If x = $$-$$4 is a root of $$\left| {\matrix{ x & 2 & 3 \cr 1 & x & 1 \cr 3 & 2 & x \cr } } \right| = 0$$, then the sum of the other two roots is

The absolute maximum value of the function $$f(x) = 4x - {1 \over 2}{x^2}$$ in the interval $$\left[ { - 2,{9 \over 2}} \right]$$ is

In a sphere of radius r, a right circular cone of height h having maximum curved surface area is inscribed. The expression for the square of curved surface of cone is

The corner points of the feasible region determined by a set of constraints (linear inequalities) are P(0, 5), Q(3, 5), R(5, 0) and S(4, 1) and the objective function is Z = ax + 2by where a, b > 0. The condition on a and b such that the maximum Z occurs at Q and S is