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Class 12 Linear Programming · Long Answer3 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Long Answer #33

Maximize Z=6*x + 3*y subject to -4*x - y ≤ -80; 3*x + 2*y ≤ 150; -x - 5*y ≤ -115; -x ≤ 0; -y ≤ 0. Solve graphically.
Coordinate-derived feasible-region graph for cbse12-math-lp-p1-short-007
Coordinate-derived feasible-region graph for cbse12-math-lp-p1-short-007

✅ Step-by-Step Solution

Final Answer: max Z=285 at (40,15).
  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: (2,72), (15,20), (40,15).
  3. Evaluate the objective at every vertex: (2,72) → 228; (15,20) → 150; (40,15) → 285.
  4. The region is bounded. max Z=285 at (40,15).

📋 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: (2,72), (15,20), (40,15).
  • [1 mark] Evaluate the objective at every vertex: (2,72) → 228; (15,20) → 150; (40,15) → 285.
  • [0.5 marks] The region is bounded. max Z=285 at (40,15).

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