Class CBSE Class 12 Mathematics Linear Programming Q #682
COMPETENCY BASED
APPLY
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
The corner points of the feasible region of a Linear Programming Problem are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). If \(Z=ax+by;\) (a, \(b>0)\) be the objective function, and maximum value of Z is obtained at (0, 2) and (3, 0), then the relation between a and b is:
(A) \(a=b\)
(B) \(a=3b\)
(C) \(b=6a\)
(D) \(3a=2b\)
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Correct Answer: D

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Step-by-Step Solution

Let \(Z = ax + by\). The maximum value of Z is obtained at (0, 2) and (3, 0). This means the value of Z at these two points is the same.

At (0, 2), \(Z = a(0) + b(2) = 2b\).

At (3, 0), \(Z = a(3) + b(0) = 3a\).

Since the maximum value is the same at both points, we have \(2b = 3a\).

Therefore, the relation between a and b is \(3a = 2b\).

Correct Answer: 3a=2b

AI Suggestion: Option D

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires students to apply their understanding of linear programming and objective functions to determine the relationship between the coefficients 'a' and 'b' based on the given corner points and the condition that the maximum value of Z occurs at two points.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding the concepts of linear programming, feasible region, objective function, and how the corner points relate to the maximum value of the objective function. It's not just about recalling facts but understanding the underlying principles.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It assesses the student's ability to apply the concepts of Linear Programming to solve a problem, rather than just recalling definitions or theorems.

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