| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Show convergence to specific root |
| Difficulty | Standard +0.3 This is a standard A-level fixed point iteration question with routine components: sketching graphs to show existence of a root, verifying bounds by substitution, algebraic rearrangement to show convergence to the correct root, and applying iteration. All steps follow predictable patterns with no novel insight required, making it slightly easier than average. |
| Spec | 1.09a 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 |
| Sketch a relevant graph, e.g. \(y = e^x - 3\); should cut vertical axis at \((0,-2)\) and have increasing gradient | B1 | |
| Sketch a second relevant graph, e.g. \(y = \sqrt{x}\), and justify the given statement; \(y=\sqrt{x}\) should start at \((0,0)\) and have reducing grading | B1 | Ignore anything outside 1st and 4th quadrants. For second B1 need to mark intersection with a dot, a cross, or say root at point of intersection, or equivalent. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Calculate the values of a relevant expression or pair of expressions at \(x=1\) and \(x=2\) | M1 | |
| Complete the argument correctly with correct calculated values | A1 | e.g. \(1 > -0.28...\), \(1.41 < 4.39...\); \(1.28 > 0\), \(-2.98 < 0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(x = \ln(3+\sqrt{x})\) and rearrange to the given equation \(\sqrt{x} = e^x - 3\) | B1 | Or rearrange \(\sqrt{x} = e^x - 3\) to \(x = \ln(3+\sqrt{x})\) and state iterative formula \(x_{n+1} = \ln(3+\sqrt{x_n})\). AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use the iterative process correctly at least once | M1 | |
| Obtain final answer \(1.43\) | A1 | |
| Show sufficient iterations to at least 4 d.p. to justify \(1.43\) to 2 d.p. or show there is a sign change in the interval \((1.425, 1.435)\); condone recovery and small differences in the final figure in the iteration | A1 | e.g. \(1,\ 1.3864,\ 1.4297,\ 1.4341,\ldots\); \(1.5,\ 1.4210,\ 1.4332,\ 1.4344,\ 1.4345,\ldots\); \(2,\ 1.4848,\ 1.4395,\ 1.4350,\ 1.4346,\ 1.4345,\ldots\) |
## Question 8(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch a relevant graph, e.g. $y = e^x - 3$; should cut vertical axis at $(0,-2)$ and have increasing gradient | **B1** | |
| Sketch a second relevant graph, e.g. $y = \sqrt{x}$, and justify the given statement; $y=\sqrt{x}$ should start at $(0,0)$ and have reducing grading | **B1** | Ignore anything outside 1st and 4th quadrants. For second B1 need to mark intersection with a dot, a cross, or say root at point of intersection, or equivalent. |
---
## Question 8(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Calculate the values of a relevant expression or pair of expressions at $x=1$ and $x=2$ | **M1** | |
| Complete the argument correctly with correct calculated values | **A1** | e.g. $1 > -0.28...$, $1.41 < 4.39...$; $1.28 > 0$, $-2.98 < 0$ |
---
## Question 8(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $x = \ln(3+\sqrt{x})$ and rearrange to the given equation $\sqrt{x} = e^x - 3$ | **B1** | Or rearrange $\sqrt{x} = e^x - 3$ to $x = \ln(3+\sqrt{x})$ and state iterative formula $x_{n+1} = \ln(3+\sqrt{x_n})$. AG |
---
## Question 8(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use the iterative process correctly at least once | **M1** | |
| Obtain final answer $1.43$ | **A1** | |
| Show sufficient iterations to at least 4 d.p. to justify $1.43$ to 2 d.p. or show there is a sign change in the interval $(1.425, 1.435)$; condone recovery and small differences in the final figure in the iteration | **A1** | e.g. $1,\ 1.3864,\ 1.4297,\ 1.4341,\ldots$; $1.5,\ 1.4210,\ 1.4332,\ 1.4344,\ 1.4345,\ldots$; $2,\ 1.4848,\ 1.4395,\ 1.4350,\ 1.4346,\ 1.4345,\ldots$ |
---8
\begin{enumerate}[label=(\alph*)]
\item By sketching a suitable pair of graphs, show that the equation
$$\sqrt { x } = \mathrm { e } ^ { x } - 3$$
has only one root.
\item Show by calculation that this root lies between 1 and 2 .
\item Show that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \ln \left( 3 + \sqrt { x _ { n } } \right)$$
converges, then it converges to the root of the equation in (a).
\item Use the iterative formula to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q8 [8]}}