| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive stationary point equation |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring quotient rule differentiation, setting the derivative to zero and rearranging (which is guided by 'show that'), then applying fixed point iteration with a given formula. All steps are standard techniques with clear instructions, making it slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07q Product and quotient rules: differentiation1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use quotient rule (or equivalent) to find first derivative | M1 | |
| Obtain \(\dfrac{dy}{dx} = \dfrac{3\ln x - \dfrac{1}{x}(3x+2)}{(\ln x)^2}\) | A1 | OE |
| Equate first derivative to zero and confirm \(x = \dfrac{3x+2}{3\ln x}\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Consider \(x - \dfrac{3x+2}{3\ln x}\) or equivalent for values 3 and 4 | M1 | M0 if using \(\dfrac{dy}{dx}\) |
| Obtain \(-0.33...\) and \(0.63...\) or equivalents and justify conclusion | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use iteration process correctly at least once | M1 | |
| Obtain final answer \(3.3223\) | A1 | Answer required to exactly 5 s.f. |
| Show sufficient iterations to 7 s.f. to justify answer or show sign change in the interval \([3.32225,\ 3.32235]\) | A1 |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use quotient rule (or equivalent) to find first derivative | M1 | |
| Obtain $\dfrac{dy}{dx} = \dfrac{3\ln x - \dfrac{1}{x}(3x+2)}{(\ln x)^2}$ | A1 | OE |
| Equate first derivative to zero and confirm $x = \dfrac{3x+2}{3\ln x}$ | A1 | AG |
---
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Consider $x - \dfrac{3x+2}{3\ln x}$ or equivalent for values 3 and 4 | M1 | M0 if using $\dfrac{dy}{dx}$ |
| Obtain $-0.33...$ and $0.63...$ or equivalents and justify conclusion | A1 | AG |
---
## Question 5(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iteration process correctly at least once | M1 | |
| Obtain final answer $3.3223$ | A1 | Answer required to exactly 5 s.f. |
| Show sufficient iterations to 7 s.f. to justify answer or show sign change in the interval $[3.32225,\ 3.32235]$ | A1 | |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{2d6fc4c5-70ec-4cd8-9b48-59d5ce0e39b7-08_575_618_262_762}
The diagram shows the curve with equation $y = \frac { 3 x + 2 } { \ln x }$. The curve has a minimum point $M$.
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and show that the $x$-coordinate of $M$ satisfies the equation $x = \frac { 3 x + 2 } { 3 \ln x }$. [3]
\item Use the equation in part (a) to show by calculation that the $x$-coordinate of $M$ lies between 3 and 4.
\item Use an iterative formula, based on the equation in part (a), to find the $x$-coordinate of $M$ correct to 5 significant figures. Give the result of each iteration to 7 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q5 [8]}}