Random Questions from 2016 Prelims #11

TPJC/P2/4

(a) The complex number w is given by 3+3\sqrt{3}i
(i) Find the modulus and argument of w, giving your answer in exact form.
(ii) Without using a calculator, find the smallest positive integer value of n for which (\frac{w^3}{w^*})^n is a real number.

(b) The complex number z is such that z^5 = - 4 \sqrt{2}
(i) Find the value of z in the form re^{i\theta}, where r > 0 and - \pi \textless \theta \le \pi.
(ii) Show the roots on an argand diagram.
(iii) The roots represented by z_1 and z_2 are such that 0 \textless arg({z_1}) \textless arg({z_2}) \textless \pi. The locus of all points z such that |z - z_1| = |z-z_2| intersects the line segment joining points representing z_1 and z_2 at the point P. P represents the complex number p. Find, in exact form, the modulus and argument of p.

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