If `|z/| barz |- barz |=1+|z|,` then prove that `z` is a purely imaginary number.

Z_1!=Z_2 are two points in an Argand plane. If a|Z_1|=b|Z_2|, then prove that (a Z_1-b Z_2)/(a Z_1+b Z_2) is purely imaginary.

If z_1, z_2 are two complex numbers (z_1!=z_2) satisfying |z1^2-z2^2|=| z 1^2+ z 2 ^2-2( z )_1( z )_2| , then a. (z_1)/(z_2) is purely imaginary b. (z_1)/(z_2) is purely real c. |a r g z_1-a rgz_2|=pi d. |a r g z_1-a rgz_2|=pi/2

Consider the equation 10 z^2-3i z-k=0,w h e r ez is a following complex variable and i^2=-1. Which of the following statements ils true? (a) For real complex numbers k , both roots are purely imaginary. (b) For all complex numbers k , neither both roots is real. (c) For all purely imaginary numbers k , both roots are real and irrational. (d) For real negative numbers k , both roots are purely imaginary.

If (z - 1)/(z + 1) is purely imaginary, then |z| is

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

Prove that z=i^i, where i=sqrt-1, is purely real.

If z=x+i ya n dw=(1-i z)//(z-i) , then show that |w|=1 z is purely real.

If arg(z - 1) = pi/6 and arg(z + 1) = 2 pi/3 , then prove that |z| = 1 .