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.

This ICSE Class 12 Mathematics practice quiz has 26 multiple-choice questions on Linear Programming – Formulation & Graphical Solutions, and 26 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz 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
   Show answer

   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
   Show answer

   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
   Show answer

   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

More questions in this quiz

Answers and explanations are in the full quiz.

1. Consider the feasible region defined by the inequalities x + y <= 5, 2x + y <= 8, x >= 0, y >= 0. What are the corner points of this feasible region?
2. For the objective function Z = 3x + 5y, find the maximum value in the feasible region defined by x + y <= 4, x + 2y <= 6, x >= 0, y >= 0.
3. Which of the following is NOT a basic feasible solution for the system of equations: x + y + s1 = 10, x + 2y + s2 = 12, where x, y, s1, s2 >= 0?
4. What is the nature of the feasible region for a linear programming problem if the constraints are x >= 0, y >= 0 and x + y <= 0?
5. Consider the feasible region described by x >= 0, y >= 0, x + y <= 10, and 2x + y >= 8. If the objective function is Z = 3x + 4y, what is the minimum value of Z?
6. For the feasible region defined by x >= 0, y >= 0, x + y <= 6, and x + 2y <= 8, find the maximum value of the objective function Z = 2x + 3y.
7. If the feasible region of an LPP is unbounded, what can be said about the optimal solution?
8. Consider the LPP: Minimize Z = -3x + 4y subject to x + y <= 5, 2x + y <= 8, x >= 0, y >= 0. Which of the following points is NOT in the feasible region?
9. Consider the feasible region defined by x + y <= 6, 2x + y <= 10, x >= 0, y >= 0. What are the coordinates of the intersection point of the lines x + y = 6 and 2x + y = 10?
10. For the LPP: Maximize Z = 4x + 3y subject to x + y <= 5, x - y <= 1, x >= 0, y >= 0. Which of the following is a corner point of the feasible region?
11. What is the feasible region for the constraints x >= 0, y >= 0, x + y <= 10, and 2x + 3y <= 20?
12. Determine the type of objective function for which an LPP might have infinitely many optimal solutions.
13. The graphical solution of an LPP with constraints x >= 0, y >= 0, x + y <= 4, 2x + y <= 6, shows that the objective function Z = 3x + 2y is maximized at the intersection of x + y = 4 and 2x + y = 6. What is the maximum value of Z?
14. If the objective function of an LPP is to be minimized and the feasible region is defined by x + y <= 10, 2x + y >= 8, x >= 0, y >= 0, which statement is true about the minimum value?
15. Consider the feasible region defined by x + y <= 10, x >= 0, y >= 0. If the objective function is Z = 2x + 3y, which of the following is the maximum value?
16. A farmer wants to maximize the yield of crops by planting wheat and rice. Each acre of wheat requires 2 hours of labor and 3 units of fertilizer, yielding 10 quintals. Each acre of rice requires 3 hours of labor and 2 units of fertilizer, yielding 12 quintals. The farmer has 120 hours of labor and 150 units of fertilizer. Formulate the LPP to maximize the yield.
17. If the feasible region of an LPP is a triangle with vertices (0,0), (5,0), and (0,5), and the objective function is Z = 2x + 3y, what is the minimum value of Z?
18. For the constraints x + y <= 5, 2x + y <= 8, x >= 0, y >= 0, if the objective function is Z = 50x + 40y, what is the maximum value?
19. Which of the following constraints would make the feasible region for Z = 3x + 2y, with x>=0, y>=0, and x+y<=6, unbounded?
20. Which of the following sets of constraints defines a feasible region that is a single point?
21. Consider the feasible region defined by x + y <= 10, x >= 0, y >= 0. If the objective function is Z = 3x + 2y, what is the minimum value of Z?

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