Complex Number

If $z_1, z_2, \ldots . . z_n$ are complex numbers such that $\left|z_1\right|=\left|z_2\right|=\ldots . .=\left|z_n\right|=1$, then $\left|z_1+z_2+\ldots . .+z_n\right|$ is equal to

If $\left|z_1\right|=2,\left|z_2\right|=3,\left|z_3\right|=4$ and $\left|2 z_1+3 z_2+4 z_3\right|=$ 4 , then absolute value of $8 z_2 z_3+27 z_3 z_1+$ $64 z_1 z_2$ equals

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

$i^{57}+\frac{1}{i^{25}}$, when simplified has the value

The smallest positive integral value of $n$ such that $\left[\frac{1+\sin \frac{\pi}{8}+i \cos \frac{\pi}{8}}{1+\sin \frac{\pi}{8}-i \cos \frac{\pi}{8}}\right]^n$ is purely imaginary, is equal to

If $|w|=2$, then the set of points $z=w-\frac{1}{w}$ is contained in or equal to the set of points $z$ satisfying

$\text { If } z_{1}=\sqrt{2}\left[\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right] \text { and } z_{2}=\sqrt{3}\left[\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right]$ then $\left|z_{1} z_{2}\right|$ is equal to $\sqrt{m}$.…

If $z$ is a complex number such that $z^{2}=(\bar{z})^{2}$, then

If $\left|z_{1}\right|=\left|z_{2}\right|=\ldots \ldots \ldots . .\left|z_{n}\right|=1,$ then the value of $\left|z_{1}+z_{2}+\ldots \ldots . z_{n}\right|-$ $\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\ldots \ldots…

If ${z}_{1}$ and ${\vec{z}}_{1}$ represents adjacent vertices of a regular polygon of $n$ sides and if $\frac{Im({z}_{1})}{Re({z}_{1})}=\sqrt{2}-1$, then $n$ is equal to

The value of $λ$ for which the loci $\text{arg} z=\frac{π}{6}$ and $|z-2\sqrt{3}i|=λ$ on the argand plane touch each other is

The ${n}^{\text{th} }$ roots of unity are in