| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from integral condition |
| Difficulty | Standard +0.3 Part (a) requires standard integration of logarithmic forms and algebraic manipulation to reach the given result—routine A-level techniques. Part (b) is straightforward application of fixed-point iteration with a given formula and initial value. The question tests competent execution of standard methods rather than problem-solving insight, making it slightly easier than average. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks |
|---|---|
| 5 | Where a candidate has misread a number in the question and used that value consistently throughout, provided that number does not alter the difficulty or |
| Answer | Marks | Guidance |
|---|---|---|
| 5(a) | Integrate to obtain form k ln(1+2x)+k lnx | |
| 1 2 | *M1 | k 0, k 0. |
| Answer | Marks |
|---|---|
| Obtain correct 2ln(1+2x)+3lnx | A1 |
| Use limits correctly and equate to ln10 | DM1 |
| Answer | Marks | Guidance |
|---|---|---|
| a3 =... | DM1 | |
| Confirm given result a=390(1+2a)−2 with sufficient detail | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| 5(b) | Use iteration process correctly at least once | M1 |
| Obtain final answer 1.68 | A1 | Answer required to exactly 3 sf. |
| Answer | Marks |
|---|---|
| change in the interval [1.675, 1.685] | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 5:
5 | Where a candidate has misread a number in the question and used that value consistently throughout, provided that number does not alter the difficulty or
the method required, award all marks earned and deduct just 1 mark for the misread.
--- 5(a) ---
5(a) | Integrate to obtain form k ln(1+2x)+k lnx
1 2 | *M1 | k 0, k 0.
1 2
Obtain correct 2ln(1+2x)+3lnx | A1
Use limits correctly and equate to ln10 | DM1
Apply relevant logarithm properties correctly and arrange as far as
a3 =... | DM1
Confirm given result a=390(1+2a)−2 with sufficient detail | A1 | AG
5
--- 5(b) ---
5(b) | Use iteration process correctly at least once | M1 | Need to see 1.6848 .
Obtain final answer 1.68 | A1 | Answer required to exactly 3 sf.
Show sufficient iterations to 5 sf to justify answer or show a sign
change in the interval [1.675, 1.685] | A1
3
Question | Answer | Marks | GuidanceIt is given that $\int_1^a \left(\frac{4}{1 + 2x} + \frac{3}{x}\right) dx = \ln 10$, where $a$ is a constant greater than 1.
\begin{enumerate}[label=(\alph*)]
\item Show that $a = \sqrt{90(1 + 2a)^{-2}}$. [5]
\item Use an iterative formula, based on the equation in (a), to find the value of $a$ correct to 3 significant figures. Use an initial value of 1.7 and give the result of each iteration to 5 significant figures. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2023 Q5 [8]}}