Assertion (A): Every point of the feasible region of a Linear Programming Problem is an optimal solution.
Reason (R): The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region.
Reason (R): The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region.
Show Hint
- Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
- Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- Assertion (A) is true but Reason (R) is false.
- Assertion (A) is false but Reason (R) is true.
The Correct Option is D
Solution and Explanation
The solution to the given problem involves understanding the concepts of Linear Programming Problems (LPP) and feasible regions.
Analysis of the Assertion (A)
Assertion (A): Every point of the feasible region of a Linear Programming Problem is an optimal solution.
The feasible region of an LPP is the set of all points that satisfy the constraints of the problem. However, for an LPP, the optimal solution does not occur at every point within this region. Instead, it is typically found at the vertices, or corner points, of the feasible region. Therefore, the assertion that every point is an optimal solution is incorrect.
Analysis of the Reason (R)
Reason (R): The optimal solution for a Linear Programming Problem exists only at one or more corner point(s) of the feasible region.
This statement is true. In the context of LPPs, according to the "Fundamental Theorem of Linear Programming," if there exists an optimal solution, it will occur at one of the corner points (vertices) of the feasible region. Therefore, this reason accurately describes where optimal solutions are located within the feasible region.
Conclusion
Upon examination:
- Assertion (A) is false—it is not true that every point of the feasible region is an optimal solution.
- Reason (R) is true—the optimal solution exists at corner points.
The correct option is: Assertion (A) is false but Reason (R) is true.
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- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
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A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs17.50 per package on nuts and Rs7.00 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 12 hours a day?
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