A wholesale merchant wants to start the business of cereal with $₹2400$. Wheat is $₹400$ per quintal and rice is $₹600$ per quintal. He has capacity to store $200$ quintal cereal. He earns the profit $₹25$ per quintal on wheat and $₹40$ per quintal on rice. If he stores $x$ quintal rice and $y$ quintal wheat, then for maximum profit, the objective function is

Linear inequalities for which the shaded region of the given figure is the solution set, are

Which of the following is/are true?

$\left(\mathrm{i}\right)$ The graphical solution of the system of inequalities $3x+2y\le 12, x\ge 1, y\ge 2$ is

$\left(\mathrm{ii}\right)$ The region represented by the solution set of the inequalities $2x+y\ge 6,$ $3x+4y\le 12$ is bounded.

$\left(\mathrm{iii}\right)$ The solution set of the inequalities $x+y\ge 4$, $2x-y>0$ is

There is a factory located at each of the two places $P$ and $Q$. From these locations, a certain commodity is delivered to each of the three depots situated at $A,B$ and $C$. The weekly requirements of the depots are respectively, $5, 5$ and $4$ units commodity, while the production capacity of the factories at $P$ and $Q$ are $8$ and $6$ units, respectively. The cost of transportation per unit is given below:

Formulate the $\mathrm{L}.\mathrm{P}.\mathrm{P}.$, so that the units transported from each factory to each depot in such an order that the transportation cost is minimum.

A manufacturer has three machines $\mathrm{I}, \mathrm{II}$ and $\mathrm{III}$ installed in his factory. Machines $\mathrm{I}$ and $\mathrm{II}$ are capable of being operated for utmost $12$ hours  whereas, machine $\mathrm{III}$ 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 $\mathrm{I}$ unit of each of $M$ and $N$ on the three machines are given in the following table:

She makes a profit of $₹600$ and $₹400$ on items $M$ and $N$, respectively,

Then, to maximise the profit, number of units of item $M$, the manufacturer has to produce, is