Class 12 Mathematics Chapter 12 Notes
Chapter 12 Mathematics Class 12 notes are based on the students’ prior knowledge of linear equations and inequalities. Students will have to solve several real-life problems with the help of linear programming. The Chapter is essential, as well as scoring from the point of view of the CBSE board exam. With the help of the Class 12 Mathematics Chapter 12 notes, students can make last-minute preparations and get maximum scores. Extramarks provide detailed information and thorough conceptual clarity of CBSE syllabus along with revision notes to help students study efficiently and excel in academics.
NCERT Class 12 Mathematics Chapter 12 Notes: Key Topics
Some of the topics introduced in Class 12 Mathematics notes Chapter 12 are as follows:
- Definition of LPP
- Method of formulating the LPP using graphical method
- Important mathematical formulas and concepts including Optimal value, Objective function, Constraints, and, Optimization problems.
- Applications of Linear Programming problems
- Theorems of LPP
Introduction:
Linear programming problems (LLP) are one of the essential Classes of optimisation problems. It is a mathematical modelling technique where the linear function is maximised or minimised under various constraints.
In general, an LPP is specified as follows:
Consider:
(i) n variables x1, x2, x3,… ,xn.
(ii) m linear inequalities in these variables.
(iii) Linear objective function.
Find values for that xi’s which satisfy the given constraints and maximise or minimise the objective function.
What is a Linear Programming Problem?
An LPP is based on finding an optimal value of a linear function of several variables with respect to the conditions which the variable satisfies. ‘Linear’ means mathematical relations which are used in problems are linear relations, and ‘Programming’ means a method of determining a plan of action.
Important definitions:
Objective function: The linear function Z = ax + by where, a,b are constants has to be minimised or maximised is termed as the Objective function. The variables x and y are known as the decision variables.
Constraints: The inequalities or equations on the variables of an LPP are called constraints. Conditions x ≥ 0, y ≥ 0 are known as non-negative constraints.
Optimization problem: It is a problem which needs to maximise or minimise a given linear function with respect to certain constraints determined by the linear inequalities.
Solutions: The values of the decision variables x and y which satisfy the given constraints and restrictions of an LPP, are termed as a solution of the LPP.
Optimal Solutions: A feasible solution that can optimise the objective function is defined as the optimal solution to the linear programming problem.
Feasible region: The point in a region which represents a suitable choice is termed the feasible region. It is a common region which determines the non-negative constraints. In other words, a solution which satisfies the non-negativity restrictions of the LPP is the feasible solution. Also, the set of all feasible solutions is termed the feasible region. Every point belonging (within or on the boundary) to this region is known as a feasible solution. The region outside the feasible region is the infeasible region.
Graphical method to solve the linear programming problems:
The problems which involve only two variables can be solved using the graphical method of linear programming problems. It is based on the extreme point theorem.
Corner Point Method:
It comprises of the following steps:
- Find the feasible region of the LPP. Determine its corner points either by inspection method or by solving the two equations intersecting at a point.
- Evaluate objective function at every corner point. Consider M and m to be the largest and smallest values of these points, respectively.
- (i) When the feasible region is bounded, then we say that M and m are maximum and minimum values of the objective function.
(ii) If the feasible region is not bounded, we have
- (a) M is the max value of Z if and only if the open half-plane determined by ax + by > M has no common point with the feasible region. Or else, maximum value M does not exist.
- (b) Similarly, m is the min value of Z if and only if the open half-plane determined by ax + by < m has no common point with the feasible region. Or else, minimum value M does not exist.
The Class 12 Mathematics Chapter 12 notes include several problems for students to practice.
Types of Linear Programming Problems:
-
Manufacturing problems:
In these types of problems, the number of units of different products is to be determined. These units will be produced and sold by a firm, and every product will require fixed manpower, hours for the machine, labour hours per unit of product, warehouse space, etc., to make maximum profit.
2. Diet problems:
In these types of problems, the number of different kinds of constituents or nutrients is to be determined. This amount will be included in a diet in such a way that the cost of the desired diet is minimised. However, the diet must contain a particular minimum amount of each constituent or nutrient.
3. Transportation problems:
In these types of problems, the transportation schedule is to be determined. This schedule will help to find a way of transporting a product at minimum cost from plants or factories situated at different locations to the various markets.
Theorem 1:
If R is the feasible region or a convex polygon for a linear programming problem and Z = ax + by is the objective function, then when Z has a maximum or minimum value (optimal value), where the decision variables x and y have constraints described by the linear inequalities or equations, the optimal value Z must occur at a corner point or vertex of the feasible region.
Theorem 2:
If R is the feasible region or a convex polygon for a linear programming problem and Z = ax + by is the objective function. Let R is bounded; then, the objective function Z has both a maximum as well as minimum value on R. Each optimal value occurs at the corner point or vertex of R.
Class 12 Mathematics Chapter 12 Notes: Exercise & Solutions
To help students gain in-depth knowledge of the Chapter, Extramarks provides well-organised and detailed Class 12 Mathematics Chapter 12 notes. Solutions to all important questions, exercise problems, solved examples, and some CBSE extra questions are included in the notes. These notes enable quick revision as it provides all important definitions, derivations, formulas, and theorems in one place. With the help of the Class 12 Mathematics Chapter 12 notes, students will learn to practice, interpret and solve complex questions in no time. Students are advised to refer to the step-by-step solutions to understand each and every step of linear programming problems. It boosts confidence and empowers students to pursue their dream careers.
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NCERT Class 12 Mathematics Chapter 12 notes: Key Features
- Class 12 Mathematics Chapter 12 notes are focused on strengthening the fundamental basic concepts for students to understand the complex concepts well.
- The notes strictly adhere to the latest CBSE Syllabus for 2021-2022.
- It is prepared by the in-house team of experts in Mathematics at Extramarks.
- Students can depend on these notes as it includes authentic and apt information.
- It prepares students for board exams as well as other national level competitive examinations such as JEE Mains, etc.
- With the help of Class 12 Mathematics notes Chapter 12, students can practice unlimited questions to reduce the chances of making simple mistakes and boost confidence.