| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | November |
| 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+2) and y = 4e^(-x) to show intersection, substituting values to locate the root, and applying a given iterative formula. All steps are routine with no novel insight required, making it slightly easier than average. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.06g Equations with exponentials: solve a^x = b1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sketch a relevant graph, e.g. \(y = \ln(x+2)\) | B1 | |
| Sketch a second relevant graph, e.g. \(y = 4e^{-x}\), and justify the given statement | B1 | Consideration of behaviour for \(x < 0\) is needed for the second B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Calculate the values of a relevant expression or pair of expressions at \(x=1\) and \(x=1.5\) | M1 | |
| Complete the argument correctly with correct calculated values | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the iterative formula correctly at least twice using output from a previous iteration | M1 | |
| Obtain final answer 1.23 | A1 | |
| Show sufficient iterations to 4 d.p. to justify 1.23 to 2 d.p., or show there is a sign change in the interval \((1.225, 1.235)\) | A1 |
## Question 5(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch a relevant graph, e.g. $y = \ln(x+2)$ | B1 | |
| Sketch a second relevant graph, e.g. $y = 4e^{-x}$, and justify the given statement | B1 | Consideration of behaviour for $x < 0$ is needed for the second B1 |
## Question 5(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Calculate the values of a relevant expression or pair of expressions at $x=1$ and $x=1.5$ | M1 | |
| Complete the argument correctly with correct calculated values | A1 | |
## Question 5(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the iterative formula correctly at least **twice** using output from a previous iteration | M1 | |
| Obtain final answer 1.23 | A1 | |
| Show sufficient iterations to 4 d.p. to justify 1.23 to 2 d.p., or show there is a sign change in the interval $(1.225, 1.235)$ | A1 | |5 (i) By sketching a suitable pair of graphs, show that the equation $\ln ( x + 2 ) = 4 \mathrm { e } ^ { - x }$ has exactly one real root.\\
(ii) Show by calculation that this root lies between $x = 1$ and $x = 1.5$.\\
(iii) Use the iterative formula $x _ { n + 1 } = \ln \left( \frac { 4 } { \ln \left( x _ { n } + 2 \right) } \right)$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.\\
\hfill \mbox{\textit{CAIE P3 2019 Q5 [7]}}