If `alpha and beta` are imaginary cube roots of unity, then find the value of `alpha^(4) + beta^(4) + 1//(alpha beta)`

If sinalpha=15/17 , cosbeta=12/13 and alpha and beta are in the first quadrant , find the values of sin(alpha+beta) , cos(alpha+beta) , and tan(alpha+beta)

If alpha, beta, gamma are the roots of the equation x^(3)+4x+1=0 , then find the value of (alpha+beta)^(-1) +(beta +gamma)^(-1) + (gamma+ alpha)^(-1)

If alpha, beta are the roots of x^(2)-a(x-1)+b=0 , then find the value of (1)/(alpha^(2)-a alpha)+(1)/(beta^(2)-a beta)+(2)/(a+b)

If alpha and beta are the roots of the equation x^(2)-7x+1=0 , then the value of 1/(alpha-7)^(2)+1/(beta-7)^(2) is :

If alpha and beta are the roots of 4x^(2) + 2x - 1 = 0 , then beta is equal to

If alpha, beta , gamma are the cube roots of unity, then the value of the determinant |{:(e^a,e^(2a),(e^(3a)-1)), (e^beta,e^(2 beta), (e^(2 beta)-1)), (e^gamma,e^(2gamma), (e^(3 gamma)-1)):}| is equal to