For any two complex numbers `z_1` and `z_2`, prove that `Re (z_1 z_2) = Re z_1 Re z_2 - Imz_1 Imz_2`

Define addition and multiplication of two complex numbers z_1 and z_2 . Hence show that : R_e (z_1 z_2)=R_e (z_1)R_e (z_2)- I_m (z_1)I_m(z_2) .

Define addition and multiplication of two complex numbers z_1 and z_2 . Hence show that : R_e (z_1+ z_2)=R_e (z_1)+R_e (z_2) .

Define addition and multiplication of two complex numbers z_1 and z_2 . Hence show that : I_m (z_1 z_2)=R_e (z_1)I_m (z_2)- I_m (z_1)R_e(z_2) .

Define addition and multiplication of two complex numbers z_1 and z_2 . Hence show that : I_m (z_1+ z_2)=I_m (z_1)+I_m(z_2) .

If z_1 and z_2 (ne 0) are two complex numbers, prove that : |z_1 z_2|= |z_1||z_2| .

If two complex numbers z_1,z_2 are such that |z_1|= |z_2| , is it then necessary that z_1 = z_2 ?

For any two complex numbers z_(1) "and" z_(2) , abs(z_(1)-z_(2)) ge {{:(abs(z_(1))-abs(z_(2))),(abs(z_(2))-abs(z_(1))):} and equality holds iff origin z_(1) " and " z_(2) are collinear and z_(1),z_(2) lie on the same side of the origin . If abs(z-(1)/(z))=2 and sum of greatest and least values of abs(z) is lambda , then lambda^(2) , is

For any two complex numbers z_(1) "and" z_(2) , abs(z_(1)-z_(2)) ge {{:(abs(z_(1))-abs(z_(2))),(abs(z_(2))-abs(z_(1))):} and equality holds iff origin z_(1) " and " z_(2) are collinear and z_(1),z_(2) lie on the same side of the origin . If abs(z-(2)/(z))=4 and sum of greatest and least values of abs(z) is lambda , then lambda^(2) , is

For any two complex numbers z_(1) "and" z_(2) , abs(z_(1)-z_(2)) ge {{:(abs(z_(1))-abs(z_(2))),(abs(z_(2))-abs(z_(1))):} and equality holds iff origin z_(1) " and " z_(2) are collinear and z_(1),z_(2) lie on the same side of the origin . If abs(z-(3)/(z))=6 and sum of greatest and least values of abs(z) is 2lambda , then lambda^(2) , is