| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| 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 involves straightforward integration of an exponential function, algebraic rearrangement to derive the required equation, and mechanical application of an iterative formula. Part (i) requires integrating 3e^(x/2) + 1 and rearranging to the given form—standard C3/C4 techniques with no novel insight. Part (ii) is purely computational iteration. While it combines multiple skills, each step is routine and the question provides the iterative formula explicitly, making it slightly easier than average. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Obtain integral of form \(ke^{1x} + mx\) | M1 | |
| Obtain correct \(6e^{1x} + x\) | A1 | |
| Apply limits and obtain correct \(6e^{1a} + a - 6\) | A1 | |
| Equate to 10 and introduce natural logarithm correctly | DM1 | |
| Obtain given answer \(a = 2\ln\left(\frac{16-a}{6}\right)\) correctly | A1 | [5] |
| (ii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 1.732 | A1 | |
| Show sufficient iterations to justify accuracy to 3 d.p. or show sign change in interval (1.7315, 1.7325) | A1 | [3] |
(i) Obtain integral of form $ke^{1x} + mx$ | M1 |
Obtain correct $6e^{1x} + x$ | A1 |
Apply limits and obtain correct $6e^{1a} + a - 6$ | A1 |
Equate to 10 and introduce natural logarithm correctly | DM1 |
Obtain given answer $a = 2\ln\left(\frac{16-a}{6}\right)$ correctly | A1 | [5]
(ii) Use the iterative formula correctly at least once | M1 |
Obtain final answer 1.732 | A1 |
Show sufficient iterations to justify accuracy to 3 d.p. or show sign change in interval (1.7315, 1.7325) | A1 | [3]
---5 (i) Given that $\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 1 \right) \mathrm { d } x = 10$, show that the positive constant $a$ satisfies the equation
$$a = 2 \ln \left( \frac { 16 - a } { 6 } \right)$$
(ii) Use the iterative formula $a _ { n + 1 } = 2 \ln \left( \frac { 16 - a _ { n } } { 6 } \right)$ with $a _ { 1 } = 2$ to find the value of $a$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
\hfill \mbox{\textit{CAIE P2 2015 Q5 [8]}}