Number Theory T...

If $$z_1, z_2, z_3$$ are three points lying on the circle |z| =2, then the minimum value of $$|z_1 + z_2|^2 + | z_2 + z_3|^2 + | z_3 + z_1|^2$$ is equal to

The modulus of $$\overline { 6+{ i }^{ 3 } } +\overline { 6+{ i } }+\overline { 6+{ i }^{ 2 } } $$ is

If the expression $${(1+ir)}^{3}$$ is of the form of $$s(1+i)$$ for some real $$s$$ where $$r$$ is also real and $$i=\sqrt{-1}$$, then the value of $$r$$ can be

If $$\tan^{-1}(\alpha+ i\beta) = x+iy,$$ then $$x$$ is equal to

If $$(\dfrac{3-z_{1}}{2-z_{1}})(\dfrac{2-z_{2}}{3-z_{2}})=k$$, then point $$A(z_{1}, z_{2}),  C(3, 0)$$ and $$D(2, 0)$$ (taken in clockwise sense ) will

If $$\left| z \right| =1$$ and $$\left| \omega -1 \right| =1$$ where $$z,\omega \in C$$ then the largest set of values of $${ \left| 2z-1 \right|  }^{ 2 }+{ \left| 2\omega -1 \right|  }^{ 2 }$$ equals 

If $$\dfrac {3+2i \sin x}{1-2i \sin x}$$ is purely imaginary then $$x=$$ ?

If $$\alpha$$ and $$\beta$$ are different complex number with $$|\beta|=1$$, then $$\left |\dfrac {\beta-\alpha}{1-\overline {\alpha }\beta}\right|$$ is equal to

Which of the following pairs are twin primes?

If $$z^4+1=\sqrt{3}$$i then?