| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Challenging +1.2 This is a standard Further Maths polar coordinates question covering routine techniques: sketching a cardioid, converting to Cartesian form using standard substitutions, and applying memorized formulas for area (∫½r²dθ) and arc length. While it requires multiple steps and Further Maths content, each part follows textbook methods without requiring novel insight or particularly challenging integration. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
The polar equation of a curve $C$ is $r = a ( 1 + \cos \theta )$ for $0 \leqslant \theta < 2 \pi$, where $a$ is a positive constant.\\
(i) Sketch $C$.\\
(ii) Show that the cartesian equation of $C$ is
$$x ^ { 2 } + y ^ { 2 } = a \left( x + \sqrt { } \left( x ^ { 2 } + y ^ { 2 } \right) \right)$$
(iii) Find the area of the sector of $C$ between $\theta = 0$ and $\theta = \frac { 1 } { 3 } \pi$.\\
(iv) Find the arc length of $C$ between the point where $\theta = 0$ and the point where $\theta = \frac { 1 } { 3 } \pi$.\\
\hfill \mbox{\textit{CAIE FP1 2017 Q11 OR}}