| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Sketch graphs to show root existence |
| Difficulty | Standard +0.3 This is a straightforward fixed-point iteration question requiring standard techniques: sketching graphs to show roots exist, applying a given iterative formula (no rearrangement needed), and using symmetry to find the second root. The modulus function adds minimal complexity since the graphs are simple to sketch, and all steps are routine A-level procedures with no novel problem-solving required. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02q Use intersection points: of graphs to solve equations1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Draw recognisable sketch of \(y = 16 - x^3\); Draw recognisable sketch of \(y = \ | 3x\ | \); Indicate in some way the two points of intersection |
| Use iterative process correctly at least once; Obtain final answer 1.804; Show sufficient iterations to justify answer or show sign change in the interval \((1.8035, 1.8045)\) | M1, A1, A1 | [3] |
| State \((1.804, 5.412)\); State \((-1.804, 5.412)\), following their first point | B1, B1✓ | [2] |
| Answer | Marks | Guidance |
|--------|-------|----------|
| Draw recognisable sketch of $y = 16 - x^3$; Draw recognisable sketch of $y = \|3x\|$; Indicate in some way the two points of intersection | B1, B1, B1 depBB | [3] |
| Use iterative process correctly at least once; Obtain final answer 1.804; Show sufficient iterations to justify answer or show sign change in the interval $(1.8035, 1.8045)$ | M1, A1, A1 | [3] |
| State $(1.804, 5.412)$; State $(-1.804, 5.412)$, following their first point | B1, B1✓ | [2] |
---\begin{enumerate}[label=(\roman*)]
\item By sketching a suitable pair of graphs, show that the equation
$$|3x| = 16 - x^4$$
has two real roots. [3]
\item Use the iterative formula $x_{n+1} = \sqrt[4]{16 - 3x_n}$ to find one of the real roots correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]
\item Hence find the coordinates of each of the points of intersection of the graphs $y = |3x|$ and $y = 16 - x^4$, giving your answers correct to 3 decimal places. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2015 Q5 [12]}}