| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 18 |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Standard +0.3 This is a multi-part question covering standard Further Maths techniques: finding a normal equation (routine differentiation and line equation), calculating a triangular area, locating roots by sign change, rearranging equations, and applying iterative methods with cobweb diagrams. All parts are textbook exercises requiring competent execution of familiar methods rather than problem-solving insight. Slightly easier than average due to straightforward structure and generous mark allocation. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations1.08e Area between curve and x-axis: using definite integrals1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams1.09d Newton-Raphson method |
A curve $C$ has equation $y = f(x)$ where
$$f(x) = x + 2\ln(e - x)$$
\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Show that the equation of the normal to $C$ at the point where $C$ crosses the $y$-axis is given by
$$y = \left(\frac{e}{2-e}\right)x + 2$$ [6 marks]
\item Find the exact area enclosed by the normal and the coordinate axes.
Fully justify your answer. [3 marks]
\end{enumerate}
\end{enumerate}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item The equation $f(x) = 0$ has one positive root, $\alpha$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\alpha$ lies between 2 and 3
Fully justify your answer. [3 marks]
\item Show that the roots of $f(x) = 0$ satisfy the equation
$$x = e - e^{-\frac{x}{2}}$$ [2 marks]
\item Use the recurrence relation
$$x_{n+1} = e - e^{-\frac{x_n}{2}}$$
with $x_1 = 2$ to find the values of $x_2$ and $x_3$ giving your answers to three decimal places. [2 marks]
\item Figure 1 below shows a sketch of the graphs of $y = e - e^{-\frac{x}{2}}$ and $y = x$, and the position of $x_1$
On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of $x_2$ and $x_3$ on the $x$-axis. [2 marks]
\includegraphics{figure_1}
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q9 [18]}}