| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2003 |
| Session | June |
| Marks | 8 |
| 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 requiring standard techniques: sketching y=ln(x) and y=2-x², substituting boundary values to verify sign change, and applying a given iterative formula. All steps are routine with no novel problem-solving required, making it slightly easier than average for A-level. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.06d Natural logarithm: ln(x) function and properties1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Make recognizable sketch over the given range of two suitable graphs, e.g. \(y = \ln x\) and \(y = 2 - x^2\) | B1+B1 | |
| State or imply link between intersections and roots and justify given answer | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Consider sign of \(\ln x - (2 - x^2)\) at \(x = 1\) and \(x = 1.4\), or equivalent | M1 | |
| Complete the argument correctly with appropriate calculation | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use the given iterative formula correctly with \(1 \leq x_n \leq 1.4\) | M1 | |
| Obtain final answer \(1.31\) | A1 | |
| Show sufficient iterations to justify its accuracy to 2d.p., or show there is a sign change in the interval \((1.305, 1.315)\) | A1 |
# Question 5(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Make recognizable sketch over the given range of two suitable graphs, e.g. $y = \ln x$ and $y = 2 - x^2$ | B1+B1 | |
| State or imply link between intersections and roots and justify given answer | B1 | |
**Total: [3]**
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# Question 5(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Consider sign of $\ln x - (2 - x^2)$ at $x = 1$ and $x = 1.4$, or equivalent | M1 | |
| Complete the argument correctly with appropriate calculation | A1 | |
**Total: [2]**
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# Question 5(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use the given iterative formula correctly with $1 \leq x_n \leq 1.4$ | M1 | |
| Obtain final answer $1.31$ | A1 | |
| Show sufficient iterations to justify its accuracy to 2d.p., or show there is a sign change in the interval $(1.305, 1.315)$ | A1 | |
**Total: [3]**
---5 (i) By sketching a suitable pair of graphs, show that the equation
$$\ln x = 2 - x ^ { 2 }$$
has exactly one root.\\
(ii) Verify by calculation that the root lies between 1.0 and 1.4 .\\
(iii) Use the iterative formula
$$x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)$$
to determine the root correct to 2 decimal places, showing the result of each iteration.
\hfill \mbox{\textit{CAIE P2 2003 Q5 [8]}}