If `z_1` is a root of the equation `a_0z^n+a_1z^(n-1)+........+(a_(n-1)z+a_n=3,` where `|a_1| lt 2` for `i=0,1,.....n,` then (a). `|z|>1/3` (b). `|z| lt 1/4` (c). `|z| gt 1/4` (d). `|z| lt 1/3`

zo is one of the roots of the equation z^n cos theta0+ z^(n-1) cos theta2 +. . . . . . + z cos theta(n-1) + cos theta(n) = 2, where theta in R , then (A) |z0| 1/2 (C) |z0| = 1/2

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

If z_1a n dz_2 are the complex roots of the equation (x-3)^3+1=0,t h e nz_1+z_2 equal to a. 1 b. 3 c. 5 d. 7

Lt a be a complex number such that |a|<1a n dz_1, z_2z_, be the vertices of a polygon such that z_k=1+a+a^2+...+a^(k-1) for all k=1,2,3, T h e nz_1, z_2 lie within the circle (a) |z-1/(1-a)|=1/(|a-1|) (b) |z+1/(a+1)|=1/(|a+1|) (c) |z-1/(1-a)|=|a-1| (d) |z+1/(a+1)|=|a+1|

The common roots of the equation Z^3+2Z^2+2Z+1=0 and Z^1985+Z^100+1=0 are

If Z is an idempotent matrix, then (I+Z)^n I+2^n Z b. I+(2^n-1)Z c. I-(2^n-1)Z d. none of these

If z_(1)=4+5i and z_(2)=-3+2i that (z_(1))/(z_(2)) is :

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

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 .