1.Formulate the objective function to maximize profit $Z$ if item $X$ yields ?40 and item $Y$ yields ?50 per unit.
2.Write the constraint for the statement: "The total weight of $x$ units of food A and $y$ units of food B cannot exceed 100 kg". (Assuming each unit weighs 1 kg).
3.Write the constraint for the statement: "A machine has at most 15 hours available. Product $x$ takes 2 hours and product $y$ takes 3 hours".
4.Express the condition: "The investment cannot exceed ?50,000" if item $x$ costs ?500 and item $y$ costs ?200.
5.Write the constraint: "Production of item $x$ must be at least twice the production of item $y$".
6.A diet must contain at least 40 units of Vitamin C. Food I contains 4 units/kg and Food II contains 5 units/kg. Formulate this constraint.
7.Write the non-negativity constraints for an LPP involving variables $x$ and $y$.
8.A tailor makes tables ($x$) and chairs ($y$). A table requires 2 hours of carpentry and 1 hour of painting. A chair requires 1 hour of carpentry and 2 hours of painting. Formulate the carpentry constraint if total carpentry time available is 24 hours.
9.Find the corner points of the feasible region bounded by $x + y \le 4$, $x \ge 0$, $y \ge 0$.
10.Find the corner points of the feasible region bounded by $2x + y \le 6$, $x \ge 0$, $y \ge 0$.
11.Find the point of intersection of the lines $x + y = 5$ and $x - y = 1$.
12.A feasible region is bounded by $x + 2y \le 8$, $3x + y \le 9$, $x \ge 0, y \ge 0$. Find the corner point formed by the intersection of the two main constraints.
13.Find all the corner points for the constraints given in Question 12.
14.Maximize $Z = 3x + 4y$ at the corner points $(0,0)$, $(4,0)$, $(2,3)$, and $(0,5)$. What is the maximum value?
15.Minimize $Z = 20x + 10y$ at the corner points $(0,5)$, $(4,3)$, and $(6,0)$. What is the minimum value?
16.Maximize $Z = 5x + 3y$ subject to $x + y \le 10$, $x \ge 0$, $y \ge 0$.
17.Minimize $Z = x + 2y$ subject to $2x + y \ge 3$, $x + 2y \ge 6$, $x \ge 0, y \ge 0$. Evaluate at corner points $(0,3)$ and $(6,0)$.
18.Evaluate $Z = 300x + 190y$ at $(0, 10)$ and $(5, 5)$. Which point yields a higher value?
19.Solve graphically to maximize $Z = x + y$ subject to $x - y \le -1, -x + y \le 0, x, y \ge 0$. Is there a feasible region?
20.Maximize $Z = 3x + 2y$ subject to $x \le 2$, $y \le 3$, $x \ge 0, y \ge 0$. Find the maximum value.
21.If the feasible region is unbounded and $m = 12$ is the minimum value of $Z = 2x + y$ at a corner point, we must graph the half-plane $2x + y < 12$. If this open half-plane has NO common points with the feasible region, is $m=12$ the true minimum?
22.If a maximum value $M$ occurs at an unbounded region, we graph $ax + by > M$. If this half-plane shares points with the feasible region, what can we conclude about the maximum value?
23.Minimize $Z = 3x + 5y$ subject to $x + y \ge 2$, $x \ge 0$, $y \ge 0$. The corner points are $(2,0)$ and $(0,2)$. Find the minimum value of Z.
24.For the constraints $x + y \ge 8$ and $x, y \ge 0$, what type of feasible region is formed (Bounded or Unbounded)?
25.In an LPP, the maximum of $Z = 4x + 6y$ occurs at both $(0, 2)$ and $(3, 0)$ yielding $Z = 12$. How many optimal solutions exist?
26.Identify the case: The constraint lines are parallel and their respective half-planes face away from each other, resulting in no overlapping area.
27.True or False: If the objective function line is parallel to one of the binding constraint lines, the LPP will have infinitely many optimal solutions.
28.A region is defined by $x \ge 2$ and $y \ge 3$. Does this region contain the origin $(0,0)$?
29.Evaluate $Z = -x + 2y$ subject to $x \ge 3, x + y \ge 5, x, y \ge 0$. The corner point is $(3, 2)$. What is the value of Z?
30.If an LPP has no feasible region, what is its maximum or minimum value?
