| 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 | Sketch graphs to show root existence |
| Difficulty | Standard +0.3 This is a standard fixed-point iteration question requiring routine sketching to show uniqueness of a root, algebraic manipulation to verify the iteration converges to that root, and mechanical application of the formula. All techniques are textbook exercises for P3 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.02q 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 |
|---|---|---|
| 3 (i) | Sketch a relevant graph, e.g. y=x3 | 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 |
| Answer | Marks |
|---|---|
| 3(ii) | ( ) ( ) |
| State or imply the equation x= 2x3 +3 / 3x2 +1 | B1 |
| Rearrange this in the form x3 =3−x, or commence work vice versa | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 3(iii) | Use the iterative formula correctly at least once | M1 |
| Obtain final answer 1.213 | A1 |
| Answer | Marks |
|---|---|
| sign change in the interval (1.2125, 1.2135) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 3:
--- 3 (i) ---
3 (i) | Sketch a relevant graph, e.g. y=x3 | 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
2
--- 3(ii) ---
3(ii) | ( ) ( )
State or imply the equation x= 2x3 +3 / 3x2 +1 | B1
Rearrange this in the form x3 =3−x, or commence work vice versa | B1
2
Question | Answer | Marks | Guidance
--- 3(iii) ---
3(iii) | 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
Question | Answer | Marks | Guidance\begin{enumerate}[label=(\roman*)]
\item By sketching a suitable pair of graphs, show that the equation $x^3 = 3 - x$ has exactly one real root. [2]
\item Show that if a sequence of real values given by the iterative formula
$$x_{n+1} = \frac{2x_n^3 + 3}{3x_n^2 + 1}$$
converges, then it converges to the root of the equation in part (i). [2]
\item Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q3 [7]}}