If $z$ is a complex number such that $| z | = 4$ and $arg \,(z) = \frac {5\pi }{6}$ , then $z$ is equal to
$ - 2\sqrt 3 + 2i$
$2\sqrt 3 + i$
$2\sqrt 3 - 2i$
$ - \sqrt 3 + i$
If $z_1 = a + ib$ and $z_2 = c + id$ are complex numbers such that $| z_1 | = | z_2 |=1$ and $R({z_1}\overline {{z_2}} ) = 0$, then the pair of complex numbers $w_1 = a + ic$ and $w_2 = b + id$ satisfies
If $z = 3 + 5i,\,\,{\rm{then }}\,{z^3} + \bar z + 198 = $
If $\frac{3+i \sin \theta}{4-i \cos \theta}, \theta \in[0,2 \pi],$ is a real number, then an argument of $\sin \theta+\mathrm{i} \cos \theta$ is
If $z_{1}=2-i, z_{2}=1+i,$ find $\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+1}\right|$
If $|{z_1} + {z_2}| = |{z_1} - {z_2}|$, then the difference in the amplitudes of ${z_1}$ and ${z_2}$ is