If the roots of the equation `x^(3) -10 x + 11 = 0` are u, v, and w, then the value of `3 cosec^(2) (tan^(-1) u + tan^(-1) v + tan^(-1 w)` is ____

Answer

6

Explanation

Let `tan^(-1) u = alpha " or " tan alpha = u`
`tan^(-1) v = beta " or " tan beta = v`
`tan^(-1) w = gamma " or " tan gamma = w`
`tan (alpha + beta + gamma) = (s_(1) -s_(3))/(1 - s_(2)) = (0 -(-11))/(1-(-10)) = (11)/(11) = 1`
`:. alpha + beta + gamma = tan^(-1) (1) = (pi)/(4)`
`rArr 3 cosec^(2) (tan^(-1) u + tan^(-1) v + tan^(-1) w) = 6`