All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
z^3 = (1 + ic )^3

= 1 + 3ic - 3c^2 - ic^3

= 1 - 3c^2 + i(3c - c^3)

(ii)
Given z^3 is real,

\Rightarrow 3c - c^3 = 0

c = 0 \mathrm{~or~} c^2 = 3

\therefore, ~ c = \pm \sqrt{3} since c is non-zero.

z = 1 \pm i \sqrt{3}

(iii)
|z| = 2

|z^n| = |z|^n = 2^n,

2^n > 1000

n > 9.96

Least n = 10.

arg(z^10) = 10 arg(z) = \frac{2}{3} \pi

KS Comments:

For (i), students should use binomial expansion to make quick work of it.
Students must remember to adjust the final argument found in (iii) to ensure that it is within the defined range.

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