Properties of Conjugate, Modulus and Argument of Complex Numbers

If $\arg \left(z\right)<0,$ then $\arg \left(-z\right)-\arg \left(z\right)=$

The value of $\mathrm{amp}\left(i\omega \right)+\mathrm{amp}\left(i{\omega }^2\right)$, where $i=\sqrt{-1}$ and $\omega =\sqrt[3]{1}=$ non-real is

If $\left|z-4+3i\right|\le 1$ and $\alpha$ and $\beta$ are the least and greatest value of $\left|z\right|$ and $k$ be the least value of  $\frac{x^4+x^2+4}{x}$ on the interval $\left(0,\infty \right)$, then $k$ is equal to -

A complex number $z=1-i$.

The argument of $\overline{z}$ is

$arg\left(\frac{1-i\sqrt{3}}{1+i\sqrt{3}}\right)=$

If $z=\frac{1}{{\left(2+3 i\right)}^2}$ then $\left|z\right|=$

If $i{z}^3+z^2-z+i=0$ (where $z$ is a complex number), then the value of $\left|z\right|$ is

The locus of a point $P\left(z\right)$ satisfying $|z+3|+|z –3|=10$ is (where $z$ is a complex number)

Suppose $z$ is any root of $11z^8+21i{z}^7+10iz-22=0$ where $i=\sqrt{-1}.$ Then, $S=|z{|}^2+|z|+1$ satisfies

If $z_1, z_2, z_3, z_4$ be the vertices of a quadrilateral taken in order such that $z_1+z_3=z_2+z_4$ and $\left|z_1-z_3\right|=\left|z_2-z_4\right|$, then $\arg \left(\frac{z_1-z_2}{z_3-z_2}\right)=$

Let $z, w$ be two complex numbers such that $\left|z\right|=1$ and $\frac{w-1}{w+1}={\left(\frac{z-1}{z+1}\right)}^2$. Then maximum value of $\left|w+1\right|$ is

The modulus of the complex number $\frac{(1-i\sqrt{3})(\cos \theta +i\sin \theta )}{(1+i)(\cos \theta -i\sin \theta )}$ is

Statement I  Both $z_1$ and $z_2$ are purely real , if  $arg (z_1 z_2) = 2\pi$  ($z_1$ and $z_2$ have principle arguments).
Statement II Principle arguments of complex number lies between $(-\pi , \pi ].$

If $z$ be any complex number such that $\left|3z-2\right|+\left|3z+2\right|=4$, then find the locus of $z$.

The maximum value of $\left|z\right|$, if $\left|z^2-3\right|=3\left|z\right|$, is

If $z_1, z_2$ and $z_3$ are three points lying on the circle $\left|z\right|=2$ then the minimum value of ${\left|z_1+z_2\right|}^2+{\left|z_2+z_3\right|}^2+{\left|z_3+z_1\right|}^2$ is 

Let $\left|z_1\right|=1, \left|z_2\right|=2, \left|z_3\right|=3$ and $z_1+z_2+z_3=3+\sqrt{5}i$, then the value of $Re\left(z_1{\overline{z}}_2+z_2{\overline{z}}_3+z_3{\overline{z}}_1\right)$ is equal to

(where $z_1, z_2$ and $z_3$ are complex numbers)

The minimum value of the expression $\left|3z-3\right|+\left|2z-4\right|$ is equal to (where, $z$ is a complex number)

If $z$ be a complex number satisfying $|z –4 + 8i|=4,$ then the least and the greatest value of $|z + 2|$ are respectively (where $i=\sqrt{-1}$ )

Consider a square $OABC$ in argand plane, where $O$ is origin and $A$ be complex number $z_0$. Then the equation of the circle that can be inscribed in this square is (Vertices of square are given in anticlockwise order and $i=\sqrt{-1}$)