| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2014 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Moderate -0.3 This is a straightforward fixed point iteration question requiring polynomial division to find a factor, algebraic rearrangement to obtain the iterative form (which is essentially given), and then mechanical application of the iteration formula. The steps are routine and clearly signposted, making it slightly easier than average for A-level, though the polynomial division and cube root iterations require care. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Identify \(x - 3\) as divisor | B1 | |
| Divide by linear expression at least as far as \(x\) term | M1 | |
| Obtain quotient \(x^2 + 3x - 16\) | A1 | |
| Obtain zero remainder with no errors in the division | A1 | |
| Equate quotient to zero and confirm \(x = \sqrt[4]{16 - 3x}\) (AG) | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Use iteration process correctly at least once | M1 | |
| Obtain final answer 2.13 | A1 | |
| Show sufficient iterations to 4 decimal places or show a sign change in the interval (2.125, 2.135) | A1 | [3] |
**(i)**
Identify $x - 3$ as divisor | B1 |
Divide by linear expression at least as far as $x$ term | M1 |
Obtain quotient $x^2 + 3x - 16$ | A1 |
Obtain zero remainder with no errors in the division | A1 |
Equate quotient to zero and confirm $x = \sqrt[4]{16 - 3x}$ (AG) | A1 | [5]
**(ii)**
Use iteration process correctly at least once | M1 |
Obtain final answer 2.13 | A1 |
Show sufficient iterations to 4 decimal places or show a sign change in the interval (2.125, 2.135) | A1 | [3]6\\
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The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = x ^ { 4 } - 3 x ^ { 3 } + 3 x ^ { 2 } - 25 x + 48 .$$
The diagram shows the curve $y = \mathrm { p } ( x )$ which crosses the $x$-axis at ( $\alpha , 0$ ) and ( 3,0 ).\\
(i) Divide $\mathrm { p } ( x )$ by a suitable linear factor and hence show that $\alpha$ is a root of the equation $x = \sqrt [ 3 ] { } ( 16 - 3 x )$.\\
(ii) Use the iterative formula $x _ { n + 1 } = \sqrt [ 3 ] { } \left( 16 - 3 x _ { n } \right)$ to find $\alpha$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2014 Q6 [8]}}