The value of `|{:(betagamma,betagamma'+beta'gamma,beta'gamma'),(gammaalpha,gammaalpha'+gamma'alpha,gamma'alpha'),(alphabeta,alphabeta'+alpha'beta,alpha'beta'):}|` is

If alpha,beta,gamma are the roots of px^(2)+qx^(2)+r=0 then the value of the determinant det[[alpha beta,beta gamma,gamma alphabeta gamma,gamma alpha,alpha betagamma alpha,alpha beta,beta gamma]] is p b.q c.0 d.r

Prove that |{:(2,alpha+beta+gamma+delta,alphabeta +gammadelta),(alpha+beta+gamma+delta,2(alpha+beta)(gamma+delta),alphabeta(gamma+delta)+gammadelta(alpha+beta)),(alphabeta+gammadelta,alphabeta(gamma+delta)+gammadelta(alpha+beta),2alphabetagamma delta):}| =0

Prove that |2alpha+beta+gamma+deltaalphabeta+gammadeltaalpha+beta+gamma+delta2(alpha+beta)(gamma+delta)alphabeta(gamma+delta)+gammadelta(alpha+beta)alphabeta+gammadeltaalphabeta(gamma+delta)+gammadelta(alpha+beta)2alphabetagammadelta|=0

Prove the following : |{:(alpha,alpha^(2),beta+gamma),(beta,beta^(2),gamma+alpha),(gamma,gamma^(2),alpha+beta):}|=|{:(alpha,beta, gamma),(alpha^(2),beta^(2),gamma^(2)),(beta+gamma,gamma+alpha, alpha+beta):}|=(beta-gamma)(gamma-alpha)(alpha-beta)(alpha+beta+gamma)

Prove that |[alpha,beta,gamma] ,[alpha^2,beta^2,gamma^2] , [beta+gamma, gamma+alpha, beta+alpha]| = (alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)

Prove that: | alpha beta gamma alpha^(2)beta^(2)gamma^(2)beta+gamma gamma+alpha alpha+beta|=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)

|[1,alpha,alpha^3],[1,beta,beta^3],[1,gamma,gamma^3]|=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+ beta+gamma)