The corner points of the feasible region of an LPP are $$(0,2),(3,0),(6,0),(6,8)$$ and $$(0,5)$$, then the minimum value of $$z=4 x+6 y$$ occurs at

A dietician has to develop a special diet using two foods $$X$$ and $$Y$$. Each packet (containing $$30 \mathrm{~g}$$ ) of food. $$X$$ contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Y contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least 240 units of calcium, atleast 460 units of iron and atmost 300 units of cholesterol. The corner points of the feasible region are

The shaded region is the solution set of the inequalities

Corner points of the feasible region determined by the system of linear constraints are $$(0,3),(1,1)$$ and $$(3,0)$$. Let $$z=p x=q y$$, where, $$p, q>0$$. Condition on $$p$$ and $$q$$, so that the minimum of $$z$$ occurs at $$(3,0)$$ and $$(1,1)$$ is