91Ó°ÊÓ

\(\bar{z}\) is the conjugate of \(z\) : (a) \(|\bar{z}|=|z|\) (b) \(\arg z=\arg \bar{z}\) (c) \(z \bar{z}\) is real (d) \(z / z\) is real (e) \(\bar{z}\) is the mirror image of \(z\) in the \(y\)-axis.

(a) Given that the complex number \(z\) and its conjugate \(\bar{z}\) satisfy the equation $$ z \bar{z}+2 \mathrm{i} z=12+6 \mathrm{i} $$ find the possible values of \(z\). (b) Mark in an Argand diagram the points representing the complex numbers \(4+3 i, 4-3 i, \frac{4+3 i}{4-3 i}\).

If \(z\) is any cube root of unity, the value of \(1+z+z^{2}\) can be: (a) 0 (b) 1 (c) 2 (d) 3 (e) \(-1\).