If `f(x) =|(x+alpha,x^2+1,a),(x+beta,2x^4-2,a),(x+gamma,(3x^4+7),a)|` , where `a != 0 and f'(x) = 0 AA x in R`, then `alpha, beta, gamma` are in

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 f(x) = |(cos(x + alpha), cos(x+beta), cos(x + gamma)),(sin(x + alpha), sin(x+beta), sin(x + gamma)), (sin(beta - gamma), sin(gamma - alpha), sin(alpha - beta))| and f(2) = -2, then |sum_(r=1)^20 f(r)| equals

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 :

|[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"

If alpha, beta, gamma are the roots of x^(3) + x^(2) + x + 1 =0 then alpha^(4) + beta^(4) + gamma^(4) =

Let f (x) = x cos x, x in [(3pi)/(2), 2pi] and g (x) be its inverse. If int _(0)^(2pi)g (x) dx = api ^(2) + beta pi+ gamma, where alpha, beta and gamma in R, then find the value of 2 (alpha + beta+ gamma).