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NCERT Class 12 · Ch 12 Exercise 12.1 · Q9

NCERT Class 12 Maths · Chapter 12 Linear Programming · Exercise 12.1 Question 9

Maximize Z=−x+2y subject to x≥3,x+y≥5,x+2y≥6,y≥0. Solve graphically.
Graphical construction for exercise 12.1 question 9
Graphical construction for exercise 12.1 question 9

💡 Concept Hints

  1. Convert the inequalities to boundary lines and identify the common feasible region.
  2. Evaluate the objective at all feasible vertices, then check directions of escape if the region is unbounded.

✅ Step-by-Step Solution

Final Answer: No finite maximum: along (3,2)+t(0,1), t≥0, Z=1+2t grows without bound.
  1. Plot each boundary and intersect its indicated closed half-plane with x,y≥0.
  2. Solving boundary pairs and checking every inequality gives vertices (3,2), (4,1), (6,0).
  3. The objective values at these vertices are (3,2) → 1; (4,1) → -2; (6,0) → -6.
  4. The region is unbounded. Feasible recession directions are (0,1), (1,0), with objective changes 2, -1. Check these signs before declaring a finite optimum.
  5. No finite maximum: along (3,2)+t(0,1), t≥0, Z=1+2t grows without bound.

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