| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from area/geometry |
| Difficulty | Moderate -0.3 This is a straightforward fixed-point iteration question requiring standard application of a given iterative formula. Parts (i) and (ii) involve simple algebraic manipulation and substitution, while part (iii) is mechanical calculator work with no conceptual challenges. Slightly easier than average due to the routine nature of all steps and clear guidance provided. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct sector formula at least once and form an equation in \(r\) and \(x\) | M1 | |
| Obtain a correct equation in any form | A1 | |
| Rearrange in the given form | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Calculate values of a relevant expression or expressions at \(x = 1\) and \(x = 1.5\) | M1 | |
| Complete the argument correctly with correct calculated values | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(1.374\) | A1 | |
| Show sufficient iterations to 5 d.p. to justify \(1.374\) to 3 d.p., or show there is a sign change in the interval \((1.3745, 1.3755)\) | A1 |
## Question 5(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct sector formula at least once and form an equation in $r$ and $x$ | M1 | |
| Obtain a correct equation in any form | A1 | |
| Rearrange in the given form | A1 | |
**Total: 3**
---
## Question 5(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Calculate values of a relevant expression or expressions at $x = 1$ and $x = 1.5$ | M1 | |
| Complete the argument correctly with correct calculated values | A1 | |
**Total: 2**
---
## Question 5(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer $1.374$ | A1 | |
| Show sufficient iterations to 5 d.p. to justify $1.374$ to 3 d.p., or show there is a sign change in the interval $(1.3745, 1.3755)$ | A1 | |
**Total: 3**
---(i) Show that $x$ satisfies the equation $x = \frac { 1 } { 3 } ( \pi + \sin x )$.\\
(ii) Verify by calculation that $x$ lies between 1 and 1.5.\\
(iii) Use an iterative formula based on the equation in part (i) to determine $x$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.\\
\hfill \mbox{\textit{CAIE P3 2017 Q5 [8]}}