A quadratic equation f(x)=0 has exactly ...

A quadratic equation `f(x)=0` has exactly one root between `a` and `b` if `f(a)` and `f(b)` have opposite signs . Let `g(x)` be a cubic equation with real roots `alpha , beta, gamma` such that `alphaltbetaltgamma`.Number roots of equation `g\'(x)` between `(beta+gamma)/2 and (2beta+gamma)/3` is a then the value of the determinant of order 3 having all its elements 1 and a is (A) 1 (B) -1 (C) `a+1` (D) none of these

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