If `z_r=cos(pi/(3_r))+isin(pi/(3_r)),r=1,2,3, ,` prove that `z_1z_2z_3 z_oo=idot`

Solve the equation |z|=z+1+2idot

If z_r =cos (ralpha)/(n^2)+isin (ralpha)/(n^2) ​ , where r=1,2,3,....n, then lim_(n→∞)(z_1.z_2.....z_n) is equal to

Let z_(1),z_(2) and z_(3) be three non-zero complex numbers and z_(1) ne z_(2) . If |{:(abs(z_(1)),abs(z_(2)),abs(z_(3))),(abs(z_(2)),abs(z_(3)),abs(z_(1))),(abs(z_(3)),abs(z_(1)),abs(z_(2))):}|=0 , prove that (i) z_(1),z_(2),z_(3) lie on a circle with the centre at origin. (ii) arg(z_(3)/z_(2))=arg((z_(3)-z_(1))/(z_(2)-z_(1)))^(2) .

If z_1 =-3i, z_2= 3 +4i and z_3=2- 3i , verify that z_1(z_2+z_3)=z_1z_2+z_1z_3 .

If |z_1|=|z_2|= ...... |z_n|=1 , prove that : |z_1+z_2+ ........ z_n|= |1/z_1+1/z_2+..........+1/z_n| .

If |z_(1)| = |z_(2)| = ….. |z_(n)| = 1 , prove that |z_(1) + z_(2) + …….. z_(n)| = |(1)/(z_(1)) + (1)/(z_(2)) + ……….. + (1)/(z_(n))| .

If theta_(i)in[0,pi//6],i=1,2,3,4,5,and sintheta_(1)z^(4)+sin theta_(2)z^(3)+sintheta_(3)z^(2)+sintheta_(4)z+sintheta_(5)=2, show that (3)/(4)lt|Z|lt1.

Let z_1=2- 3i and z_2= 5 + 12i , verify that : |z_2-z_1| >|z_2|-|z_1| .

Let z_1=2- 3i and z_2= 5 + 12i , verify that : |z_1+z_2| <|z_1|+|z_2| .

If x=r sin theta cos phi, y= r sin theta sin phi, z=r cos theta , prove that x^2+y^2+z^2=r^2 .