If `sum_(n=1)^nalpha_n=a n^2+b n ,w h e r ea ,b` are constants and `alpha_1,alpha_2alpha_3 in {12,39}a n d25alpha_137alpha_2,49alpha_3` be three digit number, then prove that `|alpha_1alpha_2alpha_3 5 7 9 25alpha_1 37alpha_2 49alpha_3|=0`

If sum_(n=1)^nalpha_n=a n^2+b n ,w h e r ea ,b are constants and alpha_1,alpha_2,alpha_3 in {1,2,3,......,9}a n d25alpha_1,37alpha_2,49alpha_3 be three digit number, then prove that |alpha_1alpha_2alpha_3 5 7 9 25alpha_1 37alpha_2 49alpha_3|=0

If 1,alpha_1, alpha_2, …alpha_(n-1) be n, nth roots of unity show that (1-alpha_1)(1-alpha_2).(1-alpha(n-1)=m

If 1,alpha_(1),alpha_(2),alpha_(3),...,alpha_(n-1) are n, nth roots of unity, then (1-alpha_(1))(1-alpha_(2))(1-alpha_(3))...(1-alpha_(n-1)) equals to

If the roots of equation x^3+a x^2+b=0a r ealpha_1,alpha_2 and alpha_3(a ,b!=0) , then find the equation whose roots are (alpha_1alpha_2+alpha_2alpha_3)/(alpha_1alpha_2alpha_3),(alpha_2alpha_3+alpha_3alpha_1)/(alpha_1alpha_2alpha_3),(alpha_1alpha_3+alpha_1alpha_2)/(alpha_1alpha_2alpha_3)

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)) .

If alpha_(1),alpha_(2)………., alpha_(n) are real numbers show that (cos alpha_(1)+cos alpha_(2)+…..+cos alpha_(n))^(2)+(sin alpha_(1)+……+sin alpha_(n))^(2) le n^(2)

If f(alpha)=[[1,alpha,alpha^2],[alpha,alpha^2,1],[alpha^2,1,alpha]] then find the value of f(3^(1/3))