| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Sketch graphs to show root existence |
| Difficulty | Moderate -0.3 This is a straightforward application of standard techniques: sketching y=x³ and y=11-2x to show intersection (basic curve sketching), then applying a given iterative formula with no derivation required. The iteration converges easily and requires only careful arithmetic. Slightly easier than average due to the routine nature of both parts and explicit instructions. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02q 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 | Mark | Guidance |
| Draw sketch of \(y = x^3\) | \*B1 | May be implied by part graph in first quadrant |
| Draw straight line with negative gradient crossing positive \(y\)-axis and indicate one intersection | DB1 | dep \*B |
| Total: | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use iterative formula correctly at least once | M1 | |
| Obtain final answer 1.926 | A1 | |
| Show sufficient iterations to justify 4 sf or show sign change in interval \((1.9255, 1.9265)\) | A1 | |
| Total: | 3 |
## Question 3(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Draw sketch of $y = x^3$ | \*B1 | May be implied by part graph in first quadrant |
| Draw straight line with negative gradient crossing positive $y$-axis and indicate one intersection | DB1 | dep \*B |
| **Total:** | **2** | |
---
## Question 3(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iterative formula correctly at least once | M1 | |
| Obtain final answer 1.926 | A1 | |
| Show sufficient iterations to justify 4 sf or show sign change in interval $(1.9255, 1.9265)$ | A1 | |
| **Total:** | **3** | |
---3 (i) By sketching a suitable pair of graphs, show that the equation
$$x ^ { 3 } = 11 - 2 x$$
has exactly one real root.\\
(ii) Use the iterative formula
$$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 11 - 2 x _ { n } \right)$$
to find the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.\\
\hfill \mbox{\textit{CAIE P2 2017 Q3 [5]}}