Single Best Answer MCQs for Sub-Topics of Topic 13: Linear Programming

Question 1. Which of the following best describes the purpose of Linear Programming?

(A) To solve non-linear equations.

(B) To find the optimal solution (maximum or minimum) for a problem with linear relationships and constraints.

(C) To analyze statistical data.

(D) To simulate complex systems.

Question 2. In a Linear Programming Problem (LPP), the function to be maximized or minimized is called the:

(A) Constraint function.

(B) Objective function.

(C) Decision variable.

(D) Feasible region.

Question 3. Which of the following is NOT a characteristic of a Linear Programming Problem?

(A) Linear relationships between variables.

(B) Presence of an objective function.

(C) Presence of constraints.

(D) Non-linear objective function or constraints.

Question 4. In a manufacturing problem, if $x_1$ represents the number of chairs produced and $x_2$ represents the number of tables produced, what are $x_1$ and $x_2$ called?

(A) Objective values.

(B) Constraint values.

(C) Decision variables.

(D) Optimal solutions.

Question 5. Constraints in an LPP represent:

(A) The function to be optimized.

(B) Limitations or restrictions on the resources or conditions.

(C) The desired outcomes of the problem.

(D) The variables to be determined.

Question 6. A common restriction in most LPPs is the non-negativity restriction. What does it imply?

(A) Decision variables must be positive integers.

(B) Decision variables must be less than or equal to zero.

(C) Decision variables must be greater than or equal to zero.

(D) Decision variables must be fractions.

Question 7. If the objective function is $Z = 5x_1 + 3x_2$ and we want to maximize $Z$, what does this function represent?

(A) Total cost.

(B) Total profit.

(C) Resource usage.

(D) Minimum value.

Question 8. A constraint like $2x_1 + 4x_2 \leq 100$ could represent:

(A) The minimum profit required.

(B) The maximum availability of a resource.

(C) The exact cost incurred.

(D) The number of units produced.

Question 9. In an LPP where a company is trying to minimize production costs, the objective function would typically represent:

(A) Revenue.

(B) Cost.

(C) Profit.

(D) Demand.

Question 10. Decision variables in an LPP must be:

(A) Only quantities to be produced.

(B) Quantities that can be controlled or decided upon.

(C) Fixed values given in the problem.

(D) Only positive integers.

Question 11. The non-negativity restriction $x_i \geq 0$ is imposed because:

(A) It simplifies the mathematical calculations.

(B) In most real-world problems, quantities or activities cannot be negative.

(C) It guarantees a unique solution.

(D) It ensures the objective function is always positive.

Question 12. What does 'linear' in Linear Programming refer to?

(A) The solution method involves drawing lines.

(B) The objective function and all constraints are linear functions of the decision variables.

(C) The feasible region is always a straight line.

(D) The problem can only be solved graphically.

Question 13. If a factory has a maximum of 1000 labour hours available per week, this represents a:

(A) Decision variable.

(B) Objective function coefficient.

(C) Constraint.

(D) Optimal value.

Question 14. The coefficients in the objective function represent:

(A) The amount of resource consumed per unit of decision variable.

(B) The contribution (e.g., profit or cost) per unit of decision variable to the objective.

(C) The maximum availability of resources.

(D) The minimum requirements.

Question 15. What is the primary goal when formulating a real-world problem as an LPP?

(A) To make the problem mathematically complicated.

(B) To identify and express the objective and constraints as linear equations or inequalities in terms of decision variables.

(C) To avoid using graphs.

(D) To find a solution before formulation.

Question 16. Which symbol is typically used to denote the objective function in an LPP?

(A) C (for Constraints)

(B) D (for Decision variables)

(C) R (for Resources)

(D) Z (commonly used)

Question 17. If an LPP involves minimizing production costs subject to meeting minimum production requirements, the constraints would likely be:

(A) Less than or equal to ($\leq$).

(B) Greater than or equal to ($\geq$).

(C) Equal to ($=$).

(D) A mix, but minimum requirements often use $\geq$.

Question 18. Consider a diet problem where we need to meet minimum nutritional requirements. The objective is usually to:

(A) Maximize nutrient intake.

(B) Minimize the cost of the diet while meeting minimum nutritional requirements.

(C) Maximize the quantity of food consumed.

(D) Minimize calorie intake.

Question 19. In the context of LPP, the term 'optimal' refers to:

(A) Any solution that satisfies the constraints.

(B) A solution that satisfies all constraints and yields the best possible value for the objective function.

(C) A solution found graphically.

(D) The non-negative solution.

Question 20. A problem involves deciding how many units of two products to produce to maximize profit, given limited machine hours and raw material. This is a classic example of a:

(A) Diet problem.

(B) Transportation problem.

(C) Manufacturing/Production problem.

(D) Assignment problem.

Question 1. The standard mathematical form of an LPP with maximization objective is generally stated as:

(A) Minimize $Z = c_1 x_1 + c_2 x_2 + ... + c_n x_n$ subject to $A\mathbf{x} \leq \mathbf{b}$, $\mathbf{x} \geq 0$.

(B) Maximize $Z = c_1 x_1 + c_2 x_2 + ... + c_n x_n$ subject to $A\mathbf{x} \geq \mathbf{b}$, $\mathbf{x} \geq 0$.

(C) Maximize $Z = c_1 x_1 + c_2 x_2 + ... + c_n x_n$ subject to $A\mathbf{x} \leq \mathbf{b}$, $\mathbf{x} \geq 0$.

(D) Minimize $Z = c_1 x_1 + c_2 x_2 + ... + c_n x_n$ subject to $A\mathbf{x} = \mathbf{b}$, $\mathbf{x} \geq 0$.

Question 2. Which part of the mathematical formulation represents the restrictions or limitations?

(A) The objective function.

(B) The decision variables.

(C) The constraint equations/inequalities.

(D) The optimal value.

Question 3. When formulating an LPP with two variables, $x$ and $y$, which of the following is a typical component?

(A) Objective function like $Z = ax^2 + by^2$.

(B) Constraint like $cx + dy + e \geq 0$.

(C) Constraint like $\sqrt{x} + \sqrt{y} \leq k$.

(D) Non-negativity restrictions $x \geq 0, y \geq 0$.

Question 4. A company makes two products, A and B. Product A requires 2 hours of labour and 3 kg of material. Product B requires 3 hours of labour and 2 kg of material. There are 60 hours of labour and 65 kg of material available. If $x_A$ is the number of units of A and $x_B$ is the number of units of B, which constraint represents the labour limit?

(A) $2x_A + 3x_B \leq 60$.

(B) $3x_A + 2x_B \leq 65$.

(C) $2x_A + 3x_B \geq 60$.

(D) $3x_A + 2x_B \geq 65$.

Question 5. In the previous question, if product A gives a profit of $\textsf{₹}40$ per unit and product B gives $\textsf{₹}50$ per unit, what is the objective function to maximize profit?

(A) Maximize $Z = 60x_A + 65x_B$.

(B) Minimize $Z = 40x_A + 50x_B$.

(C) Maximize $Z = 40x_A + 50x_B$.

(D) Minimize $Z = 2x_A + 3x_B + 3x_A + 2x_B$.

Question 6. When translating a real-world problem into an LPP, the first step is usually to:

(A) Solve the problem graphically.

(B) Identify the decision variables.

(C) Find the optimal solution.

(D) Evaluate the objective function.

Question 7. Which of the following expressions is NOT allowed in the mathematical formulation of an LPP?

(A) $3x_1 + 2x_2 \leq 100$.

(B) $5x_1 - 7x_2 = 50$.

(C) $Z = 4x_1 + 6x_2$.

(D) $x_1 x_2 \leq 200$.

Question 8. In the mathematical formulation, the terms on the right-hand side of the constraint inequalities/equations represent:

(A) The coefficients of the decision variables.

(B) The constants representing resource limits or minimum requirements.

(C) The variables.

(D) The objective function value.

Question 9. A diet needs to contain at least 20 units of nutrient A and 30 units of nutrient B. Food X costs $\textsf{₹}10$ per kg and provides 2 units of A and 3 units of B. Food Y costs $\textsf{₹}12$ per kg and provides 4 units of A and 2 units of B. Let $x$ and $y$ be the quantities (in kg) of Food X and Food Y respectively. Which constraint represents the minimum requirement for nutrient A?

(A) $2x + 4y \leq 20$.

(B) $2x + 4y \geq 20$.

(C) $10x + 12y \geq 20$.

(D) $3x + 2y \geq 30$.

Question 10. In the diet problem from the previous question, what is the objective function to minimize the cost?

(A) Minimize $Z = 2x + 3y$.

(B) Minimize $Z = 20x + 30y$.

(C) Minimize $Z = 10x + 12y$.

(D) Maximize $Z = 10x + 12y$.

Question 11. The constraint $x_1 - x_2 \leq 0$ can be rewritten as:

(A) $x_1 \geq x_2$.

(B) $x_1 \leq x_2$.

(C) $x_1 = x_2$.

(D) $x_1 + x_2 \leq 0$.

Question 12. Which of the following sets of equations/inequalities, along with non-negativity, constitutes a valid LPP formulation?

(A) Maximize $Z = x + y^2$; $x+y \leq 10$, $x,y \geq 0$.

(B) Minimize $Z = \sqrt{x} + y$; $2x+3y \geq 12$, $x,y \geq 0$.

(C) Maximize $Z = 2x + 5y$; $x+y \leq 5$, $2x-y \leq 4$, $x,y \geq 0$.

(D) Minimize $Z = xy$; $x+y=7$, $x,y \geq 0$.

Question 13. The general structure of an LPP involves optimizing a linear objective function subject to:

(A) Only equality constraints.

(B) Only inequality constraints ($\leq$ or $\geq$).

(C) A set of linear equality and/or inequality constraints and non-negativity of variables.

(D) Non-linear constraints.

Question 14. If a constraint represents a minimum requirement, the inequality sign will be:

(A) $\leq$.

(B) $\geq$.

(C) $=$.

(D) $<$.

Question 15. If a constraint represents maximum availability of a resource, the inequality sign will be:

(A) $\leq$.

(B) $\geq$.

(C) $=$.

(D) $>$.

Question 16. When formulating an LPP, the coefficients in the constraints are derived from:

(A) The profit or cost per unit.

(B) The resource consumption or requirement per unit of decision variable.

(C) The total available resources.

(D) The optimal values.

Question 17. Which of the following could NOT be a constraint in a standard LPP?

(A) $3x_1 + 2x_2 = 50$.

(B) $x_1 - x_2 \leq 10$.

(C) $x_1 / x_2 \geq 5$.

(D) $x_1 \geq 0, x_2 \geq 0$.

Question 18. Consider a transportation problem where a company needs to transport goods from warehouses to destinations. Decision variables would typically represent:

(A) The quantity of goods produced at each warehouse.

(B) The demand at each destination.

(C) The quantity of goods transported from each source to each destination.

(D) The transportation cost per unit.

Question 19. In formulating a production problem, if $x_i$ is the quantity of product $i$ produced, the objective function to maximize profit would be of the form $Z = \sum c_i x_i$. What does $c_i$ represent?

(A) Total resource used for product $i$.

(B) Cost of resources for product $i$.

(C) Profit per unit of product $i$.

(D) Demand for product $i$.

Question 20. Suppose you are formulating an LPP to minimize the cost of a blend of two chemicals, X and Y, to meet a minimum viscosity requirement. If $x$ is the amount of chemical X and $y$ is the amount of chemical Y, and the required viscosity is represented by an inequality, the decision variables are:

(A) Viscosity and cost.

(B) Amounts $x$ and $y$.

(C) Minimum viscosity requirement.

(D) Cost coefficients.

Question 1. A problem that involves determining the optimal production levels for different products to maximize profit, given resource constraints like labour, machine hours, and raw materials, is classified as a:

(A) Diet problem.

(B) Transportation problem.

(C) Manufacturing/Production problem.

(D) Investment problem.

Question 2. In a typical Diet Problem LPP, the objective is usually to:

(A) Maximize the total quantity of food.

(B) Minimize the cost of the diet while meeting minimum nutritional requirements.

(C) Maximize the calorie intake.

(D) Minimize the variety of food items.

Question 3. A problem where the goal is to determine the most economical way to ship goods from various sources (factories, warehouses) to different destinations (markets, distributors) is known as a:

(A) Production problem.

(B) Diet problem.

(C) Transportation problem.

(D) Blending problem.

Question 4. Constraints in a Transportation Problem typically include:

(A) Minimum nutrient requirements.

(B) Total available supply at each source and total demand at each destination.

(C) Maximum machine hours available.

(D) Profit per unit of product.

Question 5. What is a key characteristic of a Blending Problem LPP?

(A) Deciding which machine to use for which product.

(B) Determining the proportion of different components to mix to achieve desired properties at minimum cost.

(C) Assigning tasks to workers.

(D) Scheduling production over time.

Question 6. An LPP is formulated to decide how much money to invest in different schemes to maximize return, subject to risk limits and minimum investment amounts in certain schemes. This falls under the category of:

(A) Production problem.

(B) Diet problem.

(C) Financial/Investment problem.

(D) Transportation problem.

Question 7. In a Manufacturing Problem, decision variables usually represent:

(A) Number of units of each product to be produced.

(B) Amount of raw material consumed.

(C) Total profit earned.

(D) Labour cost.

Question 8. A furniture company makes tables and chairs. Each table requires 5 units of wood and 10 labour hours. Each chair requires 3 units of wood and 8 labour hours. Total wood available is 1000 units, and total labour hours are 1200. The profit per table is $\textsf{₹}1500$ and per chair is $\textsf{₹}1000$. This scenario is best modelled as a:

(A) Transportation problem.

(B) Diet problem.

(C) Manufacturing problem.

(D) Blending problem.

Question 9. Which type of LPP often involves constraints related to supply from origins and demand at destinations?

(A) Diet problem.

(B) Production problem.

(C) Transportation problem.

(D) Financial problem.

Question 10. A farmer wants to decide how many acres of wheat and rice to plant to maximize profit, given limits on land, water, and fertilizer. This is an example of a:

(A) Diet problem.

(B) Transportation problem.

(C) Agricultural problem (similar to production).

(D) Assignment problem.

Question 11. In a Diet Problem, the constraints usually specify:

(A) Maximum cost allowed.

(B) Exact quantities of each food item to consume.

Question 12. If an LPP involves minimizing the total distance traveled to deliver goods from multiple warehouses to multiple stores, it's likely a:

(A) Production problem.

(B) Transportation problem.

(C) Assignment problem.

(D) Blending problem.

Question 13. A company wants to allocate its marketing budget among different advertising channels (TV, radio, online) to maximize reach, subject to budget limits and minimum spending requirements on certain channels. This is an example of:

(A) Production problem.

(B) Diet problem.

(C) Resource allocation problem.

Question 14. Which of the following LPP types focuses on mixing ingredients?

(A) Manufacturing.

(B) Diet.

(C) Transportation.

(D) Blending.

Question 15. A problem of assigning workers to jobs to maximize overall efficiency is an example of a/an:

(A) Transportation problem.

(B) Assignment problem.

(C) Production problem.

(D) Diet problem.

Question 16. In a typical Transportation Problem, the objective function aims to:

(A) Maximize profit.

(B) Minimize total transportation cost.

(C) Maximize the quantity transported.

(D) Minimize production cost.

Question 17. A company needs to schedule production of different models of a product on different machines over several shifts to meet demand while minimizing production cost. This is a type of:

(A) Diet problem.

(B) Scheduling problem.

(C) Transportation problem.

(D) Blending problem.

Question 18. What distinguishes a Diet Problem from a general Production Problem?

(A) Diet problems only involve two variables.

(B) Diet problems always minimize cost, while production problems always maximize profit.

(C) Diet problems typically have constraints representing minimum requirements (often $\geq$), while production problems often have constraints representing maximum resource availability (often $\leq$).

(D) Diet problems have non-linear objective functions.

Question 19. In a Blending Problem, the properties of the final mixture are determined by:

(A) The total cost of ingredients.

(B) The proportion of each ingredient used.

(C) The transportation cost.

Question 20. Which LPP type involves flows from origins to destinations?

(A) Diet problem.

(B) Manufacturing problem.

(C) Transportation problem.

(D) Blending problem.

Question 1. The graphical representation of a linear inequality in two variables, say $ax + by \leq c$, is:

(A) A point.

(B) A line segment.

(C) A half-plane, including the boundary line.

Question 2. The Feasible Region of an LPP is the set of all points that:

(A) Satisfy only the objective function.

(B) Satisfy only the non-negativity restrictions.

(C) Satisfy all the constraints simultaneously.

(D) Give the maximum value of the objective function.

Question 3. When graphing linear inequalities like $x \geq 0$ and $y \geq 0$, the feasible region is restricted to which quadrant of the coordinate plane?

(A) First Quadrant.

(B) Second Quadrant.

(C) Third Quadrant.

(D) Fourth Quadrant.

Question 4. The feasible region for a set of linear constraints in two variables is typically a:

(A) Circle.

(B) Parabola.

(C) Polygon (possibly unbounded).

(D) Single point.

Question 5. If the feasible region of an LPP can be enclosed within a circle, it is called a:

(A) Unbounded feasible region.

(B) Infeasible region.

(C) Bounded feasible region.

(D) Optimal region.

Question 6. What happens if the constraints of an LPP are contradictory (e.g., $x \leq 2$ and $x \geq 5$)?

(A) The feasible region is a single point.

(B) The feasible region is bounded.

(C) There is no feasible region (it's an infeasible problem).

(D) The feasible region is unbounded.

Question 7. A point $(x, y)$ is in the feasible region if it satisfies:

(A) The objective function.

(B) At least one constraint.

(C) All constraints.

(D) The maximum value of the objective function.

Question 8. In graphical method, the region represented by $x+y \leq 5$ and $x,y \geq 0$ is:

(A) A triangle in the first quadrant.

(B) A rectangle in the first quadrant.

(C) A half-plane.

(D) A line segment.

Question 9. The boundaries of the feasible region are formed by:

(A) The non-negativity restrictions only.

(B) The lines corresponding to the constraint equations.

(C) The objective function line.

(D) The optimal solution point.

Question 10. If the feasible region extends infinitely in one or more directions, it is called a:

(A) Bounded region.

(B) Infeasible region.

(C) Unbounded region.

(D) Single point.

Question 11. The intersection of all the half-planes represented by the constraints of an LPP is the:

(A) Objective function.

(B) Feasible region.

(C) Optimal solution.

(D) Set of infeasible points.

Question 12. Consider the constraints $x \geq 2$, $y \geq 3$, $x, y \geq 0$. The feasible region is:

(A) Bounded.

(B) Unbounded.

(C) A single point.

Question 13. If the feasible region is a single point, it means:

(A) There is no solution.

(B) There is exactly one feasible solution.

(C) There are multiple optimal solutions.

(D) The problem is unbounded.

Question 14. A point $(x, y)$ is outside the feasible region if it:

(A) Satisfies all constraints.

(B) Satisfies at least one constraint.

(C) Violates at least one constraint.

(D) Lies on the boundary of a constraint.

Question 15. For a constraint $ax + by \leq c$, the corresponding half-plane is on the side of the line $ax + by = c$ containing the origin $(0,0)$, provided $c$ is positive. This is a useful trick for graphical representation. What happens if $c$ is negative?

(A) The half-plane still contains the origin.

(B) The half-plane is on the side opposite to the origin.

(C) The line passes through the origin.

(D) The feasible region is empty.

Question 16. If the feasible region is empty, the LPP has:

(A) A unique optimal solution.

(B) Multiple optimal solutions.

(D) An unbounded solution.

Question 17. Consider the constraints $x \leq 10$, $y \leq 12$, $x+y \leq 15$, $x \geq 0$, $y \geq 0$. The feasible region will be:

(A) Unbounded.

(B) Bounded.

(C) A single point.

(D) An empty set.

Question 18. A vertex or corner point of the feasible region is a point where:

(A) The objective function is zero.

(B) At least two boundary lines of the feasible region intersect.

(C) The non-negativity constraints are not satisfied.

(D) The region is unbounded.

Question 19. If the feasible region is unbounded, can an optimal solution exist?

(A) Yes, for maximization problems only.

(B) Yes, for minimization problems only.

(C) Yes, an optimal solution can exist for both maximization and minimization problems, or it might not exist (unbounded solution).

(D) No, an optimal solution never exists for an unbounded feasible region.

Question 20. The feasible region is a convex set. What does this property mean in the context of LPP?

(A) Any line segment connecting two points in the region lies entirely within the region.

(B) The region is circular.

(C) The region is always bounded.

(D) The region has smooth boundaries.

Question 1. A point $(x, y)$ in the solution space is called a Feasible Solution if it:

(A) Maximizes the objective function.

(B) Minimizes the objective function.

(C) Satisfies all the given constraints, including non-negativity.

(D) Lies on the boundary of the feasible region.

Question 2. An Infeasible Solution is a point that:

(A) Gives a negative value for the objective function.

(B) Lies within the feasible region.

(C) Satisfies all constraints.

(D) Does not satisfy at least one of the constraints.

Question 3. An Optimal Feasible Solution is a feasible solution that:

(A) Is a vertex of the feasible region.

(B) Maximizes the objective function (for a maximization problem) or minimizes it (for a minimization problem).

(C) Has integer values for decision variables.

(D) Satisfies only the non-negativity constraints.

Question 4. The vertices or corner points of the feasible region are important because:

(A) They always give integer solutions.

(B) They are easy to calculate.

(C) The optimal solution, if it exists and the feasible region is a convex polygon, occurs at one of these points.

(D) They are the only feasible solutions.

Question 5. If a point $(x, y)$ is feasible, it must satisfy:

(A) $Z = cx + dy$ (the objective function).

(B) $x \geq 0$ and $y \geq 0$ only.

(C) All given linear inequalities/equations.

(D) Only the inequalities.

Question 6. Consider an LPP with constraints $x+y \leq 4$, $x \geq 0$, $y \geq 0$. Which of the following points is a feasible solution?

(A) $(5, 0)$.

(B) $(2, 3)$.

(C) $(4, 1)$.

(D) $(1, 2)$.

Question 7. In the previous question, which of the points is an infeasible solution?

(A) $(0, 0)$.

(B) $(1, 2)$.

(C) $(2, 1)$.

(D) $(3, 3)$.

Question 8. The feasible region of an LPP is the set of all:

(A) Optimal solutions.

(B) Infeasible solutions.

(D) Corner points.

Question 9. If an LPP has no feasible region, then it has:

(A) A unique optimal solution at the origin.

(B) Multiple optimal solutions.

(C) No solution (feasible or optimal).

(D) An unbounded solution.

Question 10. The points lying on the boundary of the feasible region are:

(A) Always optimal solutions.

(B) Always infeasible solutions.

(C) Feasible solutions.

(D) Not considered in LPP.

Question 11. How many corner points can a bounded feasible region in a 2-variable LPP have?

(A) Exactly one.

(B) Exactly two.

(C) At least three.

(D) Any number greater than or equal to three (unless it's a line segment or a point).

Question 12. If the optimal solution occurs at two adjacent corner points of the feasible region, it means:

(A) There is a unique optimal solution.

(B) There are multiple optimal solutions along the line segment connecting these two points.

(C) The problem is infeasible.

(D) The feasible region is unbounded.

Question 13. Any point within the feasible region, not on the boundary, is a feasible solution, but is it usually the optimal solution?

(A) Yes, always.

(B) No, the optimal solution for a linear objective function over a convex feasible region occurs at a corner point.

(C) Only for minimization problems.

(D) Only if the feasible region is unbounded.

Question 14. If an LPP has an unbounded feasible region, what can we say about the existence of an optimal solution?

(A) An optimal solution always exists.

(B) An optimal solution never exists.

(C) An optimal solution may or may not exist. If it exists, it will be at a corner point.

(D) The objective function must be zero at the optimal solution.

Question 15. Consider the LPP: Maximize $Z = 2x+3y$ subject to $x+y \leq 1$, $x \geq 0$, $y \geq 0$. The feasible region is a triangle with vertices (0,0), (1,0), and (0,1). Which of these points is a feasible solution?

(A) Only (0,0).

(B) Only (1,0) and (0,1).

(C) (0,0), (1,0), (0,1), and any point inside the triangle.

(D) Only the point that maximizes Z.

Question 16. A degenerate feasible solution is a basic feasible solution in which one or more basic variables are zero. While this concept is more advanced, in graphical terms for a 2-variable problem, it might relate to:

(A) An unbounded feasible region.

(B) An infeasible region.

(C) A corner point where more lines intersect than is typical for a simple vertex.

(D) A feasible region that is a single point.

Question 17. If the feasible region is the set of points $(x,y)$ such that $x+y=5$, $x \geq 0$, $y \geq 0$, how many corner points does it have?

(A) One.

(B) Two.

(C) Infinitely many.

(D) Zero.

Question 18. Every corner point of the feasible region is a:

(A) Optimal solution.

(B) Infeasible solution.

(C) Feasible solution.

(D) Unique solution.

Question 19. If the feasible region is a single point, say (a, b), then the optimal solution is:

(A) Non-existent.

(B) Unique, and it is the point (a, b).

(C) Multiple.

(D) Unbounded.

Question 20. The collection of all feasible solutions is known as the:

(A) Optimal set.

(B) Solution space.

(C) Feasible region.

(D) Constraint set.

Question 1. The Corner Point Theorem states that if an optimal solution to an LPP exists, it must occur:

(A) At the origin (0,0).

(B) At any point within the feasible region.

(C) At one or more corner points (vertices) of the feasible region.

(D) On the boundary of the feasible region, but not necessarily at a corner point.

Question 2. According to the Corner Point Method, to find the optimal solution for a bounded feasible region, you must:

(A) Check any feasible point.

(B) Evaluate the objective function at all corner points of the feasible region.

(C) Evaluate the objective function at the center of the feasible region.

(D) Check points on the boundary but not the corners.

Question 3. If the objective function is $Z = 5x + 2y$ and a corner point of the feasible region is (3, 4), what is the value of Z at this point?

(A) 15.

(B) 8.

(C) 23.

(D) 20.

Question 4. For a maximization problem with a bounded feasible region, the optimal value of the objective function is the:

(A) Smallest value among the values at the corner points.

(B) Largest value among the values at the corner points.

(C) Average value at the corner points.

(D) Value at the origin.

Question 5. If an LPP has multiple optimal solutions, these solutions lie on:

(A) A single corner point.

(B) The feasible region's interior.

(C) A line segment connecting two adjacent corner points.

(D) All corner points simultaneously.

Question 6. For a minimization problem with a bounded feasible region, the optimal value of the objective function is the:

(A) Smallest value among the values at the corner points.

(B) Largest value among the values at the corner points.

(C) Value at the origin.

(D) Value that is closest to zero.

Question 7. If the feasible region is unbounded for a maximization problem, and the objective function can increase indefinitely over the feasible region, then:

(A) A unique optimal solution exists.

(B) Multiple optimal solutions exist.

(C) The problem has an unbounded solution (no maximum value).

(D) The optimal solution is at the origin.

Question 8. If the feasible region is unbounded for a minimization problem, and the objective function can decrease indefinitely, then:

(A) A unique optimal solution exists.

(B) Multiple optimal solutions exist.

(C) The problem has an unbounded solution (no minimum value).

(D) The optimal solution is at a corner point.

Question 9. The Corner Point Method is applicable to LPPs with:

(A) Any number of variables and constraints.

(B) Only two variables and a finite number of linear constraints.

(C) Non-linear objective functions.

(D) Integer decision variables only.

Question 10. If the feasible region is empty, the LPP has:

(A) A unique optimal solution at the origin.

(B) Multiple optimal solutions.

(C) No feasible solution, hence no optimal solution.

(D) An unbounded optimal solution.

Question 11. Suppose the maximum value of $Z = 3x + 4y$ for a feasible region with corner points (0,0), (5,0), (3,2), (0,4) is found to be 16 at point (0,4). This means:

(A) (0,4) is a feasible solution, and 16 is a feasible value of Z.

(B) (0,4) is the unique optimal solution, and 16 is the maximum value of Z.

(C) (0,4) is an infeasible solution.

(D) There must be an error, as the optimal solution is always in the interior.

Question 12. For an LPP with a bounded feasible region, how many optimal solutions can exist?

(A) Exactly one.

(B) Exactly two.

(C) Either exactly one or infinitely many.

(D) Zero, one, or infinitely many.

Question 13. If the objective function line is parallel to one of the boundary lines of the feasible region, and this boundary line corresponds to the optimal value, what can be inferred?

(A) There is a unique optimal solution at one corner point.

(B) There are no feasible solutions.

(C) There are multiple optimal solutions along that boundary line segment.

(D) The problem has an unbounded solution.

Question 14. When checking for an optimal solution in an unbounded feasible region for a maximization problem, what must be considered besides evaluating corner points?

(A) Checking the origin.

(B) Checking points in the interior of the feasible region.

(C) Checking if the objective function can be made arbitrarily large within the feasible region.

(D) Checking all boundary points.

Question 15. The principle behind the Corner Point Method relies on the fact that the feasible region is a convex polygon (in 2D), and a linear function over a convex set attains its extrema (maximum or minimum) at:

(A) The center of the set.

(B) The interior points.

(C) The extreme points (vertices/corner points).

(D) Any point on the boundary.

Question 16. If the feasible region is a single point, say (a,b), does the Corner Point Theorem still apply?

(A) No, it only applies to regions with multiple corners.

(B) Yes, the single point is the only corner point, and it is the optimal solution.

(C) Yes, but the optimal solution is unbounded.

(D) Yes, but it implies no feasible solution.

Question 17. If you are minimizing $Z = 10x + 15y$ subject to a bounded feasible region with corner points (1,5), (3,2), (6,0), (0,0), which point gives the minimum value of Z?

(A) (1,5) -> $10(1) + 15(5) = 10 + 75 = 85$

(B) (3,2) -> $10(3) + 15(2) = 30 + 30 = 60$

(C) (6,0) -> $10(6) + 15(0) = 60 + 0 = 60$

(D) (0,0) -> $10(0) + 15(0) = 0 + 0 = 0$

Question 18. In the previous question, what is the minimum value of Z?

(A) 85.

(B) 60.

(C) 0.

(D) There is no minimum.

Question 19. If the feasible region is unbounded for a minimization problem, and the objective function values increase as you move away from the origin within the feasible region, does an optimal solution (minimum) exist?

(A) Yes, the minimum will be at a corner point.

(B) No, the minimum is unbounded.

(C) Yes, at the origin.

(D) Not enough information.

Question 20. For a maximization problem with an unbounded feasible region, if the objective function value can be increased indefinitely, this indicates:

(A) A unique optimal solution.

(B) Multiple optimal solutions.

(C) An unbounded solution.

(D) The feasible region is actually bounded.

Question 1. The graphical method is suitable for solving LPPs with:

(A) Any number of variables.

(B) Exactly two decision variables.

(C) Non-linear constraints.

(D) Only integer solutions.

Question 2. The first step in the graphical method is to:

(A) Find the corner points.

(B) Evaluate the objective function.

(C) Graph the constraints and find the feasible region.

(D) Formulate the LPP.

Question 3. To graph a linear inequality like $2x + 3y \leq 6$, you first graph the line $2x + 3y = 6$. How do you determine which side of the line represents the inequality?

(A) Always shade the side towards the origin.

(B) Always shade the side away from the origin.

(C) Test a point (like the origin (0,0), if it's not on the line) to see if it satisfies the inequality.

(D) The $\leq$ sign always means shading below the line.

Question 4. After finding the feasible region, the next step in the Corner Point Method using graphical approach is to:

(A) Check a point inside the feasible region.

(B) Identify the corner points of the feasible region.

(C) Draw the objective function line.

(D) Determine if the region is bounded.

Question 5. To solve a maximization problem graphically, after finding the corner points and evaluating the objective function at each, you select the point that gives:

(A) The minimum value of the objective function.

(B) The maximum value of the objective function.

(C) A value of zero for the objective function.

(D) The average value.

Question 6. If, when solving an LPP graphically, the feasible region is empty, what is the conclusion?

(A) The problem has a unique optimal solution.

(B) The problem has multiple optimal solutions.

(C) The problem has no feasible solution.

(D) The problem has an unbounded solution.

Question 7. If, in a maximization problem with an unbounded feasible region, the objective function can be made infinitely large, the problem has:

(A) A unique optimal solution.

(B) Multiple optimal solutions.

(C) No feasible solution.

(D) An unbounded solution.

Question 8. Consider the objective function $Z = 2x + y$. If you draw lines representing $Z = k$ for different values of $k$, how do these lines relate to each other?

(A) They are perpendicular.

(B) They are parallel.

(C) They intersect at the origin.

(D) They form a circle.

Question 9. In the Iso-profit (or Iso-cost) line method, the optimal solution is found by moving the objective function line parallel to itself in the direction of optimization (increasing for max, decreasing for min) until:

(A) It passes through the origin.

(B) It touches the feasible region at the last possible point(s).

(C) It passes outside the feasible region.

(D) It is parallel to a constraint line.

Question 10. If, when solving a minimization problem graphically with an unbounded feasible region, the objective function value increases as you move away from the origin, where will the minimum value occur?

(A) In the interior of the feasible region.

(B) At one of the corner points.

(C) At infinity (unbounded solution).

(D) At the origin.

Question 11. Suppose the feasible region is a line segment. How many corner points does it have?

(A) One.

(B) Two.

(C) Infinitely many.

(D) Zero.

Question 12. If the feasible region is a single point, the graphical method will identify:

(A) No feasible solution.

(B) That single point as the unique optimal solution.

(C) Multiple optimal solutions.

(D) An unbounded solution.

Question 13. In the graphical method, finding the intersection points of the boundary lines helps in identifying the:

(A) Feasible region shape.

(B) Objective function coefficients.

(C) Corner points of the feasible region.

(D) Slope of the objective function.

Question 14. What indicates an unbounded solution graphically in a maximization problem?

(A) A bounded feasible region.

(B) The objective function line can be moved indefinitely far from the origin while still intersecting the feasible region.

(C) The feasible region is a single point.

(D) The feasible region is empty.

Question 15. If the LPP constraints are $x+y \leq 1$, $x+y \geq 3$, $x \geq 0$, $y \geq 0$, the graphical method would show:

(A) A bounded feasible region.

(B) An unbounded feasible region.

(C) No feasible region.

(D) A single feasible point.

Question 16. The final step in solving a bounded LPP graphically using the Corner Point Method is to:

(A) Graph the objective function.

(B) Identify the feasible region.

(C) Compare the objective function values at the corner points and select the optimal one.

Question 17. If the Iso-profit line coincides with one entire edge of the bounded feasible region at the optimal value, this indicates:

(A) An error in calculation.

(B) A unique optimal solution at one endpoint of the edge.

(C) Multiple optimal solutions along that edge.

(D) An unbounded solution.

Question 18. The graphical method provides a visual understanding of:

(A) The simplex algorithm.

(B) How the feasible region and objective function interact to determine the optimum.

(C) The duality principle.

(D) Integer programming.

Question 19. For a minimization problem, using the Iso-cost line method, you move the line $Z = k$ parallel to itself:

(A) Away from the origin to find the smallest Z.

(B) Towards the origin to find the smallest Z.

(C) Away from the origin to find the largest Z.

(D) Towards the origin to find the largest Z.

Question 20. The graphical method can handle LPPs with:

(A) Two or three variables.

(B) Any number of variables.

(C) Exactly two variables.

Question 21. When graphing the inequality $ax + by \geq c$ where $b > 0$, the half-plane lies:

(A) Below the line $ax + by = c$.

(B) Above the line $ax + by = c$.

(C) To the left of the line $ax + by = c$.

(D) To the right of the line $ax + by = c$.

Question 22. If the feasible region is unbounded in a minimization problem, and the objective function $Z=ax+by$ (with $a,b>0$) is being minimized, the minimum value will:

(A) Not exist (unbounded solution).

(B) Be at a corner point.

(C) Be zero at the origin.

(D) Occur in the interior of the feasible region.

Question 23. The feasible region in a 2-variable LPP is formed by the intersection of:

(A) Lines.

(B) Half-planes.

(C) Circles.

(D) Points.

Question 24. When solving an LPP graphically, the number of corner points of a bounded feasible region is always:

(A) Exactly 3.

(B) Exactly 4.

(C) Finite.

(D) Infinite.