| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Iterative formula with sketch analysis |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring iterative calculations, graphical interpretation of convergence/divergence via staircase diagrams, algebraic manipulation, Newton-Raphson implementation, and analysis of failure conditions. While the individual techniques are standard FP2 content, the combination of four parts testing different aspects of numerical methods, plus the conceptual understanding required for parts (ii) and (iv), places this above average difficulty for A-level. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Get 5.242, 5.239, 5.237 | B1\(\checkmark\) | Any 3 (minimum) correct from previous value |
| Get 5.24 | B1 | Allow one B1 for 5.24 seen if 2 d.p. used |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show reasonable staircase for any region | B1 | Drawn curve to line |
| Describe any one of the three cases | B1 | |
| Describe all three cases | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Reasonable attempt to use log/expo. rules | M1 | Allow derivation either way |
| Clearly get A.G. | A1 | |
| Attempt \(f'(x)\) and use at least once in correct N-R formula | M1 | |
| Get answers that lead to 1.31 | A1 | Minimum of 2 answers; allow truncation/rounding to at least 3 d.p. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show \(f'(\ln 36) = 0\) | B1 | |
| Explain why N-R would not work | B1 | Tangent parallel to \(Ox\) would not meet \(Ox\) again, or dividing by 0 gives an error |
# Question 7:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Get 5.242, 5.239, 5.237 | B1$\checkmark$ | Any 3 (minimum) correct from previous value |
| Get 5.24 | B1 | Allow one B1 for 5.24 seen if 2 d.p. used |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show reasonable staircase for any region | B1 | Drawn curve to line |
| Describe any one of the three cases | B1 | |
| Describe all three cases | B1 | |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Reasonable attempt to use log/expo. rules | M1 | Allow derivation either way |
| Clearly get A.G. | A1 | |
| Attempt $f'(x)$ and use at least once in correct N-R formula | M1 | |
| Get answers that lead to 1.31 | A1 | Minimum of 2 answers; allow truncation/rounding to at least 3 d.p. |
## Part (iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show $f'(\ln 36) = 0$ | B1 | |
| Explain why N-R would not work | B1 | Tangent parallel to $Ox$ would not meet $Ox$ again, or dividing by 0 gives an error |
---7\\
\includegraphics[max width=\textwidth, alt={}, center]{074597e7-5bb1-4249-9cfa-784974a6fd2b-3_531_1065_1208_539}
The line $y = x$ and the curve $y = 2 \ln ( 3 x - 2 )$ meet where $x = \alpha$ and $x = \beta$, as shown in the diagram.\\
(i) Use the iteration $x _ { n + 1 } = 2 \ln \left( 3 x _ { n } - 2 \right)$, with initial value $x _ { 1 } = 5.25$, to find the value of $\beta$ correct to 2 decimal places. Show all your working.\\
(ii) With the help of a 'staircase' diagram, explain why this iteration will not converge to $\alpha$, whatever value of $x _ { 1 }$ (other than $\alpha$ ) is used.\\
(iii) Show that the equation $x = 2 \ln ( 3 x - 2 )$ can be rewritten as $x = \frac { 1 } { 3 } \left( \mathrm { e } ^ { \frac { 1 } { 2 } x } + 2 \right)$. Use the NewtonRaphson method, with $\mathrm { f } ( x ) = \frac { 1 } { 3 } \left( \mathrm { e } ^ { \frac { 1 } { 2 } x } + 2 \right) - x$ and $x _ { 1 } = 1.2$, to find $\alpha$ correct to 2 decimal places. Show all your working.\\
(iv) Given that $x _ { 1 } = \ln 36$, explain why the Newton-Raphson method would not converge to a root of $\mathrm { f } ( x ) = 0$.
\hfill \mbox{\textit{OCR FP2 2010 Q7 [11]}}