| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and related rates |
| Difficulty | Standard +0.3 This is a straightforward two-part question combining related rates (requiring chain rule differentiation) and volume of revolution (requiring standard integration). Part (i) involves basic implicit differentiation with dy/dx = -8/x² + 2, then substituting x=1 and dx/dt=0.04. Part (ii) is a routine volume of revolution calculation using V = π∫y²dx from x=2 to x=5. Both parts are standard textbook exercises requiring only direct application of learned techniques with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08h Integration by substitution4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{dy}{dx} = \frac{-8}{x^2} + 2\) (– 6 at \(A\)) | M1, A1 [2] | Attempt at differentiation. algebraic – unsimplified. |
| \(\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt} \to -0.24\) | M1, A1 [4] | Ignore notation – needs product of 0.04 and 'his' \(\frac{dy}{dx}\). |
| (ii) \(\int y^2 = \int\frac{64}{x^2} + 4x^2 + 32\) | M1 [6] | Use of integral of \(y^2\) (ignore \(\pi\)) |
| \(= \left(-\frac{64}{x} + \frac{4x^3}{3} + 32x\right)\) | A3, 2, 1 [6] | 3 terms \(\to\) –1 each error. |
| Limits 2 to 5 used correctly \(\to 271.2\pi\) or 852 (allow \(271\pi\) or 851 to 852) | DM1, A1 [6] | Uses correct limits correctly. (omission of \(\pi\) loses last mark) |
$y = \frac{8}{x} + 2x$
(i) $\frac{dy}{dx} = \frac{-8}{x^2} + 2$ (– 6 at $A$) | M1, A1 [2] | Attempt at differentiation. algebraic – unsimplified.
$\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt} \to -0.24$ | M1, A1 [4] | Ignore notation – needs product of 0.04 and 'his' $\frac{dy}{dx}$.
(ii) $\int y^2 = \int\frac{64}{x^2} + 4x^2 + 32$ | M1 [6] | Use of integral of $y^2$ (ignore $\pi$)
$= \left(-\frac{64}{x} + \frac{4x^3}{3} + 32x\right)$ | A3, 2, 1 [6] | 3 terms $\to$ –1 each error.
Limits 2 to 5 used correctly $\to 271.2\pi$ or 852 (allow $271\pi$ or 851 to 852) | DM1, A1 [6] | Uses correct limits correctly. (omission of $\pi$ loses last mark)9\\
\includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-4_584_670_881_740}
The diagram shows part of the curve $y = \frac { 8 } { x } + 2 x$ and three points $A , B$ and $C$ on the curve with $x$-coordinates 1, 2 and 5 respectively.\\
(i) A point $P$ moves along the curve in such a way that its $x$-coordinate increases at a constant rate of 0.04 units per second. Find the rate at which the $y$-coordinate of $P$ is changing as $P$ passes through $A$.\\
(ii) Find the volume obtained when the shaded region is rotated through $360 ^ { \circ }$ about the $x$-axis.
\hfill \mbox{\textit{CAIE P1 2013 Q9 [10]}}