The vertices of the feasible region for the constraints $x+y \leq 4$,$x \leq 2$,$y \leq 1$,$x+y \geq 1$,$x, y \geq 0$ are

The graphical solution set of the system of in-equations $x+y \leq 70, x+2y \leq 100, 2x+y \leq 120, x \geq 0, y \geq 0$ is given by:

The objective function $z=x_1+x_2$,subject to $x_1+x_2 \leq 10, -2x_1+3x_2 \leq 15, x_1 \leq 6, x_1, x_2 \geq 0$,has a maximum value at:

$A$ company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type $A$ require $5 \text{ minutes}$ each for cutting and $10 \text{ minutes}$ each for assembling. Souvenirs of type $B$ require $8 \text{ minutes}$ each for cutting and $8 \text{ minutes}$ each for assembling. There are $3 \text{ hours } 20 \text{ minutes}$ available for cutting and $4 \text{ hours}$ for assembling. The profit is $Rs. 5$ each for type $A$ and $Rs. 6$ each for type $B$ souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profit?

In $L.P.P.$,the maximum value of the objective function $Z = 6x + 3y$ subject to the constraints $x + y \leq 5$,$x + 2y \geq 4$,$4x + y \leq 12$,$x, y \geq 0$ is:

The minimum value of the objective function $Z = 5x + 8y$,subject to the constraints $x + y \geq 5$,$x \leq 4$,$y \leq 2$,$x \geq 0$,and $y \geq 0$,occurs at the point: