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Class 12 Linear Programming · Long Answer3 Marksgraphical_optimization

Class 12 Maths · Linear Programming · Long Answer #27

Minimize Z=13*x - 15*y subject to x + y ≤ 7; -2*x + 3*y ≤ 6; -x ≤ 0; -y ≤ 0. Solve graphically.
Coordinate-derived feasible-region graph for cbse12-math-lp-p1-short-001
Coordinate-derived feasible-region graph for cbse12-math-lp-p1-short-001

✅ Step-by-Step Solution

Final Answer: min Z=-30 at (0,2).
  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), (0,2), (3,4), (7,0).
  3. Evaluate the objective at every vertex: (0,0) → 0; (0,2) → -30; (3,4) → -21; (7,0) → 91.
  4. The region is bounded. min Z=-30 at (0,2).

📋 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), (0,2), (3,4), (7,0).
  • [1 mark] Evaluate the objective at every vertex: (0,0) → 0; (0,2) → -30; (3,4) → -21; (7,0) → 91.
  • [0.5 marks] The region is bounded. min Z=-30 at (0,2).

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