| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2011 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Moderate -0.3 This is a structured, multi-part question on fixed point iteration with clear guidance at each step. Part (i) is routine substitution, part (ii) is algebraic rearrangement with the target form given, and part (iii) is mechanical iteration. While it requires several techniques, each step is standard and the question provides significant scaffolding, making it slightly easier than a typical A-level question. |
| 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 |
|---|---|---|
| (i) Consider sign of \(x^3 - 2x^2 + 5x - 3\) at \(x = 0.7\) and \(x = 0.8\) | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | [2] |
| (ii) Rearrange equation to given equation or vice versa | B1 | |
| State \(a = 2\) and \(b = 5\) | B1 | [2] |
| (iii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 0.74 | A1 | |
| Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval (0.735, 0.745) | B1 | [3] |
**(i)** Consider sign of $x^3 - 2x^2 + 5x - 3$ at $x = 0.7$ and $x = 0.8$ | M1 |
Complete the argument correctly with appropriate calculations | A1 | [2]
**(ii)** Rearrange equation to given equation or vice versa | B1 |
State $a = 2$ and $b = 5$ | B1 | [2]
**(iii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 0.74 | A1 |
Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval (0.735, 0.745) | B1 | [3]6 (i) Verify by calculation that the cubic equation
$$x ^ { 3 } - 2 x ^ { 2 } + 5 x - 3 = 0$$
has a root that lies between $x = 0.7$ and $x = 0.8$.\\
(ii) Show that this root also satisfies an equation of the form
$$x = \frac { a x ^ { 2 } + 3 } { x ^ { 2 } + b }$$
where the values of $a$ and $b$ are to be found.\\
(iii) With these values of $a$ and $b$, use the iterative formula
$$x _ { n + 1 } = \frac { a x _ { n } ^ { 2 } + 3 } { x _ { n } ^ { 2 } + b }$$
to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2011 Q6 [7]}}