| 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 | Sketch graphs to show root existence |
| Difficulty | Moderate -0.3 This is a standard A-level question on numerical methods requiring routine graph sketching, interval verification by substitution, and straightforward iteration. While it involves multiple parts, each step follows textbook procedures with no novel insight required—slightly easier than average due to the mechanical nature of the tasks. |
| 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 recognisable sketch of a relevant graph, e.g. \(y = \sin x\) or \(y = \frac{1}{x}\) | B1 | |
| Sketch a second relevant graph and justify the given statement | B1 | [2] |
| (ii) Consider sign of \(\frac{1}{x} - \sin x\) at \(x = 1.1\) and \(x = 1.2\), or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | [2] |
| (iii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(1.11\) | A1 | |
| Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval \((1.105, 1.115)\) | B1 | [3] |
**(i)** Make a recognisable sketch of a relevant graph, e.g. $y = \sin x$ or $y = \frac{1}{x}$ | B1 |
Sketch a second relevant graph and justify the given statement | B1 | [2] |
**(ii)** Consider sign of $\frac{1}{x} - \sin x$ at $x = 1.1$ and $x = 1.2$, or equivalent | M1 |
Complete the argument correctly with appropriate calculations | A1 | [2] |
**(iii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer $1.11$ | A1 |
Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change in the interval $(1.105, 1.115)$ | B1 | [3] |5 (i) By sketching a suitable pair of graphs, show that the equation
$$\frac { 1 } { x } = \sin x$$
where $x$ is in radians, has only one root for $0 < x \leqslant \frac { 1 } { 2 } \pi$.\\
(ii) Verify by calculation that this root lies between $x = 1.1$ and $x = 1.2$.\\
(iii) Use the iterative formula $x _ { n + 1 } = \frac { 1 } { \sin x _ { n } }$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2011 Q5 [7]}}