| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from integral condition |
| Difficulty | Challenging +1.2 This question requires expanding and integrating an exponential expression (standard C3/C4 technique), algebraic manipulation to derive the fixed point equation, and applying iterative methods. While multi-step, each component is routine: the integration follows standard patterns, the rearrangement is algebraic manipulation, and the iteration is mechanical application. The most challenging aspect is the careful algebra in part (i), but this remains a standard exam question testing well-practiced techniques rather than requiring novel insight. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.08d Evaluate definite integrals: between limits1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Rewrite integrand as \(1 + 2e^{\frac{1}{2}x} + e^x\) | B1 | |
| Integrate to obtain form \(x + k_1 e^{\frac{1}{2}x} + k_2 e^x\) | M1 | |
| Obtain \(x + 4e^{\frac{1}{2}x} + e^x\) | A1 | |
| Use limits to obtain \(a + 4e^{\frac{1}{2}a} + e^a - 5 = 10\) | A1 | |
| Rearrange as far as \(e^{\frac{1}{2}a} = ...\) including use of \(4e^{\frac{1}{2}a} + e^a = e^{\frac{1}{2}a}(4 + e^{\frac{1}{2}a})\) | M1 | |
| Confirm \(a = 2\ln\left(\frac{15-a}{4+e^{\frac{1}{2}a}}\right)\) | A1 | AG; necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Consider sign of \(a - 2\ln\left(\frac{15-a}{4+e^{\frac{1}{2}a}}\right)\) for 1.5 and 1.6 or equivalent | M1 | |
| Obtain \(-0.08...\) and \(0.06...\) or equivalents and justify conclusion | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use iterative process correctly at least once | M1 | |
| Obtain final answer 1.56 | A1 | |
| Show sufficient iterations to 5 sf to justify answer or show sign change in interval \((1.555, 1.565)\) | A1 |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Rewrite integrand as $1 + 2e^{\frac{1}{2}x} + e^x$ | B1 | |
| Integrate to obtain form $x + k_1 e^{\frac{1}{2}x} + k_2 e^x$ | M1 | |
| Obtain $x + 4e^{\frac{1}{2}x} + e^x$ | A1 | |
| Use limits to obtain $a + 4e^{\frac{1}{2}a} + e^a - 5 = 10$ | A1 | |
| Rearrange as far as $e^{\frac{1}{2}a} = ...$ including use of $4e^{\frac{1}{2}a} + e^a = e^{\frac{1}{2}a}(4 + e^{\frac{1}{2}a})$ | M1 | |
| Confirm $a = 2\ln\left(\frac{15-a}{4+e^{\frac{1}{2}a}}\right)$ | A1 | AG; necessary detail needed |
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Consider sign of $a - 2\ln\left(\frac{15-a}{4+e^{\frac{1}{2}a}}\right)$ for 1.5 and 1.6 or equivalent | M1 | |
| Obtain $-0.08...$ and $0.06...$ or equivalents and justify conclusion | A1 | |
## Question 6(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use iterative process correctly at least once | M1 | |
| Obtain final answer 1.56 | A1 | |
| Show sufficient iterations to 5 sf to justify answer or show sign change in interval $(1.555, 1.565)$ | A1 | |6 It is given that $\int _ { 0 } ^ { a } \left( 1 + \mathrm { e } ^ { \frac { 1 } { 2 } x } \right) ^ { 2 } \mathrm {~d} x = 10$, where $a$ is a positive constant.\\
(i) Show that $a = 2 \ln \left( \frac { 15 - a } { 4 + \mathrm { e } ^ { \frac { 1 } { 2 } a } } \right)$.\\
(ii) Use the equation in part (i) to show by calculation that $1.5 < a < 1.6$.\\
(iii) Use an iterative formula based on the equation in part (i) to find the value of $a$ correct to 3 significant figures. Give the result of each iteration to 5 significant figures.\\
\hfill \mbox{\textit{CAIE P2 2018 Q6 [11]}}