Linear Programming – Formulation & Graphical Solutions Quiz - ICSE Class 12 Mathematics

Free practice quiz: Linear Programming – Formulation & Graphical Solutions (ICSE Class 12 Mathematics). 26 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

Sample questions

1. A company manufactures two types of products, A and B. Each unit of product A requires 2 hours of machine time and 3 hours of labor. Each unit of product B requires 4 hours of machine time and 2 hours of labor. The company has a maximum of 120 hours of machine time and 90 hours of labor available per week. If the profit per unit of product A is Rs. 50 and per unit of product B is Rs. 40, formulate the linear programming problem to maximize the profit.

   1. Maximize Z = 50x + 40y subject to 2x + 4y <= 120, 3x + 2y <= 90, x >= 0, y >= 0
   2. Maximize Z = 40x + 50y subject to 4x + 2y <= 120, 2x + 3y <= 90, x >= 0, y >= 0
   3. Maximize Z = 50x + 40y subject to 2x + 4y >= 120, 3x + 2y >= 90, x >= 0, y >= 0
   4. Maximize Z = 50x + 40y subject to 2x + 3y <= 120, 4x + 2y <= 90, x >= 0, y >= 0
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   Answer: A. Maximize Z = 50x + 40y subject to 2x + 4y <= 120, 3x + 2y <= 90, x >= 0, y >= 0

   Let x be the number of units of product A and y be the number of units of product B. The objective is to maximize profit Z = 50x + 40y. The constraints are based on machine time (2x + 4y <= 120) and labor hours (3x + 2y <= 90). Since the number of products cannot be negative, we have x >= 0 and y >= 0.

2. If a linear programming problem has an objective function Z = ax + by and the feasible region is a convex polygon, where is the optimal solution (maximum or minimum) guaranteed to occur?

   1. At any point within the feasible region
   2. At the centroid of the feasible region
   3. At one of the vertices (corner points) of the feasible region
   4. On the boundary of the feasible region
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   Answer: C. At one of the vertices (corner points) of the feasible region

   This is a fundamental theorem in linear programming. The optimal solution (either maximum or minimum) of a linear programming problem, if it exists, occurs at one of the vertices (corner points) of the feasible region. The feasible region in linear programming problems is always a convex polygon (or unbounded region where an optimum might not exist).

3. A manufacturer produces two types of fertilizer, A and B. Type A requires 10 kg of raw material and 5 hours of processing. Type B requires 15 kg of raw material and 8 hours of processing. The total available raw material is 300 kg and total processing time is 160 hours. The profit for type A is Rs. 150 per unit and for type B is Rs. 200 per unit. Formulate an LPP to maximize profit.

   1. Maximize Z = 150x + 200y subject to 10x + 15y <= 300, 5x + 8y <= 160, x >= 0, y >= 0
   2. Maximize Z = 200x + 150y subject to 15x + 10y <= 300, 8x + 5y <= 160, x >= 0, y >= 0
   3. Maximize Z = 150x + 200y subject to 10x + 5y <= 300, 15x + 8y <= 160, x >= 0, y >= 0
   4. Maximize Z = 150x + 200y subject to 10x + 15y >= 300, 5x + 8y >= 160, x >= 0, y >= 0
   Show answer

   Answer: A. Maximize Z = 150x + 200y subject to 10x + 15y <= 300, 5x + 8y <= 160, x >= 0, y >= 0

   Let x be the number of units of fertilizer A and y be the number of units of fertilizer B. The objective is to maximize profit, so Z = 150x + 200y. The constraints are based on raw material and processing time. Raw material: 10 kg for A and 15 kg for B, with a total of 300 kg available. So, 10x + 15y <= 300. Processing time: 5 hours for A and 8 hours for B, with a total of 160 hours available. So, 5x + 8y <= 160. Also, the number of units cannot be negative, hence x >= 0 and y >= 0.

4. Consider the inequalities: x + y <= 7, 2x + 3y <= 15, x >= 0, y >= 0. Find the maximum value of Z = 5x + 4y.

   1. 35
   2. 27
   3. 28
   4. 30
   Show answer

   Answer: A. 35

   Find the corner points of the feasible region. Constraints: x >= 0

5. Which of the following inequalities represents a region that is unbounded in the first quadrant?

   1. x + y <= 5
   2. x + 2y >= 8
   3. x <= 5
   4. y <= 5
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   Answer: B. x + 2y >= 8

   The constraints x >= 0 and y >= 0 restrict the region to the first quadrant. Let's analyze each inequality in conjunction with these: - x + y <= 5: With x>=0

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