The minimum value of $Z = 3x + y$, subject to the constraints $2x + 3y \leq 6$, $x + y \geq 1$, $x \geq 0$, and $y \geq 0$ is:

The corner points of the feasible region determined by the system of linear constraints are $(0, 10)$, $(5, 5)$, $(15, 15)$, and $(0, 20)$. Let $z = px + qy$ where $p, q > 0$. The condition on $p$ and $q$ such that the maximum value of $z$ occurs at both points $(15, 15)$ and $(0, 20)$ is

Cake-$A$ requires $200\, g$ of flour and $25\, g$ of fat. Cake-$B$ requires $100\, g$ of flour and $50\, g$ of fat. Find the maximum number of cakes which can be made from $5\, kg$ of flour and $1\, kg$ of fat. The mathematical form of this $LPP$ is $.....$

$A$ manufacturing company produces two items,$A$ and $B$. Each item must be processed by two machines,$I$ and $II$. Machine $I$ can be operated for a maximum of $10$ hours $40$ minutes ($640$ minutes). It takes $20$ minutes for an item $A$ and $15$ minutes for an item $B$. Machine $II$ can be operated for a maximum of $8$ hours $20$ minutes ($500$ minutes). It takes $5$ minutes for an item $A$ and $8$ minutes for an item $B$. The profit per item of $A$ is ₹ $25$ and per item of $B$ is ₹ $18$. The formulation of an $L.P.P.$ to maximize the profit (where $x$ is the number of items $A$ and $y$ is the number of items $B$) is . . . . . . .

The maximum value of the objective function $z=4x+5y$ subject to the constraints $2x+3y \leq 12$,$2x+y \leq 8$ and $x \geq 0, y \geq 0$ is:

$A$ dietician has to develop a special diet using two foods $X$ and $Y$. Each packet (containing $30 \ g$) 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,at least $460$ units of iron and at most $300$ units of cholesterol. The corner points of the feasible region are: