Complex Numbers...

Solve $$i^{57}+\dfrac{1}{i^{125}}$$

If $$z=(3+7i)(p+iq)$$ where $$p,q\in I-\left\{ 0 \right\} $$, is purely imaginary then minimum value of $${ \left| z \right|  }^{ 2 }$$ is

The value of $$2x^{4}+5x^{3}+7x^{2}-x+41$$, when $$x=-2-\sqrt{3i}$$ is:

If $$Re(\dfrac{z+2i}{z+4})=0$$ then z lies on a circle with center:

If $$z_{1}=8 +4i,\ z_{2}=6+4i$$ and $$arg \left(\dfrac {z-z_{1}}{z-z_{2}}\right)=\dfrac {\pi}{4}$$, then $$z$$ satisfy 

The argument of the complex number $$\sin \dfrac{{6\pi }}{5} + i\left( {1 + \cos \dfrac{{6\pi }}{5}} \right)$$ is 

If $$a, b \notin R$$, then $$|e^{a + ib}| $$ is equal to

If $$z_{1} and z_{2} are on straight line$$ $$\left| \frac { 1 } { 2 } \left( z _ { 1 } + z _ { 2 } \right) + \sqrt { z _ { 1 } z _ { 2 } } \right| + \left| \frac { 1 } { 2 } \left( z _ { 1 } + z _ { 2 } \right) - \sqrt { z _ { 1 } z _ { 2 } } \right| =$$

If $$z_{1}$$ and $$z_{2}$$ two non-zero complex number such that $$|z_{1}+z_{2}|=|z_{1}|+|z_{2}|$$, then $$arg z_{1}-arg z_{2}$$ is equal to

$$z$$ is a complex number. If $$a = | x | + | y |$$ and
$$b = \sqrt { 2 } | x + i y |$$ then which of the following is
true