If `underset(xrarroo)(lim)xlog_(e)(|(alpha//x,1,gamma),(0,1//x,beta),(1,0,1//x)|)=-5.` where `alpha, beta, gamma` are finite real numbers, then

If lim_(xrarroo) xlog_(e)(|(alpha//x,1,gamma),(0,1//x,beta),(1,0,1//x)|)=-5. where alpha, beta, gamma are finite real numbers, then

If A^(-1)=[(sin^2alpha,0,0),(0,sin^2 beta,0), (0,0,sin^2gamma)] and B^(-1)=[(cos^2alpha,0,0),(0,cos^2 beta,0), (0,0,cos^2gamma)] where alpha, beta, gamma are any real numbers and C=A^(-5)+B^(-5)+5A^(-1)B^(-1)(A^(-3) +B^(-3))+10 A^(-2)B^(-2)(A^(-1)+B^(-1)) then find |C|

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 AP

Let alpha,beta, gamma be three real numbers satisfying [(alpha,beta,gamma)][(2,-1,1),(-1,-1,-2),(-1,2,1)]=[(0,0,0)] . If the point A(alpha, beta, gamma) lies on the plane 2x+y+3z=2 , then 3alpha+3beta-6gamma is equal to

If (1+alpha)/(1-alpha),(1+beta)/(1-beta), (1+gamma)/(1-gamma) are the cubic equation f(x) = 0 where alpha,beta,gamma are the roots of the cubic equation 3x^3 - 2x + 5 =0 , then the number of negative real roots of the equation f(x) = 0 is :

If (1+alpha)/(1-alpha),(1+beta)/(1-beta), (1+gamma)/(1-gamma) are the cubic equation f(x) = 0 where alpha,beta,gamma are the roots of the cubic equation 3x^3 - 2x + 5 =0 , then the number of negative real roots of the equation f(x) = 0 is :

If alpha, beta. gamma are the roots of x^3 + px^2 + q = 0, where q=0, ther Delta=[(1/alpha,1/beta,1/gamma),(1/beta,1/gamma,1/alpha),(1/gamma,1/alpha,1/beta)] equals (A) alpha beta gamma (B) alpha +beta + gamma (C) 0 (D) none of these

If alpha,beta,gamma, in (0,pi/2) , then prove that (s i(alpha+beta+gamma))/(sinalpha+sinbeta+singamma)<1

If alpha , beta, gamma are the roots of x^3+x^2-5x-1=0 then alpha+beta+gamma is equal to

The minimum value of the expression sin alpha + sin beta+ sin gamma , where alpha,beta,gamma are real numbers satisfying alpha+beta+gamma=pi is