| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show root in interval |
| Difficulty | Moderate -0.8 This is a straightforward application of standard numerical methods. Part (i) requires simple substitution to find sign changes, and part (ii) is mechanical iteration with a given formula until convergence. No conceptual insight or problem-solving is needed—just careful arithmetic and following a routine procedure. |
| Spec | 1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Evaluate, or consider the sign of, \(x^3 - 8x - 13\) for two integer values of \(x\), or equivalent | M1 | |
| Conclude \(x = 3\) and \(x = 4\) with no errors seen | A1 | [2] |
| (ii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 3.43 | A1 | |
| Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval (3.425, 3.435) | A1 | [3] |
**(i)** Evaluate, or consider the sign of, $x^3 - 8x - 13$ for two integer values of $x$, or equivalent | M1 |
Conclude $x = 3$ and $x = 4$ with no errors seen | A1 | [2]
**(ii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 3.43 | A1 |
Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in the interval (3.425, 3.435) | A1 | [3]2 The equation $x ^ { 3 } - 8 x - 13 = 0$ has one real root.\\
(i) Find the two consecutive integers between which this root lies.\\
(ii) Use the iterative formula
$$x _ { n + 1 } = \left( 8 x _ { n } + 13 \right) ^ { \frac { 1 } { 3 } }$$
to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2009 Q2 [5]}}