| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2018 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with trigonometric functions |
| Difficulty | Standard +0.3 This is a straightforward volumes of revolution question requiring the standard formula V = π∫y²dx. Since y² is already given as 3tan(x/2), the integration becomes π∫3tan(x/2)dx, which is a standard integral (rewrite tan as sin/cos, use substitution). The question guides students to the answer form, making it slightly easier than average for a C3/C4 question. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((V) = \pi\displaystyle\int_{\pi/3}^{\pi/2} 3\tan\!\left(\dfrac{x}{2}\right)dx\) | B1 | Need \(\pi\) and correct limits; limits and \(\pi\) may be implied by later working; condone omission of \(dx\) |
| \(= (\pi)\left[-6\ln\cos\!\left(\dfrac{x}{2}\right)\right]_{\pi/3}^{\pi/2}\) or \((\pi)\left[6\ln\sec\!\left(\dfrac{x}{2}\right)\right]_{\pi/3}^{\pi/2}\) | M1 A1 | M1: achieves \(k\ln\cos(x/2)\) or \(k\ln\sec(x/2)\); A1: cao for \(-6\ln\cos\!\left(\dfrac{x}{2}\right)\) or \(6\ln\sec\!\left(\dfrac{x}{2}\right)\) |
| \(= (\pi)\left[-6\ln\!\left(\dfrac{1}{\sqrt{2}}\right) + 6\ln\!\left(\dfrac{\sqrt{3}}{2}\right)\right]\) | dM1 | Dependent on first M1; substitutes limits and subtracts |
| \(= (\pi)\left[6\ln\!\left(\dfrac{\sqrt{6}}{2}\right)\right] = 3\pi\ln\!\left(\dfrac{3}{2}\right)\) | A1 | cao; depends on both M marks being evidenced |
## Question 6:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(V) = \pi\displaystyle\int_{\pi/3}^{\pi/2} 3\tan\!\left(\dfrac{x}{2}\right)dx$ | B1 | Need $\pi$ and correct limits; limits and $\pi$ may be implied by later working; condone omission of $dx$ |
| $= (\pi)\left[-6\ln\cos\!\left(\dfrac{x}{2}\right)\right]_{\pi/3}^{\pi/2}$ or $(\pi)\left[6\ln\sec\!\left(\dfrac{x}{2}\right)\right]_{\pi/3}^{\pi/2}$ | M1 A1 | M1: achieves $k\ln\cos(x/2)$ or $k\ln\sec(x/2)$; A1: cao for $-6\ln\cos\!\left(\dfrac{x}{2}\right)$ or $6\ln\sec\!\left(\dfrac{x}{2}\right)$ |
| $= (\pi)\left[-6\ln\!\left(\dfrac{1}{\sqrt{2}}\right) + 6\ln\!\left(\dfrac{\sqrt{3}}{2}\right)\right]$ | dM1 | Dependent on first M1; substitutes limits and subtracts |
| $= (\pi)\left[6\ln\!\left(\dfrac{\sqrt{6}}{2}\right)\right] = 3\pi\ln\!\left(\dfrac{3}{2}\right)$ | A1 | cao; depends on both M marks being evidenced |
---6.
\begin{figure}[h]
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\includegraphics[alt={},max width=\textwidth]{7d07e1ad-d87a-4eb5-a15e-05b927892915-14_768_712_212_616}
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\caption{Figure 2}
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\end{figure}
The curve shown in Figure 2 has equation
$$y ^ { 2 } = 3 \tan \left( \frac { x } { 2 } \right) , \quad 0 < x < \pi , \quad y > 0$$
The finite region $R$, shown shaded in Figure 2, is bounded by the curve, the line with equation $x = \frac { \pi } { 3 }$ the $x$-axis and the line with equation $x = \frac { \pi } { 2 }$\\
The region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis to generate a solid of revolution.\\
Show that the exact value of the volume of the solid generated may be written as $A \ln \left( \frac { 3 } { 2 } \right)$, where $A$ is a constant to be found.
\begin{center}
\end{center}
\hfill \mbox{\textit{Edexcel C34 2018 Q6 [5]}}