For `m , n gt 0`, let `alpha(m,n)=int_0^2 t^m(1+3t)^n dt`. If `11 alpha(10,6)+18alpha(11,5)=p(14)^6`, then p is equal to ________.

If alpha=e^((i2 pi)/(n)), then n(11-alpha)(11-alpha^(2))(11-alpha^(n-1))=

Let alpha, beta be the roots of the equation x^2 - sqrt 6 x 3 = 0 such that Im (alpha) >Im (beta) . Let a, b be integers not divisible by 3 and n be a natural number such that frac{alpha^(99)}{beta} alpha^(98) = 3^n (a ib) , i = sqrt(-1) Then n a b is equal to _______.

If int _(0) ^(pi) ( sin ^(3) x) e^(- sin^(2)x)dx = alpha - (beta)/( e) int _(0)^(1) sqrt(t) e^(t) " dt , then " alpha + beta is equal to _____

lim_ (n rarr oo) (n ^ (alpha) sin ^ (2) n!) / (n + 1), 0