Class CBSE Class 12 Mathematics Linear Programming Q #685
KNOWLEDGE BASED
APPLY
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
The corner points of the feasible region in graphical representation of a L.P.P. are \((2, 72)\), \((15, 20)\) and \((40, 15)\). If \(Z = 18x + 9y\) be the objective function, then
(A) \(Z\) is maximum at \((2, 72)\), minimum at \((15, 20)\)
(B) \(Z\) is maximum at \((15, 20)\), minimum at \((40, 15)\)
(C) \(Z\) is maximum at \((40, 15)\), minimum at \((15, 20)\)
(D) \(Z\) is maximum at \((40, 15)\), minimum at \((2, 72)\)
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Correct Answer: C
Explanation
To find the maximum and minimum values of the objective function \(Z = 18x + 9y\), we must evaluate \(Z\) at each of the given corner points of the feasible region.The corner points \((x, y)\) are: \((2, 72)\), \((15, 20)\), and \((40, 15)\).


At \(\mathbf{(2, 72)}\):\[Z = 18(2) + 9(72)\]\[Z = 36 + 648 = \mathbf{684}\]At \(\mathbf{(15, 20)}\):\[Z = 18(15) + 9(20)\]\[Z = 270 + 180 = \mathbf{450}\]At \(\mathbf{(40, 15)}\):\[Z = 18(40) + 9(15)\]\[Z = 720 + 135 = \mathbf{855}\]Maximum Value: \(855\), which occurs at the point \(\mathbf{(40, 15)}\).Minimum Value: \(450\), which occurs at the point \(\mathbf{(15, 20)}\).


The correct conclusion is: \(Z\) is maximum at \((40, 15)\), minimum at \((15, 20)\).

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Step-by-Step Solution

  1. Calculate the value of \(Z\) at each corner point:
    • At \((2, 72)\): \(Z = 18(2) + 9(72) = 36 + 648 = 684\)
    • At \((15, 20)\): \(Z = 18(15) + 9(20) = 270 + 180 = 450\)
    • At \((40, 15)\): \(Z = 18(40) + 9(15) = 720 + 135 = 855\)
  2. Compare the values of \(Z\) at each point:
    • Maximum value of \(Z\) is 855 at \((40, 15)\)
    • Minimum value of \(Z\) is 450 at \((15, 20)\)

Correct Answer: Z is maximum at (40, 15), minimum at (15, 20)

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to apply the concept of finding the maximum and minimum values of an objective function at the corner points of a feasible region.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure: evaluating the objective function at each corner point to determine the maximum and minimum values.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of the standard method for solving linear programming problems.

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