If a,b and c are complex numbers and z satisfies `az^(2)+bz+c=0`, prove that `abs(a)abs(b)=sqrt(a(bar(b))^(2)c)` and `abs(a)=abs(c) Leftrightarrow abs(z)=1`.

Numbers of complex numbers z, such that abs(z)=1 and abs((z)/bar(z)+bar(z)/(z))=1 is

a,b,c are positive numbers. Prove that, a^(2) + b^(2) + c^(2) gt ab + bc + ca

If alpha and beta are roots of the equation acostheta+bsintheta=c then prove that, sin(alpha+beta)=(2ab)/(a^(2)+b^(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) .

For every real number c ge 0, find all complex numbers z which satisfy the equation abs(z)^(2)-2iz+2c(1+i)=0 , where i=sqrt(-1) and passing through (-1,4).

If A and B are two matrices of the same order , then (AB) =A'B.

If a,b,c are real numbers such that 3(a^(2)+b^(2)+c^(2)+1)=2(a+b+c+ab+bc+ca) , than a,b,c are in

If z is any complex number satisfying abs(z-3-2i) le 2 , where i=sqrt(-1) , then the minimum value of abs(2z-6+5i) , is

If z=x+iy and x^(2)+y^(2)=16 , then the range of abs(abs(x)-abs(y)) 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