NCERT Class 12 Mathematics Mathematics Part-II: Chapter 6 — MATHEMATICS
This chapter introduces Linear Programming (LP) problems, a class of optimisation problems seeking to maximise or minimise a specific objective, such as profit or cost. It builds upon previous knowledge of linear equations and inequalities. The chapter uses a real-life example of a furniture dealer to illustrate the concept. The dealer wants to maximise profit by deciding how many tables and chairs to buy, given constraints on investment (Rs 50,000) and storage space (60 pieces). The chapter explains the mathematical formulation of such problems, defining variables, objective functions, and constraints. It sets up the initial inequalities for the furniture dealer problem, focusing on non-negativity, investment, and storage constraints. This chapter is crucial for students learning to apply mathematical concepts to solve practical, real-world optimisation challenges in various fields like industry and commerce.
Quick info
| Board | CBSE / NCERT |
|---|---|
| Class | Class 12 |
| Subject | Mathematics |
| Book | Mathematics Part-II |
| Chapter | Chapter 6 — MATHEMATICS |
| Language | English |
| PDF type | NCERT Textbook |
| Session | CBSE 2026 |
| Reading time | 4 minutes |
| Word count | 704 |
Learning outcomes
- Understand the concept of optimisation problems and linear programming.
- Formulate real-world problems into mathematical linear programming models.
- Identify objective functions and constraints in a linear programming problem.
- Apply non-negativity, investment, and storage constraints to a problem.
- Recognize the applicability of linear programming in various fields.
Vocabulary
| Word | Meaning |
|---|---|
| Linear Programming | A mathematical method for determining a way to achieve the best outcome in a mathematical model that identifies with its processes a set of variables to determine its best outcome. |
| Optimisation problems | Problems that seek to maximise or minimise a certain quantity, such as profit or cost. |
| Constraints | Limitations or restrictions that must be satisfied in an optimisation problem, such as budget or storage space. |
| Objective function | A mathematical expression representing the quantity to be maximised or minimised in an optimisation problem. |
| Mathematical formulation | The process of translating a real-world problem into mathematical equations and inequalities. |
| Non-negative constraints | Constraints that require variables to be greater than or equal to zero, as quantities cannot be negative. |
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Practice questions
- What is the main goal of an optimisation problem? Answer: To maximise or minimise a certain quantity, like profit or cost.
- What are the two main types of constraints mentioned in the furniture dealer example? Answer: Investment constraint and storage constraint.
- Why are the variables (number of tables and chairs) considered non-negative? Answer: Because you cannot buy a negative number of items.
Practice MCQs
Q1. Linear programming is a special class of which type of problems?
Explanation: Linear programming is specifically designed to solve optimisation problems, aiming to find the best possible outcome under given constraints.
Q2. In the furniture dealer example, what is the investment constraint?
Explanation: The investment constraint represents the total cost of tables (2500x) and chairs (500y) not exceeding the available capital of Rs 50,000.
Q3. What does 'x' represent in the mathematical formulation of the furniture dealer problem?
Explanation: In the problem's formulation, 'x' is defined as the number of tables the dealer buys.
Q4. What does 'y' represent in the mathematical formulation of the furniture dealer problem?
Explanation: In the problem's formulation, 'y' is defined as the number of chairs the dealer buys.
Q5. The condition that the number of tables and chairs must be non-negative are called:
Explanation: Constraints that ensure variables are greater than or equal to zero are known as non-negative constraints, as quantities like items cannot be negative.
Frequently asked questions
What is a Linear Programming Problem (LPP)?
An LPP is a type of optimisation problem that involves finding the best outcome (maximum profit or minimum cost) in a mathematical model where the objective function and constraints are linear.
What is the purpose of mathematical formulation in linear programming?
It involves translating a real-world problem into a set of mathematical equations and inequalities, defining variables, the objective function, and constraints.
What are the 'constraints' in a linear programming problem?
Constraints are limitations or restrictions, such as budget, time, or resources, that must be satisfied while trying to achieve the objective.
Why are non-negative constraints important in LPPs?
Non-negative constraints (like x ≥ 0, y ≥ 0) are crucial because variables often represent physical quantities that cannot be negative, such as the number of items produced or sold.
What is an 'optimisation problem'?
An optimisation problem is a problem where the goal is to find the best possible solution from a set of available options, typically by maximising a benefit or minimising a cost.
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NCERT Class 12 Mathematics — Mathematics Part-II — Chapter 6 — MATHEMATICS. Verified by NCERT Help Editorial Team. Reviewed on 29 Jul 2026. Last updated 10 Aug 2026.