NCERT Solutions for Class 12 Maths Chapter 12 Exercise 12.1 Linear Programming - PDF Download

1. Linear Programming Problem and its Mathematical Formulation

In this section, you will get to learn some important terms.

Optimisation Problems - A question that seeks to maximise (or, minimise) profit (or, cost) form a general class of problems called optimisation problems. 

Optimal Value - It is the maximum or minimum value of a linear function.

Objective Function - Linear function which is written as Z = ax + by where a and b are constant that need to be maximised or minimised is known as linear objective function.

• Mathematical formulation of the problem

This mathematical formulation of the problem has essential terms included.

(i)  Linear Programming Problems - There are some problems where we have to maximise or minimise the linear function Z subject to certain conditions determined by a set of linear inequalities with variables as non-negative. Such problems are called linear programming problems.

(ii) Linear Constraints - subject to the conditions that the variables are non-negative and satisfy a set of linear inequalities called linear constraints.

(iii)  Decision Variables - Variables x and y are called the decision variables. These variables decide the conclusion of linear programming questions. 

(iv) constraints - The linear inequalities or equations or restrictions on the variables of a linear programming problem are called constraints. 

• Graphical method of solving linear programming problems

This is the simplest method to solve linear programming questions. In this method we use graphs to solve given linear programming problems. 

The graph of this system (shaded region) consists of the points common to all half planes determined by the inequalities. Each point in this region represents a feasible choice open to the dealer for investing in tables and chairs. This region is called the feasible region. The region other than feasible region is called an infeasible region. Feasible solutions points within and on the boundary of the feasible region represent feasible solutions of the constraints. Any point outside the feasible region is called an infeasible solution.

Optimal (feasible) solution -  Any point in the feasible region that gives the optimal value (maximum or minimum) of the objective function is called an optimal solution.

2. Theorems

Theorem 1 - Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point* (vertex) of the feasible region.

Theorem 2 -  Let R be the feasible region for a linear programming problem, and let Z = ax + by be the objective function. If R is bounded**, then the objective function Z has both a maximum and a minimum value on R and each of these occurs at a corner point (vertex) of R.

If R is unbounded, then a maximum or a minimum value of the objective function may not exist. However, if it exists, it must occur at a corner point of R. (By Theorem 1).

In ex 12.1 class 12 maths chapter 12, there is a corner point method to solve linear programming problems. This method includes steps to solve linear programming sums which is elaborated in NCERT textbook of mathematics for class 12 maths.

Tips for Solving Exercise 12.1 Class 12 Chapter 12 Linear programming

This exercise consists of questions related to linear programming that can be solved by following the tips provided by subject experts of eSaral. You can check these tips below.

1. To solve questions of ex 12.1 class 12 maths chapter 12, you must understand the concept of maximise and minimise the value of linear programming functions.

2. There are two important theorems that must be learnt by you for deep comprehension of a topic to score high marks in exams.

3. Questions can be solved if you go through the corner point method that includes steps to solve linear programming questions.

4. NCERT solutions for ex 12.1 class 12 maths ch 12 has some essential terms, definitions and formulas that you need to learn before solving es 12.1.
   

Importance of Solving Ex 12.1 Class 12 Maths Chapter 12 Linear programming

There are a lot of benefits of solving ex 12.1 class 12 maths chapter 12 Linear Programming.

1. NCERT solutions for class 12 maths chapter 12 ex 12.1 are prepared by the best teachers of eSaral in a simple and precise manner to help you achieve the best score in exams.

2. All the questions of ex 12.1 are solved in detailed format and stepwise method to help you understand the concepts quickly.

3. NCERT solutions for ex 12.1 class 12 maths are available in PDF where you can find answers to all the questions and examples that aid you in preparing for exams.

4. Practicing questions in ex 12.1 class 12 maths chapter 12 will improve your mathematical skills and boost your self-confidence.

Frequently Asked Questions

Question 1. What is a linear programming problem?

Answer 1. A linear programming problem is one that is concerned with finding the optimal value (maximum or minimum) of a linear function of several variables.

Question 2. What are the two methods discussed in chapter 12 exercise 12.1 class 12 maths?

Answers 2. In chapter 12 exercise 12.1 class 12 maths, there are two methods such as graphical method and corner point method to solve linear programming problems.