`|{:(1,alpha,alpha^3),(1,beta,beta^3),(1,gamma,gamma^3):}|=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)`

|{:(1,alpha,alpha^2),(1,beta,beta^2),(1,gamma,gamma^2):}|=(alpha-beta)(beta-gamma)(gamma-alpha)

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)

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

prove that, |{:(sin^2alpha,sin alpha cosalpha,cos^2alpha),(sin^2beta,sinbetacosbeta,cos^2beta),(sin^2gamma,singammacosgamma,cos^2gamma):}|=-sin(alpha-beta)sin(beta-gamma)sin(gamma-alpha)

Evaluate: |[alpha, beta,gamma],[alpha^2,beta^2,gamma^2],[beta+gamma,gamma+alpha,alpha+beta]|

Prove that |{:(betagamma,betagamma'+beta'gamma,beta'gamma'),(gammaalpha,gammaalpha'+gamma'alpha,gamma'alpha'),(alphabeta,alphabeta'+alpha'beta,alpha'beta'):}| =(alphabeta'-alpha'beta)(betagamma'-beta'gamma)(gammaalpha'-gamma'alpha) .

Prove that |((beta+gamma-alpha-delta)^4,(beta+gamma-alpha-delta)^2,1),((gamma+alpha-beta-delta)^4,(gamma+alpha-beta-delta)^2,1),((alpha+beta-gamma-delta)^4,(alpha+beta-gamma-delta)^2,1)|= -64(alpha-beta)(alpha-gamma)(alpha-delta)(beta-delta)(gamma-delta)(gamma-beta)

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+gammadeltaalphabeta(gamma+delta)+gammadelta(alpha+beta)2alphabetagammadelta|=0

If alpha,beta,gamma are the angles of a triangle and system of equations cos(alpha-beta)x+cos(beta-gamma)y+cos(gamma-alpha)z=0 cos(alpha+beta)x+cos(beta+gamma)y+cos(gamma+alpha)z=0 sin(alpha+beta)x+sin(beta+gamma)y+sin(gamma+alpha)z=0 has non-trivial solutions, then triangle is necessarily a. equilateral b. isosceles c. right angled "" d. acute angled

If alpha,beta,gamma are different from 1 and are the roots of a x^3+b x^2+c x+d=0a n d(beta-gamma)(gamma-alpha)(alpha-beta)=(25)/2 , then prove that |alpha/(1-alpha)beta/(1-beta)gamma/(1-gamma)alphabetagammaalpha^2beta^2gamma^2|=(25 d)/(2(a+b+c+d))