Let S be the set of complex number a which satisfyndof `log_(1/3) { log_(1/2) (|z|^2+4|z|+3)} lt 0`, then S is (where `i=sqrt-1`)

For a complex z , let Re(z) denote the real part of z . Let S be the set of all complex numbers z satisfying z^(4)- |z|^(4) = 4 iz^(2) , where I = sqrt(-1) . Then the minimum possible value of |z_(1)-z_(2)|^(2) where z_(1),z_(2) in S with Re (z_(1)) gt 0 and Re (z_(2)) lt 0 , is ....

For a complex number z ,let Re(z) denote the real part of z . Let S be the set of all complex numbers z satisfying z^(4)-|z|^(4)=4iz^(2) ,where i=sqrt(-1) . Then the minimum possible value of |z_(1)-z_(2)|^(2) ,where, z_(1),z_(2) in S with Re(z_(1))>0 and Re(z_(2))<0, is

Let s be the set of all complex numbers Z satisfying |z^(2)+z+1|=1 . Then which of the following statements is/are TRUE?

The complex number z which satisfy the equations |z|=1 and |(z-sqrt(2)(1+i))/(z)|=1 is: (where i=sqrt(-1) )

If log_(1/3) |z+1| > log_(1/3) |z-1| then prove that Re(z) < 0.

If log_((1)/(sqrt(3))){(|z|^(2)-|z|+1)/(2+|z|)}gt -2 , then z lies inside

If log_(tan30^@)[(2|z|^(2)+2|z|-3)/(|z|+1)] lt -2 then |z|=