| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Session | Specimen |
| Marks | 7 |
| 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 multi-part question on numerical methods requiring standard techniques: sketching y=ln(x) and y=4-x/2 to show intersection, sign-change verification by substitution, and applying a given iterative formula. All steps are routine A-level procedures with no novel problem-solving or proof required, making it slightly easier than average. |
| Spec | 1.06i Exponential growth/decay: in modelling context1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Make a recognisable sketch of \(y = \ln x\) | B1 | |
| Draw straight line with negative gradient crossing positive \(y\)-axis and justify one real root | B1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Consider sign of \(\ln x + \frac{1}{2}x - 4\) at \(4.5\) and \(5.0\) or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer \(4.84\) | A1 | |
| Show sufficient iterations to justify accuracy to 2 d.p. or show sign change in interval \((4.835, 4.845)\) | A1 | |
| Total: 3 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Make a recognisable sketch of $y = \ln x$ | B1 | |
| Draw straight line with negative gradient crossing positive $y$-axis and justify one real root | B1 | |
| **Total: 2** | | |
---
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Consider sign of $\ln x + \frac{1}{2}x - 4$ at $4.5$ and $5.0$ or equivalent | M1 | |
| Complete the argument correctly with appropriate calculations | A1 | |
| **Total: 2** | | |
---
## Question 4(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use the iterative formula correctly at least once | M1 | |
| Obtain final answer $4.84$ | A1 | |
| Show sufficient iterations to justify accuracy to 2 d.p. or show sign change in interval $(4.835, 4.845)$ | A1 | |
| **Total: 3** | | |
---4 (i) By sketching a suitable pair of graphs, show that the equation
$$\ln x = 4 - \frac { 1 } { 2 } x$$
has exactly one real root, $\alpha$.\\
(ii) Verify by calculation that $4.5 < \alpha < 5.0$.\\
(iii) Use the iterative formula $x _ { n + 1 } = 8 - 2 \ln x _ { n }$ to find $\alpha$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.\\
\hfill \mbox{\textit{CAIE P2 Q4 [7]}}