| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with numerical methods |
| Difficulty | Standard +0.3 Part (i) is a straightforward application of the trapezium rule with only two intervals on a simple function. Part (ii) requires setting up and evaluating ∫(tan 2x)² dx, which involves a standard trigonometric identity (sec²θ - 1) and integration techniques covered in P2. While it combines numerical methods with volumes of revolution, both components are routine applications of standard techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08c 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 | Marks | Guidance |
| State or imply correct \(y\)-values \(0,\ \tan\frac{1}{6}\pi,\ \tan\frac{2}{6}\pi\) | B1 | Some candidates have their calculator in degree mode when working out \(\tan\frac{\pi}{6}\) etc. this gives 0.00915 and 0.0183. Allow B1 |
| Use correct formula, or equivalent, with \(h = \frac{1}{12}\pi\) and \(y\)-values | M1 | Must be convinced they have considered 3 values for \(y\) for M1 |
| Obtain 0.378 | A1 | |
| Total: | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(\pi\int(\sec^2 2x - 1)\,dx\) | B1 | |
| Integrate to obtain \(k_1\tan 2x + k_2x\), any non-zero constants including \(\pi\) or not | M1 | |
| Obtain \(\frac{1}{2}\tan 2x - x\) or \(\pi\left(\frac{1}{2}\tan 2x - x\right)\) | A1 | |
| Obtain \(\pi\left(\frac{1}{2}\sqrt{3} - \frac{1}{6}\pi\right)\) or equivalent | A1 |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply correct $y$-values $0,\ \tan\frac{1}{6}\pi,\ \tan\frac{2}{6}\pi$ | B1 | Some candidates have their calculator in degree mode when working out $\tan\frac{\pi}{6}$ etc. this gives 0.00915 and 0.0183. Allow B1 |
| Use correct formula, or equivalent, with $h = \frac{1}{12}\pi$ and $y$-values | M1 | Must be convinced they have considered 3 values for $y$ for M1 |
| Obtain 0.378 | A1 | |
| **Total:** | **3** | |
## Question 6(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $\pi\int(\sec^2 2x - 1)\,dx$ | B1 | |
| Integrate to obtain $k_1\tan 2x + k_2x$, any non-zero constants including $\pi$ or not | M1 | |
| Obtain $\frac{1}{2}\tan 2x - x$ or $\pi\left(\frac{1}{2}\tan 2x - x\right)$ | A1 | |
| Obtain $\pi\left(\frac{1}{2}\sqrt{3} - \frac{1}{6}\pi\right)$ or equivalent | A1 | |
**Total: 4**
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{6295873e-7db4-4e7e-8dcd-912ad9c41675-06_561_542_260_799}
The diagram shows the curve $y = \tan 2 x$ for $0 \leqslant x \leqslant \frac { 1 } { 6 } \pi$. The shaded region is bounded by the curve and the lines $x = \frac { 1 } { 6 } \pi$ and $y = 0$.\\
(i) Use the trapezium rule with two intervals to find an approximation to the area of the shaded region, giving your answer correct to 3 significant figures.\\
(ii) Find the exact volume of the solid formed when the shaded region is rotated completely about the $x$-axis.\\
\hfill \mbox{\textit{CAIE P2 2017 Q6 [7]}}