| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Parametric integration |
| Type | Parametric arc length calculation |
| Difficulty | Challenging +1.2 This is a Further Maths FP2 parametric arc length question requiring differentiation of a composite logarithmic/trigonometric expression and then arc length integration. Part (a) involves careful chain rule application with trigonometric identities (showing dx/dt = sin t tan t), while part (b) requires computing √((dx/dt)² + (dy/dt)²) and integrating—the algebra simplifies nicely to give a clean logarithmic answer. While technically demanding with multiple steps, it follows standard FP2 techniques without requiring novel insight, making it moderately above average difficulty. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.07r Chain rule: dy/dx = dy/du * du/dx and connected rates4.07d Differentiate/integrate: hyperbolic functions8.06a Reduction formulae: establish, use, and evaluate recursively |
6
\begin{enumerate}[label=(\alph*)]
\item Given that
$$x = \ln ( \sec t + \tan t ) - \sin t$$
show that
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = \sin t \tan t$$
\item A curve is given parametrically by the equations
$$x = \ln ( \sec t + \tan t ) - \sin t , \quad y = \cos t$$
The length of the arc of the curve between the points where $t = 0$ and $t = \frac { \pi } { 3 }$ is denoted by $s$.
Show that $s = \ln p$, where $p$ is an integer.
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2011 Q6 [10]}}