| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2013 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Moderate -0.5 This is a standard fixed-point iteration question requiring routine algebraic rearrangement, verification by substitution, and mechanical application of an iterative formula. The rearrangement from x⁴ + 2x - 9 = 0 to the given form is straightforward, and the iteration itself requires only calculator work with no conceptual challenges or novel problem-solving. |
| Spec | 1.09a 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 |
|---|---|---|
| (i) Consider sign of \(x^4 + 2x - 9\) at \(x = 1.5\) and \(x = 1.6\) | M1 | |
| Complete the argument correctly with appropriate calculations (\(f(1.5) = -0.9375, f(1.6) = 0.7536\)) | A1 | [2] |
| (ii) Rearrange \(x^4 + 2x - 9 = 0\) to given equation or vice versa | B1 | [1] |
| (iii) Use the iterative formula correctly at least once | M1 | |
| Obtain final answer 1.56 | A1 | |
| Show sufficient iterations to justify its accuracy to 2 d.p. | B1 | [3] |
| \(x_0 = 1.5\) | \(x_0 = 1.55\) | \(x_0 = 1.6\) |
| 1.5874 | 1.5614 | 1.5362 |
| 1.5424 | 1.5556 | 1.5685 |
| 1.5653 | 1.5520 | |
| 1.5536 | 1.5604 | |
| 1.5595 | 1.5561 | |
| 1.5565 | ||
| or show there is a sign change in the interval \((1.555, 1.565)\) | [3] |
**(i)** Consider sign of $x^4 + 2x - 9$ at $x = 1.5$ and $x = 1.6$ | M1 |
Complete the argument correctly with appropriate calculations ($f(1.5) = -0.9375, f(1.6) = 0.7536$) | A1 | [2]
**(ii)** Rearrange $x^4 + 2x - 9 = 0$ to given equation or vice versa | B1 | [1]
**(iii)** Use the iterative formula correctly at least once | M1 |
Obtain final answer 1.56 | A1 |
Show sufficient iterations to justify its accuracy to 2 d.p. | B1 | [3]
| $x_0 = 1.5$ | $x_0 = 1.55$ | $x_0 = 1.6$ |
|---|---|---|
| 1.5874 | 1.5614 | 1.5362 |
| 1.5424 | 1.5556 | 1.5685 |
| 1.5653 | | 1.5520 |
| 1.5536 | | 1.5604 |
| 1.5595 | | 1.5561 |
| 1.5565 | | |
or show there is a sign change in the interval $(1.555, 1.565)$ | [3]2\\
\includegraphics[max width=\textwidth, alt={}, center]{a3e778cb-9f95-4750-ba49-a57ee22af018-2_449_639_388_753}
The diagram shows the curve $y = x ^ { 4 } + 2 x - 9$. The curve cuts the positive $x$-axis at the point $P$.\\
(i) Verify by calculation that the $x$-coordinate of $P$ lies between 1.5 and 1.6.\\
(ii) Show that the $x$-coordinate of $P$ satisfies the equation
$$x = \sqrt [ 3 ] { \left( \frac { 9 } { x } - 2 \right) }$$
(iii) Use the iterative formula
$$x _ { n + 1 } = \sqrt [ 3 ] { \left( \frac { 9 } { x _ { n } } - 2 \right) }$$
to determine the $x$-coordinate of $P$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\hfill \mbox{\textit{CAIE P2 2013 Q2 [6]}}