| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | November |
| 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 convergence theory, algebraic manipulation to verify the iteration formula converges to the correct root, and careful numerical computation. While the sketching in part (i) is routine, part (ii) requires formal proof of convergence to a specific root (not just any fixed point), which goes beyond standard textbook exercises. Part (iii) is computational but requires precision. The multi-step reasoning and theoretical depth place this moderately above average difficulty. |
| Spec | 1.02p Interpret algebraic solutions: graphically1.02q Use intersection points: of graphs to solve equations1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sketch a relevant graph, e.g. \(y = x^3\) | B1 | |
| Sketch a second relevant graph, e.g. \(y = 3 - x\), and justify the given statement | B1 | Consideration of behaviour for \(x < 0\) is needed for the second B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply the equation \(x = (2x^3 + 3)/(3x^2 + 1)\) | B1 | |
| Rearrange this in the form \(x^3 = 3 - x\), or commence work *vice versa* | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(1.213\) | A1 | |
| Show sufficient iterations to 5 d.p. or more to justify \(1.213\) to 3 d.p., or show there is a sign change in the interval \((1.2125, 1.2135)\) | A1 |
## Question 3(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch a relevant graph, e.g. $y = x^3$ | B1 | |
| Sketch a second relevant graph, e.g. $y = 3 - x$, and justify the given statement | B1 | Consideration of behaviour for $x < 0$ is needed for the second B1 |
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## Question 3(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the equation $x = (2x^3 + 3)/(3x^2 + 1)$ | B1 | |
| Rearrange this in the form $x^3 = 3 - x$, or commence work *vice versa* | B1 | |
---
## Question 3(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer $1.213$ | A1 | |
| Show sufficient iterations to 5 d.p. or more to justify $1.213$ to 3 d.p., or show there is a sign change in the interval $(1.2125, 1.2135)$ | A1 | |
---3 (i) By sketching a suitable pair of graphs, show that the equation $x ^ { 3 } = 3 - x$ has exactly one real root.\\
(ii) Show that if a sequence of real values given by the iterative formula
$$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 3 } { 3 x _ { n } ^ { 2 } + 1 }$$
converges, then it converges to the root of the equation in part (i).\\
(iii) Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.\\
\hfill \mbox{\textit{CAIE P3 2018 Q3 [7]}}