| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | March |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Standard +0.3 This is a straightforward fixed-point iteration question requiring routine algebraic rearrangement, sign-change verification, and iterative calculation. While it involves multiple steps, each component is standard A-level technique with no novel insight required—slightly easier than average due to the guided structure and computational nature. |
| 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) Consider sign of \(x^2 - 3x^3 + x^2 - 4\) at \(x = 1\) and \(x = 2\), or equivalent | M1 | |
| Complete the argument correctly with correct calculated values | A1 | [2] |
| (ii) Rearrange the given quintic equation in the given form, or work vice versa | B1 | [1] |
| (iii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 1.78 | A1 | |
| Show sufficient iterations to 4 d.p. to justify 1.78 to 2 d.p., or show there is a sign change in the interval (1.775, 1.785) | A1 | [3] |
(i) Consider sign of $x^2 - 3x^3 + x^2 - 4$ at $x = 1$ and $x = 2$, or equivalent | M1 |
Complete the argument correctly with correct calculated values | A1 | [2]
(ii) Rearrange the given quintic equation in the given form, or work vice versa | B1 | [1]
(iii) Use the iterative formula correctly at least once | M1 |
Obtain final answer 1.78 | A1 |
Show sufficient iterations to 4 d.p. to justify 1.78 to 2 d.p., or show there is a sign change in the interval (1.775, 1.785) | A1 | [3]3 The equation $x ^ { 5 } - 3 x ^ { 3 } + x ^ { 2 } - 4 = 0$ has one positive root.\\
(i) Verify by calculation that this root lies between 1 and 2 .\\
(ii) Show that the equation can be rearranged in the form
$$\left. x = \sqrt [ 3 ] { ( } 3 x + \frac { 4 } { x ^ { 2 } } - 1 \right)$$
(iii) Use an iterative formula based on this rearrangement to determine the positive root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P3 2016 Q3 [6]}}