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]