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

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

Find the locus of point z if z ,i ,a n di z , are collinear.

If alpha , beta are the roots of the equation ax^(2)+bx+c=0 and S_(n)=alpha^(n)+beta^(n) , then aS_(n+1)+bS_(n)+cS_(n-1)=(n ge 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 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

Let n in Z and DeltaABC be a right tirangle with angle at C . If sin A and sin B are the roots of the equadratic equation (5n + 8) x^(2) - (7n - 20) x + 120 = 0 , then find the value of n.

Let z_1a n dz_2 b be toots of the equation z^2+p z+q=0, where he coefficients pa n dq may be complex numbers. Let Aa n dB represent z_1a n dz_2 in the complex plane, respectively. If /_A O B=theta!=0a n dO A=O B ,w h e r eO is the origin, prove that p^2=4q"cos"^2(theta//2)dot

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

If alpha_1, ,alpha_2, ,alpha_n are the roots of equation x^n+n a x-b=0, show that (alpha_1-alpha_2)(alpha_1-alpha_3).......(alpha_1-alpha_n)=n(alpha_1^(n-1)+a)

Show that: sum_(r=0)^(n-1)|z_1+alpha^r z_2|^2=n(|z_1|^2+|z_2|^2),w h e r ealpha^r ; r=0,1,2, ,(n-1) , are nth roots of unity and z_1, z_2 are an tow complex numbers.