| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| 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 iterative formula (calculator work) and recognition that at convergence x_{n+1} = x_n = α, leading to a simple equation α = α/ln(2α) that rearranges to ln(2α) = 1. While it involves logarithms, the conceptual leap is minimal and this is a standard textbook exercise on iterative sequences, 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 |
|---|---|---|
| Answer | Mark | Guidance |
| Use iteration correctly at least once | M1 | Must see correct attempt at \(x_3\) |
| Obtain final answer 1.359 | A1 | |
| Show sufficient iterations to 6 sf to justify answer or show sign change in interval [1.3585, 1.3595] | A1 | Answer required to exactly 4 sf. Must see to at least \(x_5\) |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Form correct equation in \(x\) (or \(\alpha\)) | B1 | \(x = \dfrac{x}{\ln 2x}\) OE |
| Obtain \(\frac{1}{2}e\) | B1 | |
| Total: 2 |
## Question 4:
### Part 4(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iteration correctly at least once | M1 | Must see correct attempt at $x_3$ |
| Obtain final answer 1.359 | A1 | |
| Show sufficient iterations to 6 sf to justify answer or show sign change in interval [1.3585, 1.3595] | A1 | Answer required to exactly 4 sf. Must see to at least $x_5$ |
| **Total: 3** | | |
### Part 4(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Form correct equation in $x$ (or $\alpha$) | B1 | $x = \dfrac{x}{\ln 2x}$ OE |
| Obtain $\frac{1}{2}e$ | B1 | |
| **Total: 2** | | |
---4 The sequence $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ defined by
$$x _ { 1 } = 1 , \quad x _ { n + 1 } = \frac { x _ { n } } { \ln \left( 2 x _ { n } \right) }$$
converges to the value $\alpha$.\\
(i) Use the iterative formula to find the value of $\alpha$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures.\\
(ii) State an equation satisfied by $\alpha$ and hence determine the exact value of $\alpha$.\\
\hfill \mbox{\textit{CAIE P2 2019 Q4 [5]}}