G Tewani Solutions for Chapter: Complex Numbers, Exercise 4: DPP

Let $z_1$and $z_2$ be two complex numbers with $\alpha$ and $\beta$ be their principal arguments, such that $\alpha +\beta >\pi$. Then principal $arg\left(z_1{z}_2\right)$ is given by

Let arg$\left(z_k\right) = \frac{\left(2k+1\right)\pi }{n}$ where $k=1, 2, 3, 4...,n$. If arg$\left(z_1{z}_2{z}_3...z_n\right)=\pi$, then $n$ must be of form $\left(m\in \mathrm{I}\right)$

Suppose two complex numbers $z=a+ib$ ; $w=c+id$ satisfy the equation $\frac{z+w}{z}=\frac{{\displaystyle w}}{{\displaystyle z+w}}$, then

If $\left|z\right|=1 \mathrm{and} z\ne \pm 1$, then one of the possible values of  $arg\left(z\right)-arg\left(z+1\right)-arg\left(z-1\right)$, is

If $\arg \left(z^{\frac{3}{8}}\right)=\frac{1}{2}\arg \left(z^2+\overline{z}{z}^{\frac{1}{2}}\right),$ then which of the following is not possible?

$z_1,z_2$ are two distinct points in complex plane such that $2\left|z_1\right|=3\left|z_2\right| \mathrm{and} z\in C$be any point $z=\frac{2z_1}{3z_2}+\frac{3z_2}{2z_1}$, then

Let $z_1=x_1+i{y}_1, z_2=x_2+i{y}_2$ be complex numbers in fourth quadrant of Argand plane and $\left|z_1\right|=\left|z_2\right|=1, Re\left(z_1{z}_2\right)= 0$. The complex numbers  $z_3=x_1+i{x}_{2,} z_4=y_1+i{y}_{2,} z_5=x_1+i{y}_{2, }{z}_6=x_2+i{y}_{1,}$ will always satisfy

If the imaginary part of $\frac{\mathrm{z}-3}{{\mathrm{e}}^{\mathrm{i\theta }}}+\frac{{\mathrm{e}}^{\mathrm{i\theta }}}{\mathrm{z}-3}$ is zero, then $\mathrm{z}$ can lie on