Consider \(\omega= \cos \left( {\frac{{2\pi }}{3}} \right) + {\text{i}}\sin \left( {\frac{{2\pi }}{3}} \right)\).

(a)     Show that

  (i)     \({\omega^3} = 1;\)

  (ii)     \(1 + \omega+ {\omega^2} = 0\)

(b)     (i)     Deduce that \({{\text{e}}^{{\text{i}}\theta }} + {{\text{e}}^{{\text{i}}\left( {\theta + \frac{{2\pi }}{3}} \right)}} + {{\text{e}}^{{\text{i}}\left( {\theta + \frac{{4\pi }}{3}} \right)}} = 0\).

  (ii)     Illustrate this result for \(\theta = \frac{\pi }{2}\) on an Argand diagram. 

(c)     (i)     Expand and simplify \(F(z) = (z - 1)(z - \omega)(z - {\omega^2})\) where z is a complex number.

  (ii)     Solve \(F(z) = 7\), giving your answers in terms of \(\omega\).

(a)     (i)     \({\omega^3} = {\left( {\cos \left( {\frac{{2\pi }}{3}} \right) + {\text{i}}\sin \left( {\frac{{2\pi }}{3}} \right)} \right)^3}\)

\( = \cos \left( {x \times \frac{{2\pi }}{3}} \right) + {\text{i}}\sin \left( {3 \times \frac{{2\pi }}{3}} \right)\)     (M1)

\( = \cos 2\pi + {\text{i}}\sin 2\pi \)     A1

\( = 1\)     AG

 

(ii)     \(1 + \omega+ {\omega^2} = 1 + \cos \left( {\frac{{2\pi }}{3}} \right) + {\text{i}}\sin \left( {\frac{{2\pi }}{3}} \right) + \cos \left( {\frac{{4\pi }}{3}} \right) + {\text{i}}\sin \left( {\frac{{4\pi }}{3}} \right)\)     M1A1

\( = 1 + - \frac{1}{2} + {\text{i}}\frac{{\sqrt 3 }}{2} - \frac{1}{2} - {\text{i}}\frac{{\sqrt 3 }}{2}\)     A1

\( = 0\)     AG

[5 marks]

 

(b)     (i)     \({{\text{e}}^{{\text{i}}\theta }} + {{\text{e}}^{{\text{i}}\left( {\theta + \frac{{2\pi }}{3}} \right)}} + {{\text{e}}^{{\text{i}}\left( {\theta + \frac{{4\pi }}{3}} \right)}}\)

\( = {{\text{e}}^{{\text{i}}\theta }} + {{\text{e}}^{{\text{i}}\theta }}{{\text{e}}^{{\text{i}}\left( {\frac{{2\pi }}{3}} \right)}} + {{\text{e}}^{{\text{i}}\theta }}{{\text{e}}^{{\text{i}}\left( {\frac{{4\pi }}{3}} \right)}}\)     (M1)

\( = \left( {{{\text{e}}^{{\text{i}}\theta }}\left( {1 + {{\text{e}}^{{\text{i}}\left( {\frac{{2\pi }}{3}} \right)}} + {{\text{e}}^{{\text{i}}\left( {\frac{{4\pi }}{3}} \right)}}} \right)} \right)\)

\( = {{\text{e}}^{{\text{i}}\theta }}(1 + \omega+ {\omega^2})\)     A1

\( = 0\)     AG

 

(ii)

    A1A1

 

Note: Award A1 for one point on the imaginary axis and another point marked with approximately correct modulus and argument. Award A1 for third point marked to form an equilateral triangle centred on the origin.

 

[4 marks]

 

(c)     (i)     attempt at the expansion of at least two linear factors     (M1)

\((z - 1){z^2} - z(\omega+ {\omega^2}) + {\omega^3}\)   or equivalent     (A1)

use of earlier result     (M1)

\(F(z) = (z - 1)({z^2} + z + 1) = {z^3} - 1\)     A1

 

(ii)     equation to solve is \({z^3} = 8\)     (M1)

\(z = 2,{\text{ }}2\omega,{\text{ }}2{\omega^2}\)     A2

Note: Award A1 for 2 correct solutions.

 

[7 marks]

 

Total [16 marks]

Most candidates were able to make a meaningful start to part (a) with many fully correct answers seen. Part (b) was the exact opposite with the majority of candidates not knowing what was required and failing to spot the connection to part (a). Candidates made a reasonable start to part (c), but often did not recognise the need to use the result that \(1 + \omega+ {\omega ^2} = 0\). This meant that most candidates were unable to make any progress on part (c) (ii).