Prove that there exists no complex number z such that `|z| < 1/3 and sum_(n=1)^n a_r z^r =1`, where `| a_r|< 2`.

If z is any complex number such that |z+4|lt=3, then find the greatest value of |z+1|dot

Find the Area bounded by complex numbers arg|z|lepi/4 and |z-1|lt|z-3|

For any complex number z prove that |R e(z)|+|I m(z)|<=sqrt(2)|z|

Let z and omega be two complex numbers such that |z|lt=1,|omega|lt=1 and |z-iomega|=|z-i bar omega|=2, then z equals (a) 1ori (b). ior-i (c). 1or-1 (d). ior-1

Let za n dw be two nonzero complex numbers such that |z|=|w|a n d arg(z)+a r g(w)=pidot Then prove that z=- bar w dot

Let z_1z_2, z_3 z_n be the complex numbers such that |z_1|=|z_2||z_n|=1. If z=(sum_(k=1)^n z_k)(sum_(k=1)^n1/(z_k)) then proves that z is a real number

Find the complex number z satisfying R e(z^2) =0,|z|=sqrt(3.)

If z_1a n dz_2 are two complex numbers such that |z_1|=|z_2|a n d arg(z_1)+a r g(z_2)=pi , then show that z_1,=-( barz )_2dot

If z_(1) , z_(2) are two complex numbers such that |(z_(1)-z_(2))/(z_(1)+z_(2))|=1 and iz_(1)=Kz_(2) , where K in R , then the angle between z_(1)-z_(2) and z_(1)+z_(2) is