| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2016 |
| Session | Specimen |
| Topic | Polar coordinates |
| Type | Arc length of polar curve |
| Difficulty | Challenging +1.8 This is a challenging Further Maths question requiring derivation of a reduction formula via integration by parts, then applying it to find arc length in polar coordinates. The reduction formula derivation is non-trivial, and connecting it to the polar arc length formula requires recognizing that the integral matches the required form. However, it's a structured multi-part question with clear guidance, making it more accessible than truly open-ended problems. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve8.06a Reduction formulae: establish, use, and evaluate recursively |
12 (i) Let $I _ { n } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x$, for $n \geqslant 0$. Show that, for $n \geqslant 2$,
$$( n + 2 ) I _ { n } = 125 \times 3 ^ { n - 1 } - 16 ( n - 1 ) I _ { n - 2 }$$
(ii) A curve has polar equation $r = \frac { 1 } { 4 } \theta ^ { 4 }$ for $0 \leqslant \theta \leqslant 3$.
\begin{enumerate}[label=(\alph*)]
\item Sketch this curve.
\item Find the exact length of the curve.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2016 Q12}}