Embibe Experts Solutions for Chapter: Complex Numbers, Exercise 1: EXERCISE-1

If $\omega$ is an imaginary cube root of unity, then ${\left(1+\omega -{\omega }^2\right)}^7$ equals

If $i=\sqrt{-1}$, then $4+5{\left(-\frac{1}{2}+\frac{i\sqrt{3}}{2}\right)}^{334}+3{\left(-\frac{1}{2}+\frac{i\sqrt{3}}{2}\right)}^{365}$ is equal to :

The set of points on an Argand plane which satisfy both $\left|z\right|\le 4 \&$ $\arg \left(z\right)=\frac{\pi }{3}$ are lying on:

The origin and the roots of the equation $z^2+pz+q=0, p,q\in R$ form an equilateral triangle in argand plane if -

If $z_1$ and $z_2$ are two non-zero complex numbers satisfying $\left|\frac{z_1-i{z}_2}{z_1+i{z}_2}\right|=1$, then $\frac{z_1}{z_2}$ is

${\left[\frac{-1+i\sqrt{3}}{2}\right]}^6+{\left[\frac{-1-i\sqrt{3}}{2}\right]}^6+{\left[\frac{-1+i\sqrt{3}}{2}\right]}^5+{\left[\frac{-1-i\sqrt{3}}{2}\right]}^5$ is equal to

If $z$ and $\omega$ are two non-zero complex numbers such that $\left|z\omega \right|=1$ and $Arg\left(z\right)-Arg\left(\omega \right)=\frac{\pi }{2},$ then $\overline{z}\omega$ is equal to

$z_1,z_2,z_3$ are three points lying on the circle $\left|z\right|=1$, then maximum value of ${\left|z_1-z_2\right|}^2+{\left|z_2-z_3\right|}^2+{\left|z_3-z_1\right|}^2$ is equal to -