| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from integral condition |
| Difficulty | Standard +0.3 This question requires straightforward integration of polynomial and exponential terms, followed by algebraic rearrangement to isolate the required form. The iterative part is mechanical application of fixed point iteration with no conceptual challenges. While it combines integration and iteration, both components are routine A-level techniques with clear pathways. |
| Spec | 1.08d Evaluate definite integrals: between limits1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate to obtain form \(k_1x + k_2x^2 + k_3e^{3x}\) for non-zero constants | M1 | |
| Obtain \(x + x^2 + e^{3x}\) | A1 | |
| Apply both limits to obtain \(a + a^2 + e^{3a} - 1 = 250\) or equivalent | A1 | |
| Apply correct process to reach form without e involved | M1 | |
| Confirm given \(a = \frac{1}{3}\ln(251 - a - a^2)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use iterative process correctly at least once | M1 | |
| Obtain final answer \(1.835\) | A1 | |
| Show sufficient iterations to 6 sf to justify answer or show sign change in interval \((1.8345, 1.8355)\) | A1 |
## Question 5(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate to obtain form $k_1x + k_2x^2 + k_3e^{3x}$ for non-zero constants | M1 | |
| Obtain $x + x^2 + e^{3x}$ | A1 | |
| Apply both limits to obtain $a + a^2 + e^{3a} - 1 = 250$ or equivalent | A1 | |
| Apply correct process to reach form without e involved | M1 | |
| Confirm given $a = \frac{1}{3}\ln(251 - a - a^2)$ | A1 | |
## Question 5(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iterative process correctly at least once | M1 | |
| Obtain final answer $1.835$ | A1 | |
| Show sufficient iterations to 6 sf to justify answer or show sign change in interval $(1.8345, 1.8355)$ | A1 | |
---5 It is given that $a$ is a positive constant such that
$$\int _ { 0 } ^ { a } \left( 1 + 2 x + 3 \mathrm { e } ^ { 3 x } \right) \mathrm { d } x = 250$$
(i) Show that $a = \frac { 1 } { 3 } \ln \left( 251 - a - a ^ { 2 } \right)$.\\
(ii) Use an iterative formula based on the equation in part (i) to find the value of $a$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures.\\
\hfill \mbox{\textit{CAIE P2 2017 Q5 [8]}}