If `z!=0` is a complex number, then prove that `R e(z)=0 rArr Im(z^2)=0.`

For any complex number z prove that |R e(z)|+|I m(z)|<=sqrt(2)|z|

If z_1,z_2,z_3 are three complex number then prove that z_1Im(barz_2.z_3)+z_2Im(barz_3.z_1)+z_3Im(barz_1.z_2)=0

For any complex number z,prove that |Re(z)|+|I_m(z)|le\sqrt 2 |z|

If |z+i|-|z-1|=|z|-2=0 for a complex number z, then z =

If z_1a n dz_2 are two complex numbers and c >0 , then prove that |z_1+z_2|^2lt=(1+c)|z_1|^2+(1+c^(-1))|z_2|^2dot

The three angular points of a triangle are given by Z=alpha, Z=beta,Z=gamma,w h e r ealpha,beta,gamma are complex numbers, then prove that the perpendicular from the angular point Z=alpha to the opposite side is given by the equation R e((Z-alpha)/(beta-gamma))=0

If Z is a non-real complex number, then find the minimum value of | (Imz^5)/(Im^5z) |

If z_(1), z_(2),z_(3) ,are three complex numbers, prove that z_(1).Im(barz_(2)z_(3))+z_(2).Im(barz_(3)z_(1))+z_(3)Im(bar z_(1)z_(2))=0 where Im(W) = imaginary part of W,where W is a complex number.

If z_(1),z_(2) are two complex numbers , prove that , |z_(1)+z_(2)|le|z_(1)|+|z_(2)|

If z_(1) , z_(2) are complex numbers such that Re(z_(1))=|z_(1)-2| , Re(z_(2))=|z_(2)-2| and arg(z_(1)-z_(2))=pi//3 , then Im(z_(1)+z_(2))=