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Consider the following linear programming problem (LPP).
Maximize z = 2x + 3y
Subject to
2x + 3y ≤ 12
2x - 3y ≤ 0
y ≤ 2
x, y ≥ 0
It has
- a)no feasible solution
- b)unique optimal solution
- c)alternative optimal solutions
- d)unbounded solution
Correct answer is option 'B'. Can you explain this answer?
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Consider the following linear programming problem (LPP).Maximize z = ...
Problem:
Maximize z = 2x - 3y
Subject to
2x - 3y ≤ 12
2x - 3y ≤ 0
y ≤ 2x
y ≥ 0
Feasible Region:
To find the feasible region, we need to graph the given inequalities and identify the region that satisfies all the constraints.
The first constraint, 2x - 3y ≤ 12, can be rewritten as y ≥ (2/3)x - 4. We can graph this line by plotting the points (0, -4) and (6, 0) and drawing a line through them.
The second constraint, 2x - 3y ≤ 0, can be rewritten as y ≥ (2/3)x. We can graph this line by plotting the points (0, 0) and (3, 2) and drawing a line through them.
The third constraint, y ≤ 2x, can be graphed by plotting the points (0, 0) and (1, 2) and drawing a line through them.
Finally, the constraint y ≥ 0 represents the non-negative region of the y-axis.
After graphing all the constraints, we can shade the region that satisfies all the constraints. The feasible region is the intersection of all the shaded regions.
Analysis:
In this case, the feasible region is a bounded region consisting of a triangle. Since the objective function is linear, it will have a maximum value at one of the corner points of the feasible region.
Corner Points:
To find the corner points, we need to find the intersections of the lines representing the constraints.
- Intersection of (2/3)x - 4 and (2/3)x: (6, 0)
- Intersection of (2/3)x - 4 and 2x: (3, 2)
- Intersection of (2/3)x and 2x: (0, 0)
Objective Function:
Now, we evaluate the objective function z = 2x - 3y at each of the corner points.
- At (6, 0): z = 2(6) - 3(0) = 12
- At (3, 2): z = 2(3) - 3(2) = 0
- At (0, 0): z = 2(0) - 3(0) = 0
Optimal Solution:
The maximum value of z is 12, which occurs at the corner point (6, 0). Therefore, the LPP has a unique optimal solution.
Conclusion:
The correct answer is option 'B', which states that the LPP has a unique optimal solution.
Maximize z = 2x - 3y
Subject to
2x - 3y ≤ 12
2x - 3y ≤ 0
y ≤ 2x
y ≥ 0
Feasible Region:
To find the feasible region, we need to graph the given inequalities and identify the region that satisfies all the constraints.
The first constraint, 2x - 3y ≤ 12, can be rewritten as y ≥ (2/3)x - 4. We can graph this line by plotting the points (0, -4) and (6, 0) and drawing a line through them.
The second constraint, 2x - 3y ≤ 0, can be rewritten as y ≥ (2/3)x. We can graph this line by plotting the points (0, 0) and (3, 2) and drawing a line through them.
The third constraint, y ≤ 2x, can be graphed by plotting the points (0, 0) and (1, 2) and drawing a line through them.
Finally, the constraint y ≥ 0 represents the non-negative region of the y-axis.
After graphing all the constraints, we can shade the region that satisfies all the constraints. The feasible region is the intersection of all the shaded regions.
Analysis:
In this case, the feasible region is a bounded region consisting of a triangle. Since the objective function is linear, it will have a maximum value at one of the corner points of the feasible region.
Corner Points:
To find the corner points, we need to find the intersections of the lines representing the constraints.
- Intersection of (2/3)x - 4 and (2/3)x: (6, 0)
- Intersection of (2/3)x - 4 and 2x: (3, 2)
- Intersection of (2/3)x and 2x: (0, 0)
Objective Function:
Now, we evaluate the objective function z = 2x - 3y at each of the corner points.
- At (6, 0): z = 2(6) - 3(0) = 12
- At (3, 2): z = 2(3) - 3(2) = 0
- At (0, 0): z = 2(0) - 3(0) = 0
Optimal Solution:
The maximum value of z is 12, which occurs at the corner point (6, 0). Therefore, the LPP has a unique optimal solution.
Conclusion:
The correct answer is option 'B', which states that the LPP has a unique optimal solution.
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Consider the following linear programming problem (LPP).Maximize z = 2x + 3ySubject to2x + 3y ≤ 122x - 3y ≤ 0y ≤ 2x, y ≥ 0It hasa)no feasible solutionb)unique optimal solutionc)alternative optimal solutionsd)unbounded solutionCorrect answer is option 'B'. Can you explain this answer?
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Consider the following linear programming problem (LPP).Maximize z = 2x + 3ySubject to2x + 3y ≤ 122x - 3y ≤ 0y ≤ 2x, y ≥ 0It hasa)no feasible solutionb)unique optimal solutionc)alternative optimal solutionsd)unbounded solutionCorrect answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Consider the following linear programming problem (LPP).Maximize z = 2x + 3ySubject to2x + 3y ≤ 122x - 3y ≤ 0y ≤ 2x, y ≥ 0It hasa)no feasible solutionb)unique optimal solutionc)alternative optimal solutionsd)unbounded solutionCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following linear programming problem (LPP).Maximize z = 2x + 3ySubject to2x + 3y ≤ 122x - 3y ≤ 0y ≤ 2x, y ≥ 0It hasa)no feasible solutionb)unique optimal solutionc)alternative optimal solutionsd)unbounded solutionCorrect answer is option 'B'. Can you explain this answer?.
Consider the following linear programming problem (LPP).Maximize z = 2x + 3ySubject to2x + 3y ≤ 122x - 3y ≤ 0y ≤ 2x, y ≥ 0It hasa)no feasible solutionb)unique optimal solutionc)alternative optimal solutionsd)unbounded solutionCorrect answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Consider the following linear programming problem (LPP).Maximize z = 2x + 3ySubject to2x + 3y ≤ 122x - 3y ≤ 0y ≤ 2x, y ≥ 0It hasa)no feasible solutionb)unique optimal solutionc)alternative optimal solutionsd)unbounded solutionCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following linear programming problem (LPP).Maximize z = 2x + 3ySubject to2x + 3y ≤ 122x - 3y ≤ 0y ≤ 2x, y ≥ 0It hasa)no feasible solutionb)unique optimal solutionc)alternative optimal solutionsd)unbounded solutionCorrect answer is option 'B'. Can you explain this answer?.
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