| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find equation satisfied by limit |
| Difficulty | Moderate -0.3 This is a straightforward fixed point iteration question requiring routine application of the iterative formula (part i) and simple algebraic manipulation to find the limit equation (part ii). Both parts are mechanical with no conceptual challenges—students substitute repeatedly, then set x_{n+1} = x_n = α and rearrange. Slightly easier than average due to the direct nature of both tasks. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
2 The sequence defined by
$$x _ { 1 } = 3 , \quad x _ { n + 1 } = \sqrt [ 3 ] { 31 - \frac { 5 } { 2 } x _ { n } }$$
converges to the number $\alpha$.\\
(i) Find the value of $\alpha$ correct to 3 decimal places, showing the result of each iteration.\\
(ii) Find an equation of the form $a x ^ { 3 } + b x + c = 0$, where $a$, $b$ and $c$ are integers, which has $\alpha$ as a root.
\hfill \mbox{\textit{OCR C3 2008 Q2 [6]}}