| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | January |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric integration |
| Type | Parametric volume of revolution |
| Difficulty | Standard +0.8 This question combines standard parametric techniques (finding normals, Cartesian equations) with a more challenging volume of revolution calculation requiring parametric integration with substitution. Part (c) demands careful setup of the integral V = π∫y²dx with limits converted to parameter values, then integration by substitution—this multi-step process with logarithmic bounds elevates it above typical C4 questions. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| \(\frac{dx}{dt}=\frac{1}{t},\quad \frac{dy}{dt}=2t\) | ||
| \(\frac{dy}{dx}=2t^2\) | M1 A1 | |
| Using \(mm'=-1\), at \(t=3\): \(m'=-\frac{1}{18}\) | M1 A1 | |
| \(y-7=-\frac{1}{18}(x-\ln 3)\) | M1 A1 | (6) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| \(x=\ln t \Rightarrow t=e^x\) | B1 | |
| \(y=e^{2x}-2\) | M1 A1 | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| \(V=\pi\int(e^{2x}-2)^2\,dx\) | M1 | |
| \(\int(e^{2x}-2)^2\,dx=\int(e^{4x}-4e^{2x}+4)\,dx\) | M1 | |
| \(=\frac{e^{4x}}{4}-\frac{4e^{2x}}{2}+4x\) | M1 A1 | |
| \(\pi\left[\frac{e^{4x}}{4}-\frac{4e^{2x}}{2}+4x\right]_{\ln2}^{\ln4}=\pi\left[(64-32+4\ln4)-(4-8+4\ln2)\right]\) | M1 | |
| \(=\pi(36+4\ln2)\) | A1 | (6) [15] |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| \(V=\pi\int(t^2-2)^2\frac{dx}{dt}\,dt\) | M1 | |
| \(\int\left((t^2-2)^2\times\frac{1}{t}\right)dt=\int\left(t^3-4t+\frac{4}{t}\right)dt\) | M1 | |
| \(=\frac{t^4}{4}-2t^2+4\ln t\) | M1 A1 | |
| Limits \(t=2\) and \(t=4\) | B1 | |
| \(\pi\left[\frac{t^4}{4}-2t^2+4\ln t\right]_2^4=\pi\left[(64-32+4\ln4)-(4-8+4\ln2)\right]\) | M1 | |
| \(=\pi(36+4\ln2)\) | A1 | (6) |
## Question 6:
### Part (a):
| Working | Marks | Notes |
|---------|-------|-------|
| $\frac{dx}{dt}=\frac{1}{t},\quad \frac{dy}{dt}=2t$ | | |
| $\frac{dy}{dx}=2t^2$ | M1 A1 | |
| Using $mm'=-1$, at $t=3$: $m'=-\frac{1}{18}$ | M1 A1 | |
| $y-7=-\frac{1}{18}(x-\ln 3)$ | M1 A1 | (6) |
### Part (b):
| Working | Marks | Notes |
|---------|-------|-------|
| $x=\ln t \Rightarrow t=e^x$ | B1 | |
| $y=e^{2x}-2$ | M1 A1 | (3) |
### Part (c):
| Working | Marks | Notes |
|---------|-------|-------|
| $V=\pi\int(e^{2x}-2)^2\,dx$ | M1 | |
| $\int(e^{2x}-2)^2\,dx=\int(e^{4x}-4e^{2x}+4)\,dx$ | M1 | |
| $=\frac{e^{4x}}{4}-\frac{4e^{2x}}{2}+4x$ | M1 A1 | |
| $\pi\left[\frac{e^{4x}}{4}-\frac{4e^{2x}}{2}+4x\right]_{\ln2}^{\ln4}=\pi\left[(64-32+4\ln4)-(4-8+4\ln2)\right]$ | M1 | |
| $=\pi(36+4\ln2)$ | A1 | (6) **[15]** |
**Alternative using parameters:**
| Working | Marks | Notes |
|---------|-------|-------|
| $V=\pi\int(t^2-2)^2\frac{dx}{dt}\,dt$ | M1 | |
| $\int\left((t^2-2)^2\times\frac{1}{t}\right)dt=\int\left(t^3-4t+\frac{4}{t}\right)dt$ | M1 | |
| $=\frac{t^4}{4}-2t^2+4\ln t$ | M1 A1 | |
| Limits $t=2$ and $t=4$ | B1 | |
| $\pi\left[\frac{t^4}{4}-2t^2+4\ln t\right]_2^4=\pi\left[(64-32+4\ln4)-(4-8+4\ln2)\right]$ | M1 | |
| $=\pi(36+4\ln2)$ | A1 | (6) |
---\begin{enumerate}
\item The curve $C$ has parametric equations
\end{enumerate}
$$x = \ln t , \quad y = t ^ { 2 } - 2 , \quad t > 0$$
Find\\
(a) an equation of the normal to $C$ at the point where $t = 3$,\\
(b) a cartesian equation of $C$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a3ece8a8-8107-4c3a-a6a9-c19b5e35ec5a-10_579_759_740_571}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
The finite area $R$, shown in Figure 1, is bounded by $C$, the $x$-axis, the line $x = \ln 2$ and the line $x = \ln 4$. The area $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis.\\
(c) Use calculus to find the exact volume of the solid generated.
\hfill \mbox{\textit{Edexcel C4 2011 Q6 [15]}}