If `x=a+b,y=a gamma+b beta and z=abeta+ b gamma` , where `gamma and beta` are the imaginary cube roots ofunity, 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 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 A(alpha, (1)/(alpha)), B(beta, (1)/(beta)), C(gamma,(1)/(gamma)) be the vertices of a Delta ABC , where alpha, beta are the roots of x^(2)-6ax+2=0, beta, gamma are the roots of x^(2)-6bx+3=0 and gamma, alpha are the roots of x^(2)-6cx + 6 =0 , a, b, c being positive. The value of a+b+c is

alpha and beta are the roots of the equation x^(2)-3x+a=0 and gamma and delta are the roots of the equation x^(2)-12x+b=0 .If alpha,beta,gamma, and delta form an increasing GP., then (a,b)-

x^(3)-x^(2)+beta x+gamma=0 where beta and gamma are real If the roots of the equation are in A.P then the mean than the mean root is

If alpha,beta and gamma are angles such that tan alpha+tan beta+tan gamma=tan alpha tan beta tan gamma and x=cos alpha+i sin alpha,y=cos beta+i sin beta and z=cos gamma+i sin gamma then the value of xyz is