| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2003 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and tangent/normal |
| Difficulty | Standard +0.3 This is a straightforward volumes of revolution question combined with basic tangent finding. Part (i) requires standard differentiation and straight line geometry. Part (ii) is a routine integration of a rational function for volume of revolution. All techniques are standard P1/C2 level with no novel insights required, making it slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Cuts \(y=0\) when \(x=4\frac{2}{3}\) | M1, A1, A1 [6] | Use of fn of fn. Needs \(\times 3\) for M mark. Co. Use of line form with \(\frac{dy}{dx}\). Must use calculus. \(\surd\) on his \(\frac{dy}{dx}\). Normal M0. Needs \(y=0\) and \(\frac{1}{2}bh\) for M mark. (beware fortuitous answers). |
| Area of Q \(= \frac{1}{2} \times 2\frac{2}{3} \times 1 = \frac{4}{3}\) | M1, A1, ∇ [6] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow 8\pi\) | M1, A1, A1, DM1, A1 [5] | Uses \(\int y^2 +\) some integration \(\rightarrow (3x+2)^k\). A1 without the \(-3\). A1 for \(-3\) and \(\pi\). DMO if 0 ignored. Co. Beware fortuitous answers. |
**(i)** $\frac{dy}{dx} = \frac{-24}{(3x+2)^2}$
Eqn of tangent: $y-1=\frac{3}{8}(x-2)$
Cuts $y=0$ when $x=4\frac{2}{3}$ | M1, A1, A1 [6] | Use of fn of fn. Needs $\times 3$ for M mark. Co. Use of line form with $\frac{dy}{dx}$. Must use calculus. $\surd$ on his $\frac{dy}{dx}$. Normal M0. Needs $y=0$ and $\frac{1}{2}bh$ for M mark. (beware fortuitous answers).
Area of Q $= \frac{1}{2} \times 2\frac{2}{3} \times 1 = \frac{4}{3}$ | M1, A1, ∇ [6] |
**(ii)** Vol $=\pi \int y^2dx = \pi[64(3x+2)^{-2} \times 3]$ limits from 0 to 2
$\rightarrow 8\pi$ | M1, A1, A1, DM1, A1 [5] | Uses $\int y^2 +$ some integration $\rightarrow (3x+2)^k$. A1 without the $-3$. A1 for $-3$ and $\pi$. DMO if 0 ignored. Co. Beware fortuitous answers.9\\
\includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-4_563_679_938_733}
The diagram shows points $A ( 0,4 )$ and $B ( 2,1 )$ on the curve $y = \frac { 8 } { 3 x + 2 }$. The tangent to the curve at $B$ crosses the $x$-axis at $C$. The point $D$ has coordinates $( 2,0 )$.\\
(i) Find the equation of the tangent to the curve at $B$ and hence show that the area of triangle $B D C$ is $\frac { 4 } { 3 }$.\\
(ii) Show that the volume of the solid formed when the shaded region $O D B A$ is rotated completely about the $x$-axis is $8 \pi$.
\hfill \mbox{\textit{CAIE P1 2003 Q9 [11]}}