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 alpha, beta, gamma are the roots of the equatiion x^(3)-px^(2)+qx-r=0 find sumalpha^(2) beta

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 alpha, beta gamma are the real roots of the equation x^(3)-3px^(2)+3qx-1=0 , then find the centroid of the triangle whose vertices are (alpha, (1)/(alpha)), (beta, (1)/(beta)) and (gamma, (1)/(gamma)) .

If alpha,beta,gamma are the roots of x^(3)+2x^(2)-x-3=0 The value of |{:(alpha, beta ,gamma),(gamma,alpha ,beta),(beta,gamma ,alpha):}| is equal to

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

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

If alpha, beta are the roots of the equationn x^(2)-3x+5=0 and gamma, delta are the roots of the equation x^(2)+5x-3=0 , then the equation whose roots are alpha gamma+beta delta and alpha delta+beta gamma is

If alpha, beta are the roots of the quadratic equation x^2 + bx - c = 0 , the equation whose roots are b and c , is a. x^(2)+alpha x- beta=0 b. x^(2)-[(alpha +beta)+alpha beta]x-alpha beta( alpha+beta)=0 c. x^(2)+[(alpha + beta)+alpha beta]x+alpha beta(alpha + beta)=0 d. x^(2)+[(alpha +beta)+alpha beta)]x -alpha beta(alpha +beta)=0

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)

Prove that |[[alpha,beta,gamma],[alpha^2,beta^2,gamma^2],[alpha^3,beta^3,gamma^3]]|= alpha beta gamma (alpha-beta)(beta-gamma)(gamma-alpha)