| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find equation satisfied by limit |
| Difficulty | Standard +0.3 This is a straightforward fixed point iteration question requiring routine application of the formula (part i) and recognizing that at convergence x_{n+1} = x_n = α to form an equation (part ii). The algebra to solve the resulting cubic is manageable. Slightly easier than average as it's a standard textbook exercise with clear methodology. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(2.289\) | A1 | |
| Show sufficient iterations to justify accuracy to 3 d.p. or show sign change in interval (\(2.2885, 2.2895\)) | A1 | [3] |
| (ii) State \(x = 2 + \frac{4}{x^2 + 2x + 4}\) or equivalent | B1 | |
| Obtain \(\sqrt[3]{12}\) | B1 | [2] |
(i) Use the iterative formula correctly at least once | M1 |
Obtain final answer $2.289$ | A1 |
Show sufficient iterations to justify accuracy to 3 d.p. or show sign change in interval ($2.2885, 2.2895$) | A1 | [3]
(ii) State $x = 2 + \frac{4}{x^2 + 2x + 4}$ or equivalent | B1 |
Obtain $\sqrt[3]{12}$ | B1 | [2]
---2 The sequence of values given by the iterative formula
$$x _ { n + 1 } = 2 + \frac { 4 } { x _ { n } ^ { 2 } + 2 x _ { n } + 4 }$$
with initial value $x _ { 1 } = 2$, converges to $\alpha$.\\
(i) Determine the value of $\alpha$ correct to 3 decimal places, giving the result of each iteration to 5 decimal places.\\
(ii) State an equation satisfied by $\alpha$ and hence find the exact value of $\alpha$.
\hfill \mbox{\textit{CAIE P2 2015 Q2 [5]}}