D.P. GUPTA, Preetima Bajpai and, Sanjeev Kumar Jha Solutions for Chapter: Complex Numbers, Exercise 1: Exercise 1

If $z$be a complex number satisfying $z^4+z^3+2z^2+z+1=0$, then $\left|z\right|$ is equal to

The value of ${\left(1+\omega -{\omega }^2\right)}^7$ is, where $\omega$ is complex cube root of unity

Let, $z=1-t+i\sqrt{\left(t^2+t+2\right)}$, where $t$ is a real parameter. The locus of $z$ in the Argand plane is

If $\omega$ is imaginary cube root of unity, then $\sin \left\{\left({\omega }^{13}+{\omega }^2\right)\pi +\frac{\pi }{4}\right\}$ is equal to

If $\left|z_1\right|=\left|z_2\right|=\dots \dots \left|z_n\right|=1,$ then the value of $\left|z_1+z_2+\dots \dots z_n\right|-\left|\frac{1}{z_1}+\frac{1}{z_2}+\dots \dots +\frac{1}{z_n}\right|$ is

For all complex numbers $z_1,z_2$ satisfying $\left|z_1\right|=12$ and $\left|z_2-3-4i\right|=5,$ then the minimum value of $\left|z_1-z_2\right|$ is

Complex number $z$ satisfies $\left|z-a+ia\right|=1$ and has the least absolute value. Its absolute value is

$z_1, z_2, z_3$ are the affixes of the vertices of a triangle having it circumcentre at the origin. If $z$ is the affix of its orthocentre, then