| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| 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 iteration question with routine parts: sketching graphs to show uniqueness, verifying a root interval by substitution, proving convergence to the root algebraically, and applying the iteration formula. All techniques are textbook exercises requiring no novel insight, though part (iii) requires careful algebraic manipulation. Slightly easier than average due to straightforward execution. |
| Spec | 1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases1.09c 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. \(y = \ln x\) or \(y = 2 - x^2\) | B1 | |
| Sketch a second relevant graph and justify the given statement | B1 | [2] |
| (ii) Consider sign of \(\ln x - (2 - x^2)\) at \(x = 1.3\) and \(x = 1.4\), or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | [2] |
**(i)** Make a recognisable sketch of a relevant graph, e.g. $y = \ln x$ or $y = 2 - x^2$ | B1 |
Sketch a second relevant graph and justify the given statement | B1 | [2]
**(ii)** Consider sign of $\ln x - (2 - x^2)$ at $x = 1.3$ and $x = 1.4$, or equivalent | M1 |
Complete the argument correctly with appropriate calculations | A1 | [2](i) By sketching a suitable pair of graphs, show that the equation
$$\ln x = 2 - x ^ { 2 }$$
has only one root.\\
(ii) Verify by calculation that this root lies between $x = 1.3$ and $x = 1.4$.\\
(iii) Show that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)$$
converges, then it converges to the root of the equation in part (i).\\
(iv) Use the iterative formula $x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2010 Q6 [8]}}