| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| 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 question involves straightforward integration of exponential and polynomial functions, followed by algebraic rearrangement to isolate 'a'. The iterative part is mechanical application of a given rearrangement. While it requires multiple steps, each individual step uses standard C3 techniques without requiring novel insight or complex problem-solving. |
| 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 |
|---|---|---|
| (i) Obtain integral of form \(k_1 e^{2x} + k_2 x^2\) | M1 | any non-zero constants \(k_1, k_2\) |
| Obtain correct \(3e^{2x} + \frac{1}{3}x^3\) | A1 | |
| Obtain \(3e^{2x} + \frac{1}{3}a^3 - 3\) | A1 | |
| Equate definite integral to 42 and attempt rearrangement | M1 | using sound processes |
| Confirm \(a = \frac{1}{3}\ln(15 - \frac{1}{3}a^3)\) | A1 | 5 AG; necessary detail required |
| (ii) Obtain correct first iterate \(1.348…\) | B1 | |
| Attempt correct process to find at least 2 iterates | M1 | |
| Obtain at least 3 correct iterates | A1 | |
| Obtain \(1.344\) | A1 | 4 answer required to exactly 3 d.p.; allow recovery after error |
(i) Obtain integral of form $k_1 e^{2x} + k_2 x^2$ | M1 | any non-zero constants $k_1, k_2$
Obtain correct $3e^{2x} + \frac{1}{3}x^3$ | A1 |
Obtain $3e^{2x} + \frac{1}{3}a^3 - 3$ | A1 |
Equate definite integral to 42 and attempt rearrangement | M1 | using sound processes
Confirm $a = \frac{1}{3}\ln(15 - \frac{1}{3}a^3)$ | A1 | 5 AG; necessary detail required
(ii) Obtain correct first iterate $1.348…$ | B1 |
Attempt correct process to find at least 2 iterates | M1 |
Obtain at least 3 correct iterates | A1 |
Obtain $1.344$ | A1 | 4 answer required to exactly 3 d.p.; allow recovery after error
[$1 \to 1.34844 \to 1.34382 \to 1.34389$]
---6 (i) Given that $\int _ { 0 } ^ { \mathrm { a } } \left( 6 \mathrm { e } ^ { 2 \mathrm { x } } + \mathrm { x } \right) \mathrm { dx } = 42$, show that $\mathrm { a } = \frac { 1 } { 2 } \ln \left( 15 - \frac { 1 } { 6 } \mathrm { a } ^ { 2 } \right)$.\\
(ii) Use an iterative formula, based on the equation in part (i), to find the value of a correct to 3 decimal places. Use a starting value of 1 and show the result of each iteration.
\hfill \mbox{\textit{OCR C3 2007 Q6 [9]}}