| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2002 |
| Session | June |
| Marks | 5 |
| 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 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 = 2α² + 2 is easily derived and solved, making this slightly easier than average despite being a two-part question. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use the formula correctly at least once | M1 | |
| State \(\alpha = 1.26\) as final answer | A1 | |
| Show sufficient iterations to justify \(\alpha = 1.26\) to 2d.p., or show there is a sign change in the interval \((1.255, 1.265)\) | A1 | Total: 3 marks |
| (ii) State any suitable equation in one unknown e.g. \(x = \frac{3}{2}\left(x+\frac{1}{x^2}\right)\) | B1 | |
| State exact value of \(\alpha\) (or \(x\)) is \(\sqrt[3]{2}\) or \(2^{\frac{1}{3}}\) | B1 | Total: 2 marks |
**(i)** Use the formula correctly at least once | M1 |
State $\alpha = 1.26$ as final answer | A1 |
Show sufficient iterations to justify $\alpha = 1.26$ to 2d.p., or show there is a sign change in the interval $(1.255, 1.265)$ | A1 | **Total: 3 marks**
**(ii)** State any suitable equation in one unknown e.g. $x = \frac{3}{2}\left(x+\frac{1}{x^2}\right)$ | B1 |
State exact value of $\alpha$ (or $x$) is $\sqrt[3]{2}$ or $2^{\frac{1}{3}}$ | B1 | **Total: 2 marks**
---4 The sequence of values given by the iterative formula
$$x _ { n + 1 } = \frac { 2 } { 3 } \left( x _ { n } + \frac { 1 } { x _ { n } ^ { 2 } } \right)$$
with initial value $x _ { 1 } = 1$, converges to $\alpha$.\\
(i) Use this formula to find $\alpha$ correct to 2 decimal places, showing the result of each iteration.\\
(ii) State an equation satisfied by $\alpha$, and hence find the exact value of $\alpha$.
\hfill \mbox{\textit{CAIE P3 2002 Q4 [5]}}