| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Maximum/minimum distance from pole or line |
| Difficulty | Challenging +1.8 This Further Maths polar coordinates question requires finding maximum perpendicular distance (involving calculus and trigonometric manipulation), solving a transcendental equation, finding curve intersections, and computing areas with polar integrals. The derivation of 2θ tan θ = 1 requires geometric insight about perpendicular distance in polar form, and part (iv) involves setting up and evaluating a non-trivial polar area integral. While systematic, it demands strong technical facility across multiple topics and extended multi-step reasoning beyond standard textbook exercises. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
The curve $C _ { 1 }$ has polar equation $r ^ { 2 } = 2 \theta$, for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.\\
(i) The point on $C _ { 1 }$ furthest from the line $\theta = \frac { 1 } { 2 } \pi$ is denoted by $P$. Show that, at $P$,
$$2 \theta \tan \theta = 1$$
and verify that this equation has a root between 0.6 and 0.7 .\\
The curve $C _ { 2 }$ has polar equation $r ^ { 2 } = \theta \sec ^ { 2 } \theta$, for $0 \leqslant \theta < \frac { 1 } { 2 } \pi$. The curves $C _ { 1 }$ and $C _ { 2 }$ intersect at the pole, denoted by $O$, and at another point $Q$.\\
(ii) Find the exact value of $\theta$ at $Q$.\\
(iii) The diagram below shows the curve $C _ { 2 }$. Sketch $C _ { 1 }$ on this diagram.\\
(iv) Find, in exact form, the area of the region $O P Q$ enclosed by $C _ { 1 }$ and $C _ { 2 }$.\\
\hfill \mbox{\textit{CAIE FP1 2019 Q11 EITHER}}