Embibe Experts Solutions for Chapter: Complex Numbers, Exercise 1: CG PET 2019

If $x=9^{1/3} 9^{1/9} 9^{1/27}\mathrm{...}\infty$

$y=4^{1/3} 4^{-1/9} 4^{1/27}\mathrm{...}\infty$

$z=\sum _{r=1}^{\infty }{\left(1+i\right)}^{-r}$,

then $\arg \left(x+yz\right)$ will be

If $z=x+iy$ is a complex number where $x$ and $y$ are integers. Then, the area of the rectangle whose vertices are the roots of the equation $z{\overline{z}}^3+\overline{z}{z}^3=350$, is

If $e^{i\theta }=\cos \theta +i\sin \theta$, the the value of $\frac{{\left(1+i\right)}^2}{1-i}$ is

The value of ${\left(1+i\right)}^8+{\left(1-i\right)}^8$ is

If $\left|z_1\right|=\left|z_2\right|$ and $\arg \left(z_1\right)+\arg \left(z_2\right)=0$, then

If $z$ is a point on the argand plane such that $\left|z-1\right|=1,$ then  $\frac{z-2}{z}$ is equal to

If for complex numbers $z_1$ and $z_2$, arg $\left(\frac{z_1}{z_2}\right)=0$, then $\left|z_1-z_2\right|$ is equal to

If $\omega$ is a non-real cube root of unity, then the expression $\left(1-\omega \right)\left(1-{\omega }^2\right)\left(1+{\omega }^4\right)\left(1+{\omega }^8\right)$ is equal to