| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show convergence to specific root |
| Difficulty | Standard +0.3 This is a standard fixed-point iteration question requiring sign-change identification, algebraic manipulation to show convergence to the root, and numerical iteration. All steps are routine A-level techniques with no novel insight required, making it slightly easier than average. |
| 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 |
|---|---|---|
| Answer | Marks | Guidance |
| Evaluate, or consider the sign of, \(x^3 - x^2 - 6\) for two integer values of \(x\), or equivalent | M1 | |
| Obtain the pair \(x = 2\) and \(x = 3\), with no errors seen | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State a suitable equation, e.g. \(x = \sqrt{(x + (6/x))}\) | B1 | |
| Rearrange this as \(x^3 - x^2 - 6 = 0\), or work *vice versa* | B1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(2.219\) | A1 | |
| Show sufficient iterates to 5 d.p. to justify \(2.219\) to 3 d.p., or show there is a sign change in the interval \((2.2185, 2.2195)\) | A1 | |
| Total: 3 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Evaluate, or consider the sign of, $x^3 - x^2 - 6$ for two integer values of $x$, or equivalent | M1 | |
| Obtain the pair $x = 2$ and $x = 3$, with no errors seen | A1 | |
| **Total: 2** | | |
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State a suitable equation, e.g. $x = \sqrt{(x + (6/x))}$ | B1 | |
| Rearrange this as $x^3 - x^2 - 6 = 0$, or work *vice versa* | B1 | |
| **Total: 2** | | |
## Question 4(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer $2.219$ | A1 | |
| Show sufficient iterates to 5 d.p. to justify $2.219$ to 3 d.p., or show there is a sign change in the interval $(2.2185, 2.2195)$ | A1 | |
| **Total: 3** | | |4 The equation $x ^ { 3 } - x ^ { 2 } - 6 = 0$ has one real root, denoted by $\alpha$.\\
(i) Find by calculation the pair of consecutive integers between which $\alpha$ lies.\\
(ii) Show that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \sqrt { } \left( x _ { n } + \frac { 6 } { x _ { n } } \right)$$
converges, then it converges to $\alpha$.\\
(iii) Use this iterative formula to determine $\alpha$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.\\
\hfill \mbox{\textit{CAIE P3 Q4 [7]}}