| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2002 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Challenging +1.2 This is a multi-part polar coordinates question requiring standard techniques (asymptote verification using limits, sketching, area formula, arc length formula) with straightforward calculus. The asymptote proof is guided, and all parts follow textbook methods without requiring novel insight, though the topic itself (polar coordinates in Further Maths) places it above average difficulty. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
5 The curve $C$ has polar equation $r \theta = 1$, for $0 < \theta \leqslant 2 \pi$.\\
(i) Use the fact that $\frac { \sin \theta } { \theta }$ tends to 1 as $\theta$ tends to 0 to show that the line with carresian equation $y = 1$ is an asymptote to $C$.\\
(ii) Sketch $C$.
The points $P$ and $Q$ on $C$ correspond to $\theta = \frac { 1 } { 6 } \pi$ and $\theta = \frac { 1 } { 3 } \pi$ respectively.\\
(iii) Find the area of the sector $O P Q$, where $O$ is the origin.\\
(iv) Show that the length of the are $P Q$ is
$$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { \sqrt { } \left( 1 + \theta ^ { 2 } \right) } { \theta ^ { 2 } } \mathrm {~d} \theta$$
\hfill \mbox{\textit{CAIE FP1 2002 Q5 [8]}}