Show that `|{:(x-3,x-4,x-alpha),(x-2,x-3,x-beta),(x-1,x-2,x-gamma):}|=0,where" "alpha,beta,gamma` are in A.P

If alpha,beta,gamma are in A.P. then show that , |[x-3,x-4,x-alpha],[x-2,x-3 ,x-beta],[x-1,x-2,x-gamma]|=0

Without expanding,show that abs((x+1,x+2,x+a),(x+2,x+3,x+b),(x+3,x+4,x+c)) =0,Where a ,b,c are in A.P

Let x-ysin alpha-zsin beta=0 , xsin alpha+zsin gamma-y=0 and xsin beta+ysin gamma-z=0 be the equations of the planes such that alpha+beta+gamma=pi//2 (where alpha,beta and gamma!=0)dot Then show that there is a common line of intersection of the three given planes.

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))

If alpha,beta,gamma are the roots of a x^3+b x^2+cx+d=0 and |[alpha,beta,gamma],[beta,gamma,alpha],[gamma,alpha,beta]|=0, alpha!=beta!=gamma then find the equation whose roots are alpha+beta-gamma,beta+gamma-alpha , and gamma+alpha-beta .

Find the value of x such that ((x+alpha)^n-(x+beta)^n)/(alpha-beta)=(sin (ntheta))/(sin^n theta) , where alpha and beta are the roots of the equation t^2-2t+2=0 .

If f(x)=(x-1)(x-2)(x-3)(x-4) , show that the equation f'(x)=0 has three roots alpha, beta and gamma where 1 lt alpha lt 2, 2 lt beta lt 3 and 3 lt gamma lt 4 .

If alpha,beta,gamma are such that alpha+beta+gamma =4, alpha^2+beta^2+gamma^2 =6, alpha^3+beta^3+gamma^3 =8, then the value of [alpha^4+beta^4+gamma^4] must be equal to (where[.] denotes the greatest integer function)

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)

Let alpha,beta,gamma be the roots of f(x)=0, where f(x)=x^3+x^2-5x-1 . Then [alpha]+[beta]+[gamma] is, where [.] is greatest integer function