G Tewani Solutions for Chapter: Complex Numbers, Exercise 5: DPP

Number of imaginary complex numbers satisfying the equation, $z^2=\overline{z}.2^{1-\left|z\right|}$is

The least positive argument of the $4^{\mathrm{th}}$ root of the complex number $2-i\sqrt{12}$ is

Number of $n^{\mathrm{th}}$ roots of unity that are also the roots of the equation $z^2+az+b=0$, for some integers $a$ and $b$, is

If $z$ is a complex number satisfying the equation $z^6+z^3+1=0$ and if this equation has a root $r{e}^{\mathit{i}\mathit{\theta }}$ with $90°<\theta <180°,$ then the value of $\theta$ is

Suppose $A$ is a complex number and  $n\in N$ such that $A^n={\left(A+1\right)}^n=1$, then the least value of $n$ is

If $z_1, z_2, z_3, ... z_n$ are in $\mathrm{GP}$ with the first term as unity such that $z_1+z_2+z_3+ ...+z_n=0$ and if $z_1, z_2, z_3, ... z_n$ represent the vertices of an $n$ sided-polygon, then the distance between the incentre and circumcentre of the polygon is

If $\alpha$ is the fifth root of unity, then

If $z_1, z_2, z_3$ are any three roots of the equation $z^6={\left(z+1\right)}^6$, then $\arg \left(\frac{z_1-z_3}{z_2-z_3}\right)$ can be equal to