Assertion-Reason MCQs for Sub-Topics of Topic 13: Linear Programming

Question 1. Assertion (A): In a Linear Programming Problem, the objective function must always be a linear equation.

Reason (R): The term 'Linear' in Linear Programming implies that all relationships, including the objective function and constraints, are linear functions of the decision variables.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): Decision variables in an LPP can take any real value, positive or negative.

Reason (R): Real-world problems often deal with physical quantities or activities which cannot be negative, hence non-negativity restrictions ($x_i \geq 0$) are frequently imposed.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): Constraints in an LPP represent the maximum possible value of the objective function.

Reason (R): Constraints define the limitations or restrictions within which the optimization problem must be solved, shaping the feasible region.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): An LPP can have only one objective function.

Reason (R): Linear Programming is designed to optimize a single, well-defined goal (profit, cost, etc.) under given limitations.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The coefficients in the objective function are always positive.

Reason (R): Coefficients in the objective function represent the contribution (positive or negative) of each decision variable to the total objective value (e.g., profit per unit can be positive, cost per unit could effectively be seen with negative contribution in some formulations or just positive for minimization).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): When formulating a constraint about maximum resource availability, a 'greater than or equal to' ($\geq$) inequality is used.

Reason (R): A maximum limit on resources means the total usage must be less than or equal to the available amount.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): The coefficients in the constraints represent the unit contribution of each variable to the objective function.

Reason (R): Coefficients in the constraints represent the amount of resource consumed or requirement met per unit of the corresponding decision variable.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): An expression like $x_1^2 + x_2 \leq 10$ can be a constraint in an LPP.

Reason (R): The mathematical formulation of an LPP requires all constraints to be linear functions of the decision variables.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The non-negativity restrictions ($x_i \geq 0$) are always explicitly written as part of the LPP formulation.

Reason (R): These restrictions are fundamental assumptions in most standard LPP models, reflecting that quantities are typically non-negative.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): When formulating a diet problem aiming to meet minimum nutritional needs, the constraints will typically involve 'greater than or equal to' ($\geq$) inequalities.

Reason (R): Minimum requirements mean the total amount of a nutrient obtained must be at least the specified level.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): A Diet Problem is typically a maximization problem.

Reason (R): The usual objective in a Diet Problem is to minimize the cost while satisfying minimum nutritional requirements.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): Transportation Problems always have constraints involving minimum requirements at sources and maximum availability at destinations.

Reason (R): Transportation Problems involve shipping from sources with supply capacities to destinations with demand requirements.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): A Production Problem's constraints usually involve limits on resources like labour or machine time.

Reason (R): Production processes consume limited resources, which creates restrictions on the quantities that can be produced.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): Blending Problems are concerned with determining the optimal proportion of different components to mix.

Reason (R): The objective in Blending Problems is usually to minimize the cost of the mixture while meeting certain quality or property specifications.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): All real-world problems can be formulated as LPPs.

Reason (R): LPP requires linearity in the objective function and constraints, which is not always the case in real-world situations.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): A point is in the feasible region if it satisfies the objective function.

Reason (R): The feasible region consists of all points that satisfy *all* the constraints of the LPP simultaneously.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): The feasible region of an LPP is always a bounded polygon.

Reason (R): The feasible region is the intersection of half-planes, which is always a convex set, but it can be unbounded or even empty.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): If the constraints of an LPP are contradictory, the feasible region is empty.

Reason (R): An empty feasible region means there is no point that satisfies all constraints simultaneously.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): The region represented by $x \geq 0, y \geq 0$ in the Cartesian plane is the first quadrant, which is an unbounded feasible region.

Reason (R): This region extends infinitely in the positive x and y directions.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): Any point on the boundary of the feasible region is also a feasible solution.

Reason (R): The boundary lines are defined by the constraint equations, and points on these lines satisfy the constraints as equalities.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): Every feasible solution is also an optimal solution.

Reason (R): An optimal solution is a feasible solution that yields the best value of the objective function, but not all feasible solutions do this.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): An infeasible solution satisfies at least one constraint.

Reason (R): An infeasible solution violates at least one constraint of the LPP.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): If an LPP has an optimal solution, it must be unique.

Reason (R): An LPP can have a unique optimal solution, multiple optimal solutions, or no optimal solution.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): Corner points of the feasible region are important because they are always optimal solutions.

Reason (R): If an optimal solution exists for an LPP with a convex feasible region, it occurs at one or more of the corner points.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): If the feasible region of an LPP is empty, then it has no optimal solution.

Reason (R): An optimal solution must be a feasible solution, and if no feasible solutions exist, no optimal solution can exist.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): For a bounded feasible region, the optimal solution to an LPP always occurs at a corner point.

Reason (R): The Corner Point Theorem states that the optimal value of a linear objective function over a non-empty bounded convex feasible region is attained at a corner point.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): If a maximization LPP has an unbounded feasible region, it will always have an unbounded solution.

Reason (R): An unbounded feasible region only means that variables can be arbitrarily large. The objective function might still be bounded over this region.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): If the objective function has the same optimal value at two adjacent corner points, then any point on the line segment joining these points is also an optimal solution.

Reason (R): This indicates the objective function line is parallel to the edge connecting the two corner points at the optimal value.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): To find the optimal solution for a bounded feasible region, one must evaluate the objective function at all interior points.

Reason (R): The optimal solution for a bounded feasible region lies at one or more corner points.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): The Corner Point Method can be used for LPPs with three or more variables.

Reason (R): The graphical representation and identification of corner points become complex or impossible in dimensions higher than two or three.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 1. Assertion (A): The graphical method is suitable for solving LPPs with three decision variables.

Reason (R): The graphical method involves plotting constraints and the feasible region on a 2D coordinate plane, limiting it primarily to problems with two variables.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 2. Assertion (A): To determine the feasible region for an inequality like $ax + by \leq c$, one can test the origin $(0,0)$.

Reason (R): If the origin satisfies the inequality and is not on the line $ax+by=c$, then the feasible half-plane is the one containing the origin.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 3. Assertion (A): If the feasible region is empty when using the graphical method, the LPP has an unbounded solution.

Reason (R): An empty feasible region means there are no points that satisfy all constraints, indicating no feasible or optimal solution exists.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 4. Assertion (A): In a maximization problem, the optimal value of Z is found graphically at the corner point furthest from the origin.

Reason (R): The objective function line is moved parallel outwards, and its maximum value is attained at the last point it touches the feasible region, which is usually a corner point, but not necessarily the one furthest from the origin in Euclidean distance.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.

Question 5. Assertion (A): If the feasible region is unbounded in a minimization problem with positive coefficients in the objective function, a minimum value always exists at a corner point.

Reason (R): If the objective function increases as variables increase towards infinity in the feasible region, the lowest value must be attained at a boundary point, specifically a corner point for a linear objective function.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

(E) Both A and R are false.