| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Surface area of revolution: parametric curve |
| Difficulty | Challenging +1.2 This is a Further Maths surface area of revolution question with parametric equations. Part (a) is straightforward differentiation and algebraic verification. Part (b) requires applying the formula S = 2π∫y√((dx/dt)² + (dy/dt)²)dt, but the algebra simplifies nicely using part (a), making the integration routine. Standard FP2 material with no novel insights required, but slightly above average due to the parametric context and multi-step nature. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| \(x^2 + y^2 = (1-t^2)^2 + 4t^2 = (1+t^2)^2\) | B1, M1, A1 | AG; must be intermediate line |
| Answer | Marks | Guidance |
|---|---|---|
| \(= \frac{256\pi}{15}\) | M1A1, m1, A1F, A1F | Must be correct substitutions for M1. Allow if one term integrated correctly. Any form. |
**(a)** $x = 1 - t^2, y = 2t$
$x^2 + y^2 = (1-t^2)^2 + 4t^2 = (1+t^2)^2$ | B1, M1, A1 | AG; must be intermediate line
**(b)** $S = 2\pi\int_1^2 \sqrt{(1+t^2)} \, t^2 \, dt$
$= 2\pi\left[\frac{t^3}{3} + \frac{t^2}{5}\right]_1^2$
$= 2\pi\left[\frac{8}{3} + \frac{32}{5} - \frac{1}{3} - \frac{1}{5}\right]$
$= \frac{256\pi}{15}$ | M1A1, m1, A1F, A1F | Must be correct substitutions for M1. Allow if one term integrated correctly. Any form.
**Total: 8 marks**
---2 A curve has parametric equations
$$x = t - \frac { 1 } { 3 } t ^ { 3 } , \quad y = t ^ { 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + t ^ { 2 } \right) ^ { 2 }$$
\item The arc of the curve between $t = 1$ and $t = 2$ is rotated through $2 \pi$ radians about the $x$-axis.
Show that $S$, the surface area generated, is given by $S = k \pi$, where $k$ is a rational number to be found.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2006 Q2 [8]}}