Show that the roots of equation `(1+z)^n=(1-z)^n are i tan (rpi)/n, r=0,1,2,,…………,(n-1)` excluding the vlaue when n is even and `r=n/2`

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

Show that the roots of the equation (1+x)^(2n)+(1-x)^(2n)=0 are given by x=+-i tan{(2k-1)(pi)/(2n)) where k=1,2,3,...n

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=

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

Show that the function f:N rarr Z, defined by f(n)=(1)/(2)(n-1) when n is odd ;-1/2 n when n is even is both one-one and onto

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 .

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

If n be integer gt1, then prove that sum_(r=1)^(n-1) cos (2rpi)/n=-1

The distance of the roots of the equation tan theta_(0) z^(n) + tan theta_(1) z^(n-1) + …+ tan theta_(n) = 3 from z=0 , where theta_(0) , theta_(1) , theta_(2),…, theta_(n) in [0, (pi)/(4)] satisfy

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)