| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2007 |
| 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 simple algebraic manipulation to find the limit equation by setting x_{n+1} = x_n = α (part ii). The equation 3α³ = 2α³ + 12 is easily solved. While it involves multiple steps, each is standard procedure with no novel insight required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 2.29 | A1 | |
| Show sufficient iterations to justify its accuracy to 2 d.p. (must be working to 4 d.p.) – 3 iterations are sufficient | B1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| State equation \(x = \frac{2}{3}x + \frac{4}{x^2}\), or equivalent | B1 | |
| Derive the exact answer \(\alpha\) (or \(x\)) \(= \sqrt{12}\), or equivalent | B1 | [2] |
**(i)**
Use the iterative formula correctly at least once | M1 |
Obtain final answer 2.29 | A1 |
Show sufficient iterations to justify its accuracy to 2 d.p. (must be working to 4 d.p.) – 3 iterations are sufficient | B1 | [3]
**(ii)**
State equation $x = \frac{2}{3}x + \frac{4}{x^2}$, or equivalent | B1 |
Derive the exact answer $\alpha$ (or $x$) $= \sqrt{12}$, or equivalent | B1 | [2]2 The sequence of values given by the iterative formula
$$x _ { n + 1 } = \frac { 2 x _ { n } } { 3 } + \frac { 4 } { x _ { n } ^ { 2 } }$$
with initial value $x _ { 1 } = 2$, converges to $\alpha$.\\
(i) Use this iterative formula to determine $\alpha$ correct to 2 decimal places, giving the result of each iteration to 4 decimal places.\\
(ii) State an equation that is satisfied by $\alpha$ and hence find the exact value of $\alpha$.
\hfill \mbox{\textit{CAIE P2 2007 Q2 [5]}}