points of hyperbolic curves |
| Difficulty | Challenging +1.2 Part (i) involves standard hyperbolic function manipulation requiring algebraic skill to derive a quartic in e^x, then factorization and exact coordinate extraction - moderately challenging but systematic. Part (ii) appears to contain an error (the integral diverges at x=0), but if taken at face value requires integration by parts or substitution - this would be routine Further Maths content. Overall slightly above average difficulty due to the multi-step hyperbolic algebra and exact value work. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07b Hyperbolic graphs: sketch and properties4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 14.08c Improper integrals: infinite limits or discontinuous integrands |
In this question you must show detailed reasoning.
\begin{enumerate}[label=(\roman*)]
\item The curves with equations
$$y = \frac{3}{4}\sinh x \text{ and } y = \tanh x + \frac{1}{5}$$
intersect at just one point $P$
\begin{enumerate}[label=(\alph*)]
\item Use algebra to show that the $x$ coordinate of $P$ satisfies the equation
$$15e^{4x} - 48e^{3x} + 32e^x - 15 = 0$$
[3]
\item Show that $e^x = 3$ is a solution of this equation. [1]
\item Hence state the exact coordinates of $P$. [1]
\end{enumerate}
\item Show that
$$\int_{-4}^{0} \frac{e^x}{x^2} dx = e^{-\frac{1}{4}}$$
[4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2026 Q4 [9]}}