Class 12 Linear Programming · Long Answer3 Marksgraphical_optimization
Class 12 Maths · Linear Programming · Long Answer #55
Maximize Z=3*x + y subject to x + 2*y ≤ 2; 2*x + y ≤ 2; -x ≤ 0; -y ≤ 0. Solve graphically.
✅ Step-by-Step Solution
Final Answer: max Z=3 at (1,0).
- Draw each boundary ax+by=c and retain the closed half-plane ax+by≤c; intersect all constraints, including x,y≥0.
- Solving pairs of boundary equations and rejecting points violating any inequality gives vertices: (0,0), (0,1), (2/3,2/3), (1,0).
- Evaluate the objective at every vertex: (0,0) → 0; (0,1) → 1; (2/3,2/3) → 8/3; (1,0) → 3.
- The region is bounded. max Z=3 at (1,0).
📋 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), (0,1), (2/3,2/3), (1,0).
- [1 mark] Evaluate the objective at every vertex: (0,0) → 0; (0,1) → 1; (2/3,2/3) → 8/3; (1,0) → 3.
- [0.5 marks] The region is bounded. max Z=3 at (1,0).
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