Linear Programming Problem — Practice Quiz
CBSE · Class 12 · Applied Mathematics
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Quick Quiz: Linear Programming Problem
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A company produces chairs and tables. Each chair requires 2 hours of labor and each table requires 3 hours. If 60 hours are available and x chairs and y tables are produced, which constraint represents this limitation?
Maximize Z = 3x + 2y subject to x + y ≤ 5, x ≤ 3, y ≤ 4, x ≥ 0, y ≥ 0. What is the maximum value of Z?
In an LPP, if the feasible region is unbounded, the objective function:
A farmer has 100 acres to plant wheat (x) and corn (y). Wheat gives profit of Rs. 200 per acre, corn gives Rs. 300 per acre. If he wants to plant at least 20 acres of wheat, the objective function to maximize profit is:
Sample Questions
Find the corner points of the feasible region defined by: x + y ≤ 4, x ≤ 3, y ≤ 2, x ≥ 0, y ≥ 0
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(0, 0), (3, 0), (2, 2), (0, 2), (3, 1)
Corner points are intersections of boundary lines: Origin (0,0), x-axis intersections (3,0), y-axis intersections (0,2), and intersections of constraint lines like x+y=4 with x=3 giving (3,1), and x+y=4 with y=2 giving (2,2).
Which of the following are essential components of a Linear Programming Problem?
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Decision variables, Objective function, Linear constraints, Non-negativity restrictions
An LPP requires: decision variables (x, y, etc.), an objective function to optimize (Z = ax + by), linear constraints (inequalities/equations), and non-negativity restrictions (x ≥ 0, y ≥ 0). Quadratic terms make it non-linear.
Solve: Minimize Z = 2x + 3y subject to x + y ≥ 4, x ≥ 1, y ≥ 1. Find the minimum value of Z.
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7
Corner points of feasible region: (1,3), (3,1), and intersection points. At (1,3): Z = 2(1) + 3(3) = 11. At (3,1): Z = 2(3) + 3(1) = 9. The point where x + y = 4 intersects the boundary gives minimum at (1,3) where Z = 7.
The constraint x ≥ 0, y ≥ 0 in an LPP ensures that:
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Variables take only non-negative values
Non-negativity constraints x ≥ 0, y ≥ 0 restrict the feasible region to the first quadrant only, ensuring variables cannot take negative values. This is essential for most real-world problems where negative quantities don't make sense.
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Sources & Official References
- NCERT Official — ncert.nic.in
- CBSE Academic — cbseacademic.nic.in
- CBSE Official — cbse.gov.in
- National Education Policy 2020 — education.gov.in
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