Polar Coordinates (Edexcel A Level Further Maths: Core Pure): Exam Questions

Figure 1 shows a sketch of two curves $C_1$ and $C_2$ with polar equations

$C_1:r=(1+\sin \theta )0\le \theta <2\pi$

$C_2:r=3(1-\sin \theta )0\le \theta <2\pi$

The region $R$ lies inside $C_1$ and outside $C_2$ and is shown shaded in Figure 1.

Show that the area of $R$ is

$p\sqrt{3}-q\pi$

where$p$ and $q$ are integers to be determined.

The curve $C$ shown in Figure 1 has polar equation

$r=4+\cos 2\theta$      $0\le \theta \le \frac{\mathrm{\pi }}{2}$

At the point $A$ on $C$, the value of $r$ is $\frac{9}{2}$

The point $N$ lies on the initial line and $AN$ is perpendicular to the initial line.

The finite region $R,$ shown shaded in Figure 1, is bounded by the curve $C$, the initial line and the line $AN$.

Find the exact area of the shaded region $R$, giving your answer in the form  $p\pi +q\sqrt{3}$ where $p$ and $q$ are rational numbers to be found.

(i) Show on an Argand diagram the locus of points given by the values of $z$ satisfying

$|z-4-3\mathrm{i}|=5$

Taking the initial line as the positive real axis with the pole at the origin and given that $\theta \in \left[\alpha ,\alpha +\mathrm{\pi }\right],$where $\alpha =-\arctan \left(\frac{4}{3}\right),$

(ii) show that this locus of points can be represented by the polar curve with equation

$r=8\cos \theta +6\sin \theta$

The set of points $A$ is defined by

$A=\left\{z:0\le \arg z\le \frac{\pi }{3}\right\}\cap \left\{z:\left|z-4-3\mathrm{i}\right|\le 5\right\}$

(i) Show, by shading on your Argand diagram, the set of points $A$.

(ii) Find the exact area of the region defined by $A$, giving your answer in simplest form.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a design for a flower bed. A circular fountain of radius 2 m has centre $O$. The fountain is surrounded by a planted region $T$, shown shaded in Figure 1.

The region $T$ is bounded by the fountain and by a curved outer edge. The outer edge is modelled by the curve with polar equation

$r=5+a\text{cos}2\theta ,0\le \theta <2\pi$

where $a$ is a positive constant.

Given that the shortest distance between the fountain and the outer edge of the flower bed is 1 m,

determine the value of $a$.

Hence, using algebraic integration, determine, according to the model, the exact area of $T$.

The locus $C$ is given by

$|z-2\text{i}|=2$

The locus $D$ is given by

$\arg z=\frac{\pi }{6}$

Sketch, on the same Argand diagram, the locus $C$ and the locus $D$.

The set of points $A$ is defined by

$A=\left\{z:|z-2\text{i}|\le 2\right\}\cap \left\{z:\frac{\pi }{6}\le \arg z\le \frac{\pi }{2}\right\}$

Show, by shading on your Argand diagram, the set of points $A$.

Find the area of the region defined by $A$, giving your answer in the form $p\pi +q\sqrt{3}$ where $p$ and $q$ are constants to be determined.