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Class 12 Linear Programming · Long Answer3 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Long Answer #29

Maximize Z=80*x + 120*y subject to 3*x + 4*y ≤ 60; x + 3*y ≤ 30; -x ≤ 0; -y ≤ 0. Solve graphically.
Coordinate-derived feasible-region graph for cbse12-math-lp-p1-short-003
Coordinate-derived feasible-region graph for cbse12-math-lp-p1-short-003

✅ Step-by-Step Solution

Final Answer: max Z=1680 at (12,6).
  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), (0,10), (12,6), (20,0).
  3. Evaluate the objective at every vertex: (0,0) → 0; (0,10) → 1200; (12,6) → 1680; (20,0) → 1600.
  4. The region is bounded. max Z=1680 at (12,6).

📋 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), (0,10), (12,6), (20,0).
  • [1 mark] Evaluate the objective at every vertex: (0,0) → 0; (0,10) → 1200; (12,6) → 1680; (20,0) → 1600.
  • [0.5 marks] The region is bounded. max Z=1680 at (12,6).

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