| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from integral condition |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring routine integration, algebraic manipulation to rearrange for the given form, interval verification by substitution, and application of a provided iterative formula. All techniques are standard P2 content with no novel insight required—slightly easier than average due to the scaffolded structure and explicit guidance at each step. |
| Spec | 1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08c 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 |
|---|---|
| (i) Integrate to obtain \(ke^{3x} + mx^3\) | M1 |
| Apply both limits to obtain \(\frac{k}{3}e^{3t} + \frac{1}{3}a^3 - \frac{k}{3} = 10\) or equivalent | A1 |
| Rearrange to form involving natural logarithm | DM1 |
| Obtain \(a = \frac{3}{2}\ln(61 - 2a^3)\) with no errors seen (AG) | A1 [4] |
| (ii) Consider sign of \(a - \frac{3}{2}\ln(61 - 2a^3)\) for \(1.0\) and \(1.5\) or equivalent | M1 |
| Obtain \(-0.36\) and \(0.17\) or equivalent and justify conclusion | A1 [2] |
| (iii) Use iteration process correctly at least once | M1 |
| Obtain final answer \(1.343\) | A1 |
| Show sufficient iterations to 5 decimal places to justify answer or show a sign change in the interval \((1.3425, 1.3435)\) | A1 [3] |
(i) Integrate to obtain $ke^{3x} + mx^3$ | M1 |
Apply both limits to obtain $\frac{k}{3}e^{3t} + \frac{1}{3}a^3 - \frac{k}{3} = 10$ or equivalent | A1 |
Rearrange to form involving natural logarithm | DM1 |
Obtain $a = \frac{3}{2}\ln(61 - 2a^3)$ with no errors seen (AG) | A1 [4] |
(ii) Consider sign of $a - \frac{3}{2}\ln(61 - 2a^3)$ for $1.0$ and $1.5$ or equivalent | M1 |
Obtain $-0.36$ and $0.17$ or equivalent and justify conclusion | A1 [2] |
(iii) Use iteration process correctly at least once | M1 |
Obtain final answer $1.343$ | A1 |
Show sufficient iterations to 5 decimal places to justify answer or show a sign change in the interval $(1.3425, 1.3435)$ | A1 [3] |7 It is given that $\int _ { 0 } ^ { a } \left( \frac { 1 } { 2 } \mathrm { e } ^ { 3 x } + x ^ { 2 } \right) \mathrm { d } x = 10$, where $a$ is a positive constant.\\
(i) Show that $a = \frac { 1 } { 3 } \ln \left( 61 - 2 a ^ { 3 } \right)$.\\
(ii) Show by calculation that the value of $a$ lies between 1.0 and 1.5.\\
(iii) Use an iterative formula, based on the equation in part (i), 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 2014 Q7 [9]}}