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RD Sharma Class 12 Solutions Chapter 30 - Linear programming (Ex 30.2) Exercise 30.2

RD Sharma Class 12 Solutions Chapter 30 - Linear programming (Ex 30.2) Exercise 30.2

Q: 1. What is the primary method used to solve the problems in RD Sharma Class 12 Solutions for Chapter 30, Exercise 30.2?

The solutions for Exercise 30.2 primarily use the Graphical Method for solving Linear Programming Problems (LPPs). This method involves visually representing the problem's constraints as straight lines on a graph to identify a feasible region and then finding the optimal (maximum or minimum) value of the objective function within that region.

Q: 5. What is the difference between a bounded and an unbounded feasible region in the context of problems in Exercise 30.2?

A bounded feasible region is one that is completely enclosed by the constraint lines, forming a closed polygon. In this case, both a maximum and a minimum value for the objective function will exist at the corner points. An unbounded feasible region is one that extends infinitely in at least one direction. For unbounded regions, an optimal solution may or may not exist. A minimum might exist, but a maximum might not (or vice-versa), depending on the objective function.

RD Sharma Class 12 Solutions Chapter 30 - Linear programming (Ex 30.2) Exercise 30.2 - Free PDF

Free PDF download of RD Sharma Class 12 Solutions Chapter 30 - Linear programming Exercise 30.2 solved by Expert Mathematics Teachers on Vedantu.com. All Chapter 30 - Linear programming Ex 30.2 Questions with Solutions for RD Sharma Class 12 Maths to help you to revise the complete Syllabus and Score More marks. Register for online coaching for IIT JEE (Mains & Advanced) and other engineering entrance exams.

What is included in RD Sharma Class 12 Solutions Chapter 30 - Linear programming (Ex 30.2) Exercise 30.2 - Free PDF

RD Sharma Solutions for Class 12 Maths Chapter 30 Exercise 30.2 Linear Programming are provided here for the benefit of students. Students from CBSE, ICSE or other government board exams prefer RD Sharma Books to prepare for exams and other tests such as job entrance exams and various courses. The RD Sharma Class 12 solutions are specially designed by Vedantu’s team of experts.

Students can use these solutions to overcome the fear of solving maths and the solutions are designed in such a way that the students can find simple ways to solve different problems. These solutions can help students to refine their learning and problem-solving skills. Students can go through the RD Sharma solutions for class 12 chapter 30 Linear Programming Exercise 30.2 below and it will be beneficial for them.

Different Types of Linear Programming in RD Sharma Class 12 Solutions Chapter 30 - Linear programming (Ex 30.2) Exercise 30.2 - Free PDF

Different types of problems in the direct planning problem are included in the ideas of the 12th class. Of course:

(I) Production Problem - Here we increase profits with the help of less resource use.

(II) Eating Disorders - We determine the number of different nutrients in food to reduce production costs.

(III) Transportation Problem - Here we determine the plan to find the cheapest way to transport a product in a short time.

Some Important Terms of Linear Programming That You Should Know

There are certain words used in the construction and resolution of consecutive planning problems. Let's explain some of the keywords we will use here:

Objective Function: The linear function of the Z = ax + by form, where a and b are constant, which should be reduced or enlarged is called the linear objective function.

Consider for example, Z = 175x + 150y. This is the function of the line object. The variables x and y are called resolution variables.

Constraints: The linear inequalities or equations or limitations of LPP variables (Linear programming problems) are called constraints. Circumstances x ≥ 0, y ≥ 0 are called non-negative constraints.

For example, 5x + y ≤ 100; x + y ≤ 60 are obstacles.

Optimization Problem: A problem that seeks to increase or decrease linear function (say with two variables x and y) under certain constraints as determined by linear inequality set is called an optimization problem. Linear programming problems are special types of optimization problems.
Feasible Region: A feasible region determined by all given constraints including non-critical barriers (x ≥ 0, y ≥ 0) of the linear programming problem is called the potential region (or feasible region) of the problem. The region beyond the feasible region is called an infeasible region.
Feasible Solutions: These are the points within and in the scope of the space that may represent feasible solutions to the issues. Any point outside of a feasible area is called an infeasible solution.
Optimal (or feasible) Solution: Any point in a feasible region that provides the total amount (maximum or minimum) of the objective function is called the optimal solution.

FAQs on RD Sharma Class 12 Solutions Chapter 30 - Linear programming (Ex 30.2) Exercise 30.2

1. What is the primary method used to solve the problems in RD Sharma Class 12 Solutions for Chapter 30, Exercise 30.2?

The solutions for Exercise 30.2 primarily use the Graphical Method for solving Linear Programming Problems (LPPs). This method involves visually representing the problem's constraints as straight lines on a graph to identify a feasible region and then finding the optimal (maximum or minimum) value of the objective function within that region.

2. What are the essential steps to follow when using the graphical method for questions in Exercise 30.2?

To solve any LPP from Exercise 30.2 using the graphical method, you should follow these steps as per the CBSE curriculum:

Step 1: Formulate the LPP by defining the objective function (Z = ax + by) and all the constraints (linear inequalities).
Step 2: Plot each constraint as an equation on a graph (e.g., plot x + 2y = 10 for the constraint x + 2y ≤ 10).
Step 3: Identify the feasible region, which is the common area shared by all constraints, including the non-negative constraints (x ≥ 0, y ≥ 0).
Step 4: Determine the coordinates of the vertices (corner points) of this feasible region.
Step 5: Substitute these corner point coordinates into the objective function to find the optimal value. The largest value is the maximum and the smallest is the minimum.

3. How do I identify the feasible region on the graph when solving problems from Exercise 30.2?

To identify the feasible region, first plot each constraint as a line. For each constraint (e.g., ax + by ≤ c), test a point (usually the origin, (0,0), if the line doesn't pass through it) to see if it satisfies the inequality. If it does, the region containing that point is the solution for that constraint. The feasible region is the overlapping area that satisfies all the constraints simultaneously.

4. Why is the 'Corner Point Method' crucial for finding the optimal solution in these exercises?

The Corner Point Method is crucial because of a fundamental theorem in linear programming. It states that if an optimal solution (maximum or minimum) for an LPP exists, it must occur at one of the vertices (or corner points) of the feasible region. This allows us to avoid checking infinite points within the region and instead only test a finite number of corner points, which simplifies the process of finding the best possible outcome significantly.

5. What is the difference between a bounded and an unbounded feasible region in the context of problems in Exercise 30.2?

A bounded feasible region is one that is completely enclosed by the constraint lines, forming a closed polygon. In this case, both a maximum and a minimum value for the objective function will exist at the corner points. An unbounded feasible region is one that extends infinitely in at least one direction. For unbounded regions, an optimal solution may or may not exist. A minimum might exist, but a maximum might not (or vice-versa), depending on the objective function.

6. How can I determine if a Linear Programming Problem from Exercise 30.2 has multiple optimal solutions?

A Linear Programming Problem can have multiple optimal solutions if the objective function line is parallel to one of the constraint lines that forms an edge of the feasible region. When you test the corner points, you will find that two adjacent vertices give the same optimal (max or min) value. In this case, every point on the line segment connecting these two vertices is also an optimal solution.

7. What does it mean if there is no feasible region after plotting the constraints for a problem?

If there is no overlapping area that satisfies all the given constraints simultaneously, it means there is no feasible region. This indicates that the Linear Programming Problem has no solution. The constraints are contradictory, and it's impossible to find any point (x, y) that meets all the required conditions at the same time.