Solve the equation `z^2 +|z|=0` , where z is a complex number.

Number of solutions of the equation z^(2)+|z|^(2)=0 , where z in C , is

The equation z^(2)-i|z-1|^(2)=0, where i=sqrt(-1), has.

Let z_(1)andz_(2) be roots of the equation z^(2)+pz+q=0, where the coefficients p and q may be complex numbers . Let A and B represent z_(1)andz_(2) in the complex plane. If angleAOB=alphane0andOA=OB, where O is the origin, prove that p^(2)=4cos^(2)(alpha//2).

Solve the system of equation Re(z^(2))=0,|z|=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).

State true of false for the following: If z is a complex number such that z ne0 and Re(z)=0 then 1_(m)(z^(2))=0

Find the circumcentre of the triangle whose vertices are given by the complex numbers z_(1),z_(2) and z_(3) .

If z_1 and z_2 are two complex number such that |(z_1-z_2)/(z_1+z_2)|=1 , Prove that iz_1/z_2=k where k is a real number Find the angle between the lines from the origin to the points z_1 + z_2 and z_1-z_2 in terms of k

Show that the area of the triagle on the argand plane formed by the complex numbers Z, iz and z+iz is (1)/(2)|z|^(2)," where "i=sqrt(-1).

if omega is the nth root of unity and Z_1 , Z_2 are any two complex numbers , then prove that . Sigma_(k=0)^(n-1)| z_1+ omega^k z_2|^2=n{|z_1|^2+|z_2|^2} where n in N