| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Volume of revolution (parametric) |
| Difficulty | Standard +0.3 This is a standard C4 parametric curves question covering routine techniques: evaluating coordinates at boundary values, finding dy/dx using the chain rule, converting to Cartesian form, and computing a volume of revolution. All parts follow textbook methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.06d Natural logarithm: ln(x) function and properties1.07s Parametric and implicit differentiation4.08d Volumes of revolution: about x and y axes |
Fig. 7 shows the curve BC defined by the parametric equations
$$x = 5 \ln u, \quad y = u + \frac{1}{u}, \quad 1 \leq u \leq 10.$$
The point A lies on the $x$-axis and AC is parallel to the $y$-axis. The tangent to the curve at C makes an angle $\theta$ with AC, as shown.
\includegraphics{figure_7}
\begin{enumerate}[label=(\roman*)]
\item Find the lengths OA, OB and AC. [5]
\item Find $\frac{dy}{dx}$ in terms of $u$. Hence find the angle $\theta$. [6]
\item Show that the cartesian equation of the curve is $y = e^{x/5} + e^{-x/5}$. [2]
\end{enumerate}
An object is formed by rotating the region OACB through $360°$ about Ox.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Find the volume of the object. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 2013 Q7 [18]}}