| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with trigonometric functions |
| Difficulty | Standard +0.3 This is a standard volumes of revolution question with a helpful identity provided in part (i). The identity verification requires routine trigonometric manipulation (using tan²x = sec²x - 1 and the double angle formula), and the integration follows directly. Part (ii) applies the result from (i) using the standard formula V = π∫y²dx. While it involves multiple steps and trigonometric identities, all techniques are standard P2 material with no novel insight required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Replace \(\tan^2 x\) by \(\sec^2 x - 1\) | B1 | |
| Express \(\cos^2 x\) in the form \(\pm\frac{1}{2} \pm \frac{1}{2}\cos 2x\) | M1 | |
| Obtain given answer \(\sec^2 x + \frac{1}{4}\cos 2x - \frac{1}{4}\) correctly | A1 | |
| Attempt integration of expression | M1 | |
| Obtain \(\tan x + \frac{1}{4}\sin 2x - \frac{1}{4}x\) | A1 | |
| Use limits correctly for integral involving at least \(\tan x\) and \(\sin 2x\) | M1 | |
| Obtain \(\frac{3}{4} - \frac{1}{8}\pi\) or exact equivalent | A1 | [7] |
| (ii) State or imply volume is \(\int\pi(\tan x + \cos x)^2 \, dx\) | B1 | |
| Attempt expansion and simplification | M1 | |
| Integrate to obtain one term of form \(k\cos x\) | M1 | |
| Obtain \(\pi(\frac{3}{4} - \frac{1}{8}\pi) + \pi(2 - \sqrt{2})\) or equivalent | A1 | [4] |
**(i)** Replace $\tan^2 x$ by $\sec^2 x - 1$ | B1 |
Express $\cos^2 x$ in the form $\pm\frac{1}{2} \pm \frac{1}{2}\cos 2x$ | M1 |
Obtain given answer $\sec^2 x + \frac{1}{4}\cos 2x - \frac{1}{4}$ correctly | A1 |
Attempt integration of expression | M1 |
Obtain $\tan x + \frac{1}{4}\sin 2x - \frac{1}{4}x$ | A1 |
Use limits correctly for integral involving at least $\tan x$ and $\sin 2x$ | M1 |
Obtain $\frac{3}{4} - \frac{1}{8}\pi$ or exact equivalent | A1 | [7]
**(ii)** State or imply volume is $\int\pi(\tan x + \cos x)^2 \, dx$ | B1 |
Attempt expansion and simplification | M1 |
Integrate to obtain one term of form $k\cos x$ | M1 |
Obtain $\pi(\frac{3}{4} - \frac{1}{8}\pi) + \pi(2 - \sqrt{2})$ or equivalent | A1 | [4]7 (i) Show that $\tan ^ { 2 } x + \cos ^ { 2 } x \equiv \sec ^ { 2 } x + \frac { 1 } { 2 } \cos 2 x - \frac { 1 } { 2 }$ and hence find the exact value of
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 2 } x + \cos ^ { 2 } x \right) d x$$
(ii)\\
\includegraphics[max width=\textwidth, alt={}, center]{48ab71ff-c37b-4e0b-b031-d99b0cf517a8-3_550_785_1573_721}
The region enclosed by the curve $y = \tan x + \cos x$ and the lines $x = 0 , x = \frac { 1 } { 4 } \pi$ and $y = 0$ is shown in the diagram. Find the exact volume of the solid produced when this region is rotated completely about the $x$-axis.
\hfill \mbox{\textit{CAIE P2 2012 Q7 [11]}}