By expressing ${z_1}$ and ${z_2}$ in modulus-argument form write down the modulus of $w$;

By expressing ${z_1}$ and ${z_2}$ in modulus-argument form write down the argument of $w$.

Find the smallest positive integer value of $n$, such that ${w^n}$ is a real number.

${z_1} = 2{\text{cis}}\left( {\frac{\pi }{3}} \right)$ and ${z_2} = \sqrt 2 {\text{cis}}\left( {\frac{\pi }{4}} \right)$     A1A1

Note:     Award A1A0 for correct moduli and arguments found, but not written in mod-arg form.

$\left| w \right| = \sqrt 2$     A1

[3 marks]

${z_1} = 2{\text{cis}}\left( {\frac{\pi }{3}} \right)$ and ${z_2} = \sqrt 2 {\text{cis}}\left( {\frac{\pi }{4}} \right)$     A1A1

Note:     Award A1A0 for correct moduli and arguments found, but not written in mod-arg form.

$\arg w = \frac{\pi }{{12}}$     A1

Notes:     Allow FT from incorrect answers for ${z_1}$ and ${z_2}$ in modulus-argument form.

[1 mark]

EITHER

$\sin \left( {\frac{{\pi n}}{{12}}} \right) = 0$     (M1)

OR

$\arg ({w^n}) = \pi$     (M1)

$\frac{{n\pi }}{{12}} = \pi$

THEN

$\therefore n = 12$     A1

[2 marks]