The set `{R e((2i z)/(1-z^2)): z is a comp l e x n u mber ,|z|=1,z=+-1}` is________.

If w=alpha+ibeta, where beta!=0 and z!=1 , satisfies the condition that ((w- barw z)/(1-z)) is a purely real, then the set of values of z is |z|=1,z!=2 (b) |z|=1a n dz!=1 z= z (d) None of these

Let x_(1)" and "y_(1) be real numbers. If z_(1)" and "z_(2) are complex numbers such that |z_1|=|z_2|=4 , then |x_(1)z_(1)-y_(1)z_(2)|^(2)+|y_(1)z_(1)+x_(1)z_(2)|^(2)=

a. Let z_(1),z_(2) and z_(3) be complex numbers such that |z_(1)|=|z_(2)|=|z_(3)|=rgt0 and z_(1)+z_(2)+z_(3)!=0 . Prove that |(z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1))/(z_(1)+z_(2)+z_(3))|=r b. Find all cube roots of sqrt(3)+i .

Let z_1 and z_2 be two distinct complex numbers and let z=(1-t)z_1+t z_2 for some real number t with 0 (a) |z-z_1|+|z-z_2|=|z_1-z_2| (b) arg(z-z_1)=arg(z-z_2) (c) (z-z_1) bar(z_2-z_1) = bar (z-z_1)(z_2-z_1) (d) a r g(z-z_1)=a r g(z_2-z_1) .

If |z|=1 and z'=(1+z^(2))/(z) , then

If x+y+z=12a n dx^2+y^2+z^2=96a n d1/x+1/y+1/z=36 , then the value x^3+y^3+z^3 divisible by prime number is________.

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

Let |z|=2a n dw-(z+1)/(z-1),w h e r ez ,w , in C (where C is the set of complex numbers). Then product of least and greatest value of modulus of w is__________.

Let z be not a real number such that (1+z+z^2)//(1-z+z^2) in R , then prove tha |z|=1.

If (2z_1)/(3z_2) is purely imaginary then |(z_(1)-z_(2))/(z_(1)+z_(2))|