| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find constant from definite integral |
| Difficulty | Standard +0.3 This is a straightforward multi-step problem requiring standard integration of 1/(2x+1), evaluation of definite integral, algebraic rearrangement to the given form, and then iterative solution. All techniques are routine for P2 level with no novel insight required, 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 | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain integral of form \(k\ln(2x+1)\) | *M1 | |
| Obtain correct \(5\ln(2x+1)\) | A1 | |
| Apply limits correctly and equate to 7 | DM1 | |
| Apply appropriate logarithm property to reach at least \(a^3 = \ldots\) | DM1 | |
| Confirm \(a = \sqrt[3]{0.5e^{1.4}(2a+1)} - 0.5\) | A1 | AG – necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| 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 |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain integral of form $k\ln(2x+1)$ | *M1 | |
| Obtain correct $5\ln(2x+1)$ | A1 | |
| Apply limits correctly and equate to 7 | DM1 | |
| Apply appropriate logarithm property to reach at least $a^3 = \ldots$ | DM1 | |
| Confirm $a = \sqrt[3]{0.5e^{1.4}(2a+1)} - 0.5$ | A1 | AG – necessary detail needed |
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## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| 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 | |
---5 It is given that $\int _ { a } ^ { a ^ { 3 } } \frac { 10 } { 2 x + 1 } \mathrm {~d} x = 7$, where $a$ is a constant greater than 1 .
\begin{enumerate}[label=(\alph*)]
\item Show that $a = \sqrt [ 3 ] { 0.5 \mathrm { e } ^ { 1.4 } ( 2 a + 1 ) - 0.5 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-08_2718_35_107_2011}\\
\includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-09_2725_35_99_20}
\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.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q5 [8]}}