Class CBSE Class 12 Mathematics Linear Programming Q #680
COMPETENCY BASED
APPLY
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
For a Linear Programming Problem (LPP), the given objective function is \(Z=x+2y\). The feasible region PQRS determined by the set of constraints is shown as a shaded region in the graph. \(P\equiv(\frac{3}{13},\frac{24}{13})\) \(Q\equiv(\frac{3}{2},\frac{15}{4})\) \(R\equiv(\frac{7}{2},\frac{3}{4})\) \(S\equiv(\frac{18}{7},\frac{2}{7})\). Which of the following statements is correct?

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(A) Z is minimum at \(S(\frac{18}{7},\frac{2}{7})\)
(B) Z is maximum at \(R(\frac{7}{2},\frac{3}{4})\)
(C) (Value of Z at P) > (Value of Z at Q)
(D) (Value of Z at Q) < (Value of Z at R)
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Correct Answer: B

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Step-by-Step Solution

  1. Calculate the value of Z at each point:
    • At \(P(\frac{3}{13},\frac{24}{13})\), \(Z = \frac{3}{13} + 2(\frac{24}{13}) = \frac{3+48}{13} = \frac{51}{13} \approx 3.92\)
    • At \(Q(\frac{3}{2},\frac{15}{4})\), \(Z = \frac{3}{2} + 2(\frac{15}{4}) = \frac{3}{2} + \frac{15}{2} = \frac{18}{2} = 9\)
    • At \(R(\frac{7}{2},\frac{3}{4})\), \(Z = \frac{7}{2} + 2(\frac{3}{4}) = \frac{7}{2} + \frac{3}{2} = \frac{10}{2} = 5\)
    • At \(S(\frac{18}{7},\frac{2}{7})\), \(Z = \frac{18}{7} + 2(\frac{2}{7}) = \frac{18+4}{7} = \frac{22}{7} \approx 3.14\)
  2. Evaluate the options:
    • (A) Z is minimum at \(S(\frac{18}{7},\frac{2}{7})\). Z at S is approximately 3.14, which is the minimum value among the calculated values. So, this statement is correct.
    • (B) Z is maximum at \(R(\frac{7}{2},\frac{3}{4})\). Z at R is 5, while Z at Q is 9. So, this statement is incorrect.
    • (C) (Value of Z at P) > (Value of Z at Q). Z at P is approximately 3.92, and Z at Q is 9. So, this statement is incorrect.
    • (D) (Value of Z at Q) < (Value of Z at R). Z at Q is 9, and Z at R is 5. So, this statement is incorrect.

Correct Answer: Z is minimum at \(S(\frac{18}{7},\frac{2}{7})\)

APPLY|||COMPETENCY|||PROCEDURAL|||MEDIUM|||
Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concept of linear programming to find the minimum and maximum values of the objective function within the feasible region. This involves substituting the coordinates of the vertices into the objective function and comparing the results.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a specific procedure to solve: evaluating the objective function at each corner point of the feasible region and then comparing the values to determine the minimum and maximum.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question assesses the student's ability to apply the concepts of linear programming to a given scenario, rather than simply recalling definitions or formulas.

Step-by-Step Solution

  1. Calculate Z at each point:
    • P(3/13, 24/13): Z = (3/13) + 2*(24/13) = 51/13 ≈ 3.92
    • Q(3/2, 15/4): Z = (3/2) + 2*(15/4) = 3/2 + 15/2 = 18/2 = 9
    • R(7/2, 3/4): Z = (7/2) + 2*(3/4) = 7/2 + 3/2 = 10/2 = 5
    • S(18/7, 2/7): Z = (18/7) + 2*(2/7) = 22/7 ≈ 3.14
  2. Evaluate the options:
    • (A) Z is minimum at S(18/7, 2/7): Z at S is approximately 3.14, which is the smallest value. This statement is correct.
    • (B) Z is maximum at R(7/2, 3/4): Z at R is 5, but Z at Q is 9. This statement is incorrect.
    • (C) (Value of Z at P) > (Value of Z at Q): Z at P is approximately 3.92, and Z at Q is 9. This statement is incorrect.
    • (D) (Value of Z at Q) < (Value of Z at R): Z at Q is 9, and Z at R is 5. This statement is incorrect.

Correct Answer: Z is minimum at S(18/7, 2/7)

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to apply their understanding of linear programming and feasible regions to calculate the value of the objective function at different points and compare them.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a specific procedure to solve: evaluating the objective function at each corner point of the feasible region.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question assesses the student's ability to apply the concepts of linear programming to a given scenario, rather than simply recalling definitions or formulas.

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