| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Moderate -0.3 This is a standard fixed-point iteration question requiring algebraic manipulation of logarithms (routine), verification of a root interval (straightforward substitution), and applying an iterative formula (mechanical process). While it involves multiple parts, each step uses well-practiced A-level techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use the power law correctly | *M1 | |
| Use correct process to obtain equation with no logarithms | DM1 | |
| Confirm \(x = \sqrt{\frac{9}{x+2}}\) | A1 | AG; condone absence of justification for choice of positive root |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Consider sign of \(x - \sqrt{\frac{9}{x+2}}\) or equivalent for 1.5 and 2 | M1 | |
| Obtain \(-0.1\ldots\) and \(0.5\) or equivalents and justify conclusion | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use iteration process correctly at least once | M1 | |
| Obtain final answer 1.58 | A1 | Final answer required to exactly 3 sf |
| Show sufficient iterations to 5 sf to justify answer or show a sign change in interval \([1.575, 1.585]\) | A1 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use the power law correctly | *M1 | |
| Use correct process to obtain equation with no logarithms | DM1 | |
| Confirm $x = \sqrt{\frac{9}{x+2}}$ | A1 | AG; condone absence of justification for choice of positive root |
---
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Consider sign of $x - \sqrt{\frac{9}{x+2}}$ or equivalent for 1.5 and 2 | M1 | |
| Obtain $-0.1\ldots$ and $0.5$ or equivalents and justify conclusion | A1 | AG |
---
## Question 5(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use iteration process correctly at least once | M1 | |
| Obtain final answer 1.58 | A1 | Final answer required to exactly 3 sf |
| Show sufficient iterations to 5 sf to justify answer or show a sign change in interval $[1.575, 1.585]$ | A1 | |
---5
\begin{enumerate}[label=(\alph*)]
\item Given that $2 \ln ( x + 1 ) + \ln x = \ln ( x + 9 )$, show that $x = \sqrt { \frac { 9 } { x + 2 } }$.
\item It is given that the equation $x = \sqrt { \frac { 9 } { x + 2 } }$ has a single root.
Show by calculation that this root lies between 1.5 and 2.0.
\item Use an iterative formula, based on the equation in part (b), to find the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q5 [8]}}