| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | November |
| 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: (i) sign change identification using simple substitution, (ii) algebraic manipulation to show the rearrangement is equivalent to the original equation, and (iii) calculator iteration. All steps are routine textbook exercises 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 |
|---|---|
| (i) 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 |
| [2] | |
| (ii) 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 |
| [2] | |
| (iii) 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 |
| [3] |
**(i)** 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 |
| [2] |
**(ii)** 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 |
| [2] |
**(iii)** 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 |
| [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 2015 Q4 [7]}}