| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Apply iteration to find root (pure fixed point) |
| Difficulty | Standard +0.8 This is a Further Pure 2 question requiring systematic iteration, derivative calculation, and understanding of convergence theory including the relationship between successive errors. While the iteration itself is mechanical, parts (ii) and (iv) require theoretical understanding of fixed-point iteration convergence beyond standard A-level, and part (iv) involves applying the error formula across multiple iterations—more sophisticated than typical FP2 questions. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams1.09e Iterative method failure: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(x_2 = 0.1890\) | B1 | |
| \(x_3 = 0.2087\) | B1√ | From their \(x_1\) (or any other correct) |
| \(x_4 = 0.2050\) | B1√ | Get at least two others correct, all to a minimum of 4 d.p. |
| \(x_5 = 0.2057\) | ||
| \(x_6 = 0.2055\) | ||
| \(x_7 = (x_8) = 0.2056\) (to \(x_7\) minimum) | ||
| \(\alpha = 0.2056\) | B1 | cao; answer may be retrieved despite some errors |
| (ii) Attempt to diff. \(f(x)\) | M1 | \(k/(2+x)^3\) |
| Use \(\alpha\) to show \(f'(\alpha) \neq 0\) | A1√ | Clearly seen, or explain \(k/(2+x)^3 \neq 0\) as \(k \neq 0\); allow \(\pm 0.1864\) |
| SR Translate \(y=1/x^2\) | M1 | |
| State/show \(y=1/x^2\) has no TP | A1 | |
| (iii) \(\delta_5 = -0.0037\) (allow \(-0.004\)) | B1√ | Allow \(\pm\), from their \(x_4\) and \(x_3\) |
| (iv) Develop from \(\delta_{10} = f'(\alpha) \delta_i\) etc to get \(\delta_i\) or quote \(\delta_{10} = \delta_5 [f'(\alpha)]^T\) | M1 | Or any \(\delta_j\) use \(\delta_i = x_{10} - x_9\) |
| Use their \(\delta_5\) and \(f'(\alpha)\) | M1 | |
| Get \(0.000000028\) | A1 | Or answer that rounds to \(\pm 0.00000003\) |
**(i)** $x_2 = 0.1890$ | B1 |
$x_3 = 0.2087$ | B1√ | From their $x_1$ (or any other correct)
$x_4 = 0.2050$ | B1√ | Get at least two others correct, all to a minimum of 4 d.p.
$x_5 = 0.2057$ |
$x_6 = 0.2055$ |
$x_7 = (x_8) = 0.2056$ (to $x_7$ minimum) |
$\alpha = 0.2056$ | B1 | cao; answer may be retrieved despite some errors
**(ii)** Attempt to diff. $f(x)$ | M1 | $k/(2+x)^3$
Use $\alpha$ to show $f'(\alpha) \neq 0$ | A1√ | Clearly seen, or explain $k/(2+x)^3 \neq 0$ as $k \neq 0$; allow $\pm 0.1864$
SR Translate $y=1/x^2$ | M1 |
State/show $y=1/x^2$ has no TP | A1 |
**(iii)** $\delta_5 = -0.0037$ (allow $-0.004$) | B1√ | Allow $\pm$, from their $x_4$ and $x_3$
**(iv)** Develop from $\delta_{10} = f'(\alpha) \delta_i$ etc to get $\delta_i$ or quote $\delta_{10} = \delta_5 [f'(\alpha)]^T$ | M1 | Or any $\delta_j$ use $\delta_i = x_{10} - x_9$
Use their $\delta_5$ and $f'(\alpha)$ | M1 |
Get $0.000000028$ | A1 | Or answer that rounds to $\pm 0.00000003$8 The iteration $x _ { n + 1 } = \frac { 1 } { \left( x _ { n } + 2 \right) ^ { 2 } }$, with $x _ { 1 } = 0.3$, is to be used to find the real root, $\alpha$, of the equation $x ( x + 2 ) ^ { 2 } = 1$.\\
(i) Find the value of $\alpha$, correct to 4 decimal places. You should show the result of each step of the iteration process.\\
(ii) Given that $\mathrm { f } ( x ) = \frac { 1 } { ( x + 2 ) ^ { 2 } }$, show that $\mathrm { f } ^ { \prime } ( \alpha ) \neq 0$.\\
(iii) The difference, $\delta _ { r }$, between successive approximations is given by $\delta _ { r } = x _ { r + 1 } - x _ { r }$. Find $\delta _ { 3 }$.\\
(iv) Given that $\delta _ { r + 1 } \approx \mathrm { f } ^ { \prime } ( \alpha ) \delta _ { r }$, find an estimate for $\delta _ { 10 }$.
\hfill \mbox{\textit{OCR FP2 2007 Q8 [10]}}