G Tewani Solutions for Chapter: Complex Numbers, Exercise 2: DPP

The argument of $\left(z-a\right)\left(\overline{z}-b\right)$ is equal to that of $\frac{\left(\sqrt{3}+i\right)\left(1+\sqrt{3}i\right)}{1+i}$  where  $a,b$ are real numbers. If the locus of $z$ in the Argand diagram is a circle with its centre at $\frac{\left(3+i\right)}{2}$, then the ordered pair $\left(a,b\right)$ is

If a complex number $z$ satisfies ${\left|z\right|}^2+\frac{4}{|z{|}^2}-2\left(\frac{z}{\overline{z}}+\frac{\overline{z}}{z}\right)-16=0$, then the maximum value of $\left|z\right|$ is

If $\mathrm{cos\alpha }+\mathrm{cos\beta }+\mathrm{cos\gamma }=0=\mathrm{sin\alpha }+\mathrm{sin\beta }+\mathrm{sin\gamma },$ then $\frac{\sin 3\mathrm{\alpha }+\sin 3\mathrm{\beta }+\sin 3\mathrm{\gamma }}{\sin (\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })}$ is equal to

The least value of ${\left|z-3-4i\right|}^2+{\left|z+2-7i\right|}^2+{\left|z-5+2i\right|}^2$ occurs when $z$ is

The roots of the equation $x^4-2x^2+4=0$ are the vertices of a

If  $z_1, z_2$ are complex numbers such that $Re\left(z_1\right)=\left|z_1-2\right|$ , $Re\left(z_2\right)=\left|z_2-2\right|$ and $\arg \left(z_1-z_2\right)=\frac{\pi }{3},$ then $Im\left(z_1+z_2\right)=?$

If $z=e^{i\raisebox{1ex}{$2\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$, then $1+z+z^2+z^3+5z^4+4z^5+4z^6+4z^7$$+4z^8+5z^9=$

If $z=(3+7i)(a+ib),$ (where $a, b\in Z-\left\{0\right\}$) is purely imaginary then the minimum value of ${\left|z\right|}^2$ is