| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795 (Pre-U Further Mathematics) |
| Session | Specimen |
| Marks | 12 |
| Topic | Taylor series |
| Type | Differential equation given |
| Difficulty | Standard +0.8 This is a substantial multi-part Further Maths question requiring chain rule differentiation, product rule for higher derivatives, manipulation of differential equations, Maclaurin series construction from derivatives, and verification using composition of series. While each individual technique is standard, the question demands careful algebraic manipulation across multiple steps and integration of several topics, placing it moderately above average difficulty. |
| Spec | 4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n4.08g Derivatives: inverse trig and hyperbolic functions |
Given that $y = \cos\{\ln(1 + x)\}$, prove that
\begin{enumerate}[label=(\roman*)]
\item $(1 + x)\frac{\mathrm{d}y}{\mathrm{d}x} = -\sin\{\ln(1 + x)\}$, [1]
\item $(1 + x)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1 + x)\frac{\mathrm{d}y}{\mathrm{d}x} + y = 0$. [2]
\end{enumerate}
Obtain an equation relating $\frac{\mathrm{d}^3 y}{\mathrm{d}x^3}$, $\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}$ and $\frac{\mathrm{d}y}{\mathrm{d}x}$. [2]
Hence find Maclaurin's series for $y$, up to and including the term in $x^3$. [4]
Verify that the same result is obtained if the standard series expansions for $\ln(1 + x)$ and $\cos x$ are used. [3]
\hfill \mbox{\textit{Pre-U Pre-U 9795 Q13 [12]}}