| Exam Board | WJEC |
|---|---|
| Module | Further Unit 4 (Further Unit 4) |
| Year | 2024 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Direct substitution into standard series |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring Maclaurin series (straightforward recall), then using it to solve a transcendental equation (requiring iterative approximation or polynomial solving), and finally computing an area using integration. Part (a) is routine, but parts (b) and (c) require connecting multiple techniques and careful numerical work. The 13 total marks and multi-step nature place it above average difficulty, though not exceptionally hard for Further Maths students. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
The function $f$ is defined by $f(x) = \cosh\left(\frac{x}{2}\right)$.
\begin{enumerate}[label=(\alph*)]
\item State the Maclaurin series expansion for $\cosh\left(\frac{x}{2}\right)$ up to and including the term in $x^4$. [2]
\end{enumerate}
Another function $g$ is defined by $g(x) = x^2 - 2$. The diagram below shows parts of the graphs of $y = f(x)$ and $y = g(x)$.
\includegraphics{figure_2}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item The two graphs intersect at the point A, as shown in the diagram. Use your answer from part (a) to find an approximation for the $x$-coordinate of A, giving your answer correct to two decimal places. [5]
\item Using your answer to part (b), find an approximation for the area of the shaded region enclosed by the two graphs, the $x$-axis and the $y$-axis. [6]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 4 2024 Q2 [13]}}