| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and stationary points |
| Difficulty | Standard +0.3 This is a straightforward volumes of revolution question with standard calculus techniques. Part (i) requires routine differentiation and solving for a stationary point (5 marks). Part (ii) involves setting up and evaluating a standard volume integral with polynomial and reciprocal terms—algebraically involved but conceptually routine for P3 level. No novel insight required, just careful execution of standard methods. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative4.08d Volumes of revolution: about x and y axes |
\includegraphics{figure_11}
The diagram shows part of the curve $y = (x + 1)^2 + (x + 1)^{-1}$ and the line $x = 1$. The point $A$ is the minimum point on the curve.
\begin{enumerate}[label=(\roman*)]
\item Show that the $x$-coordinate of $A$ satisfies the equation $2(x + 1)^3 = 1$ and find the exact value of $\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}$ at $A$. [5]
\item Find, showing all necessary working, the volume obtained when the shaded region is rotated through $360°$ about the $x$-axis. [6]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q11 [11]}}