The constraints $-x_{1} + x_{2} \leq 1$,$-x_{1} + 3x_{2} \leq 9$,$x_{1}, x_{2} \geq 0$ define:

The maximum value of $z$ in the following equation $z=6xy+y^2$,subject to the constraints $3x+4y \leq 100$,$4x+3y \leq 75$,$x \geq 0$,and $y \geq 0$ is:

$A$ company manufactures two types of sweaters: type $A$ and type $B.$ It costs $Rs. 360$ to make a type $A$ sweater and $Rs. 120$ to make a type $B$ sweater. The company can make at most $300$ sweaters and spend at most $Rs. 72000$ a day. The number of sweaters of type $B$ cannot exceed the number of sweaters of type $A$ by more than $100.$ The company makes a profit of $Rs. 200$ for each sweater of type $A$ and $Rs. 120$ for every sweater of type $B.$ Formulate this problem as a $LPP$ to maximize the profit to the company.

$A$ cooperative society of farmers has $50$ hectares of land to grow two crops $X$ and $Y$. The profit from crops $X$ and $Y$ per hectare are estimated as Rs $10,500$ and Rs $9,000$ respectively. To control weeds,a liquid herbicide has to be used for crops $X$ and $Y$ at rates of $20$ litres and $10$ litres per hectare. Further,no more than $800$ litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximise the total profit of the society?

The feasible region of an $LPP$ is shown in the figure. If $z = 3x + 9y$,then the minimum value of $z$ occurs at

The maximum value of $z=6x+8y$ subject to the constraints $x-y \geq 0$,$x+3y \leq 12$,$x \geq 0$,$y \geq 0$ is: