Number Theory T...

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 $$ z = \dfrac {-1}{2} + i \dfrac {\sqrt3}{2} $$, then $$ 8 + 10z + 7z^2 $$ is equal to :

If the complex numbers $$z_1, z_2$$ and $$z_3$$ denote the vertices of an isosceles triangle, right angled at $$z_1$$, then $$(z_1 - z_2)^2 + (z_1 - z_3)^2$$ is equal to

If $$z$$ is a complex number such that $$z + |z| = 8 + 12i$$, then the value of $$|z^{2}|$$ is

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?

If '$$\omega$$' is a complex cube root of unity,then $$\omega ^{ \begin{pmatrix} \frac { 1 }{ 3 }  & +\frac { 2 }{ 9 } +\frac { 4 }{ 27 } ...\infty  \end{pmatrix} }+\omega^{ \begin{pmatrix} \frac { 1 }{ 2 }  & +\frac { 3 }{ 8 } +\frac { 9 }{ 32 } ...\infty  \end{pmatrix} }=$$

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

Let$$ z$$ = $$\cos\theta + i \sin\theta$$. Then the value of $$\sum\limits_{m=1}^15Im( z^{2m-1})$$ at $$\theta = 2^0$$ is 

If $$z_1, z_2$$ are two complex numbers such that $$arg(z_1+z_2)=0$$ and $$Im(z_1z_2)=0$$, then.