COMPETENCY BASED
APPLY
1 Marks
2025
AISSCE(Board Exam)
MCQ SINGLE
In a Linear Programming Problem (LPP), the objective function $Z = 2x + 5y$ is to be maximised under the following constraints :$x + y \leq 4$, $3x + 3y \geq 18$, $x, y \geq 0$
Study the graph and select the correct option.
(A)
lies in the shaded unbounded region.
(B)
lies in $\triangle AOB$.
(C)
does not exist.
(D)
lies in the combined region of $\triangle AOB$ and unbounded shaded region.
Correct Answer: C
Explanation
The feasible region of a Linear Programming Problem is the set of points that satisfy all the given constraints simultaneously.Constraint 1 requires the points to be in the region where the sum of $x$ and $y$ is less than or equal to 4.Constraint 2 requires the points to be in the region where the sum of $x$ and $y$ is greater than or equal to 6.Mathematically, a number cannot be both $\le 4$ and $\ge 6$ at the same time. Visually, looking at the graph, there is a clear gap between the shaded region $\Delta AOB$ and the shaded unbounded region above $PQ$. The two regions do not overlap.3. ConclusionSince there is no common region that satisfies all constraints, the feasible region is an empty set.Without a feasible region, there are no valid values for $x$ and $y$ to substitute into the objective function $Z$. Therefore, an optimal solution (maximum value) cannot be found.Answer:The correct option is (C) does not exist.
Step-by-Step Solution
- Analyze the Constraints:
- $x + y \leq 4$
- $3x + 3y \geq 18 \implies x + y \geq 6$
- $x, y \geq 0$
- Identify the Feasible Region: The first constraint $x + y \leq 4$ defines a region below the line $x + y = 4$. The second constraint $x + y \geq 6$ defines a region above the line $x + y = 6$.
- Check for Overlap: The lines $x + y = 4$ and $x + y = 6$ are parallel. Since $6 > 4$, there is no region where both $x + y \leq 4$ and $x + y \geq 6$ are simultaneously satisfied. Therefore, there is no feasible region.
- Conclusion: Since there is no feasible region that satisfies all the constraints, the solution to the LPP does not exist.
Correct Answer: does not exist.
Pedagogical Audit
Bloom's Analysis:
This is an APPLY question because students need to apply their knowledge of linear programming to analyze the constraints and determine if a feasible region exists.
Knowledge Dimension:
CONCEPTUAL
Justification:
The question requires understanding the concepts of constraints and feasible regions in linear programming, rather than just recalling facts or procedures.
Syllabus Audit:
In the context of CBSE Class 12, this is classified as COMPETENCY. It assesses the ability to apply the concepts of LPP to a given problem and interpret the results, rather than just testing knowledge of definitions or formulas.
Step-by-Step Solution
- Analyze the constraints:
We have the following constraints:
- $x + y \leq 4$
- $3x + 3y \geq 18$, which simplifies to $x + y \geq 6$
- $x \geq 0$
- $y \geq 0$
- Determine the feasible region: The first constraint, $x + y \leq 4$, represents the region below the line $x + y = 4$. The second constraint, $x + y \geq 6$, represents the region above the line $x + y = 6$.
- Check for overlapping region: The lines $x + y = 4$ and $x + y = 6$ are parallel lines. Since $6 > 4$, there is no region that satisfies both $x + y \leq 4$ and $x + y \geq 6$ simultaneously. Therefore, there is no feasible region.
- Conclusion: Since there is no feasible region that satisfies all the constraints, the solution to the LPP does not exist.
Correct Answer: does not exist.
AI Suggestion: Option C
AI generated content. Review strictly for academic accuracy.
Pedagogical Audit
Bloom's Analysis:
This is an APPLY question because it requires the student to apply their understanding of linear programming constraints and feasible regions to determine the solution's existence based on the given constraints.
Knowledge Dimension:
CONCEPTUAL
Justification:
The question tests the understanding of the concepts of linear programming, including objective functions, constraints, and feasible regions. It requires students to interpret the constraints and determine the feasible region.
Syllabus Audit:
In the context of CBSE Class 12, this is classified as COMPETENCY. The question requires the student to apply the concepts of LPP to analyze the constraints and determine the feasibility of the solution, going beyond rote memorization of formulas.
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