| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Apply iteration to find root (pure fixed point) |
| Difficulty | Standard +0.8 This is a standard FP2 numerical methods question involving iterative schemes and error analysis. Part (i) requires straightforward iteration with calculator work. Part (ii) involves computing error ratios and connecting them to the derivative at the fixed point—a key theoretical result in iterative methods. While this requires understanding of convergence theory beyond basic A-level, it's a well-rehearsed FP2 topic with clear structure. The multi-step nature and theoretical connection elevate it above average, but it remains a textbook-style question without novel insight required. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams1.09e Iterative method failure: convergence conditions |
It is given that $\alpha$ is the only real root of the equation $x^3 + 2x - 28 = 0$ and that $1.8 < \alpha < 2$.
\begin{enumerate}[label=(\roman*)]
\item The iteration $x_{n+1} = \sqrt[3]{28 - 2x_n}$, with $x_1 = 1.9$, is to be used to find $\alpha$. Find the values of $x_2$, $x_3$ and $x_4$, giving the answers correct to 7 decimal places. [3]
\item The error $e_n$ is defined by $e_n = \alpha - x_n$. Given that $\alpha = 1.891 574 9$, correct to 7 decimal places, evaluate $\frac{e_3}{e_2}$ and $\frac{e_4}{e_3}$. Comment on these values in relation to the gradient of the curve with equation $y = \sqrt[3]{28 - 2x}$ at $x = \alpha$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2009 Q2 [12]}}