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CBSE 12th Maths Linear Programming Formulas: Check here for all the important formulas of mathematics in Chapter 12 Linear Programming of Class 12, along with major definitions, theorems and examples.
Maths Linear Programming Formulas: The Class 12 mathematics curriculum consists of several chapters, and new concepts are introduced to students. One such topic is the Class 12 NCERT Chapter 12 Linear Programming.
While calculus is often the main focus of students, other chapters like Linear Programming shouldn’t be overlooked. It’s an important chapter from an exam perspective and also a fascinating one.
However, students need to have some idea of geometry and linear equations to better understand Linear Programming. It’s a highly scoring and easy-to-understand topic that can help you fetch good marks in the board exam. As such, we bring you the formulas of linear programming.
In light of the syllabus rationalization, we have included the formulas from the revised and deleted syllabus separately for high achievers.
These formulas and theorems can help simplify complex equations and solve problems quickly. You can check out the following CBSE Class 12 Maths Chapter 12 Linear Programming Formulas below.
Recommended:
CBSE Class 12 Maths Mind Map for Chapter 12 Linear Programming
CBSE Class 12 Maths Chapter 12 Linear Programming MCQs
CBSE Class 12 Maths Chapter 12 Linear Programming Formulas and Theorems
We have listed all the important formulas, definitions and properties of CBSE Class 12 Linear Programming here.
A linear programming problem is one that is concerned with finding the optimal value (maximum or minimum) of a linear function of several variables (called objective function) subject to the conditions that the variables are non-negative and satisfy a set of linear inequalities (called linear constraints). Variables are sometimes called decision variables and are non-negative.
Fundamental 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.
Corner Point Method
- Find the feasible region of the linear programming problem and determine its corner points (vertices) either by inspection or by solving the two equations of the lines intersecting at that point.
- Evaluate the objective function Z = ax + by at each corner point. Let M and m, respectively denote the largest and smallest values of these points.
3. (i) When the feasible region is bounded, M and m are the maximum and minimum values of Z.
(ii) In case, the feasible region is unbounded, we have:
- (a) M is the maximum value of Z, if the open half plane determined by ax + by > M has no point in common with the feasible region. Otherwise, Z has no maximum value.
(b) Similarly, m is the minimum value of Z, if the open half plane determined by ax + by < m has no point in common with the feasible region. Otherwise, Z has no minimum value.
Formulas and Definitions from Deleted Syllabus and Exemplar
Objective Function Linear function Z = ax + by, where a and b are constants, which has to be maximised or minimised is called a linear objective function.
Decision Variables In the objective function Z = ax + by, x and y are called decision variables.
Constraints The linear inequalities or restrictions on the variables of an LPP are called constraints. The conditions x ≥ 0, y ≥ 0 are called non-negative constraints.
Feasible Region The common region determined by all the constraints including non-negative constraints x ≥ 0, y ≥ 0 of an LPP is called the feasible region for the problem.
Feasible Solutions Points within and on the boundary of the feasible region for an LPP represent feasible solutions.
Infeasible Solutions Any Point outside 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.
Also Read
CBSE Class 12 Maths Syllabus 2023-24
CBSE Class 12 Maths Sample Paper 2023-24
NCERT Solutions for Class 12 Maths PDF
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