Argand Plane and Polar Representation

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 –

$3\left(\cos 300°-i\sin 30°\right)$, when expressed in polar form, is

$\left(\sin 120°-i\cos 120°\right)$, when expressed in polar form, is

If $n$ is a positive integer, then ${\left(p+iq\right)}^{\frac{1}{n}}+{\left(p-iq\right)}^{\frac{1}{n}}=$

The polar co-ordinates of a point whose cartesian co-ordinates are $\left(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)$ are

If $z=(\cos \theta ,\sin \theta )$, find $\left(z-\frac{1}{z}\right)$

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

The amplitude of $i^{17}$ is

If $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=\left|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}\right|=1, then \left|z_1+z_2+z_3\right|$ is :(where $z_1$,$z_2$ and $z_3$ are three complex numbers)

Find the point of intersection of the curves $\arg \left(z-4i\right)=\frac{3\mathrm{\pi }}{4} \& \arg \left(3z+1-3i\right)=\frac{\pi }{4}$ (where, $z$ is a complex number)

$z_1,z_2$ 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 complex numbers $w_1=1+i, w_2=1-i$ and $w_3=5-2i$ are  representing the points $A,B$ and $C$ respectively on the Argand diagram. The equations given below have one common solution.
$\left|z-w_1\right|=\left|z-w_2\right|$ and $\left|z-w_2\right|=\left|z-w_3\right|$ If that common root is plotted as a point on the Argand diagram, what is the coordinate of that point?

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

The point of intersection of the curves $\arg \left(z-4i\right)=\frac{3\pi }{4},\arg \left(3z+1-3i\right)=\frac{\pi }{4}$ is

If ${\left(1+i\sqrt{3}\right)}^{12}=a+ib$$,$ here $a$ and $b$ are real, then the value of $b$ is

${\mathrm{z}}_1=\mathrm{1}+\mathrm{i}\sqrt{3}$ 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?

If $\left|z\right|\ge 3$ , then the least value of $\left|z+\frac{1}{4}\right|$ is

If $a=\cos \frac{2\pi }{7}+i\sin \frac{2\pi }{7},$ then the quadratic equation whose roots are $\alpha =a+a^2+a^4$ and $\beta =a^3+a^5+a^6$ , is

The locus of $z$ satisfying $\left|z\right|+\left|z-1\right|=3$ is