Find the modulus and argument of z , giving the argument in degrees.

Find the cube root of z which lies in the first quadrant of the Argand diagram, giving your answer in Cartesian form.

Find the smallest positive integer n for which ${z^n}$ is a positive real number.

$\bmod (z) = 2,{\text{ }}\arg (z) = 150^\circ$     A1A1

[2 marks]

${z^{\frac{1}{3}}} = {2^{\frac{1}{3}}}(\cos 50^\circ  + {\text{i}}\sin 50^\circ )$     (M1)

$= 0.810 + 0.965{\text{i}}$     A1

[2 marks]

we require to find a multiple of 150 that is also a multiple of 360, so by any method,     M1

n = 12     A1

Note: Only award 1 mark for part (c) if n = 12 is based on $\arg (z) = - 30$ .

[2 marks]