Show that the equation `a z^3+b z^2+ barb z+ bara =0` has a root `alpha` such that `|alpha|=1,a ,b ,za n dalpha` belong to the set of complex numbers.

The equation az^(3) + bz^(2) + barbz + bara = 0 has a root alpha , where a, b,z and alpha belong to the set of complex numbers. The number value of |alpha|

If alpha is a complex constant such that alpha^(2)+z+bar(alpha)=0 has a real root then

The roots z_(1),z_(2),z_(3) of the equation z^(3)+3 alpha z^(2)+3 beta z+gamma=0 correspond to the points A,B and C on the complex plane.Find the complex number representing the centroid of the triangle ABC, and show that the triangle is equilateral if alpha^(2)=beta

If a cos alpha+i sin alpha and the equation az^(2)+z+1=0 has a pure imaginary root, then tan alpha=

If alpha is a complex constant such that alpha z^(2)+z+bar(alpha)=0 has a real root,then (a) alpha+bar(alpha)=1 (b) alpha+bar(alpha)=0( c) alpha+bar(alpha)=-1 (d)the absolute value of the real root is 1

If the complex numbers z_1,z_2,z_3 represents the vertices of a triangle ABC, where z_1,z_2,z_3 are the roots of equation z^3+3alphaz^2+3alphaz^2+3betaz+gamma=0, alpha,beta,gamma beng complex numbers and alpha^2=beta then /_\ABC is (A) equilateral (B) right angled (C) isosceles but not equilateral (D) scalene

A root of unity is a complex number that is a solution to the equation,z^(n)=1 for some positive integer nNumber of roots of unity that are also the roots of the equation z^(2)+az+b=0, for some integer a and b is

z_(1) and z_(2) are the roots of the equaiton z^(2) -az + b=0 where |z_(1)|=|z_(2)|=1 and a,b are nonzero complex numbers, then

The system of equations alpha x-y-z=alpha-1 , x-alpha y-z=alpha-1,x-y-alpha z=alpha-1 has no solution if alpha

The range of real number alpha for which the equation z+alpha|z-1|+2i=0 has a solution is :