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CBSE Class 12 Mathematics Chapter 12 Linear Programming Revision Notes: The 2024 board exams are here, and it is time to lay down the books and start revising the topics. Mathematics is a subject that requires a ton of practice and thorough revision to fully master. In these final days before the examination, students should be wise to review what they know instead of choosing to learn new concepts.
The CBSE Class 12 board exams are set to commence from February 15, 2024, and the math paper will take place on March 9. The twelfth chapter in the Class 12 math books is Linear Programming. It’s one of the most important chapters in the 12th mathematics syllabus and has a lot of importance in the final exam as well.
You can check out the CBSE Class 12 Chapter 12 Linear Programming revision notes here, along with additional study resources like mind maps and multiple-choice questions.
CBSE Class 12 Maths Chapter 12 Linear Programming Revision Notes
Basic Definitions, Theorems and Formulas
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
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.
- (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.
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.
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