The shaded area in the figure given below is a solution set of a system of inequations. The minimum value of the objective function $Z = 3x + 5y$,subject to the linear constraints given by this system of inequations,is:

The maximum value of $Z=5x+4y$,subject to the constraints $y \leq 2x$,$x \leq 2y$,$x+y \leq 3$,$x \geq 0$,$y \geq 0$ is:

$A$ manufacturer has three machines $I, II$ and $III$ installed in his factory. Machines $I$ and $II$ are capable of being operated for at most $12 \, hours$ whereas machine $III$ must be operated for at least $5 \, hours$ a day. She produces only two items $M$ and $N$ each requiring the use of all the three machines. The number of hours required for producing $1$ unit of each of $M$ and $N$ on the three machines are given in the following table:

Items	Machine $I$	Machine $II$	Machine $III$
$M$	$1$	$2$	$1$
$N$	$2$	$1$	$1.25$

She makes a profit of $Rs. \, 600$ and $Rs. \, 400$ on items $M$ and $N$ respectively. How many of each item should she produce so as to maximise her profit assuming that she can sell all the items that she produced? What will be the maximum profit?

$A$ wholesale dealer wants to start a business with $Rs. 2,40,000$. The cost price of a quintal of wheat is $Rs. 2000$ and the cost price of a quintal of rice is $Rs. 3000$. He has space capacity for $200$ quintals of grain. The profit from the sale of one quintal of wheat is $Rs. 125$ and that from one quintal of rice is $Rs. 200$. If he has $x$ quintals of rice and $y$ quintals of wheat,then the objective function for the maximum profit is $....$

The production of item $A$ is $x$ and the production of item $B$ is $y$. If the corner points of the bounded feasible region are $(1,0), (2,0), (0,2)$ and $(0,1)$,then the maximum profit $z = 2000x + 5000y$ is $\ldots \ldots$

The shaded area in the given figure is a solution set for some system of inequations. The maximum value of the function $z=10x+25y$ subject to the linear constraints given by the system is