| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from integral condition |
| Difficulty | Standard +0.3 Part (a) requires straightforward integration of an exponential function and algebraic rearrangement to reach the given form—standard P2 technique. Part (b) applies a given iterative formula with clear instructions, requiring only careful calculator work. This is a routine multi-part question slightly easier than average due to its procedural nature and explicit guidance. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.08d Evaluate definite integrals: between limits1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrate to obtain the form \(k_1e^{2x} + k_2x\) | *M1 | Where \(k_1k_2 \neq 0\) |
| Obtain correct \(\frac{3}{2}e^{2x} - x\) | A1 | |
| Use limits correctly and attempt rearrangement at least as far as \(e^{2a} = \ldots\) | DM1 | Must be equated to 12 and simplified using a correct method. Do not condone verification. |
| Confirm given result \(a = \frac{1}{2}\ln(9 + \frac{2}{3}a)\) with sufficient detail | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use iteration process correctly at least once | M1 | Need to see 1.13434 and 1.13895 |
| Obtain final answer 1.139 | A1 | Final answer needed to exactly 4 sf |
| Show sufficient iterations to 6 sf to justify answer or show a sign change in interval [1.1385, 1.1395] | A1 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate to obtain the form $k_1e^{2x} + k_2x$ | *M1 | Where $k_1k_2 \neq 0$ |
| Obtain correct $\frac{3}{2}e^{2x} - x$ | A1 | |
| Use limits correctly and attempt rearrangement at least as far as $e^{2a} = \ldots$ | DM1 | Must be equated to 12 and simplified using a correct method. Do not condone verification. |
| Confirm given result $a = \frac{1}{2}\ln(9 + \frac{2}{3}a)$ with sufficient detail | A1 | AG |
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use iteration process correctly at least once | M1 | Need to see 1.13434 and 1.13895 |
| Obtain final answer 1.139 | A1 | Final answer needed to exactly 4 sf |
| Show sufficient iterations to 6 sf to justify answer or show a sign change in interval [1.1385, 1.1395] | A1 | |3 It is given that $\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { 2 x } - 1 \right) \mathrm { d } x = 12$, where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $a = \frac { 1 } { 2 } \ln \left( 9 + \frac { 2 } { 3 } a \right)$.
\item Use an iterative formula, based on the equation in (a), to find the value of $a$ correct to 4 significant figures. Use an initial value of 1 and give the result of each iteration to 6 significant figures. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2023 Q3 [7]}}