| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and tangent/normal |
| Difficulty | Standard +0.8 This is a multi-part question requiring finding intersection points, deriving tangent equations, proving they meet on y=x (algebraically non-trivial), then computing a volume of revolution with limits requiring solving a quadratic. The proof element and algebraic manipulation elevate this above routine volume questions, but the techniques are all standard A-level methods. |
| Spec | 1.07m Tangents and normals: gradient and equations4.08d Volumes of revolution: about x and y axes |
\includegraphics{figure_11}
The diagram shows part of the curve $y = \frac{x}{2} + \frac{6}{x}$. The line $y = 4$ intersects the curve at the points $P$ and $Q$.
\begin{enumerate}[label=(\roman*)]
\item Show that the tangents to the curve at $P$ and $Q$ meet at a point on the line $y = x$. [6]
\item Find, showing all necessary working, the volume obtained when the shaded region is rotated through 360° about the $x$-axis. Give your answer in terms of $\pi$. [6]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q11 [12]}}