If w = 2 + 2i , find the modulus and argument of w.

Given \(z = \cos \left( {\frac{{5\pi }}{6}} \right) + {\text{i}}\sin \left( {\frac{{5\pi }}{6}} \right)\), find in its simplest form \({w^4}{z^6}\).

modulus \( = \sqrt 8 \)     A1

argument \( = \frac{\pi }{4}\) (accept 45°)     A1

Note: A0 if extra values given.

[2 marks]

METHOD 1

\({w^4}{z^6} - 64{e^{\pi {\text{i}}}} \times {e^{5\pi {\text{i}}}}\)     (A1)(A1) 

Note: Allow alternative notation.

\( = 64{e^{6\pi {\text{i}}}}\)     (M1)

\( = 64\)     A1

METHOD 2

\({w^4} = - 64\)     (M1)(A1)

\({z^6} = - 1\)     (A1)

\({w^4}{z^6} = 64\)     A1

[4 marks]

Those who tackled this question were generally very successful. A few, with varying success, tried to work out the powers of the complex numbers by multiplying the Cartesian form rather than using de Moivre’s Theorem.

Those who tackled this question were generally very successful. A few, with varying success, tried to work out the powers of the complex numbers by multiplying the Cartesian form rather than using de Moivre’s Theorem.