The solution set of the constraints $2x + 3y \leq 6$,$x + 4y \leq 4$,$x \geq 0$,and $y \geq 0$ includes the point $\ldots$ as a corner point.

The feasible region of the $L$.$P$.$P$. (Linear Programming Problem) to maximize $z = 70x + 50y$ subject to the constraints $8x + 5y \leq 60$,$4x + 5y \leq 40$ and $x \geq 0, y \geq 0$ is:

The shaded area in the figure below is the solution set for a certain linear programming problem. The linear constraints are given by:

$A$ 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.

$A$ dietician wishes to mix together two kinds of food $X$ and $Y$ in such a way that the mixture contains at least $10$ units of vitamin $A$,$12$ units of vitamin $B$,and $8$ units of vitamin $C$. The vitamin contents of one $kg$ of food are given below:

Food	Vitamin $A$	Vitamin $B$	Vitamin $C$
$X$	$1$	$2$	$3$
$Y$	$2$	$2$	$1$

One $kg$ of food $X$ costs $Rs. 16$ and one $kg$ of food $Y$ costs $Rs. 20$. Find the least cost of the mixture which will produce the required diet?

The difference between the maximum value and minimum value of the objective function $z = 3x + 5y$ of a linear programming problem subject to constraints $5x + 10y \leq 50$, $x + y \geq 1$, $y \leq 4$ and $x \geq 0, y \geq 0$ is $3\lambda$. Then the value of $\lambda$ is