| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | November |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Derive equation from integral condition |
| Difficulty | Standard +0.3 This question involves standard integration of 1/(2x+1), applying limits, and algebraic manipulation to reach the given form—all routine C3/P2 techniques. The iterative formula is then directly applied with no conceptual difficulty. While multi-step, each component is straightforward textbook material requiring no novel insight. |
| 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 | Guidance |
|---|---|---|
| 5(a) | Obtain integral of form kln(2x+1) | *M1 |
| Obtain correct5ln(2x+1) | A1 | |
| Apply limits correctly and equate to 7 | DM1 | |
| Apply appropriate logarithm property to reach at least a3 =... | DM1 | |
| Confirm a=30.5e1.4(2a+1)−0.5 | A1 | AG – necessary detail needed. |
| Answer | Marks | Guidance |
|---|---|---|
| 5(b) | Use iterative process correctly at least once | M1 |
| Obtain final answer 2.18 | A1 | Answer required to exactly 3 sf. |
| Answer | Marks |
|---|---|
| interval [2.175, 2.185] | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 5:
--- 5(a) ---
5(a) | Obtain integral of form kln(2x+1) | *M1
Obtain correct5ln(2x+1) | A1
Apply limits correctly and equate to 7 | DM1
Apply appropriate logarithm property to reach at least a3 =... | DM1
Confirm a=30.5e1.4(2a+1)−0.5 | A1 | AG – necessary detail needed.
5
--- 5(b) ---
5(b) | Use iterative process correctly at least once | M1
Obtain final answer 2.18 | A1 | Answer required to exactly 3 sf.
Show sufficient iterations to 5 sf to justify answer or show a sign change in the
interval [2.175, 2.185] | A1
3
Question | Answer | Marks | GuidanceIt is given that $\int_a^{a^2} \frac{10}{2x+1} dx = 7$, where $a$ is a constant greater than $1$.
\begin{enumerate}[label=(\alph*)]
\item Show that $a = \sqrt[9]{0.5e^{1.4}(2a+1) - 0.5}$. [5]
\item Use an iterative formula, based on the equation in part (a), to find the value of $a$ correct to $3$ significant figures. Use an initial value of $2$ and give the result of each iteration to $5$ significant figures. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q5 [17]}}