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

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

The region  lies inside  and outside  and is shown shaded in Figure 1.

Show that the area of  is

where and  are integers to be determined.

The curve  shown in Figure 1 has polar equation

At the point  on , the value of  is 

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

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

Find the exact area of the shaded region , giving your answer in the form  where  and  are rational numbers to be found.

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

Taking the initial line as the positive real axis with the pole at the origin and given that

      where 

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

The set of points A is defined by

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

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