If `z=x+iy (x, y in R, x !=-1/2)`, the number of values of z satisfying `|z|^n=z^2|z|^(n-2)+z |z|^(n-2)+1.` `(n in N, n>1)` 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

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

If z= x+iy find the locus of z if |2z+1| = 2 .

Find the number of complex numbers which satisfies both the equations |z-1-i|=sqrt(2)a n d|z+1+i|=2.

If n in N >1 , then the sum of real part of roots of z^n=(z+1)^n is equal to n/2 b. ((n-1))/2 c. n/2 d. ((1-n))/2

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

If n is a positive integer, prove that |I m(z^n)|lt=n|Im(z)||z|^(n-1.)

If n >1 , show that the roots of the equation z^n=(z+1)^n are collinear.

If |z|=1a n dz!=+-1, then all the values of z/(1-z^2) lie on (a)a line not passing through the origin (b) |z|=sqrt(2) (c)the x-axis (d) the y-axis