| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | November |
| Marks | 5 |
| 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 sketching of y = e^(-x/2) and y = x^5 to show intersection, followed by routine iteration to convergence. The rearrangement is given, and the iteration converges quickly with no complications—slightly easier than average A-level pure maths questions. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Draw approximately correct sketch of \(y = e^{-\frac{1}{2}x}\) | B1 | With some curve in second quadrant as well as first |
| Draw approximately correct sketch of \(y = x^5\) and confirm one root | B1 | With some curve in third quadrant as well as first |
| Alternative method: Draw approximately correct sketch of \(y = 5\ln x\) or \(y = \ln x^5\) | B1 | |
| Draw approximately correct sketch of \(y = -\frac{x}{2}\) and confirm one root | B1 | Must have intersection in the 4th quadrant |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use iteration process correctly at least once | M1 | |
| Obtain final answer \(0.9128\) | A1 | Answer required to exactly 4 s.f. |
| Show sufficient iterations to 6 s.f. to justify answer or show sign change in the interval \([0.91275,\ 0.91285]\) | A1 |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Draw approximately correct sketch of $y = e^{-\frac{1}{2}x}$ | B1 | With some curve in second quadrant as well as first |
| Draw approximately correct sketch of $y = x^5$ and confirm one root | B1 | With some curve in third quadrant as well as first |
| **Alternative method:** Draw approximately correct sketch of $y = 5\ln x$ or $y = \ln x^5$ | B1 | |
| Draw approximately correct sketch of $y = -\frac{x}{2}$ and confirm one root | B1 | Must have intersection in the 4th quadrant |
---
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iteration process correctly at least once | M1 | |
| Obtain final answer $0.9128$ | A1 | Answer required to exactly 4 s.f. |
| Show sufficient iterations to 6 s.f. to justify answer or show sign change in the interval $[0.91275,\ 0.91285]$ | A1 | |
---4
\begin{enumerate}[label=(\alph*)]
\item By sketching a suitable pair of graphs on the same diagram, show that the equation
$$\mathrm { e } ^ { - \frac { 1 } { 2 } x } = x ^ { 5 }$$
has exactly one real root.
\item Use the iterative formula $x _ { n + 1 } = \sqrt [ 5 ] { \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } }$ to determine the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2022 Q4 [5]}}