| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| 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 requiring standard techniques: finding stationary points by differentiation, solving a simple equation for x-intercepts, and applying the standard formula V = π∫y² dx. The integration involves expanding (4√x - x)² and integrating polynomial terms, which is routine for P1 level. All steps are textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.08d Evaluate definite integrals: between limits4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| \(= 0\) when \(x = 4 \to (4, 4)\) | B1 B1 B1 M1 A1 [5] | co – independent of working. B1 for each part. Sets to 0 and solves his eqn. co |
| Answer | Marks | Guidance |
|---|---|---|
| \(\pi\int(16x + x^2 - 8x^{\frac{3}{2}}) \, dx\) | M1 | Use of correct formula + attempt at integration. |
| Answer | Marks | Guidance |
|---|---|---|
| Limits 0 to 16 \(\to 136.5\pi\) (or \(137\pi\)) | A3,2,1 DM1 A1 [6] | One mark for each term – unsimplified. Correct use of his limits. co – (429 ok) |
$$y = 4\sqrt{x} - x$$
**(i)** At A, $4\sqrt{x} - x = 0 \to A(16, 0)$
$\frac{dy}{dx} = 2x^{-\frac{1}{2}} - 1$
$= 0$ when $x = 4 \to (4, 4)$ | B1 B1 B1 M1 A1 [5] | co – independent of working. B1 for each part. Sets to 0 and solves his eqn. co
**(ii)** Vol $= \pi\int y^2 \, dx =$
$\pi\int(16x + x^2 - 8x^{\frac{3}{2}}) \, dx$ | M1 | Use of correct formula + attempt at integration.
$\pi\left[8x^2 + \frac{x^3}{3} - 8\frac{x^{\frac{5}{2}}}{5}\right]$
Limits 0 to 16 $\to 136.5\pi$ (or $137\pi$) | A3,2,1 DM1 A1 [6] | One mark for each term – unsimplified. Correct use of his limits. co – (429 ok)\includegraphics{figure_11}
The diagram shows part of the curve $y = 4\sqrt{x} - x$. The curve has a maximum point at $M$ and meets the $x$-axis at $O$ and $A$.
\begin{enumerate}[label=(\roman*)]
\item Find the coordinates of $A$ and $M$. [5]
\item Find the volume obtained when the shaded region is rotated through $360°$ about the $x$-axis, giving your answer in terms of $\pi$. [6]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2011 Q11 [11]}}