| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Find equation satisfied by limit |
| Difficulty | Standard +0.3 This is a straightforward fixed-point iteration question requiring routine application of the formula (part i), simple algebraic rearrangement to find the cubic (part ii), and basic analysis of roots (part iii). All steps are standard C3 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(x_2 = \sqrt{7} = 1.9129...\) \(x_3 = 1.9517..., x_4 = 1.9346...\) \(\alpha = 1.94\) to 2dp | B1, M1, A1 | For 1.91... seen or implied; For continuing the correct process; For correct value reached, following \(x_3\) and \(x_4\) both 1.94 to 2dp |
| (ii) \(x = \sqrt[3]{(17 - 5x)} \Rightarrow x^3 + 5x - 17 = 0\) | M1, A1 | For letting \(x_n = x_{n+1} = x\) (or \(\alpha\)); For correct equation stated |
| (iii) EITHER: Graphs of \(y = x^3\) and \(y = 17 - 5x\) only cross once Hence there is only one real root | M1, A1✓ | For argument based on sketching a pair of graphs, or a sketch of the cubic by calculator; For correct conclusion for a valid reason |
| OR: \(\frac{d}{dx}(x^3 + 5x - 17) = 3x^2 + 5 > 0\) Hence there is only one real root | M1, A1✓ | For consideration of the cubic's gradient; For correct conclusion for a valid reason |
**(i)** $x_2 = \sqrt{7} = 1.9129...$ $x_3 = 1.9517..., x_4 = 1.9346...$ $\alpha = 1.94$ to 2dp | B1, M1, A1 | For 1.91... seen or implied; For continuing the correct process; For correct value reached, following $x_3$ and $x_4$ both 1.94 to 2dp
**(ii)** $x = \sqrt[3]{(17 - 5x)} \Rightarrow x^3 + 5x - 17 = 0$ | M1, A1 | For letting $x_n = x_{n+1} = x$ (or $\alpha$); For correct equation stated
**(iii)** **EITHER:** Graphs of $y = x^3$ and $y = 17 - 5x$ only cross once Hence there is only one real root | M1, A1✓ | For argument based on sketching a pair of graphs, or a sketch of the cubic by calculator; For correct conclusion for a valid reason
**OR:** $\frac{d}{dx}(x^3 + 5x - 17) = 3x^2 + 5 > 0$ Hence there is only one real root | M1, A1✓ | For consideration of the cubic's gradient; For correct conclusion for a valid reason
**Total for Question 3: 7 marks**
---3 The sequence defined by the iterative formula
$$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 17 - 5 x _ { n } \right)$$
with $x _ { 1 } = 2$, converges to $\alpha$.\\
(i) Use the iterative formula to find $\alpha$ correct to 2 decimal places. You should show the result of each iteration.\\
(ii) Find a cubic equation of the form
$$x ^ { 3 } + c x + d = 0$$
which has $\alpha$ as a root.\\
(iii) Does this cubic equation have any other real roots? Justify your answer.
\hfill \mbox{\textit{OCR C3 Q3 [7]}}