Let $z$ and $w$ be the two non-zero complex numbers such that $|z|\, = \,|w|$ and $arg\,z + arg\,w = \pi $. Then $z$ is equal to

Let $z$ satisfy $\left| z \right| = 1$ and $z = 1 - \vec z$.

Statement $1$ : $z$ is a real number

Statement $2$ : Principal argument of $z$ is $\frac{\pi }{3}$

If $5 + ix^3y^2$ and $x^3 + y^2 + 6i$ are conjugate complex numbers and arg $(x + iy) = \theta $ , then ${\tan ^2}\,\theta $ is equal to

Let $z_1$ and $z_2$ be two complex number such that $z_1$ $+z_2=5$ and $z_1^3+z_2^3=20+15 i$. Then $\left|z_1^4+z_2^4\right|$ equals-

The product of two complex numbers each of unit modulus is also a complex number, of

If $|z_1|=1, \, |z_2| =2, \,|z_3|=3$ and $|9z_1z_2 + 4z_1z_3+z_2z_3| =12$ then the value of $|z_1+z_2+z_3|$ is equal to :-