| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2008 |
| Session | November |
| Marks | 8 |
| 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 A-level iterative methods question requiring graph sketching, sign change verification, algebraic manipulation to show convergence to the correct root, and numerical iteration. All techniques are routine for P2/C3 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Make a recognizable sketch of a relevant graph, e.g. \(y = \cos x\) or \(y = 2 - 2x\) | B1 | |
| Sketch a second relevant graph and justify the given statement | B1 | [2] |
| (ii) Consider sign of \(\cos x - (2 - 2x)\) at \(x = 0.5\) and \(x = 1\), or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | [2] |
| (iii) Show that the given equation is equivalent to \(x = 1 - \frac{1}{2}\cos x\), or vice versa | B1 | [1] |
| (iv) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 0.58 | A1 | |
| Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval (0.575, 0.585) | B1 | [3] |
**(i)** Make a recognizable sketch of a relevant graph, e.g. $y = \cos x$ or $y = 2 - 2x$ | B1 |
Sketch a second relevant graph and justify the given statement | B1 | [2]
**(ii)** Consider sign of $\cos x - (2 - 2x)$ at $x = 0.5$ and $x = 1$, or equivalent | M1 |
Complete the argument correctly with appropriate calculations | A1 | [2]
**(iii)** Show that the given equation is equivalent to $x = 1 - \frac{1}{2}\cos x$, or vice versa | B1 | [1]
**(iv)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 0.58 | A1 |
Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval (0.575, 0.585) | B1 | [3]7 (i) By sketching a suitable pair of graphs, show that the equation
$$\cos x = 2 - 2 x$$
where $x$ is in radians, has only one root for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$.\\
(ii) Verify by calculation that this root lies between 0.5 and 1 .\\
(iii) Show that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = 1 - \frac { 1 } { 2 } \cos x _ { n }$$
converges, then it converges to the root of the equation in part (i).\\
(iv) Use this iterative formula, with initial value $x _ { 1 } = 0.6$, to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2008 Q7 [8]}}