Complex Numbers...

If $$f\left( z \right) =\dfrac { 1-{ z }^{ 3 } }{ 1-z } $$, where $$z=x+iy$$ with $$z\neq 1$$, then $$Re\overline { \left\{ f\left( z \right)  \right\}  } =0$$ reduces to

Let Z and w be complex numbers. If $$Re(z)=|z-2|, Re(w) = |w-z|$$ and $$arg(z-w)=\dfrac{\pi}{3}$$, then the value of $$Im(z+w)$$, is

If $$\left( \dfrac{1 + i}{1 - i} \right)^m = 1$$, then the least positive integral value of m is

If $$z=1+i$$, then the argument of $${ z }^{ 2 }{ e }^{ z-i }$$ is

If $$p, q$$ are odd integers, then the roots of the equation $$2px^{2} + (2p + q) x + q = 0$$ are

If $$iz^{3} + z^{2} - z + i = 0$$, then $$|z|$$ is equal to

The principal argument of the complex number $$z=\cfrac { 1+\sin { \cfrac { \pi  }{ 3 }  } +i\cos { \cfrac { \pi  }{ 3 }  }  }{ 1+\sin { \cfrac { \pi  }{ 3 }  } -i\cos { \cfrac { \pi  }{ 3 }  }  } $$ is?

The value of $$ \sum _{ k=0 }^{ n }{ (i^k + i^{k+1} ) } , $$ where $$ i^2 = -1 ,$$ is equal to :

If $$z_{1}$$ and $$z_{2}$$ be complex numbers such that $$z_{1} + i(\overline {z_{2}}) = 0$$ and $$arg (\overline {z_{1}}z_{2}) = \dfrac {\pi}{3}$$. Then, $$arg (\overline {z_{1}})$$ is equal to

The inequality $$\left| z-4 \right| <\left| z-2 \right| $$ represents the region given by: