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📐 Mathematics  ·  Linear Programming  ·  JEE

Two corner points (10, 0) and (0, 15) of a feasible region both give Z = 300 for the objective function Z = 30x + 20y. What does this indicate about the LPP?

Answer: The LPP has multiple optimal solutions along the segment joining the two points.

  • A There is a calculation error since two different points cannot give equal Z
  • B The LPP has multiple optimal solutions along the segment joining the two points
  • C The feasible region is empty
  • D Neither point is actually a corner point

Correct answer: B. The LPP has multiple optimal solutions along the segment joining the two points

Explanation: Z(10,0)=300, Z(0,15)=300, both equal. This happens when the objective function's line is parallel to the boundary line joining these corners, giving infinitely many optimal solutions along that edge.

Feasible Region and Corner PointsxyOABCfeasible regionZ = ax+by is evaluated ONLY at corners O, A, B, C - the optimum is always at one of these

The feasible region (shaded) is bounded by the constraint lines; the fundamental theorem of LPP guarantees the optimal value of the objective function occurs at one of the corner points (O, A, B, C), so only these need to be checked, not the entire region.

Concept context

Optimizing a linear objective function subject to linear constraints using the graphical corner point method.

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