Argument of a Complex Number

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

Let $z$and $w$be two non-zero complex numbers such that  $\left|z\right|=\left|w\right|$  and  $\arg z+\arg w=\pi$  then $z$equals –

A complex number $z=1+i\sqrt{3}$.

The general argument of $z$ is

A complex number $z=\sqrt{3}+i$.

The general argument of $z$ is $2n\mathrm{\pi }+\frac{\mathrm{\pi }}{6}$, where $n$ is an integer.

A complex number $z=1-i$.

The argument of $\overline{z}$ is

A complex number $z=3+9i$.

Find the argument of $\overline{z}$.

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

The modulus-amplitude form of $\frac{(1-i{)}^3(2-i)}{(2+i)(1+i)}$ is

The amplitude of $i^{17}$ 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 $(-\pi , \pi ].$

$z_1,z_2$ and $z_3,z_4$ are $2$ pairs of complex conjugate numbers. Find the value of $\text{arg}\left(\frac{{\text{z}}_1}{{\text{z}}_4}\right)+\text{arg}\left(\frac{{\text{z}}_2}{{\text{z}}_3}\right)$.

Let z= $\frac{(2\sqrt{3}+2i{)}^8}{(1-i{)}^6}+\frac{(1+i{)}^6}{(2\sqrt{3}-2i{)}^8}$. Let θ be the argument of z such that θ∈ (– π,π] then 4 sinθ is equal to

If $\left|\begin{array}{ccc}p& q& r\\ q& r& p\\ r& p& q\end{array}\right|$=0, where p, q, r all the moduli of non-zero complex numbers z1, z2, z3, then prove that arg $\left(\frac{z_3}{z_2}\right)$=λ arg $\left(\frac{z_3-z_1}{z_2-z_1}\right)$ find λ

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 $i=\sqrt{-1}$)

${\mathrm{z}}_1=\mathrm{1}+\mathrm{i}\sqrt{3}$ and ${\mathrm{z}}_2=-\mathrm{1}-\mathrm{i}\sqrt{3}$ are two distinct complex number, represented by two distinct points in the argand plane, then find their arguments?

The argument of $\frac{1+i\sqrt{3}}{\sqrt{3}+1}$ is equal to

If $\left|z-25i\right|\le 15$ , then $|$Maximum $arg\left(z\right)-$Minimum $arg\left(z\right)|$ equals -

If P and Q are represented by the complex numbers $z_1$ and $z_2$, such that $\left|\frac{1}{z_2}+\frac{1}{z_1}\right|= \left|\frac{1}{z_2}-\frac{1}{z_1}\right|$, then the circumcenter of $\mathrm{\Delta }OPQ$ (where O is the origin) is

The argument of $\frac{1+i\sqrt{3}}{\sqrt{3}+i}$ is equal to

If $\arg \left(z\right)<0,$then $\arg \left(-z\right)-\arg \left(z\right)$ can be equal to