| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | November |
| 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 application of differentiation to find stationary points, followed by algebraic manipulation to rearrange into the given form. The iteration in part (b) is mechanical. While it involves exponentials and logarithms, the steps are routine for A-level: differentiate, set to zero, rearrange using log laws, then iterate. Slightly easier than average due to the 'show that' structure guiding students through the algebra. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07n Stationary points: find maxima, minima using derivatives1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt use of product rule to differentiate \(xe^{2x}\) | M1 | |
| Obtain \(e^{2x} + 2xe^{2x} - 5e^{-x}\) | A1 | |
| Equate first derivative to zero and multiply by \(e^x\) to obtain an equation involving \(e^{3x}\) | M1 | |
| Obtain \(e^{3x}(1+2x) = 5\) or equivalent | A1 | |
| Confirm given result \(x = \frac{1}{3}\ln 5 - \frac{1}{3}\ln(1+2x)\) with sufficient detail | A1 | AG |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use iteration process correctly at least once | M1 | Need 0.35 and 2 correct values |
| Obtain final answer \(0.357\) | A1 | Answer required to exactly 3sf. Allow recovery |
| Show sufficient iterations to 5sf to justify answer or show sign change in interval \([0.3565,\ 0.3575]\) | A1 | |
| 3 |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt use of product rule to differentiate $xe^{2x}$ | M1 | |
| Obtain $e^{2x} + 2xe^{2x} - 5e^{-x}$ | A1 | |
| Equate first derivative to zero and multiply by $e^x$ to obtain an equation involving $e^{3x}$ | M1 | |
| Obtain $e^{3x}(1+2x) = 5$ or equivalent | A1 | |
| Confirm given result $x = \frac{1}{3}\ln 5 - \frac{1}{3}\ln(1+2x)$ with sufficient detail | A1 | AG |
| | **5** | |
---
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use iteration process correctly at least once | M1 | Need 0.35 and 2 correct values |
| Obtain final answer $0.357$ | A1 | Answer required to exactly 3sf. Allow recovery |
| Show sufficient iterations to 5sf to justify answer or show sign change in interval $[0.3565,\ 0.3575]$ | A1 | |
| | **3** | |
---4 The curve with equation $y = x \mathrm { e } ^ { 2 x } + 5 \mathrm { e } ^ { - x }$ has a minimum point $M$.
\begin{enumerate}[label=(\alph*)]
\item Show that the $x$-coordinate of $M$ satisfies the equation $x = \frac { 1 } { 3 } \ln 5 - \frac { 1 } { 3 } \ln ( 1 + 2 x )$.
\item Use an iterative formula, based on the equation in part (a), to find the $x$-coordinate of $M$ correct to 3 significant figures. Use an initial value of 0.35 and give the result of each iteration to 5 significant figures.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q4 [8]}}