Multiple Correct Answers MCQs for Sub-Topics of Topic 13: Linear Programming

Question 1. Which of the following statements accurately describe the characteristics of a Linear Programming Problem (LPP)?

(A) It involves optimizing a linear objective function.

(B) All constraints must be linear inequalities.

(C) Decision variables are typically restricted to be non-negative.

(D) The relationships between variables in the objective function and constraints are linear.

(E) It can handle quadratic relationships between variables.

Question 2. In the context of an LPP, which of the following are considered "constraints"?

(A) The function to be maximized or minimized.

(B) Limitations on available resources like raw materials or labour hours.

(C) Minimum requirements for nutrients in a diet problem.

(D) The decision variables themselves.

(E) Production capacity limits.

Question 3. Which of the following are valid types of objective functions in an LPP?

(A) Maximize $Z = 3x_1 + 5x_2$.

(B) Minimize $C = 10y_1 + 8y_2 + 12y_3$.

(C) Maximize $P = x_1 x_2$.

(D) Minimize $Cost = \sqrt{x_1} + x_2$.

(E) Maximize $Z = 2x_1 - 4x_2$.

Question 4. What do the decision variables in a typical LPP represent?

(A) The constants in the objective function.

(B) The quantities of activities or items that need to be determined to achieve the optimal solution.

(C) The total amount of resources available.

(D) The profit or cost associated with each unit of activity.

(E) The variables over which decisions are to be made.

Question 5. The non-negativity restrictions ($x_i \geq 0$) in an LPP are common because they reflect that:

(A) Quantities of physical goods or activities are typically non-negative.

(B) It simplifies the graphical representation.

(C) Negative production or consumption is usually not meaningful in real-world scenarios.

(D) It ensures the feasible region is in the first quadrant (in 2D).

(E) It guarantees a unique optimal solution.

Question 6. Consider a problem where a company wants to maximize profit from selling two types of products, A and B, given limited raw material. Which of the following would likely be decision variables and the objective function type?

(A) Decision variables: Quantity of raw material used.

(B) Decision variables: Number of units of product A and product B produced.

(C) Objective function type: Minimization of cost.

(D) Objective function type: Maximization of profit.

(E) Constraints: Raw material availability.

Question 7. Which of the following mathematical expressions could represent valid linear constraints in an LPP (assuming $x_1, x_2$ are decision variables)?

(A) $2x_1 + 5x_2 \leq 100$.

(B) $x_1 - x_2 = 0$.

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

(D) $x_1^2 + x_2^2 \leq 50$.

(E) $x_1 \geq 0$ and $x_2 \geq 0$.

Question 8. A company aims to minimize the cost of production. The objective function would represent the total cost, and the constraints might involve meeting minimum production targets and using available resources. Which statements are true?

(A) The objective function would be of the minimization type.

(B) Constraints related to minimum production targets would use $\geq$ inequalities.

(C) Constraints related to maximum resource availability would use $\leq$ inequalities.

(D) Decision variables would represent the costs of resources.

(E) The coefficients in the objective function would represent costs per unit of decision variable.

Question 9. The term "programming" in Linear Programming refers to:

(A) Writing computer code to solve the problem.

(B) Planning or scheduling activities.

(C) Graphical representation of the problem.

(D) Optimizing a process or plan.

(E) A mathematical procedure for determining optimal allocation of scarce resources.

Question 10. If an LPP has a constraint like "The total production of items A and B must be exactly 100 units", how would this be represented mathematically (let $x_A$ and $x_B$ be the number of units)?

(A) $x_A + x_B \leq 100$.

(B) $x_A + x_B \geq 100$.

(C) $x_A + x_B = 100$.

(D) $x_A \leq 100$ and $x_B \leq 100$.

(E) It is a linear constraint.

Question 11. Which of the following scenarios can potentially be formulated as an LPP?

(A) Determining the best mix of ingredients for a product to minimize cost while meeting quality standards.

(B) Finding the shortest route between two cities.

(C) Assigning tasks to workers to maximize total efficiency.

(D) Predicting stock market prices based on historical data.

(E) Deciding production quantities for multiple products to maximize profit under resource constraints.

Question 12. The purpose of the objective function in an LPP is to:

(A) Define the limitations of the problem.

(B) Specify the relationship between decision variables.

(C) Represent the goal that needs to be optimized (maximized or minimized).

(D) Ensure that the decision variables are non-negative.

(E) Provide a single numerical value to compare different feasible solutions.

Question 1. When formulating an LPP from a real-world problem, what are the essential steps involved?

(A) Identifying the decision variables.

(B) Defining the objective function.

(C) Identifying and expressing the constraints as linear equations or inequalities.

(D) Solving the problem graphically.

(E) Adding non-negativity restrictions for decision variables.

Question 2. The general mathematical structure of an LPP includes:

(A) Optimization of a linear function.

(B) Subject to a set of linear constraints (equalities or inequalities).

(C) Constraints can be linear or non-linear.

(D) Decision variables are usually restricted to be non-negative.

(E) Decision variables can be any real numbers.

Question 3. Consider a company producing two types of pens, P1 and P2. P1 takes 1 unit of plastic and 2 units of ink. P2 takes 3 units of plastic and 4 units of ink. The company has 100 units of plastic and 180 units of ink available. P1 sells for $\textsf{₹}10$ and P2 for $\textsf{₹}15$. Let $x_1$ and $x_2$ be the number of pens of type P1 and P2 respectively. Which of the following statements are correct for formulating this as a maximization LPP?

(A) Decision variables are $x_1, x_2 \geq 0$.

(B) Objective function: Maximize $Z = 10x_1 + 15x_2$.

(C) Plastic constraint: $x_1 + 3x_2 \leq 100$.

(D) Ink constraint: $2x_1 + 4x_2 \leq 180$.

(E) Objective function: Minimize $Z = 10x_1 + 15x_2$.

Question 4. In the consolidated mathematical formulation $\text{Optimize } Z = \mathbf{c}^T \mathbf{x} \text{ subject to } A\mathbf{x} \leq \mathbf{b}, \mathbf{x} \geq \mathbf{0}$, what do the components represent?

(A) $\mathbf{x}$ is the vector of decision variables.

(B) $\mathbf{c}$ is the vector of coefficients in the objective function.

(C) $A$ is the matrix of constraint coefficients.

(D) $\mathbf{b}$ is the vector of right-hand side values (resource limits or requirements).

(E) $\leq$ implies all constraints must be of the 'less than or equal to' type.

Question 5. A farm has 50 hectares of land to grow wheat and rice. Wheat requires 2 labour days per hectare and rice requires 3 labour days per hectare. Total labour days available are 120. Profit from wheat is $\textsf{₹}8000$ per hectare and from rice is $\textsf{₹}10000$ per hectare. Let $x_w$ and $x_r$ be the hectares of wheat and rice planted. Which are correct parts of the LPP formulation to maximize profit?

(A) Decision variables: $x_w, x_r \geq 0$.

(B) Land constraint: $x_w + x_r \leq 50$.

(C) Labour constraint: $2x_w + 3x_r \leq 120$.

(D) Objective function: Maximize $Z = 8000x_w + 10000x_r$.

(E) Labour constraint: $2x_w + 3x_r = 120$.

Question 6. In translating a real-world problem into an LPP, identifying the constraints involves determining:

(A) The relationships between variables that limit the possible values of the decision variables.

(B) The coefficients of the objective function.

(C) The total amount of each limited resource.

(D) The minimum or maximum requirement for certain outcomes.

(E) Which variables can be positive or negative.

Question 7. Which of the following aspects are considered in the mathematical formulation of an LPP with two variables, say $x$ and $y$?

(A) A linear objective function of the form $Z = ax + by$.

(B) Constraints which are linear equations or inequalities involving $x$ and $y$.

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

(D) The constraints must always be of the form $\leq$.

(E) The objective function must always be of the maximization type.

Question 8. A factory manufactures cycles and scooters. It takes 2 hours to make a cycle and 3 hours to make a scooter. Total manufacturing time available is 100 hours. Material cost for a cycle is $\textsf{₹}500$ and for a scooter is $\textsf{₹}800$. Total material budget is $\textsf{₹}30000$. Let $x_c$ be the number of cycles and $x_s$ be the number of scooters. Which constraints are correct?

(A) $2x_c + 3x_s \leq 100$ (Time constraint).

(B) $500x_c + 800x_s \leq 30000$ (Material cost constraint).

(C) $x_c \geq 0, x_s \geq 0$ (Non-negativity).

(D) $2x_c + 500x_c \leq 100 + 30000$ (Incorrect combination).

(E) $x_c + x_s \leq \text{Something}$ (There is no total unit constraint given).

Question 9. In the context of LPP formulation, the coefficients in the objective function and constraints represent:

(A) The unknown decision variables.

(B) Known numerical values based on the problem data (e.g., profit/unit, resource consumption/unit, resource availability).

(C) Slopes of the constraint lines.

(D) Intercepts of the constraint lines.

(E) Rates of change related to the decision variables.

Question 10. A diet plan requires at least 40 units of Vitamin A and 50 units of Vitamin B. Food X costs $\textsf{₹}50$ per kg and provides 2 units of A and 5 units of B. Food Y costs $\textsf{₹}70$ 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. Which inequalities are correct for this problem aiming to minimize cost?

(A) $2x + 4y \geq 40$ (Vitamin A requirement).

(B) $5x + 2y \geq 50$ (Vitamin B requirement).

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

(D) Objective function: Minimize $Z = 50x + 70y$.

(E) $2x + 4y \leq 40$.

Question 11. Which of the following mathematical forms correctly represents the consolidated structure of an LPP?

(A) Maximize $Z = \sum c_i x_i$ subject to $\sum a_{ij} x_j \leq b_i$ and $x_j \geq 0$.

(B) Minimize $Z = \sum c_i x_i$ subject to $\sum a_{ij} x_j \geq b_i$ and $x_j \geq 0$.

(C) Optimize $Z = \sum c_i x_i$ subject to $\sum a_{ij} x_j \{ \leq, =, \geq \} b_i$ and $x_j \geq 0$.

(D) Optimize $Z = f(x_1, ..., x_n)$ where $f$ is any function, subject to linear constraints.

(E) Optimize $Z = \mathbf{c}^T \mathbf{x}$ subject to $A\mathbf{x} \{ \leq, =, \geq \} \mathbf{b}$ and $\mathbf{x} \geq \mathbf{0}$.

Question 12. When translating a constraint like "The amount of raw material A used must not exceed the available quantity of raw material A" into an inequality, what must be identified?

(A) The decision variables related to the use of raw material A.

(B) The amount of raw material A required per unit of each product that uses it.

(C) The total available quantity of raw material A.

(D) The profit per unit of product.

(E) The cost of raw material A.

Question 1. Which of the following are commonly recognized types of LPPs based on real-world applications?

(A) Manufacturing/Production Problems.

(B) Diet Problems.

(C) Transportation Problems.

(D) Blending Problems.

(E) Regression Analysis Problems.

Question 2. A Diet Problem typically involves:

(A) Minimizing the cost of a diet.

(B) Maximizing the nutrient content of a diet.

(C) Constraints related to minimum daily requirements of various nutrients.

(D) Constraints related to maximum allowable intake of certain components (e.g., fat, calories).

(E) Determining the shortest route for food delivery.

Question 3. In a Transportation Problem, what are the key elements to consider?

(A) Sources with available supply.

(B) Destinations with specific demand.

(C) Unit transportation costs between each source and each destination.

(D) Maximizing profit from production.

(E) Minimizing the total transportation cost.

Question 4. A Manufacturing/Production Problem often aims to:

(A) Maximize total profit.

(B) Minimize total production cost.

(C) Decide the quantity of each product to manufacture.

(D) Allocate limited resources (labour, machine time, material) among different products.

(E) Determine the optimal blend of raw materials.

Question 5. Which of the following would likely be formulated as a Blending Problem?

(A) Mixing different grades of crude oil to produce gasoline with specific octane ratings at minimum cost.

(B) Combining various food ingredients to create a feed mix meeting minimum protein and vitamin levels at the lowest cost.

(C) Deciding how many units of furniture to produce from available wood and labour.

(D) Determining the quantity of goods to ship from warehouses to retail stores.

(E) Mixing different types of steel scrap to produce a certain alloy with specific properties.

Question 6. Consider a problem where a company needs to ship packets of goods from two warehouses (W1, W2) to three retail outlets (R1, R2, R3). Which aspects would be part of the Transportation Problem formulation?

(A) Decision variables representing the quantity shipped from W_i to R_j.

(B) Constraints ensuring the total quantity shipped from each warehouse does not exceed its supply capacity.

(C) Constraints ensuring the total quantity received at each retail outlet meets its demand.

(D) Objective function aiming to maximize the number of packets shipped.

(E) Objective function aiming to minimize the total shipping cost based on per-unit costs between each pair of warehouse and outlet.

Question 7. Which of the following are characteristics of a typical Manufacturing Problem LPP?

(A) Constraints often involve limits on resources like machine hours, labour hours, or raw materials.

(B) Decision variables represent the quantity of each product to manufacture.

(C) The objective function is usually to maximize profit or minimize cost.

(D) Constraints are typically $\geq$ inequalities representing minimum requirements.

(E) It always involves only two products.

Question 8. In a Diet Problem, if $x_i$ represents the quantity of food item $i$ and $c_i$ is its cost per unit, and the diet needs at least $R_j$ units of nutrient $j$, where food item $i$ contains $a_{ij}$ units of nutrient $j$, which mathematical expressions are likely to appear in the formulation?

(A) Minimize $Z = \sum c_i x_i$.

(B) Constraints $\sum_i a_{ij} x_i \geq R_j$ for each nutrient $j$.

(C) Constraints $\sum_i a_{ij} x_i \leq R_j$.

(D) $x_i \geq 0$ for all food items $i$.

(E) Maximize $Z = \sum c_i x_i$.

Question 9. Which of the following are key characteristics of a Transportation Problem?

(A) Multiple sources and multiple destinations.

(B) Objective is usually to minimize the total cost of moving goods.

(C) Supply constraints at sources are typically 'less than or equal to' (or equal to) the available supply.

(D) Demand constraints at destinations are typically 'greater than or equal to' (or equal to) the required demand.

(E) Constraints involve minimum nutrient requirements.

Question 10. Which LPP types often involve minimum requirements in their constraints?

(A) Manufacturing Problems (e.g., minimum production quantity).

(B) Diet Problems (e.g., minimum nutrient intake).

(C) Blending Problems (e.g., minimum percentage of a component).

(D) Transportation Problems (e.g., minimum demand to be met).

(E) Investment Problems (e.g., minimum investment in a safe option).

Question 11. What distinguishes a Blending Problem from a general Production Problem?

(A) Blending problems focus on mixing inputs to achieve desired output properties.

(B) Production problems focus on resource allocation to manufacture distinct items.

(C) Blending problems always involve minimizing cost.

(D) Production problems always involve maximizing profit.

(E) Blending problems use proportions or percentages as part of constraints.

Question 1. The Feasible Region of an LPP is defined as the set of all points that:

(A) Maximize or minimize the objective function.

(B) Satisfy the non-negativity constraints.

(C) Satisfy all the constraints of the problem simultaneously.

(D) Represent possible values for the decision variables that are allowed by the limitations.

(E) Lie outside the boundaries defined by the constraints.

Question 2. In a 2-variable LPP ($x, y$), the graphical representation of the feasible region involves:

(A) Drawing the lines corresponding to the constraint equations.

(B) Identifying the half-plane for each inequality constraint.

(C) Finding the intersection of all the feasible half-planes.

(D) Plotting the objective function line.

(E) Always being a bounded polygon.

Question 3. Which of the following are properties of the feasible region in an LPP?

(A) It is always a convex set.

(B) It can be empty.

(C) It can be bounded.

(D) It can be unbounded.

(E) It always contains the origin.

Question 4. A point $(x, y)$ is considered an Infeasible Solution if:

(A) It yields a poor value for the objective function.

(B) It satisfies all constraints.

(C) It lies outside the feasible region.

(D) It violates at least one of the constraints.

(E) $x < 0$ or $y < 0$ (assuming standard non-negativity).

Question 5. Consider the constraints $x+y \leq 5$, $x \geq 0$, $y \geq 0$. Which points lie within or on the boundary of the feasible region?

(A) $(0,0)$.

(B) $(2,2)$.

(C) $(3,2)$.

(D) $(5,0)$.

(E) $(4,4)$.

Question 6. If an LPP has constraints $x \geq 2$, $y \geq 1$, $x+y \leq 8$, $x \geq 0$, $y \geq 0$. Describe the feasible region.

(A) It is a convex polygon.

(B) It is unbounded.

(C) It is in the first quadrant.

(D) It is bounded by the lines $x=2, y=1, x+y=8, x=0, y=0$.

(E) It is empty.

Question 7. An unbounded feasible region means that:

(A) The problem has no feasible solution.

(B) The feasible region extends infinitely in at least one direction.

(C) An optimal solution may or may not exist.

(D) The objective function can potentially take on infinitely large or small values.

(E) The feasible region is always a polygon.

Question 8. How is the feasible region typically determined graphically in a 2-variable LPP?

(A) By finding the point where the objective function is maximized.

(B) By shading the area that satisfies all constraints simultaneously.

(C) By identifying the corner points.

(D) By drawing a single line representing the objective function.

(E) By considering the non-negativity restrictions $x \geq 0, y \geq 0$, which limit the region to the first quadrant.

Question 9. If the constraints of an LPP are $x+y \geq 10$ and $x+y \leq 5$, $x \geq 0$, $y \geq 0$, what can be said about the feasible region?

(A) It is bounded.

(B) It is unbounded.

(C) It is empty.

(D) There are no points $(x,y)$ that can satisfy both $x+y \geq 10$ and $x+y \leq 5$ simultaneously.

(E) The problem is infeasible.

Question 10. A convex set is one where for any two points within the set, the line segment connecting them lies entirely within the set. Why is the feasible region of an LPP always a convex set?

(A) Because the objective function is linear.

(B) Because the intersection of multiple convex sets is also convex.

(C) Because each linear inequality/equation defines a half-plane (or line), which is a convex set.

(D) Because it makes finding the optimal solution easier.

(E) Because of the non-negativity constraints.

Question 11. Which of the following are true about the boundaries of the feasible region?

(A) They are formed by the lines corresponding to the constraints.

(B) Points on the boundaries are considered feasible solutions.

(C) Corner points are intersection points of these boundary lines.

(D) The objective function forms a boundary.

(E) Non-negativity constraints ($x \geq 0, y \geq 0$) form two of the boundaries (the axes) in the first quadrant.

Question 1. A Feasible Solution to an LPP must satisfy which conditions?

(A) It must lie within or on the boundary of the feasible region.

(B) It must satisfy all the given linear constraints.

(C) It must satisfy the non-negativity restrictions.

(D) It must be a corner point.

(E) It must maximize or minimize the objective function.

Question 2. What characterizes an Optimal Feasible Solution?

(A) It is a point within the feasible region.

(B) It satisfies all constraints.

(C) It gives the best possible value (maximum for maximization, minimum for minimization) for the objective function among all feasible solutions.

(D) It is always unique.

(E) It is usually a corner point of the feasible region (if one exists).

Question 3. Which of the following statements about Infeasible Solutions are true?

(A) They violate at least one constraint.

(B) They lie outside the feasible region.

(C) They cannot be considered as possible optimal solutions.

(D) They are points that satisfy the objective function but not the constraints.

(E) If all solutions are infeasible, the feasible region is empty.

Question 4. What is true about the Vertices (Corner Points) of the feasible region in a 2-variable LPP?

(A) They are intersection points of the boundary lines of the feasible region.

(B) They are feasible solutions.

(C) If an optimal solution exists for a bounded region, it occurs at one or more corner points.

(D) Every feasible solution is a corner point.

(E) They are also called extreme points.

Question 5. Consider an LPP with constraints $x+y \leq 6$, $x \geq 2$, $y \geq 1$. Which points are feasible solutions?

(A) $(2,1)$.

(B) $(3,2)$.

(C) $(1,5)$.

(D) $(4,2)$.

(E) $(0,0)$.

Question 6. If an LPP has no feasible solution, what does this imply?

(A) The constraints are contradictory.

(B) The feasible region is empty.

(C) There is no point that satisfies all constraints simultaneously.

(D) An optimal solution exists but cannot be found.

(E) The objective function is irrelevant.

Question 7. If the feasible region is a line segment, then:

(A) Any point on the line segment is a feasible solution.

(B) The endpoints of the line segment are the corner points.

(C) If an optimal solution exists, it must be at one of the endpoints or along the entire segment.

(D) The feasible region is bounded.

(E) The problem is infeasible.

Question 8. Consider a maximization problem with objective function $Z = 5x + 2y$. If the feasible region is bounded and includes the points $(0,0), (5,0), (0,3), (2,4)$. Which of these are feasible solutions? What is the maximum value of Z?

(A) $(0,0)$ is feasible.

(B) $(2,4)$ is feasible.

(C) Value of Z at $(5,0)$ is $5(5) + 2(0) = 25$.

(D) Value of Z at $(2,4)$ is $5(2) + 2(4) = 10 + 8 = 18$.

(E) The maximum value of Z is 25.

Question 9. If an LPP has multiple optimal solutions, it implies that:

(A) The objective function takes the optimal value at more than one feasible point.

(B) The optimal solutions occur along a line segment on the boundary of the feasible region.

(C) There must be at least two corner points that give the same optimal value.

(D) Any convex combination of these multiple optimal solutions is also an optimal solution.

(E) The feasible region must be unbounded.

Question 10. Which of the following are true statements regarding solutions of an LPP?

(A) Every feasible solution is an optimal solution.

(B) Every optimal solution is a feasible solution.

(C) If a feasible solution exists, an optimal solution always exists.

(D) If the feasible region is empty, there are no feasible solutions and no optimal solutions.

(E) If the feasible region is a single point, that point is the unique optimal solution.

Question 1. According to the Corner Point Theorem (Fundamental Theorem of LPP), for a bounded feasible region, which statements are true?

(A) An optimal solution exists.

(B) The optimal solution occurs at least at one corner point.

(C) If the optimal solution occurs at more than one corner point, it occurs at every point on the line segment joining them.

(D) The optimal value is found by evaluating the objective function at all corner points and selecting the best value.

(E) The optimal solution can occur in the interior of the feasible region.

Question 2. Steps involved in the Corner Point Method for a bounded feasible region include:

(A) Graphing the feasible region.

(B) Identifying all the corner points of the feasible region.

(C) Evaluating the objective function at each corner point.

(D) Selecting the corner point(s) that yield the optimal value of the objective function.

(E) Checking a test point in the interior of the feasible region.

Question 3. If, for a maximization problem with a bounded feasible region, the maximum value of the objective function is found at two corner points, say A and B, what can be concluded?

(A) The problem has a unique optimal solution at point A.

(B) The problem has multiple optimal solutions.

(C) Any point on the line segment AB is also an optimal solution.

(D) The objective function line is parallel to the edge connecting A and B at the maximum value.

(E) The feasible region must be unbounded.

Question 4. Consider a minimization problem with an unbounded feasible region. If the minimum value of the objective function exists, where must it occur?

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

(B) In the interior of the feasible region.

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

(D) Along an edge of the feasible region.

(E) The minimum might not exist (unbounded solution from below).

Question 5. Which conditions are sufficient to guarantee the existence of an optimal solution for an LPP?

(A) The feasible region is bounded.

(B) The feasible region is non-empty.

(C) The objective function is linear.

(D) The decision variables are non-negative.

(E) The feasible region is bounded and non-empty.

Question 6. If an LPP has an unbounded feasible region for a maximization problem, which situations are possible?

(A) A unique optimal solution exists at a corner point.

(B) Multiple optimal solutions exist along an edge.

(C) The objective function is unbounded (can increase indefinitely), meaning no maximum exists.

(D) The objective function is bounded (has a maximum value), which occurs at a corner point.

(E) There are no feasible solutions.

Question 7. If the feasible region is unbounded in a minimization problem, and the objective function $Z = ax + by$ (where $a, b > 0$) is being minimized, what outcomes are possible?

(A) An unbounded solution (Z can be arbitrarily small/negative if coefficients allow, but here positive).

(B) A unique optimal solution at a corner point.

(C) Multiple optimal solutions along an edge.

(D) No feasible solution.

(E) The minimum value is 0 at the origin (if origin is feasible).

Question 8. Evaluating the objective function $Z = cx + dy$ at a corner point $(x_0, y_0)$ means calculating:

(A) $c \cdot x_0 + d \cdot y_0$.

(B) The slope of the objective function line.

(C) The value of Z at that specific feasible point.

(D) A potential optimal value.

(E) The coordinates of the corner point.

Question 9. Suppose the feasible region is a single point (a,b). Which statements about the optimal solution are true?

(A) The point (a,b) is a corner point.

(B) The feasible region is bounded.

(C) The point (a,b) is the unique feasible solution.

(D) The point (a,b) is the unique optimal solution for both maximization and minimization problems.

(E) No optimal solution exists.

Question 10. The Corner Point Method is particularly useful for which type of LPPs?

(A) LPPs with only equality constraints.

(B) LPPs with two decision variables that can be solved graphically.

(C) LPPs with a bounded feasible region.

(D) LPPs with integer-only solutions.

(E) LPPs with more than two variables that must be solved using software.

Question 11. If the feasible region is unbounded, and the objective function $Z = ax + by$ (where $a, b > 0$) is being maximized, which outcomes are possible?

(A) An unbounded solution (Z can be arbitrarily large).

(B) A unique optimal solution at a corner point.

(C) Multiple optimal solutions along an edge.

(D) No feasible solution.

(E) The optimal value is 0 at the origin.

Question 12. If the feasible region is unbounded in a minimization problem, and the objective function $Z = ax + by$ (where $a, b > 0$) is being minimized, what outcomes are possible?

(A) An unbounded solution (Z can be arbitrarily small/negative if coefficients allow, but here positive).

(B) A unique optimal solution at a corner point.

(C) Multiple optimal solutions along an edge.

(D) No feasible solution.

(E) The minimum value is 0 at the origin (if origin is feasible).

Question 1. The graphical method is effective for solving LPPs when:

(A) There are exactly two decision variables.

(B) The constraints are linear.

(C) The objective function is linear.

(D) The feasible region is non-empty.

(E) The problem has integer variables.

Question 2. Steps involved in the graphical method for a 2-variable LPP include:

(A) Formulating the LPP mathematically.

(B) Plotting the lines corresponding to the constraint equations.

(C) Identifying and shading the feasible region.

(D) Finding the coordinates of the corner points of the feasible region.

(E) Applying the simplex method.

Question 3. When graphing a linear inequality like $ax + by \leq c$, the boundary line is $ax + by = c$. Which statements about the half-plane are true?

(A) If $c>0$, the origin $(0,0)$ is a convenient test point.

(B) If the origin satisfies the inequality, the half-plane containing the origin is the feasible region for that constraint.

(C) If the origin does not satisfy the inequality, the half-plane opposite to the origin is the feasible region for that constraint.

(D) If $c=0$, the line passes through the origin, and another test point must be used.

(E) The inequality $\leq$ always implies shading below the line.

Question 4. In the graphical method using the Iso-profit/Iso-cost line approach, what is done?

(A) A line representing the objective function is drawn for an arbitrary value of Z.

(B) This line is moved parallel to itself.

(C) For maximization, the line is moved away from the origin as much as possible while still touching the feasible region.

(D) For minimization, the line is moved towards the origin as much as possible while still touching the feasible region.

(E) The slope of the objective function line is calculated.

Question 5. When solving a maximization problem graphically, the optimal solution occurs at a corner point where:

(A) The objective function value is minimum.

(B) The objective function value is maximum.

(C) The last point touched by the Iso-profit line as it moves away from the origin within the feasible region.

(D) The highest value of Z is achieved among all corner points.

(E) It is closest to the origin.

Question 6. Special cases that can arise when using the graphical method include:

(A) The problem has no feasible solution.

(B) The problem has an unbounded solution.

(C) The problem has multiple optimal solutions.

(D) The feasible region is a single point.

(E) The objective function is non-linear.

Question 7. If the feasible region is unbounded in a maximization problem, how do you check if the solution is unbounded?

(A) Check the values of the objective function at the corner points.

(B) Draw the objective function line and see if it can be moved indefinitely far in the maximizing direction while staying within the feasible region.

(C) If the values of the objective function increase indefinitely along an edge or ray of the feasible region.

(D) If there is no maximum value among the corner points.

(E) Check if the origin is feasible.

Question 8. Consider solving a minimization problem graphically. The steps are similar to maximization, but with key differences. Which differences apply?

(A) The objective is to find the smallest value of Z.

(B) When evaluating corner points, select the one yielding the minimum Z.

(C) Using the Iso-cost line, move it towards the origin as much as possible within the feasible region.

(D) An unbounded feasible region always means an unbounded solution for minimization.

(E) If the objective function can be made arbitrarily small (e.g., tending to $-\infty$), the solution is unbounded (from below).

Question 9. If, during the graphical method, the feasible region turns out to be empty, what should be concluded?

(A) There are no points $(x, y)$ that satisfy all the constraints simultaneously.

(B) The LPP has no feasible solution.

(C) The LPP has no optimal solution.

(D) The constraints are contradictory.

(E) The objective function is incorrectly defined.

Question 10. Which statements are true about the graphical representation of $x \geq 0$ and $y \geq 0$ constraints?

(A) They limit the feasible region to the first quadrant of the coordinate plane.

(B) The lines $x=0$ (y-axis) and $y=0$ (x-axis) are boundary lines.

(C) Any point with a negative x or y coordinate is infeasible with respect to these constraints.

(D) They do not affect the feasible region if other constraints are present.

(E) They ensure the feasible region is bounded.

Question 11. When finding the corner points of the feasible region graphically, you need to:

(A) Identify the points where two or more boundary lines intersect.

(B) Solve systems of linear equations formed by the intersecting boundary lines.

(C) Check if the intersection points are actually part of the feasible region.

(D) Find the points where the objective function intersects the constraints.

(E) Include the origin if it's a vertex of the feasible region.

Question 12. If the objective function is parallel to one of the edges of a bounded feasible region, and that edge contains optimal solutions, which of the following are true?

(A) There are infinitely many optimal solutions.

(B) The optimal solutions are all the points on that specific edge segment.

(C) The value of the objective function is the same at both corner points defining that edge.

(D) This indicates an unbounded solution.

(E) This is a case of multiple optimal solutions.

Question 13. The graphical method is limited because:

(A) It can only handle two decision variables.

(B) It cannot handle equality constraints.

(C) It is difficult to accurately read values from the graph for complex problems.

(D) It cannot identify unbounded solutions.

(E) It cannot handle more than a few constraints effectively.