| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | March |
| 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 iterative formula question requiring routine calculator work to find the limit (part i), then simple algebraic manipulation to find the equation satisfied by α (setting x_{n+1} = x_n = α and squaring). The algebra is mechanical and the concept is standard A-level material, making it slightly easier than average. |
| 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 1.516 | A1 | |
| Show sufficient iterations to justify accuracy to 3 dp or show sign change in interval (1.5155, 1.5165) | B1 | [3] |
| (ii) State equation \(x = \sqrt[3]{\frac{1}{2}x^2 + 4x^{-3}}\) or equivalent | B1 | |
| Obtain exact value \(\sqrt[3]{8}\) or \(8^{0.2}\) | B1 | [2] |
(i) Use the iterative formula correctly at least once | M1 |
Obtain final answer 1.516 | A1 |
Show sufficient iterations to justify accuracy to 3 dp or show sign change in interval (1.5155, 1.5165) | B1 | [3]
(ii) State equation $x = \sqrt[3]{\frac{1}{2}x^2 + 4x^{-3}}$ or equivalent | B1 |
Obtain exact value $\sqrt[3]{8}$ or $8^{0.2}$ | B1 | [2]4 The sequence of values given by the iterative formula
$$x _ { n + 1 } = \sqrt { } \left( \frac { 1 } { 2 } x _ { n } ^ { 2 } + 4 x _ { n } ^ { - 3 } \right)$$
with initial value $x _ { 1 } = 1.5$, converges to $\alpha$.\\
(i) Use this iterative formula to find $\alpha$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.\\
(ii) State an equation that is satisfied by $\alpha$ and hence find the exact value of $\alpha$.
\hfill \mbox{\textit{CAIE P2 2016 Q4 [5]}}