Let `f(x)=x^3-3x^2 + [a]`. If `a in [alpha, beta)` be the complete range of values of 'a' for whichf(x)=0 has three real and distinct solutions.then `(beta-alpha)` is equal to

The values of alpha and beta for which |e^(|x-beta|)-alpha|=2 has four distinct solutions are

The values of alpha and beta for which |e^(|x-beta|)-alpha|=2 has four distinct solutions are

If the roots of x^(4) - 6x^(3) +13x^2-12x + 4 = 0 are alpha, alpha , beta , beta then the value of alpha, beta are

If alpha and beta be the roots of equation x^(2) + 3x + 1 = 0 then the value of ((alpha)/(1 + beta))^(2) + ((beta)/(1 + alpha))^(2) is equal to

If alpha and beta be the roots of equation x^(2) + 3x + 1 = 0 then the value of ((alpha)/(1 + beta))^(2) + ((beta)/(1 + alpha))^(2) is equal to

If alpha and beta be the roots of equation x^(2) + 3x + 1 = 0 then the value of ((alpha)/(1 + beta))^(2) + ((beta)/(1 + alpha))^(2) is equal to

If alpha and beta be the roots of equation x^(2) + 3x + 1 = 0 then the value of ((alpha)/(1 + beta))^(2) + ((beta)/(1 + alpha))^(2) is equal to

Let f(x)=ax^3+bx^2+cx+1 has exterma at x=alpha,beta such that alpha beta < 0 and f(alpha) f(beta) < 0 f . Then the equation f(x)=0 has three equal real roots one negative root if f(alpha) (a) 0 and f(beta) (b) one positive root if f(alpha) (c) 0 and f(beta) (d) none of these

Let f(x)=ax^3+bx^2+cx+1 has exterma at x=alpha,beta such that alpha beta < 0 and f(alpha) f(beta) < 0 f . Then the equation f(x)=0 has three equal real roots one negative root if f(alpha) (a) 0 and f(beta) (b) one positive root if f(alpha) (c) 0 and f(beta) (d) none of these

If alpha , beta in C are the distinct roots of the equation x^2-x+1 = 0 , then (alpha)^101+(beta)^107 is equal to