If`z_(1),z_(2)inC,z_(1)^(2)+z_(2)^(2)inR,z_(1)(z_(1)^(2)-3z_(2)^(2))=2` and `z_(2)(3z_(1)^(2)-z_(2)^(2))=11,` the value of `z_(1)^(2)+z_(2)^(2)` is

If |z_(1)| = 1, |z_(2)| =2, |z_(3)|=3, and |9z_(1)z_(2) + 4z_(1)z_(3)+z_(2)z_(3)|= 12 , then find the value of |z_(1) + z_(2) + z_(3)| .

If |z_(1)|=|z_(2)|andamp(z_(1))+amp(z_(2))=0, then

If |z_(1)|=2,|z_(2)|=3,|z_(3)|=4and|z_(1)+z_(2)+z_(3)|=5. "then"|4z_(2)z_(3)+9z_(3)z_(1)+16z_(1)z_(2)| is

If z_1,z_2 in C , show that (z_1+ z_2)^2= z_1^2+2 z_1 z_2+ z_2^2 .

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)|=|z_(2)| and arg (z_(1)//z_(2))=pi, then z 1 ​ +z 2 ​ is equal to

If x_(1) = 3y_(1) + 2y_(2) -y_(3), " " y_(1)=z_(1) - z_(2) + z_(3) x_(2) = -y_(1) + 4y_(2) + 5y_(3), y_(2)= z_(2) + 3z_(3) x_( 3)= y_(1) -y_(2) + 3y_(3)," " y_(3) = 2z_(1) + z_(2) espress x_(1), x_(2), x_(3) in terms of z_(1) ,z_(2),z_(3) .

If arg (z^(1//2))=(1)/(2)arg(z^(2)+barzz^(1//3)), find the value of |z|.

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

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