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Class 12 Linear Programming · Long Answer3 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Long Answer #37

Maximize Z=22*x + 18*y subject to 3*x + 2*y ≤ 48; x + y ≤ 20; -x ≤ 0; -y ≤ 0. Solve graphically.
Coordinate-derived feasible-region graph for cbse12-math-lp-p1-short-011
Coordinate-derived feasible-region graph for cbse12-math-lp-p1-short-011

✅ Step-by-Step Solution

Final Answer: max Z=392 at (8,12).
  1. Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
  2. Solving pairs of boundary equations and rejecting points violating any inequality gives vertices: (0,0), (0,20), (8,12), (16,0).
  3. Evaluate the objective at every vertex: (0,0) → 0; (0,20) → 360; (8,12) → 392; (16,0) → 352.
  4. The region is bounded. max Z=392 at (8,12).

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
  • [1 mark] Solving pairs of boundary equations and rejecting points violating any inequality gives vertices: (0,0), (0,20), (8,12), (16,0).
  • [1 mark] Evaluate the objective at every vertex: (0,0) → 0; (0,20) → 360; (8,12) → 392; (16,0) → 352.
  • [0.5 marks] The region is bounded. max Z=392 at (8,12).

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