| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Exact value from iterative result |
| Difficulty | Standard +0.3 This is a straightforward Newton-Raphson application with a simple function. Part (i) is routine iteration requiring only differentiation and substitution. Part (ii) requires recognizing that f(x)=0 gives x²=7, so α=√7 (trivial algebra). Part (iii) involves basic error calculations and verification of a given relationship. All steps are mechanical with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.09d Newton-Raphson method1.09e Iterative method failure: convergence conditions |
| Answer | Marks |
|---|---|
| Integrate \(k_1 e^{pt}\) to obtain \(k_2 e^{pt}\) | M1 |
| Obtain correct indefinite integral of their \(k_1 e^{pt}\) | A1 |
| Substitute limits to obtain \(\frac{1}{5}(e^3 - 1)\) or \(\frac{1}{6}(e^3 - 1)\) | A1 |
| Integrate \(k(2x - 1)^n\) to obtain \(k'(2x - 1)^{n+1}\) | M1 |
| Obtain correct indefinite integral of their \(k(2x - 1)^n\) | A1 |
| Substitute limits to obtain \(\frac{1}{18}\pi\) or \(\frac{1}{18}\) | A1 |
| Apply formula \(\int \pi y^2 dx\) at least once | B1 |
| Subtract, correct way found, attempts at volumes | M1 |
| Obtain \(\frac{1}{3}\pi e^3 - \frac{2}{3}\pi\) | A1 |
| | |
|---|---|
| Integrate $k_1 e^{pt}$ to obtain $k_2 e^{pt}$ | M1 |
| Obtain correct indefinite integral of their $k_1 e^{pt}$ | A1 |
| Substitute limits to obtain $\frac{1}{5}(e^3 - 1)$ or $\frac{1}{6}(e^3 - 1)$ | A1 |
| Integrate $k(2x - 1)^n$ to obtain $k'(2x - 1)^{n+1}$ | M1 |
| Obtain correct indefinite integral of their $k(2x - 1)^n$ | A1 |
| Substitute limits to obtain $\frac{1}{18}\pi$ or $\frac{1}{18}$ | A1 |
| Apply formula $\int \pi y^2 dx$ at least once | B1 |
| Subtract, correct way found, attempts at volumes | M1 |
| Obtain $\frac{1}{3}\pi e^3 - \frac{2}{3}\pi$ | A1 |6 It is given that $\mathrm { f } ( x ) = 1 - \frac { 7 } { x ^ { 2 } }$.\\
(i) Use the Newton-Raphson method, with a first approximation $x _ { 1 } = 2.5$, to find the next approximations $x _ { 2 }$ and $x _ { 3 }$ to a root of $\mathrm { f } ( x ) = 0$. Give the answers correct to 6 decimal places. [3]\\
(ii) The root of $\mathrm { f } ( x ) = 0$ for which $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$ are approximations is denoted by $\alpha$. Write down the exact value of $\alpha$.\\
(iii) The error $e _ { n }$ is defined by $e _ { n } = \alpha - x _ { n }$. Find $e _ { 1 } , e _ { 2 }$ and $e _ { 3 }$, giving your answers correct to 5 decimal places. Verify that $e _ { 3 } \approx \frac { e _ { 2 } ^ { 3 } } { e _ { 1 } ^ { 2 } }$.
\hfill \mbox{\textit{OCR FP2 2008 Q6 [7]}}