| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Staircase/cobweb diagram |
| Difficulty | Standard +0.3 This is a straightforward application of fixed point iteration requiring routine calculations (finding x₂, x₃), a standard error ratio calculation to estimate F'(α), and a basic sketch showing convergence type. All techniques are standard FP2 material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x_1 = 3.1 \Rightarrow x_2 = 3.13140\) | B1 | For correct \(x_2\) |
| \(x_3 = 3.14148\) | B1 | For correct \(x_3\) |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(F'(\alpha) \approx \frac{e_3}{e_2} = \frac{0.00471}{0.01479} = 0.318\ (0.31846)\) | M1, A1 | For dividing \(e_3\) by \(e_2\); for estimate of \(F'(\alpha)\) |
| \(F'(\alpha) = \frac{1}{\alpha} = 0.3178\ (0.31784)\) | B1 | For true \(F'(\alpha)\) obtained from \(\frac{d}{dx}(2 + \ln x)\) |
| Total: 3 | TMDP anywhere in (i)(ii) deduct 1 once (but answers must round to given values or A0) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Diagram with \(y = x\) and \(y = F(x)\) drawn, crossing as shown | B1 | For \(y = x\) and \(y = F(x)\) drawn, crossing as shown |
| Lines drawn to illustrate iteration (Min 2 horizontal and 2 vertical seen) | B1 | For lines drawn to illustrate iteration |
| Staircase | B1 | For stating "staircase" |
| Total: 3 |
# Question 3(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x_1 = 3.1 \Rightarrow x_2 = 3.13140$ | B1 | For correct $x_2$ |
| $x_3 = 3.14148$ | B1 | For correct $x_3$ |
| **Total: 2** | | |
---
# Question 3(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $F'(\alpha) \approx \frac{e_3}{e_2} = \frac{0.00471}{0.01479} = 0.318\ (0.31846)$ | M1, A1 | For dividing $e_3$ by $e_2$; for estimate of $F'(\alpha)$ |
| $F'(\alpha) = \frac{1}{\alpha} = 0.3178\ (0.31784)$ | B1 | For true $F'(\alpha)$ obtained from $\frac{d}{dx}(2 + \ln x)$ |
| **Total: 3** | | **TMDP anywhere in (i)(ii) deduct 1 once (but answers must round to given values or A0)** |
---
# Question 3(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Diagram with $y = x$ and $y = F(x)$ drawn, crossing as shown | B1 | For $y = x$ and $y = F(x)$ drawn, crossing as shown |
| Lines drawn to illustrate iteration (Min 2 horizontal and 2 vertical seen) | B1 | For lines drawn to illustrate iteration |
| Staircase | B1 | For stating "staircase" |
| **Total: 3** | | |
---3 It is given that $\mathrm { F } ( x ) = 2 + \ln x$. The iteration $x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)$ is to be used to find a root, $\alpha$, of the equation $x = 2 + \ln x$.\\
(i) Taking $x _ { 1 } = 3.1$, find $x _ { 2 }$ and $x _ { 3 }$, giving your answers correct to 5 decimal places.\\
(ii) The error $e _ { n }$ is defined by $e _ { n } = \alpha - x _ { n }$. Given that $\alpha = 3.14619$, correct to 5 decimal places, use the values of $e _ { 2 }$ and $e _ { 3 }$ to make an estimate of $\mathrm { F } ^ { \prime } ( \alpha )$ correct to 3 decimal places. State the true value of $\mathrm { F } ^ { \prime } ( \alpha )$ correct to 4 decimal places.\\
(iii) Illustrate the iteration by drawing a sketch of $y = x$ and $y = \mathrm { F } ( x )$, showing how the values of $x _ { n }$ approach $\alpha$. State whether the convergence is of the 'staircase' or 'cobweb' type.
\hfill \mbox{\textit{OCR FP2 2011 Q3 [8]}}