`|1alphaalpha1betaF|=(alpha-P)(beta-alpha)`

Without expanding, prove the following |(1,1,1),(alpha,beta,gamma),(betagamma,gammaalpha,alphabeta)|=(beta-alpha)(gamma-alpha)(alpha-beta)

alpha beta x ^(alpha-1) e^(-beta x ^(alpha))

If the curves : alpha x^(2)+betay^(2)=1 and alpha'x^(2)+beta'y^(2)=1 intersect orthogonally, prove that (alpha-alpha')beta beta'=(beta-beta')alpha alpha' .

Prove that |1alphaalpha^2alphaalpha^2 1alpha^2 1alpha|=-(1-alpha^3)^2dot

Prove that |1alphaalpha^2alphaalpha^2 1alpha^2 1alpha|=-(1-alpha^3)^2dot

If |alpha|,|beta| lt 1, s _(1) =1- alpha +alpha^(2) -alpha^(3)+…oo and s_(2)=1-beta +beta ^(2) -beta ^(3)+…oo then (1-alpha beta +alpha^(2) beta ^(2) -alpha ^(3) beta ^(3) +…oo) is equal to

If alpha and beta are roots of x^2+px+2=0 and 1/alpha,1/beta are the roots of 2x^2+2qx+1=0 . Find the value of (alpha-1/alpha)(beta-1/beta)(alpha+1/beta)(beta+1/alpha)

If alpha and beta are roots of the equation x^(2)+px+2=0 and (1)/(alpha)and (1)/(beta) are the roots of the equation 2x^(2)+2qx+1=0 , then (alpha-(1)/(alpha))(beta-(1)/(beta))(alpha+(1)/(beta))(beta+(1)/(alpha)) is equal to :

If the roots of the equation ax^(2)-bx+c=0 are alpha,beta, then the roots of the equation b^(2)cx^(2)-ab^(2)x+a^(3)=0 are (1)/(alpha^(3)+alpha beta),(1)/(beta^(3)+alpha beta) b.(1)/(alpha^(2)+alpha beta),(1)/(beta^(2)+alpha beta) c.(1)/(alpha^(4)+alpha beta),(1)/(beta^(4)+alpha beta) d.none of these