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

If $|z+\omega {|}^2=|z{|}^2+|\omega {|}^2$, where $z$ and $\omega$ are complex numbers, then

If $z_1, z_2$ be two complex numbers satisfying the equation $\left|\frac{z_1+z_2}{z_1-z_2}\right|=1$, then

If $\left|\frac{2i{z}_1-z_1-z_2}{2i{z}_1+z_1+z_2}\right|=\left|\frac{\cos 2\varphi +i\sin 2\varphi }{\cos 2\varphi -i\sin 2\varphi }\right|$ and $z_1, z_2$ are non-zero complex numbers, then

Let complex number $z$ satisfy $\left|z-\frac{2}{z}\right|=1,$ then $\left|z\right|$ can take all values except

If $z\in C$ and $|z{|}^2-2|z|-3<0,$ then the possible value(s) of $\left|z^2+2z\cos \theta \right|$ for all values of $\theta$ is(are)

If $z\in C$ and $|z{|}^2-2|z|-3<0$, then the possible value(s) of $\left|z^2+2z\cos \theta \right|$ for all values of $\theta$ is(are)

If $z$ is a complex number and the minimum value of $\left|z\right|+|z-1|+|2z-3|$ is $k$ if $\beta =2\left[\alpha \right]+3=3[\alpha -k]$ then (where $[.]$ denotes the greatest integer function)

If $z_1,z_2\in C,z_1^2+z_2^2\in R,z_1\left(z_1^2-3z_2^2\right)=2$ and $z_2\left(3z_1^2-z_2^2\right)=11$, then the value of $z_1^2+z_2^2$ is