If `alpha,beta` are the roots of the equation `x^(2)+4x+p=0,` where `=Sigma_(r-0)^(n) ""^(n)C_(r)(1+rt)/((1+nt)^(r ))(-1)^(r )`, then `|alpha-beta|` is equal to:

Answer

D

Explanation

`p=underset(r=0)overset(n)Sigma.^(n)(1+rt)/(1+nt)^(r) (-1)^(r)=`
`underset(r=0)overset(n)Sigma.^(n)C_(r)((-1)/(1+nt))^(r)+ntunderset(r=0)overset(n)Sigma.^(n-1)C_(r-1)((-1)/(1+nt))^(r)`
`=(1-(1)/(1+nt))^(n)-(nt)/(1+nt)underset(r=0)overset(n)sum.^(n-1)C_(r-1)((-1)/(1+nt))^(r-1)`
`=((nt)/(1+nt))^(n)-(nt)/(1+nt)(1-(1)/(1+nt))^(n-1)=0`
`implies x^(2)+4 x=0 implies x=0,-4implies|alpha-beta|=4`