| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| 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 graph sketching to show existence of a root, algebraic manipulation to verify the iteration converges to that root, and numerical computation. All steps are routine for P3 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Make recognizable sketch of a relevant graph | B1 | |
| Sketch the other relevant graph and justify the given statement | B1 | |
| Total | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State \(x = \frac{1}{2}\ln(25/x)\) | B1 | |
| Rearrange in the form \(5e^{-x} = \sqrt{x}\) | B1 | |
| Total | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use iterative formula correctly at least once | M1 | |
| Obtain final answer 1.43 | A1 | |
| Show sufficient iterations to 4 d.p. to justify 1.43 to 2 d.p., or show sign change in interval \((1.425, 1.435)\) | A1 | |
| Total | [3] |
## Question 6:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Make recognizable sketch of a relevant graph | B1 | |
| Sketch the other relevant graph and justify the given statement | B1 | |
| **Total** | **[2]** | |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| State $x = \frac{1}{2}\ln(25/x)$ | B1 | |
| Rearrange in the form $5e^{-x} = \sqrt{x}$ | B1 | |
| **Total** | **[2]** | |
### Part (iii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use iterative formula correctly at least once | M1 | |
| Obtain final answer 1.43 | A1 | |
| Show sufficient iterations to 4 d.p. to justify 1.43 to 2 d.p., or show sign change in interval $(1.425, 1.435)$ | A1 | |
| **Total** | **[3]** | |
---6 (i) By sketching a suitable pair of graphs, show that the equation
$$5 \mathrm { e } ^ { - x } = \sqrt { } x$$
has one root.\\
(ii) Show that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 25 } { x _ { n } } \right)$$
converges, then it converges to the root of the equation in part (i).\\
(iii) Use this iterative formula, with initial value $x _ { 1 } = 1$, to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2016 Q6 [7]}}