Vardaan Learning Institute

If the optimal value of the objective function $Z = px + qy$ occurs at two adjacent corner points $(x_1, y_1)$ and $(x_2, y_2)$ of a feasible region, find the necessary mathematical condition relating $p, q, x_1, x_2, y_1,$ and $y_2$.

The objective function $Z = 3x + py$ has a maximum value at both the points $(3, 5)$ and $(4, 3)$. Find the value of $p$.

A feasible region is defined by $2x + y \le 10$, $x + 3y \le 15$, $x \ge 0, y \ge 0$. Find the condition on $a$ and $b$ such that the maximum of $Z = ax + by$ occurs exclusively at the point $(3, 4)$.

If the half-plane $ax + by > M$ shares no points with the feasible region, what does $M$ represent for the objective function $Z = ax + by$?

Determine if the region defined by $|x - 2| + |y - 3| \le 4$ is a convex set. Is it bounded or unbounded?

Let the constraints of an LPP be $x + 2y \ge 10$, $3x + y \ge 15$, $x, y \ge 0$. If $Z = px + qy$ ($p,q>0$) is to be minimized, find the ratio $p/q$ for which the minimum occurs at infinitely many points.

Minimize $Z = -x + 2y$ subject to $x \ge 3$, $x + y \ge 5$, $x + 2y \ge 6$, $y \ge 0$. After finding the minimum at a corner point, perform the half-plane check and state the final minimum value.

Maximize $Z = 2x + 3y$ subject to $x + y \ge 4$, $x \le 5$, $y \ge 0$. Analyze the unboundedness and determine if a maximum value exists.

Solve the LPP: Minimize $Z = x + y$ subject to $3x + 2y \ge 12$, $x + 3y \ge 11$, $x \ge 0, y \ge 0$. Evaluate the objective function and perform the rigorous open half-plane check.

Consider the constraints: $x - y \le 1$, $x + y \ge 3$, $x, y \ge 0$. Find the minimum value of $Z = 4x + 6y$ if it exists.

Show graphically or analytically that the LPP: Maximize $Z = 3x + 4y$ subject to $x - y \ge 0$, $-x + 3y \le 3$, $x \ge 0, y \ge 0$ has an unbounded solution.

If the constraints are $2x + 3y \le 6$ and $4x + 6y \ge 24$ ($x, y \ge 0$), find the optimal value of $Z = x + y$.

Find the vertices of the feasible region bounded by $x + 2y \le 10$, $3x + y \le 15$, $x, y \ge 0$. Evaluate $Z = 5x + 3y$ to find the optimal solution.

A company produces two types of goods A and B. A requires 2 hours on machine I and 2 hours on machine II. B requires 3 hours on machine I and 1 hour on machine II. Machine I is available for 12 hours and machine II for 8 hours. If profit on A is ?400 and on B is ?500, formulate and find the maximum profit.

An LPP is defined as: Maximize $Z = 2x + 5y$ subject to $2x + 4y \le 8$, $3x + y \le 6$, $x + y \le 4$, $x \ge 0, y \ge 0$. Identify any redundant constraint(s) in this system.

Find the maximum value of $Z = 6x + 4y$ subject to $x \le 2$, $x + y \le 3$, $-2x + y \le 1$, $x \ge 0, y \ge 0$.

Minimize $Z = 5x + 10y$ subject to $x + 2y \le 120$, $x + y \ge 60$, $x - 2y \ge 0$, $x, y \ge 0$.

A dietician wishes to mix two types of foods in such a way that the vitamin contents of the mixture contain at least 8 units of Vitamin A and 10 units of Vitamin C. Food I contains 2 units/kg of A and 1 unit/kg of C. Food II contains 1 unit/kg of A and 2 units/kg of C. It costs ?50/kg to purchase Food I and ?70/kg for Food II. Formulate and solve to minimize cost.

Explain analytically why an objective function $Z = ax + by$ achieves its maximum at a corner point by relating it to the parallel shifting of the Iso-profit line $ax + by = k$.

If $Z = 3x + c \cdot y$ has multiple optimal solutions on the segment joining $(1, 4)$ and $(2, 3)$, find the exact value of $c$.

Can the optimal value of an LPP occur at a point strictly inside (interior of) the feasible region? Justify your answer mathematically.

If a feasible region has corner points $(0,0), (5,0), (4,3), (0,4)$, find the relation between $p$ and $q$ ($p,q>0$) such that the maximum of $Z = px + qy$ occurs exclusively at $(4,3)$.

Solve: Maximize $Z = x + 2y$ subject to $y \le x$, $y \ge 0$, $x \le 5$.

If the constraints $ax + by \le c$ and $dx + ey \le f$ define a feasible region, what geometric condition ensures the region is bounded in the first quadrant?

A transport company has two depots A and B with 20 and 30 trucks. Three factories C, D, E need 15, 20, 15 trucks. If the cost of transport per truck from A to C, D, E is ?50, ?60, ?40 and from B to C, D, E is ?40, ?80, ?50. Write the objective function to minimize transport cost in terms of variables $x$ and $y$.