Linear Programming is a remarkable and practical section of mathematics, especially prominent in the Class 12 curriculum. This chapter introduces students to the concept of optimizing a linear objective function, subject to a set of linear inequality and equality constraints. It's a crucial topic for Class 12 students, as it has vast applications in various fields like economics, business, engineering, and science, where making optimal decisions is essential.

Our Linear Programming Class 12 notes are crafted to provide students with a clear and concise understanding of this vital topic. These notes delve into the fundamentals of Linear Programming, explaining concepts like constraints, objective functions, and different methods of finding optimal solutions. They are designed to be comprehensive yet easy to grasp, ensuring that students can confidently tackle exam questions.

To aid in visual learning and quick revision, we've developed a mind map for Linear Programming Class 12. This mind map summarizes the key concepts and methodologies in an easy-to-understand format, helping students quickly recall important information during their study sessions and exams.

We also offer a collection of Linear Programming Class 12 MCQs (Multiple Choice Questions). These MCQs are a great tool for self-assessment and exam preparation, as they cover a wide range of topics and help students test their understanding of the subject.

For those students seeking to deepen their knowledge and challenge themselves further, we provide extra questions on Linear Programming Class 12. These questions are designed to push the boundaries of students' understanding and application skills.

In addition to the resources for Class 12 students, our general Linear Programming notes are an excellent reference for anyone interested in exploring this fascinating area of mathematics. Whether you are a student preparing for your board exams, a teacher looking for resources, or just someone interested in understanding the basics of optimization, our Linear Programming notes offer valuable insights into this practical and intriguing field.

Introduction:

Linear Programming is a significant and highly applicable part of mathematics, particularly in the Class 12 curriculum. It revolves around the concept of optimizing a specific linear objective function, subject to a set of linear constraints. This mathematical strategy is widely used in various industries, including business, economics, and operations research, to find the best possible solution in a given range of options under certain constraints. It helps in making efficient decisions and maximizing or minimizing a particular quantity, such as profit or cost.

Linear Programming Problem and its Mathematical Formulation: The Linear Programming Problem (LPP) involves finding the maximum or minimum value of a linear function, known as the objective function, subject to a set of linear inequalities called constraints. The mathematical formulation of an LPP begins with defining the objective function, which is typically in the form of $\mathrm{<span\; style="font-size:\; 18px;">�</span>}<span\; style="font-size:\; 18px;">(</span>\mathrm{<span\; style="font-size:\; 18px;">�</span>}<span\; style="font-size:\; 18px;">,</span>\mathrm{<span\; style="font-size:\; 18px;">�</span>}<span\; style="font-size:\; 18px;">)</span><span\; style="font-size:\; 18px;">=</span>\mathrm{<span\; style="font-size:\; 18px;">�</span>}\mathrm{<span\; style="font-size:\; 18px;">�</span>}<span\; style="font-size:\; 18px;">+</span>\mathrm{<span\; style="font-size:\; 18px;">�</span>}\mathrm{<span\; style="font-size:\; 18px;">�</span>}$, where $�$ and $�$ are the decision variables, and $�$ and $�$ are coefficients. The next step is to set up the constraints, which are inequalities involving the decision variables. These constraints form a feasible region, and the solution of the LPP lies at one of the vertices (corner points) of this region. The problem is solved using methods such as the graphical method or simplex method, depending on the complexity of the problem.

Linear Programming All Formulas: In Linear Programming, formulas play a crucial role in solving problems. These include the formulas for the objective function and the constraints. The objective function formula is $\mathrm{<span\; style="font-size:\; 18px;">�</span>}<span\; style="font-size:\; 18px;">=</span>\mathrm{<span\; style="font-size:\; 18px;">�</span>}\mathrm{<span\; style="font-size:\; 18px;">�</span>}<span\; style="font-size:\; 18px;">+</span>\mathrm{<span\; style="font-size:\; 18px;">�</span>}\mathrm{<span\; style="font-size:\; 18px;">�</span>}$, where $�$ is the value to be optimized. The constraint formulas are in the form of linear inequalities, such as $��+��\le �$, $��+��\ge �$, and $��+ℎ�=�$, where $�$, $�$, and $�$ are constants. Additionally, non-negativity constraints are applied, where $�\ge 0$ and $�\ge 0$, assuming that the decision variables cannot be negative. These formulas are used to construct the feasible region and determine the optimal solution.