NCERT Class 12 Mathematics Ganit-II: Chapter 6 — रैखिक प्रोग्रामन
This chapter, "Linear Programming" (रैखिक प्रोग्रामन), introduces students to optimization problems, a special type of problem focused on maximizing profits or minimizing costs. It builds upon previous knowledge of linear equations and inequalities from Class XI. The chapter presents a real-life example of a furniture dealer who wants to maximize profit by deciding how many tables and chairs to buy within investment and storage constraints. It explains the concept of mathematical formulation for such problems, defining variables (number of tables 'x' and chairs 'y') and non-negativity constraints (x ≥ 0, y ≥ 0). The chapter aims to equip students with the ability to solve linear programming problems using graphical methods, highlighting their importance in various fields like industry and management science. This foundational understanding is crucial for advanced problem-solving in mathematics and its applications.
Quick info
| Board | CBSE / NCERT |
|---|---|
| Class | Class 12 |
| Subject | Mathematics |
| Book | Ganit-II |
| Chapter | Chapter 6 — रैखिक प्रोग्रामन |
| Language | Hindi |
| PDF type | NCERT Textbook |
| Session | CBSE 2026 |
| Reading time | 4 minutes |
| Word count | 746 |
Learning outcomes
- Understand the concept of optimization problems.
- Learn to formulate real-life problems into mathematical models.
- Identify objective functions and constraints in linear programming.
- Understand the non-negativity constraints for variables.
- Prepare for solving linear programming problems using graphical methods.
Vocabulary
| Word | Meaning |
|---|---|
| रैखिक प्रोग्रामन | Linear Programming |
| इष्टतमकारी समस्याएँ | Optimization problems |
| असमिकाओं | Inequalities |
| चर राशियों | Variables |
| आलेखीय निरूपण | Graphical representation |
| गणितीय सूत्राीकरण | Mathematical formulation |
| व्यवरोधें | Constraints |
| ध्न राशि | Amount of money |
| सकल लाभ | Gross profit |
| अध्कितम | Maximum |
| न्यूनतमीकरण | Minimization |
| )णेतर | Non-negative |
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Practice questions
- What is an optimization problem? Answer: An optimization problem is one where we try to find the maximum profit or minimum cost.
- What does 'non-negative' mean for variables in linear programming? Answer: It means the variables (like the number of items) cannot be negative; they must be zero or positive.
- Give an example of a constraint in the furniture dealer problem. Answer: The limited investment amount (Rs 50,000) or the limited storage space (60 items) are constraints.
Practice MCQs
Q1. What is the main goal in an optimization problem?
Explanation: Optimization problems specifically aim to find the best possible outcome, which is either the maximum profit or the minimum cost.
Q2. In the furniture dealer example, what does 'x' represent?
Explanation: The problem statement defines 'x' as the number of tables the furniture dealer buys.
Q3. What does the constraint 'x ≥ 0' signify?
Explanation: The non-negativity constraint 'x ≥ 0' means that the number of tables purchased cannot be less than zero, allowing for zero or a positive number of tables.
Q4. Linear Programming is a type of:
Explanation: Linear Programming is described as a special but important type of optimization problem.
Q5. Which of the following is NOT a constraint in the furniture dealer problem?
Explanation: The profit per item (Rs 250 for a table, Rs 75 for a chair) is part of the objective function to be maximized, not a constraint limiting the decision.
Frequently asked questions
What is Linear Programming?
Linear Programming (LP) is a mathematical technique used to find the best possible outcome (like maximum profit or minimum cost) in a given situation, subject to certain limitations or constraints.
What is an optimization problem?
An optimization problem is a problem where the goal is to find the maximum or minimum value of a particular quantity, such as profit or cost, under specific conditions.
What are constraints in Linear Programming?
Constraints are limitations or restrictions that must be satisfied, such as limited resources (money, space, time) or requirements.
What is the objective function in LP?
The objective function is the mathematical expression that represents the quantity to be maximized or minimized, such as profit or cost.
Why are non-negativity constraints important?
Non-negativity constraints (like x ≥ 0, y ≥ 0) are important because in real-world problems, quantities like the number of items cannot be negative.
What is the mathematical formulation of a problem?
It is the process of translating a real-world problem into a mathematical model, involving defining variables, an objective function, and constraints.
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NCERT Class 12 Mathematics — Ganit-II — Chapter 6 — रैखिक प्रोग्रामन. Verified by NCERT Help Editorial Team. Reviewed on 29 Jul 2026. Last updated 10 Aug 2026.