Express w² and w³ in modulus-argument form.

Sketch on an Argand diagram the points represented by w⁰ , w¹ , w² and w³.

Show that the area of the quadrilateral Q is $\frac{21\sqrt{3}}{2}$.

Let $z=2\left(\text{cos}\frac{\pi }{n}+\text{i}\text{sin}\frac{\pi }{n}\right),n\in {\mathbb{Z}}^{+}$. The points represented on an Argand diagram by $z^0,z^1,z^2,\dots ,z^n$ form the vertices of a polygon $P_n$.

Show that the area of the polygon $P_n$ can be expressed in the form $a\left(b^n-1\right)\text{sin}\frac{\pi }{n}$, where $a,b\in \mathbb{R}$.

$w^2=4\text{cis}\left(\frac{2\pi }{3}\right)\text{;}{w}^3=8\text{cis}\left(\pi \right)$     (M1)A1A1

Note: Accept Euler form.

Note: M1 can be awarded for either both correct moduli or both correct arguments.

Note: Allow multiplication of correct Cartesian form for M1, final answers must be in modulus-argument form.

[3 marks]

     A1A1

[2 marks]

use of area = $\frac{1}{2}ab\text{sin}C$     M1

$\frac{1}{2}\times 1\times 2\times \text{sin}\frac{\pi }{3}+\frac{1}{2}\times 2\times 4\times \text{sin}\frac{\pi }{3}+\frac{1}{2}\times 4\times 8\times \text{sin}\frac{\pi }{3}$      A1A1

Note: Award A1 for $C=\frac{\pi }{3}$, A1 for correct moduli.

$=\frac{21\sqrt{3}}{2}$     AG

Note: Other methods of splitting the area may receive full marks.

[3 marks]

$\frac{1}{2}\times 2^0\times 2^1\times \text{sin}\frac{\pi }{n}+\frac{1}{2}\times 2^1\times 2^2\times \text{sin}\frac{\pi }{n}+\frac{1}{2}\times 2^2\times 2^3\times \text{sin}\frac{\pi }{n}+\dots +\frac{1}{2}\times 2^{n-1}\times 2^n\times \text{sin}\frac{\pi }{n}$      M1A1

Note: Award M1 for powers of 2, A1 for any correct expression including both the first and last term.

$=\text{sin}\frac{\pi }{n}\times \left(2^0+2^2+2^4+\dots +2^{n-2}\right)$

identifying a geometric series with common ratio 2²(= 4)     (M1)A1

$=\frac{1-2^{2n}}{1-4}\times \text{sin}\frac{\pi }{n}$     M1

Note: Award M1 for use of formula for sum of geometric series.

$=\frac{1}{3}\left(4^n-1\right)\text{sin}\frac{\pi }{n}$     A1

[6 marks]