| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2018 |
| Session | March |
| Marks | 5 |
| Topic | Fixed Point Iteration |
| Type | Apply iteration to find root (pure fixed point) |
| Difficulty | Moderate -0.5 This is a straightforward application of fixed point iteration with a simple function (x = cos x). The iteration formula is given directly, requires no rearrangement, and converges quickly. Part (i) is graphical estimation, part (ii) is mechanical calculator work with no conceptual challenges—easier than average A-level questions. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Any value in the range \([0.70, 0.77]\) | M1, A1 | or implied by answer |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| answer \(x = 0.739\) (3 sf) | M1, M1, A1 | Any starting point must be seen |
| Answer | Marks |
|---|---|
| [2] |
## 4(i)
Line $y = x$ drawn
Any value in the range $[0.70, 0.77]$ | M1, A1 | or implied by answer
| [2]
## 4(ii)
eg $\cos 0.75 = 0.731688869$
Eg $\cos0.7390791171 = 0.7390891857$
answer $x = 0.739$ (3 sf) | M1, M1, A1 | Any starting point must be seen
Any $x_{n+1} = \cos x_n$ where both $x_{n+1}$ & $x_n$ round to $0.7391$, and answer stated
| [2]
---4 The diagram shows part of the graph of $y = \cos x$, where $x$ is measured in radians.\\
\includegraphics[max width=\textwidth, alt={}, center]{6a6316e4-7b2d-4533-988a-4863d79ce668-05_609_846_294_607}\\
(i) Use the copy of this diagram in the Printed Answer Booklet to find an approximate solution to the equation $x = \cos x$.\\
(ii) Use an iterative method to find the solution to the equation $x = \cos x$ correct to 3 significant figures. You should show your first, second and last two iterations, writing down all the figures on your calculator.
\hfill \mbox{\textit{OCR H240/02 2018 Q4 [5]}}