| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with numerical methods |
| Difficulty | Standard +0.8 This question combines numerical methods (trapezium rule) with volumes of revolution requiring integration. Part (i) is routine application of trapezium rule. Part (ii) requires recognizing that y² simplifies nicely when rotated (eliminating the square root), then integrating 1 + 3cos²(x/2) using the double angle formula—a multi-step problem requiring algebraic insight and integration technique beyond standard exercises. |
| Spec | 1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(y\) values \(2, \sqrt{2.5}, 1\) or equivalents | B1 | |
| Use correct formula, or equivalent, with \(h = \frac{1}{2}\pi\) and three \(y\) values | M1 | |
| Obtain \(\frac{1}{2} \times \frac{1}{2}\pi(2 + 2\sqrt{2.5} + 1)\) or equivalent and hence 4.84 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply volume is \(\int \pi(1 + 3\cos^2\frac{1}{2}x)\,dx\) | B1 | Allow if \(\pi\) appears later; condone omission of \(dx\) |
| Use appropriate identity to express integrand in form \(k_1 + k_2\cos x\) | M1 | |
| Obtain \(\int \pi(\frac{5}{2} + \frac{3}{2}\cos x)\,dx\) or \(\int(\frac{5}{2} + \frac{3}{2}\cos x)\,dx\) | A1 | Condone omission of \(dx\) |
| Integrate to obtain \(\pi(\frac{5}{2}x + \frac{3}{2}\sin x)\) or \(\frac{5}{2}x + \frac{3}{2}\sin x\) | A1 | |
| Obtain \(\frac{5}{2}\pi^2\) with no errors seen | A1 |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $y$ values $2, \sqrt{2.5}, 1$ or equivalents | B1 | |
| Use correct formula, or equivalent, with $h = \frac{1}{2}\pi$ and three $y$ values | M1 | |
| Obtain $\frac{1}{2} \times \frac{1}{2}\pi(2 + 2\sqrt{2.5} + 1)$ or equivalent and hence 4.84 | A1 | |
---
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply volume is $\int \pi(1 + 3\cos^2\frac{1}{2}x)\,dx$ | B1 | Allow if $\pi$ appears later; condone omission of $dx$ |
| Use appropriate identity to express integrand in form $k_1 + k_2\cos x$ | M1 | |
| Obtain $\int \pi(\frac{5}{2} + \frac{3}{2}\cos x)\,dx$ or $\int(\frac{5}{2} + \frac{3}{2}\cos x)\,dx$ | A1 | Condone omission of $dx$ |
| Integrate to obtain $\pi(\frac{5}{2}x + \frac{3}{2}\sin x)$ or $\frac{5}{2}x + \frac{3}{2}\sin x$ | A1 | |
| Obtain $\frac{5}{2}\pi^2$ with no errors seen | A1 | |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{6bf7ba66-8362-4ac0-8e5c-3f88a3ccdf86-10_351_488_264_826}
The diagram shows the curve with equation $y = \sqrt { } \left( 1 + 3 \cos ^ { 2 } \left( \frac { 1 } { 2 } x \right) \right)$ for $0 \leqslant x \leqslant \pi$. The region $R$ is bounded by the curve, the axes and the line $x = \pi$.\\
(i) Use the trapezium rule with two intervals to find an approximation to the area of $R$, giving your answer correct to 3 significant figures.\\
(ii) The region $R$ is rotated completely about the $x$-axis. Without using a calculator, find the exact volume of the solid produced.\\
\hfill \mbox{\textit{CAIE P2 2018 Q6 [8]}}