| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Challenging +1.8 This is a Further Maths polar coordinates question requiring: (a) solving ln(1+sin θ)=0 for the loop endpoints, (b) applying the polar area formula ½∫r²dθ with a non-trivial integrand requiring integration by parts, and (c) a geometric argument comparing the actual radius to what a circle would require. The integration is technically demanding and the multi-part structure with proof element places it well above average difficulty, though the individual techniques are standard for Further Maths students. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09c Area enclosed: by polar curve |
3 The equation of a curve in polar coordinates is $r = \ln ( 1 + \sin \theta )$ for $\alpha \leqslant \theta \leqslant \beta$ where $\alpha$ and $\beta$ are non-negative angles. The curve consists of a single closed loop through the pole.
\begin{enumerate}[label=(\alph*)]
\item By solving the equation $r = 0$, determine the smallest possible values of $\alpha$ and $\beta$.
\item Find the area enclosed by the curve, giving your answer to 4 significant figures.
\item Hence, by considering the value of $r$ at $\theta = \frac { \alpha + \beta } { 2 }$, show that the loop is not circular.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q3 [6]}}