| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Topic | Polar coordinates |
| Type | Convert Cartesian to polar equation |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring conversion between Cartesian and polar forms, algebraic manipulation with trigonometric identities, and analysis of curve properties. While the individual steps are methodical (substitute x=r cos θ, y=r sin θ, factor out r), part (b) requires careful trigonometric manipulation and parts (c) and (e) require geometric insight about symmetry and loop formation. The question is more demanding than standard A-level but follows established techniques for Further Maths students. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta) |
7 A curve has cartesian equation $x ^ { 3 } + y ^ { 3 } = 2 x y$.\\
$C$ is the portion of the curve for which $x \geqslant 0$ and $y \geqslant 0$. The equation of $C$ in polar form is given by $r = \mathrm { f } ( \theta )$ for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
\begin{enumerate}[label=(\alph*)]
\item Find $f ( \theta )$.
\item Find an expression for $\mathrm { f } \left( \frac { 1 } { 2 } \pi - \theta \right)$, giving your answer in terms of $\sin \theta$ and $\cos \theta$.
\item Hence find the line of symmetry of $C$.
\item Find the value of $r$ when $\theta = \frac { 1 } { 4 } \pi$.
\item By finding values of $\theta$ when $r = 0$, show that $C$ has a loop.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q7 [8]}}