| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2026 |
| Session | November |
| Marks | 8 |
| Topic | Taylor series |
| Type | Hyperbolic power functions |
| Difficulty | Challenging +1.3 This is a Further Maths question requiring differentiation of hyperbolic functions, pattern recognition to find k=4, and Maclaurin series construction using the differential equation. While it involves multiple techniques and hyperbolic functions (inherently harder), the steps are methodical and follow standard procedures without requiring novel insight. The mark allocation (5+3=8 marks) suggests moderate complexity above typical A-level but not exceptionally challenging for FM students. |
| Spec | 4.07d Differentiate/integrate: hyperbolic functions4.08a Maclaurin series: find series for function |
Given that
$$y = \cos x \sinh x \quad x \in \mathbb{R}$$
\begin{enumerate}[label=(\alph*)]
\item show that
$$\frac{d^4y}{dx^4} = ky$$
where $k$ is a constant to be determined. [5]
\item Hence determine the first three non-zero terms of the Maclaurin series for $y$, giving each coefficient in simplest form. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2026 Q9 [8]}}