| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Rotation about x-axis: rational or reciprocal function |
| Difficulty | Moderate -0.3 This is a straightforward two-part question requiring basic differentiation using the chain rule and a standard volume of revolution integral. Part (i) is routine differentiation and substitution; part (ii) involves the standard formula V = π∫y²dx with a simple rational function that integrates easily. Both parts are textbook exercises with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{dy}{dx} = -6(3x - 2)^2 \times 3\) | B1 M1 | B1 (without the \(\times 3\)). Use of chain rule |
| If \(x = 2\), \(m = -1\frac{1}{8}\) (-1.125) | A1 [3] | co |
| (ii) Vol \(= \int \frac{36}{(3x-2)^2}dx\) | B1 | Attempt at \(\pi \int y^2\) - even if \(\pi\) missing |
| \(\left[\frac{-36}{(3x-2)} + 3\right]\) | B1 B1 | No need for \(\pi\) here |
| Use of limits [2] – [1] \(\rightarrow 9\pi\) | M1 A1 [5] | Correct use of correct limits. co |
(i) $\frac{dy}{dx} = -6(3x - 2)^2 \times 3$ | B1 M1 | B1 (without the $\times 3$). Use of chain rule
If $x = 2$, $m = -1\frac{1}{8}$ (-1.125) | A1 [3] | co
(ii) Vol $= \int \frac{36}{(3x-2)^2}dx$ | B1 | Attempt at $\pi \int y^2$ - even if $\pi$ missing
$\left[\frac{-36}{(3x-2)} + 3\right]$ | B1 B1 | No need for $\pi$ here
| |
Use of limits [2] – [1] $\rightarrow 9\pi$ | M1 A1 [5] | Correct use of correct limits. co9\\
\includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-3_391_595_1978_774}
The diagram shows part of the curve $y = \frac { 6 } { 3 x - 2 }$.\\
(i) Find the gradient of the curve at the point where $x = 2$.\\
(ii) Find the volume obtained when the shaded region is rotated through $360 ^ { \circ }$ about the $x$-axis, giving your answer in terms of $\pi$.
\hfill \mbox{\textit{CAIE P1 2009 Q9 [8]}}