| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Apply iteration to find root (pure fixed point) |
| Difficulty | Standard +0.3 This is a straightforward numerical methods question requiring calculator work to iterate a given formula and basic error analysis. Part (i) is pure computation with no conceptual challenge. Part (ii) involves simple arithmetic to find errors and uses the standard result that e_{n+1} ≈ g'(α)e_n, which is a direct application of taught theory. While FP2 content, this requires only routine application of the iterative methods syllabus with no problem-solving or insight needed. |
| Spec | 1.07l Derivative of ln(x): and related functions1.09d Newton-Raphson method |
It is given that $f(x) = x^2 - \sin x$.
\begin{enumerate}[label=(\roman*)]
\item The iteration $x_{n+1} = \sqrt{\sin x_n}$, with $x_1 = 0.875$, is to be used to find a real root, $\alpha$, of the equation $f(x) = 0$. Find $x_2, x_3$ and $x_4$, giving the answers correct to 6 decimal places. [2]
\item The error $e_n$ is defined by $e_n = \alpha - x_n$. Given that $\alpha = 0.876726$, correct to 6 decimal places, find $e_3$ and $e_4$. Given that $g(x) = \sqrt{\sin x}$, use $e_3$ and $e_4$ to estimate $g'(\alpha)$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2010 Q1 [5]}}