| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Moderate -0.3 This is a multi-part question requiring trapezium rule application, exact integration of a logarithmic function, and deduction by subtraction. While it tests multiple skills, each component is routine for P2 level: trapezium rule is algorithmic, the integration is standard, and part (c) simply requires subtracting numerical values. The reasoning about over/under-estimate requires understanding concavity but is straightforward. Slightly easier than average due to the guided structure and standard techniques. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use \(y\)-values \(\sqrt[3]{7}\), \(\sqrt[3]{12}\) and \(\sqrt[3]{27}\), or decimal equivalents | B1 | \(1.913, 2.289, 3\) |
| Use correct formula, or equivalent, with \(h = 1\) | M1 | May see 2 separate trapezia (\(2.1\ldots + 2.65\ldots\)). |
| Obtain \(4.75\) | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate to obtain form \(k\ln(2x+5)\) | M1 | |
| Obtain correct \(\frac{27}{2}\ln(2x+5)\) | A1 | Condone inclusion of working for area in part (a). |
| Apply limits \(0\) and \(2\) to obtain \(\frac{27}{2}\ln\frac{9}{5}\) or exact equivalent of required form and no extra terms | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain \(3.19\) | B1 FT | Following answers to (b) and (a). If incorrect answers for (a) and/or (b), then need to see *their* (b) (with no extra terms) \(-\) *their* (a). |
| Total | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State under-estimate … | *B1 | |
| … because (a) is over-estimate due to tops of trapezia being above curve | DB1 | Or similarly detailed comment. |
| Total | 2 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use $y$-values $\sqrt[3]{7}$, $\sqrt[3]{12}$ and $\sqrt[3]{27}$, or decimal equivalents | B1 | $1.913, 2.289, 3$ |
| Use correct formula, or equivalent, with $h = 1$ | M1 | May see 2 separate trapezia ($2.1\ldots + 2.65\ldots$). |
| Obtain $4.75$ | A1 | |
| **Total** | **3** | |
---
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate to obtain form $k\ln(2x+5)$ | M1 | |
| Obtain correct $\frac{27}{2}\ln(2x+5)$ | A1 | Condone inclusion of working for area in part (a). |
| Apply limits $0$ and $2$ to obtain $\frac{27}{2}\ln\frac{9}{5}$ or exact equivalent of required form and no extra terms | A1 | |
| **Total** | **3** | |
---
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $3.19$ | B1 FT | Following answers to (b) and (a). If incorrect answers for (a) and/or (b), then need to see *their* (b) (with no extra terms) $-$ *their* (a). |
| **Total** | **1** | |
---
## Question 6(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| State under-estimate … | *B1 | |
| … because (a) is over-estimate due to tops of trapezia being above curve | DB1 | Or similarly detailed comment. |
| **Total** | **2** | |
---\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule with two intervals to find an approximation to the area of region $A$. Give your answer correct to 3 significant figures.\\
\includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-10_2720_38_105_2010}\\
\includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-11_2716_29_107_22}
\item Find the exact total area of regions $A$ and $B$. Give your answer in the form $k \ln m$, where $k$ and $m$ are constants.
\item Deduce an approximation to the area of region $B$. Give your answer correct to 3 significant figures.
\item State, with a reason, whether your answer to part (c) is an over-estimate or an under-estimate of the area of region $B$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q6 [9]}}