| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show convergence to specific root |
| Difficulty | Standard +0.3 This is a standard fixed-point iteration question requiring sign change verification, algebraic manipulation to show the iteration converges to the root, and calculator work. All steps are routine textbook exercises with no novel insight required, making it slightly easier than average. |
| 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/Working | Mark | Guidance |
| Compare signs of \(x^3 - 2x - 2\) when \(x = 1\) and \(x = 2\), or equivalent | M1 | |
| Complete the argument with correct calculations | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply the equation \(x = (2x^3 + 2)/(3x^2 - 2)\) | B1 | |
| Rearrange this in the form \(x^3 - 2x - 2 = 0\), or work *vice versa* | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use the iterative formula correctly at least once with \(x_n > 0\) | M1 | |
| Obtain final answer \(1.77\) | A1 | |
| Show sufficient iterations to 4 d.p. to justify accuracy to 2 d.p., or show there is a sign change in the interval \((1.765,\ 1.775)\) | A1 |
## Question 4:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Compare signs of $x^3 - 2x - 2$ when $x = 1$ and $x = 2$, or equivalent | M1 | |
| Complete the argument with correct calculations | A1 | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply the equation $x = (2x^3 + 2)/(3x^2 - 2)$ | B1 | |
| Rearrange this in the form $x^3 - 2x - 2 = 0$, or work *vice versa* | B1 | |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use the iterative formula correctly at least once with $x_n > 0$ | M1 | |
| Obtain final answer $1.77$ | A1 | |
| Show sufficient iterations to 4 d.p. to justify accuracy to 2 d.p., or show there is a sign change in the interval $(1.765,\ 1.775)$ | A1 | |
---4 The equation $x ^ { 3 } - 2 x - 2 = 0$ has one real root.\\
(i) Show by calculation that this root lies between $x = 1$ and $x = 2$.\\
(ii) Prove that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 2 } { 3 x _ { n } ^ { 2 } - 2 }$$
converges, then it converges to this root.\\
(iii) Use this iterative formula to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2009 Q4 [7]}}