| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| 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 standard Further Maths surface area of revolution question with parametric equations. Part (a) is routine differentiation and algebraic verification. Part (b) applies the standard formula S = ∫2πy√((dx/dt)² + (dy/dt)²)dt, which simplifies nicely using part (a). The integration requires substitution (u = 1 + 1/t²) but is straightforward. While this is FP2 content, it follows a predictable template with no novel insights required, making it slightly above average difficulty overall. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian8.06b Arc length and surface area: of revolution, cartesian or parametric |
A curve $C$ is defined parametrically by
$$x = \frac{t^2 + 1}{t}, \quad y = 2\ln t$$
**(a) [4 marks]**
Show that $\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = \left(1 + \frac{1}{t^2}\right)^2$
**(b) [5 marks]**
The arc of $C$ from $t = 1$ to $t = 2$ is rotated through $2\pi$ radians about the $x$-axis. Find the area of the surface generated, giving your answer in the form $\pi(m\ln 2 + n)$, where $m$ and $n$ are integers.3 A curve $C$ is defined parametrically by
$$x = \frac { t ^ { 2 } + 1 } { t } , \quad y = 2 \ln t$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + \frac { 1 } { t ^ { 2 } } \right) ^ { 2 }$.
\item The arc of $C$ from $t = 1$ to $t = 2$ is rotated through $2 \pi$ radians about the $x$-axis. Find the area of the surface generated, giving your answer in the form $\pi ( m \ln 2 + n )$, where $m$ and $n$ are integers.\\[0pt]
[5 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2015 Q3 [9]}}