| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| 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 A-level fixed point iteration question requiring routine techniques: showing a root exists in an interval by sign change, algebraic rearrangement to verify the iteration converges to the correct root, and performing iterations to find the root. While it involves multiple parts and exponential/logarithmic manipulation, these are well-practiced skills with no novel insight required, making it slightly easier than average. |
| Spec | 1.06g Equations with exponentials: solve a^x = b1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Consider sign of \(x - 10/(e^{2x} - 1)\) at \(x = 1\) and \(x = 2\) | M1 | |
| Complete the argument correctly with correct calculated values | A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply \(\alpha = \frac{1}{2}\ln(1 + 10/\alpha)\) | B1 | |
| Rearrange this as \(\alpha = 10/(e^{2\alpha} - 1)\) or work vice versa | B1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 1.14 | A1 | |
| Show sufficient iterations to 4 d.p. to justify 1.14 to 2 d.p., or show there is a sign change in the interval (1.135, 1.145) | A1 | 3 marks |
**(i)**
Consider sign of $x - 10/(e^{2x} - 1)$ at $x = 1$ and $x = 2$ | M1 |
Complete the argument correctly with correct calculated values | A1 | 2 marks
**(ii)**
State or imply $\alpha = \frac{1}{2}\ln(1 + 10/\alpha)$ | B1 |
Rearrange this as $\alpha = 10/(e^{2\alpha} - 1)$ or work vice versa | B1 | 2 marks
**(iii)**
Use the iterative formula correctly at least once | M1 |
Obtain final answer 1.14 | A1 |
Show sufficient iterations to 4 d.p. to justify 1.14 to 2 d.p., or show there is a sign change in the interval (1.135, 1.145) | A1 | 3 marks4 The equation $x = \frac { 10 } { \mathrm { e } ^ { 2 x } - 1 }$ has one positive real root, denoted by $\alpha$.\\
(i) Show that $\alpha$ lies between $x = 1$ and $x = 2$.\\
(ii) Show that if a sequence of positive values given by the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 1 + \frac { 10 } { x _ { n } } \right)$$
converges, then it converges to $\alpha$.\\
(iii) Use this iterative formula to determine $\alpha$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2014 Q4 [7]}}