| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Topic | Fixed Point Iteration |
| Type | Show root in interval |
| Difficulty | Moderate -0.8 Part (i) is a standard 'show root in interval' question requiring simple substitution into a cubic to verify sign change. Part (ii) is routine iteration with a given formula - purely mechanical calculation with no derivation or analysis required. Both parts are below-average difficulty, involving straightforward application of basic techniques with no problem-solving insight needed. |
| Spec | 1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
**Part (i)**
- Attempt $f(0) = 2$ and $f(1) = -3$ or equiv: M1
- Conclude correctly: A1 **[2]**
**Part (ii)**
- Attempt to use iterative formula and no other method: M1
- 0.5, 0.3541666, 0.340737425, 0.339926715, 0.339879765, 0.339877052: A1
- Conclude 0.3399: A1 **[3]**
**[Total: 5]**4 (i) Show that the equation $x ^ { 3 } - 6 x + 2 = 0$ has a root between $x = 0$ and $x = 1$.\\
(ii) Use the iterative formula $x _ { n + 1 } = \frac { 2 + x _ { n } ^ { 3 } } { 6 }$ with $x _ { 0 } = 0.5$ to find this root correct to 4 decimal places, showing the result of each iteration.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2012 Q4 [5]}}