| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 9 |
| 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 fixed-point iteration question requiring sketching graphs to show uniqueness, verifying a root interval by substitution, algebraic manipulation to show convergence to the correct root, and performing iterations. All steps are routine A-level techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.06d Natural logarithm: ln(x) function and properties1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Make recognisable sketch of appropriate exponential curve, e.g. \(y = 9e^{-2x}\) | B1 | |
| Sketch appropriate second curve, e.g. \(y = x\) correctly and justify the given statement | B1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Consider sign of \(x - 9e^{-2x}\) at \(x = 1\) and \(x = 2\), or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply the equation \(x = \frac{1}{2}(\ln 9 - \ln x)\) | B1 | |
| Rearrange this in the form given in part (i), or work vice versa | B1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(x = 1.07\) | A1 | |
| Show sufficient iterations to justify accuracy to 2 d.p. or show sign change in interval \((1.065, 1.075)\) | A1 | Total: 3 |
## Question 6:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Make recognisable sketch of appropriate exponential curve, e.g. $y = 9e^{-2x}$ | B1 | |
| Sketch appropriate second curve, e.g. $y = x$ correctly and justify the given statement | B1 | Total: 2 |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Consider sign of $x - 9e^{-2x}$ at $x = 1$ and $x = 2$, or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | Total: 2 |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply the equation $x = \frac{1}{2}(\ln 9 - \ln x)$ | B1 | |
| Rearrange this in the form given in part (i), or work vice versa | B1 | Total: 2 |
### Part (iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer $x = 1.07$ | A1 | |
| Show sufficient iterations to justify accuracy to 2 d.p. or show sign change in interval $(1.065, 1.075)$ | A1 | Total: 3 |
---6 (i) By sketching a suitable pair of graphs, show that there is only one value of $x$ that is a root of the equation $x = 9 \mathrm { e } ^ { - 2 x }$.\\
(ii) Verify, by calculation, that this root lies between 1 and 2 .\\
(iii) Show that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 2 } \left( \ln 9 - \ln x _ { n } \right)$$
converges, then it converges to the root of the equation given in part (i).\\
(iv) Use the iterative formula, with $x _ { 1 } = 1$, to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2006 Q6 [9]}}