| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show root in interval |
| Difficulty | Standard +0.3 This is a standard C3 numerical methods question covering sign change for root existence, algebraic rearrangement to derive an iterative formula, and applying iteration to find a root. All three parts follow routine procedures taught in the specification with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
$$\text{f}(x) = x^2 + 5x - 2 \sec x, \quad x \in \mathbb{R}, \quad -\frac{\pi}{2} < x < \frac{\pi}{2}.$$
\begin{enumerate}[label=(\roman*)]
\item Show that the equation $\text{f}(x) = 0$ has a root, $\alpha$, such that $1 < \alpha < 1.5$ [2]
\item Show that a suitable rearrangement of the equation $\text{f}(x) = 0$ leads to the iterative formula
$$x_{n+1} = \cos^{-1} \left( \frac{2}{x_n^2 + 5x_n} \right).$$ [3]
\item Use the iterative formula in part (ii) with a starting value of 1.25 to find $\alpha$ correct to 3 decimal places. You should show the result of each iteration. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q4 [8]}}