Prove that none of the roots of the equation `z^(n) = 2(1+ z+z^2 + …...+ z^(n-1)), n gt 1`, lies outside the circle `|z|=3`.

The roots of the equation z^(n)=(z+3)^(n)

Prove that, for integral value of n ge1 , all the roots of the equation nz^(n) =1 + z+ z^2 +….+z^(n) lie within the circle |z|=(n)/(n-1) .

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

If nin Ngt1, find the sum of real parts of the roots of the equation z^(n)=(z+1)^(n).

If |z-1| + |z + 3| le 8 , then prove that z lies on the circle.

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 z_(1),z_(2),z_(3),…,z_(n-1) are the roots of the equation z^(n-1)+z^(n-2)+z^(n-3)+…+z+1=0 , where n in N, n gt 2 and omega is the cube root of unity, then

If z_1,z_2,z_3,………..z_(n-1) are the roots of the equation 1+z+z^2+…….+z^(n-1)=0, where n epsilon N, ngt2 then (A) z_1,z_2, …z_(n-1) are terms of a G.P. (B) z_1,z_2,……,z_(n-1) are terms of an A.P. (C) |z_1|=|z_2|=|z_3|=.|z_(n-1)|!=1 (D) none of these

The roots of equation z^(n)=1 are z_(0),z_(1), z_(2),...,,z_(n-1) taken in anticlockwise sense on the complex plane with z_(0)=1 ,if the angle subtended by z_(2) and z_(n-2) at the origin is pi/4 ,then n=

Let 1, z_(1),z_(2),z_(3),…., z_(n-1) be the nth roots of unity. Then prove that (1-z_(1))(1 - z_(2)) …. (1-z_(n-1))= n . Also,deduce that sin .(pi)/(n) sin.(2pi)/(pi)sin.(3pi)/(n)...sin.((n-1)pi)/(n) = (pi)/(2^(n-1))