| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths polar coordinates question requiring standard techniques: trigonometric identity (routine), conversion to Cartesian form (mechanical application of r²=x²+y², x=rcosθ, y=rsinθ), area integration using the standard polar formula (direct application), and tangent condition (requires dy/dx=0 but follows standard method). While it has multiple parts and is Further Maths content, each component uses well-practiced techniques without requiring novel insight or particularly complex reasoning. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
A curve $C$ has polar equation $r = 2 a \cos \left( 2 \theta + \frac { 1 } { 2 } \pi \right)$ for $0 \leqslant \theta < 2 \pi$, where $a$ is a positive constant.\\
(i) Show that $r = - 2 a \sin 2 \theta$ and sketch $C$.\\
(ii) Deduce that the cartesian equation of $C$ is
$$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = - 4 a x y .$$
(iii) Find the area of one loop of $C$.\\
(iv) Show that, at the points (other than the pole) at which a tangent to $C$ is parallel to the initial line,
$$2 \tan \theta = - \tan 2 \theta .$$
\hfill \mbox{\textit{CAIE FP1 2017 Q11 EITHER}}