RD Sharma Solutions for Class 12 Maths Chapter 30 - Linear programming - Free PDF Download
FAQs on RD Sharma Class 12 Maths Solutions Chapter 30 - Linear programming
1. How are the solutions for RD Sharma Class 12 Chapter 30 on Linear Programming structured?
The solutions for RD Sharma Chapter 30 are structured in a clear, step-by-step format to ensure understanding. Each problem's solution typically begins by identifying the decision variables, followed by the mathematical formulation of the objective function and all constraints. The process then moves to the graphical method, showing how to plot the feasible region and identify its corner points before finding the optimal solution.
2. What is the first step when solving a Linear Programming Problem from RD Sharma Chapter 30?
The first and most crucial step is the mathematical formulation of the problem. This involves a few key actions:
- Identifying the decision variables (e.g., the number of units to produce).
- Defining the objective function, which is the linear equation you need to maximize or minimize (e.g., profit or cost).
- Listing all constraints as linear inequalities based on the given conditions.
- Stating the non-negativity constraints (e.g., x ≥ 0, y ≥ 0), as quantities cannot be negative.
3. How do you graphically find the feasible region for a Linear Programming Problem in the RD Sharma solutions?
To graphically find the feasible region, you first treat each inequality constraint as an equation and plot its corresponding straight line. For each line, you then determine which side of it satisfies the original inequality, often by testing the origin (0,0). The common shaded area that satisfies all constraints simultaneously is the feasible region, which represents all possible solutions to the problem.
4. What are the key components of a Linear Programming Problem as per the Class 12 syllabus?
A Linear Programming Problem (LPP) consists of the following essential components:
- Decision Variables: These are the unknown quantities (like x and y) that need to be determined.
- Objective Function: This is a linear function, typically denoted by Z, that you aim to optimize (maximize or minimize).
- Constraints: These are the set of linear inequalities that represent the limitations or restrictions on the decision variables.
- Non-Negativity Restriction: This is the condition that the decision variables cannot be negative.
5. Why is the optimal solution for a Linear Programming Problem always found at a corner point of the feasible region?
This is a fundamental theorem of linear programming. The feasible region is a convex polygon, and the objective function is a linear equation. As the line representing the objective function moves across this region, its value increases or decreases. The absolute maximum or minimum value will always be reached at the farthest point the line can go while still touching the feasible region. This point will invariably be one of the vertices (corner points) of the polygon.
6. What is the difference between a bounded and an unbounded feasible region, and how does it affect the solution?
A bounded feasible region is an area enclosed on all sides by the constraint lines. In such cases, an LPP is guaranteed to have both a maximum and a minimum value. In contrast, an unbounded feasible region extends indefinitely in at least one direction. For an unbounded region, an LPP may or may not have an optimal solution. An optimal value might exist at a corner point, but it's also possible for the function to increase or decrease indefinitely.
7. How do the RD Sharma solutions help in identifying redundant constraints in an LPP?
The graphical method detailed in the RD Sharma solutions makes it visually clear when a constraint is redundant. A redundant constraint is one that does not impact the shape or size of the feasible region because other constraints are stricter. When plotting the inequalities, you'll notice that the final feasible region is formed without the boundary line of the redundant constraint playing any role in defining it.
8. What does it mean if a Linear Programming Problem has no feasible region, and how is this shown graphically?
If an LPP has no feasible region, it implies that there is no solution that can satisfy all the given constraints at the same time. This is termed an infeasible solution. Graphically, this is represented by a situation where the shaded areas corresponding to the individual inequalities do not overlap at any point, meaning there is no common area.
9. How is the objective function used to determine the optimal solution in the graphical method?
Once the feasible region and all its corner points (vertices) are identified, the Corner Point Method is applied. This involves substituting the coordinates of each corner point into the objective function (Z) to calculate its value. The largest of these calculated values is the maximum solution, and the smallest is the minimum solution, thus giving the optimal value for the problem.
10. Do the RD Sharma solutions for Linear Programming cover different real-world problem types?
Yes, Chapter 30 in RD Sharma Class 12 and its solutions cover various practical applications of LPP. The exercises and examples guide students on how to formulate and solve different types of word problems, including diet problems (minimizing food cost), manufacturing problems (maximizing production profit), and transportation problems (minimizing shipping costs), which are common in board exams.
