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_i| lt 2` for `i=0,1,.....n,` then (a). `|z|=3/2` (b). `|z| lt 1/4` (c). `|z| gt 1/4` (d). `|z| > 1/3`

If 8iz^3+12z^2-18z+27i=0, then (a). |z|=3/2 (b). |z|=2/3 (c). |z|=1 (d). |z|=3/4

zo is one of the roots of the equation z^n cos theta_0+ z^(n-1) cos theta_1 +. . . . . . + z cos theta_(n-1) + cos theta_(n) = 2 , where theta in R , then (A) |z_0| lt 1/2 (B) |z_0| gt 1/2 (C) |z_0| = 1/2 (D)None of these

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

If one root of the equation z^2-a z+a-1= 0 is (1+i), where a is a complex number then find the root.

If z_1,z_2,z_3 are any three roots of the equation z^6=(z+1)^6, then arg((z_1-z_3)/(z_2-z_3)) can be equal to

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

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

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/z_2|=1 and arg (z_1z_2)=0 , then a. z_1 = z_2 b. |z_2|^2 = z_1*z_2 c. z_1*z_2 = 1 d. none of these

Let a be a complex number such that |a|<1a n dz_1, z_2,z_3,... 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|