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 ${z^0},\,\,{z^1},\,\,{z^2},\, \ldots \,,\,\,{z^n}$ 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]

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

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

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

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]