Modulus and Argument of Complex Numbers

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 –

The number of integer solutions of the equation ${\left|1-i\right|}^x=2^x$ is

Define $f:C\to R$ by $f\left(z\right)=\left|z\right|\forall z\in C.$ Then which of the following is false?

If $z\in C,$ then the minimum value of $\left|z\right|+|2z-3|+|z-1|$ is

For two complex numbers $z_1,z_2$ the relation $|z_1+z_2|=\left|z_1\right|+\left|z_2\right|$ hold, if

In the Argand plane if O, P, Q represent respectively the origin, the complex number z and z + iz, then angle OPQ is

If $\frac{3+i\mathrm{s}\mathrm{i}\mathrm{n}\theta }{4-i\mathrm{c}\mathrm{o}\mathrm{s}\theta },\theta \in \left[0,2\pi \right],$ is a real number, then an argument of $\mathrm{s}\mathrm{i}\mathrm{n}\theta +i\mathrm{c}\mathrm{o}\mathrm{s}\theta$ 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 $(-\pi , \pi ).$

Which of the following is correct for any two complex numbers $z_1$ and $z_2$?

If $\left|z\right|=1, z^{\text{'}}=\frac{1+z^2}{z}$, then :

Let $z$ be a complex number such that $\left|\frac{z-i}{z+2i}\right|=1$ and $\left|z\right|=\frac{5}{2}$ . Then the value of $\left|z+3i\right|$ is

If $z_1, z_2, z_3$represent the vertices of an equilateral triangle such that $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|$, then

If $\left|z\right|=1, z^{\text{'}}=\frac{1+z^2}{z}$, then :

If $P$ represents $z=x+iy$ in the argand plane ${\left|z-1\right|}^2+{\left|z+1\right|}^2=4$, then locus of $P$ is (where $i=\sqrt{-1}$)

If $z_1,z_2$ and $z_3,z_4$ are two pairs of complex conjugate numbers, then $\arg \left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right)$ can be equal to

If $z=x+iy$ and $\left|z-2+2 i\right|=4$ then the locus of $z$ in the complex plane is

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

If $z$ is a purely imaginary number and $\mathrm{Im} \mathrm{z}<0$, then $\mathrm{amp} \mathrm{z}=$

The principle amplitude of $-1-i$ is