Using properties of determinants, prove the following: `|alphabetagammaalpha^2beta^2gamma^2beta+gammagamma+alphaalpha+beta|=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)`

Using properties of determinants 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 )

If alpha+beta+gamma=2 pi, then

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)

If alpha, beta, gamma are the roots of the cubic x^(3)-px^(2)+qx-r=0 Find the equations whose roots are (i) beta gamma +1/(alpha), gamma alpha+1/(beta), alpha beta+1/(gamma) (ii) (beta+gamma-alpha),(gamma+alpha-beta),(alpha+beta-gamma) Also find the valueof (beta+gamma-alpha)(gamma+alpha-beta)(alpha+beta-gamma)

If cos ^(-1) alpha+cos ^(-1) beta+cos ^(-1) gamma=3 pi then alpha(beta+gamma)+beta(gamma+alpha)+gamma(alpha+beta)=

cos alpha sin (beta-gamma)+cos beta sin (gamma-alpha) +cos gamma(sin alpha-beta)=

Reflection of the point (alpha, beta, gamma) in XY plane is

Reflexion of the point (alpha, beta, gamma) in XY plane is

Distance of the point (alpha, beta, gamma) from the y-axis is

If A=[(alpha, beta),(gamma, -alpha)] is such that A^(2)=I , then