Linear Programming — Class 12 Mathematics PYQs

The corner points of the feasible region in graphical representation of a L.P.P. are $(2,72)$, $(15,20)$ and $(40,15)$. If $Z=18x+9y$ be the objective function, then

(A) $Z$ is maximum at $(2,72)$, minimum at $(15,20)$
(B) $Z$ is maximum at $(15,20)$, minimum at $(40,15)$
(C) $Z$ is maximum at $(40,15)$, minimum at $(15,20)$
(D) $Z$ is maximum at $(40,15)$, minimum at $(2,72)$

If the feasible region of a linear programming problem with objective function $Z=ax+by$, is bounded, then which of the following is correct ?

(A) It will only have a maximum value.
(B) It will only have a minimum value.
(C) It will have both maximum and minimum values.
(D) It will have neither maximum nor minimum value.

A factory produces two products $X$ and $Y$. The profit earned by selling $X$ and $Y$ is represented by the objective function $Z=5x+7y$, where $x$ and $y$ are the number of units of $X$ and $Y$ respectively sold. Which of the following statement is correct ?

(A) The objective function maximizes the difference of the profit earned from products $X$ and $Y$.
(B) The objective function measures the total production of products $X$ and $Y$.
(C) The objective function maximizes the combined profit earned from selling $X$ and $Y$.
(D) The objective function ensures the company produces more of product $X$ than product $Y$.

Assertion (A): Every point of the feasible region of a Linear Programming Problem is an optimal solution.

Reason (R): The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region.

(A) Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(C) Assertion (A) is true but Reason (R) is false.
(D) Assertion (A) is false but Reason (R) is true.

Solve the following linear programming problem graphically :

Maximise $Z=x+2y$

Subject to the constraints :
$$x-y\ge 0$$
$$x-2y\ge -2$$
$$x\ge 0,y\ge 0$$

Solve the following linear programming problem graphically :

Maximise $Z=20x+30y$

Subject to the constraints :
$$x+y\le 80$$
$$2x+3y\ge 100$$
$$x\ge 14$$
$$y\ge 14$$

Solve the following linear programming problem graphically :

Maximise $Z=8x+9y$

Subject to the constraints :
$$2x+3y\le 6$$
$$3x-2y\le 6$$
$$y\le 1$$
$$x\ge 0,y\ge 0$$

Solve the following linear programming problem graphically :

Minimise $Z=x-5y$

Subject to the constraints :
$$x-y\ge 0$$
$$-x+2y\ge 2$$
$$x\ge 3,y\le 4,y\ge 0$$

The common region determined by all the constraints of a linear programming problem is called :

A linear programming problem deals with the optimization of a/an :