| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Volume of revolution |
| Difficulty | Standard +0.3 This is a standard C4 volume of revolution question requiring finding x-intercepts by solving a basic trigonometric equation (2cos x = 1), then applying the standard formula V = π∫y² dx with y² = (1-2cos x)² which expands to a straightforward integrand using standard identities. The integration is routine using cos²x = (1+cos 2x)/2, making this slightly easier than average for C4. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| \(\{y=0 \Rightarrow\}\ 1 - 2\cos x = 0\) | M1 | \(1-2\cos x = 0\), seen or implied. |
| At least one correct value of \(x\) | A1 | Any one of \(\frac{\pi}{3}\) or \(\frac{5\pi}{3}\) or 60 or 300 or awrt 1.05 or 5.23 or awrt 5.24. |
| \(x = \frac{\pi}{3},\ \frac{5\pi}{3}\) | A1 cso | Both \(\frac{\pi}{3}\) and \(\frac{5\pi}{3}\). [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(V = \pi\int_{\pi/3}^{5\pi/3}(1-2\cos x)^2\,dx\) | B1 | For \(\pi\int(1-2\cos x)^2\). Ignore limits and \(dx\). |
| \(\int(1-2\cos x)^2\,dx = \int(1 - 4\cos x + 4\cos^2 x)\,dx\) | ||
| \(= \int\left(1 - 4\cos x + 4\cdot\frac{1+\cos 2x}{2}\right)dx\) | M1 | \(\cos 2x = 2\cos^2 x - 1\) used. Can be implied by \(\cos^2 x = \frac{1+\cos 2x}{2}\) or \(4\cos^2 x \to 2+2\cos 2x\). |
| \(= \int(3 - 4\cos x + 2\cos 2x)\,dx\) | ||
| \(= 3x - 4\sin x + \frac{2\sin 2x}{2}\) | M1, A1 | M1: Attempts \(\int y^2\) to give any two of \(\pm A \to \pm Ax\), \(\pm B\cos x \to \pm B\sin x\), \(\pm\lambda\cos 2x \to \pm\mu\sin 2x\). A1: Correct integration \(3x - 4\sin x + \frac{2\sin 2x}{2}\). |
| Apply limits \(x=\frac{5\pi}{3}\) and \(x=\frac{\pi}{3}\), subtract correctly | ddM1 | Depends on both previous M marks. Some evidence of substituting \(x=\frac{5\pi}{3}\) and \(x=\frac{\pi}{3}\) and subtracting. Ignore \(\pi\). |
| \(= \pi\left(4\pi + 3\sqrt{3}\right)\) or \(4\pi^2 + 3\pi\sqrt{3}\) | A1 | Two term exact answer. [6] |
# Question 6:
## Part (a):
| $\{y=0 \Rightarrow\}\ 1 - 2\cos x = 0$ | M1 | $1-2\cos x = 0$, seen or implied. |
| At least one correct value of $x$ | A1 | Any one of $\frac{\pi}{3}$ or $\frac{5\pi}{3}$ or 60 or 300 or awrt 1.05 or 5.23 or awrt 5.24. |
| $x = \frac{\pi}{3},\ \frac{5\pi}{3}$ | A1 cso | Both $\frac{\pi}{3}$ and $\frac{5\pi}{3}$. **[3]** |
## Part (b):
| $V = \pi\int_{\pi/3}^{5\pi/3}(1-2\cos x)^2\,dx$ | B1 | For $\pi\int(1-2\cos x)^2$. Ignore limits and $dx$. |
| $\int(1-2\cos x)^2\,dx = \int(1 - 4\cos x + 4\cos^2 x)\,dx$ | | |
| $= \int\left(1 - 4\cos x + 4\cdot\frac{1+\cos 2x}{2}\right)dx$ | M1 | $\cos 2x = 2\cos^2 x - 1$ used. Can be implied by $\cos^2 x = \frac{1+\cos 2x}{2}$ or $4\cos^2 x \to 2+2\cos 2x$. |
| $= \int(3 - 4\cos x + 2\cos 2x)\,dx$ | | |
| $= 3x - 4\sin x + \frac{2\sin 2x}{2}$ | M1, A1 | M1: Attempts $\int y^2$ to give any two of $\pm A \to \pm Ax$, $\pm B\cos x \to \pm B\sin x$, $\pm\lambda\cos 2x \to \pm\mu\sin 2x$. A1: Correct integration $3x - 4\sin x + \frac{2\sin 2x}{2}$. |
| Apply limits $x=\frac{5\pi}{3}$ and $x=\frac{\pi}{3}$, subtract correctly | ddM1 | Depends on both previous M marks. Some evidence of substituting $x=\frac{5\pi}{3}$ and $x=\frac{\pi}{3}$ and subtracting. Ignore $\pi$. |
| $= \pi\left(4\pi + 3\sqrt{3}\right)$ or $4\pi^2 + 3\pi\sqrt{3}$ | A1 | Two term exact answer. **[6]** |6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a98d4a7f-1e6d-4294-9b5c-c945e8fbe83e-09_862_1534_219_205}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows a sketch of part of the curve with equation $y = 1 - 2 \cos x$, where $x$ is measured in radians. The curve crosses the $x$-axis at the point $A$ and at the point $B$.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $\pi$, the $x$ coordinate of the point $A$ and the $x$ coordinate of the point $B$.
The finite region $S$ enclosed by the curve and the $x$-axis is shown shaded in Figure 3. The region $S$ is rotated through $2 \pi$ radians about the $x$-axis.
\item Find, by integration, the exact value of the volume of the solid generated.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2013 Q6 [9]}}