If `S_r = alpha^r + beta^r + gamma^r` then show that `|[S_0,S_1,S_2],[S_1,S_2,S_3],[S_2,S_3,S_4]| = (alpha-beta)^2(beta-gamma)^2(gamma-alpha)^2`

if alpha,beta,gamma are the roots of x^3-3x^2 +3x + 7 =0 then (alpha-1)/(beta-1)+(beta-1)/(gamma-1)+(gamma-1)/(alpha-1)

Answer each question by selecting the proper alternative from those given below each question so as to make the statement true : If alpha, beta" and "gamma are the zeroes of the cubic polynomial ax^(3)+bx^(2)+cx+d , then (1)/(alpha beta )+(1)/(beta gamma)+(1)/(gamma alpha)="..........."

If alpha, beta, gamma are the roots of the cubic equation x^(3)+qx+r=0 then the find equation whose roots are (alpha-beta)^(2),(beta-gamma)^(2),(gamma-alpha)^(2) .

If 102!=2^(alpha)*3^(beta)*5^(gamma)*7^(delta) …, then

If alpha+beta+gamma=2pi , then show that tan.alpha/2 + tan.beta/2 + tan.gamma/2 = tan.alpha/2 tan.beta/2 tan.gamma/2 .

If alpha,beta,gamma are the roots of x^3-x^2-1=0 then the value of (1+alpha)//(1-alpha)+(1+beta)//(1-beta)+(1+gamma)//(1-gamma) is equal to

Using properties of determinants in Exercise 11 to 15 prove that |{:(alpha,alpha^2,beta+gamma),(beta,beta^2,gamma+alpha),(gamma,gamma^2,alpha+beta):}|=(beta-gamma)(gamma-alpha)(alpha+beta+gamma)(alpha-beta)

Let alpha,beta "and " gamma are three distinct roots of |{:(x-1,-6,2),(-6,x-2,-4),(2,-4,x-6):}| =0 the value of ((1)/(alpha)+(1)/(beta)+(1)/(gamma))^(-1) is

cos^(2) alpha +cos^(2) beta +cos^(2) gamma is equal to