| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show convergence to specific root |
| Difficulty | Standard +0.8 This question requires understanding of fixed point iteration, manipulation of trigonometric equations (converting cot x = 1-x to the iterative form), proving convergence conditions, and careful numerical work. The rearrangement to x_{n+1} = π + arctan(1/(1-x_n)) is non-trivial and requires insight into inverse trig functions and periodicity. While systematic, it demands more conceptual understanding than routine iteration questions. |
| Spec | 1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Calculate the value of a relevant expression or expressions at \(x = 2.5\) and at another relevant value, e.g. \(x = 3\) | M1 | |
| Complete the argument correctly with correct calculated values | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State a suitable equation, e.g. \(x = \pi + \tan^{-1}\!\left(1/(1-x)\right)\) without suffices | B1 | |
| Rearrange this as \(\cot x = 1 - x\), or commence working *vice versa* | B1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 2.576 only | A1 | |
| Show sufficient iterations to 5 d.p. to justify 2.576 to 3 d.p., or show there is a sign change in the interval \((2.5755, 2.5765)\) | A1 | |
| Total: 3 |
## Question 6(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Calculate the value of a relevant expression or expressions at $x = 2.5$ and at another relevant value, e.g. $x = 3$ | M1 | |
| Complete the argument correctly with correct calculated values | A1 | |
| **Total: 2** | | |
## Question 6(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State a suitable equation, e.g. $x = \pi + \tan^{-1}\!\left(1/(1-x)\right)$ without suffices | B1 | |
| Rearrange this as $\cot x = 1 - x$, or commence working *vice versa* | B1 | |
| **Total: 2** | | |
## Question 6(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 2.576 only | A1 | |
| Show sufficient iterations to 5 d.p. to justify 2.576 to 3 d.p., or show there is a sign change in the interval $(2.5755, 2.5765)$ | A1 | |
| **Total: 3** | | |
---6 The equation $\cot x = 1 - x$ has one root in the interval $0 < x < \pi$, denoted by $\alpha$.\\
(i) Show by calculation that $\alpha$ is greater than 2.5.\\
(ii) Show that, if a sequence of values in the interval $0 < x < \pi$ given by the iterative formula $x _ { n + 1 } = \pi + \tan ^ { - 1 } \left( \frac { 1 } { 1 - x _ { n } } \right)$ converges, then it converges to $\alpha$.\\
(iii) Use this iterative formula to determine $\alpha$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.\\
\hfill \mbox{\textit{CAIE P3 2017 Q6 [7]}}