Complex Number

If $\left|\begin{array}{ccc}6i& -3i& 1\\ 4& 3i& -1\\ 20& 3& i\end{array}\right|=x+iy,$ then

If $\left|z\right|<4,$ then $\left|iz+3-4i\right|$ is less than

If $\left|z+4\right|\le 3,$ then the greatest and the least value of $\left|z+1\right|$ are

The minimum value of $\left|z\right|+|z-i|$ is

If $a=\cos \alpha +i\sin \alpha , b=\cos \beta +i\sin \beta , c=\cos \gamma +i\sin \gamma$ and $\frac{b}{c}+\frac{c}{a}+\frac{a}{b}=1,$ then $\cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta )$ is equal to 

For any two non zero complex numbers $z_1, z_2,$ the value of $\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right|$ is

If $\alpha , \beta$ are the roots of the equation $a{x}^2+bx+c=0$ and $S_n={\alpha }^n+{\beta }^n,$ then $a{S}_{n+1}+c{S}_{n-1}=$

For all complex numbers $z_1,z_2$ satisfying $\left|z_1\right|=12$ and $\left|z_2-3-4i\right|=5,$ then the minimum value of $\left|z_1-z_2\right|$ is

If $z$ is any complex number satisfying $\left|z-1\right|=1,$ then which of the following is correct?

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

If $\left|\frac{z_1}{z_2}\right|=1$ and $\arg \left(z_1{z}_2\right)=0,$ then

If $a>0, a\in R, z=a+2i$ and $z\left|z\right|-az+1=0$, then

${\left[i^{19}+{\left(\frac{1}{i}\right)}^{25}\right]}^2$ equals

$\sqrt{-1-\sqrt{-1-\sqrt{-1- ...\infty }}}$ equals

The value of the expression $x^4-8x^3+18x^2-8x+2$ when $x=2+\sqrt{3}$ is

If $\left|z_1\right|=\left|z_2\right|=\dots .=\left|z_n\right|=1,$ then $\left|\frac{1}{z_1}+\frac{1}{z_2}+\dots \dots +\frac{1}{z_n}\right|$.

The equation $Z^3+iZ-1=0$ has

If $x+iy=\frac{3}{2+\cos \theta +i\sin \theta },$ then $x^2+y^2$ is equal to

The multiplicative inverse of $z=3-2i$ is

For any complex number $z,$ the minimum value of $\left|z\right|+\left|z-1\right|$ is