| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2014 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find normal equation at parameter |
| Difficulty | Challenging +1.2 Part (a) requires standard parametric differentiation to find dy/dx, then the normal gradient, followed by finding where the normal crosses the x-axis—routine C4 technique. Part (b) involves volume of revolution with parametric equations requiring integration and subtracting a cone volume, which is more involved but follows standard methods. The 15 marks and multi-step nature elevate it slightly above average, but all techniques are standard C4 content with no novel insights required. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation4.08d Volumes of revolution: about x and y axes |
\includegraphics{figure_4}
Figure 4 shows a sketch of part of the curve $C$ with parametric equations
$$x = 3\tan\theta, \quad y = 4\cos^2\theta, \quad 0 \leq \theta < \frac{\pi}{2}$$
The point $P$ lies on $C$ and has coordinates $(3, 2)$.
The line $l$ is the normal to $C$ at $P$. The normal cuts the $x$-axis at the point $Q$.
\begin{enumerate}
\item[(a)] Find the $x$ coordinate of the point $Q$. \hfill [6]
\end{enumerate}
The finite region $S$, shown shaded in Figure 4, is bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$. This shaded region is rotated $2\pi$ radians about the $x$-axis to form a solid of revolution.
\begin{enumerate}
\item[(b)] Find the exact value of the volume of the solid of revolution, giving your answer in the form $p\pi + q\pi^2$, where $p$ and $q$ are rational numbers to be determined.
[You may use the formula $V = \frac{1}{3}\pi r^2 h$ for the volume of a cone.] \hfill [9]
\end{enumerate}
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2014 Q7 [15]}}