If `x=a+b+c`, `y=aalpha+betab+c` and `z=abeta+balpha+c` , where `alpha` and `beta` are the complex cube roots of unity, then xyz=

If x=a+b+c,y=a alpha+beta b+c and z=a beta+b alpha+c, where alpha and beta are the complex cube roots of unity,then xyz=

If x=a+b,y=a gamma+b beta and z=a beta+b gamma, where gamma and beta are the imaginary cube roots ofunity,then xyz=

If p=a+b omega+c omega^(2);q=b+c omega+a omega^(2) and r=c+a omega+b omega^(2) where a,b,c!=0 and omega is the complex cube root of unity,then

If x=p+q,y=p omega+q omega^(2), and z=p omega^(2)+q omega where omega is a complex cube root of unity then xyz=

If alpha be the real cube root of and beta,gamma be the complex cube roots of m, a real positive number,then for any x,y,z show that (x beta+y gamma+z alpha)/(x gamma+y alpha+z beta)=omega^(2), where omega is a complex cube root of unity.

If alpha,beta are the roots of the equation ax^(2)+bx+c=0 and omega,omega^(2) are the complex roots of unity,then omega^(2)a+omega beta)(omega alpha+omega^(2)beta)=