`If n ge3` and `1,alpha_1, alpha_2, alpha_3....alpha_(n-1)` are the , nth roots of unity then find the value of `sum_(1 le i lt j le (n-1)) (alpha_i alpha_j`)

If alpha_(1), alpha_(2), alpha_(3), alpha_(4) are the roots of the equation x^(4)+(2-sqrt(3))x^(2)+2+sqrt(3)=0 , then the value of (1-alpha_(1))(1-alpha_(2))(1-alpha_(3))(1-alpha_(4)) is

If alpha , beta , gamma are the roots of the equations x^3+px^2+qx+r=0 find the value of sum1/alpha

If alpha and beta (alpha gt beta) are the roots of x^(2) + kx - 1 =0 , then find the value of tan^(-1) alpha - tan^(-1) beta

If alpha,betaa n dgamma are roots of 2x^3+x^2-7=0 , then find the value of sum(alpha/beta+beta/alpha) .

If alpha in (-(pi)/(2), 0) , then find the value of tan^(-1) (cot alpha) - cot^(-1) (tan alpha)

If 1,alpha_1,alpha_2, ,alpha_(n-1) are the n t h roots of unity, prove that (1-alpha_1)(1-alpha_2)(1-alpha_(n-1))=ndot Deduce that sinpi/nsin(2pi)/n sin((n-1)pi)/n=n/(2^(n-1))

If alpha,beta,gamma are the roots of the equation x^3+4x+1=0, then find the value of (alpha+beta)^(-1)+(beta+gamma)^(-1)+(gamma+alpha)^(-1) .

If the roots of equation x^(3) + ax^(2) + b = 0 are alpha _(1), alpha_(2), and alpha_(3) (a , b ne 0) . Then find the equation whose roots are (alpha_(1)alpha_(2)+alpha_(2)alpha_(3))/(alpha_(1)alpha_(2)alpha_(3)), (alpha_(2)alpha_(3)+alpha_(3)alpha_(1))/(alpha_(1)alpha_(2)alpha_(3)), (alpha_(1)alpha_(3)+alpha_(1)alpha_(2))/(alpha_(1)alpha_(2)alpha_(3)) .

Let alpha,beta are the roots of x^2+b x+1=0. Then find the equation whose roots are - (alpha+1//beta)and-(beta+1//alpha) .

If alpha,beta are roots of x^2-3x+a=0 , a in R and alpha <1< beta then find the value of a.