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

Find the argument of $\left(-1-2\sqrt{3}i\right)+\left(2+3\sqrt{3}i\right)$.

Find the argument of $\left(-2-2i\right)+\left(1+3i\right)$.

Find the argument of $\left(-2+2i\right)+\left(1-3i\right)$.

Find the argument of $\left(2+2i\right)+\left(-1-3i\right)$.

Find the argument of $\left(-1-2i\right)+\left(2+3i\right)$.

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 -

Find the modulus of conjugate of given complex number.

$z=3+2i$

Find the modulus of conjugate of given complex number.

$z=-2+3i$

Find the modulus of conjugate of given complex number.

$z=-2-3i$

Find the modulus of conjugate of given complex number.

$z=2-3i$

Find the modulus of conjugate of given complex number.

$z=2+3i$

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 $(-\pi , \pi ).$

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

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 :

Let $z_1=10+6 i, z_2=4+6 i$. If $z$ is any complex number such that, $\arg \left(\frac{z-z_1}{z-z_2}\right)=\frac{\pi }{4}$, then prove that $|z-7-9 i|=3\sqrt{2}$.

$w=\frac{z+1}{z-1}$ where $z$ is a complex number. If $z$ lies on the circle $|z-i|=2$ show that $w$ lies on a circle in the complex plane.