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Class XII Mathematics

Chapter 11: Linear Programming

Standard NCERT & CBSE aligned study curriculum. Master concepts, track accuracy, revise weak areas, and challenge yourself with 9 customized practice modes.

Chapter Overview

Welcome to Class XII Mathematics: Linear Programming. This chapter forms a core structural component of the math syllabus, designed to build analytical rigor and key formula models.

Use the detailed subtopic guide below to review standard definitions, key mathematical rules, and study guidelines.

Prerequisite Concepts

Linear InequalitiesLinear Equations in Two Variables

Detailed Subtopics Study Guide

Review detailed conceptual explanations, mathematical equations, and guidelines for each subtopic in this chapter:

1Linear Programming problems mathematical formulation

Concept Explanation

Formulating a Linear Programming Problem (LPP) involves defining decision variables, constructing a linear objective function to maximize or minimize, and setting up linear inequalities representing constraints.

Mathematical Representation
\text{Max/Min } Z = c_1 x + c_2 y \quad \text{subject to } a_{ij} x_j \le b_i \, \text{ and } \, x_j \ge 0
Study Guideline: Clearly define your decision variables (x and y) first, then write down the constraints from the problem's physical limitations.

2Graphical method for solving LP in two variables

Concept Explanation

The graphical method solves LPPs by plotting constraints on a coordinate grid, shading the feasible region, and finding the optimal vertex (corner point) using the Corner Point Theorem.

Mathematical Representation
\text{Optimal solution occurs at one of the corner points of the feasible region}
Study Guideline: The Corner Point Theorem guarantees that the maximum or minimum value of the objective function always lies at one of the vertices of the feasible region.

3Feasible and infeasible boundary regions

Concept Explanation

The feasible region is the common region determined by all constraints, including non-negativity constraints. If no common region satisfies all constraints simultaneously, the problem is infeasible.

Mathematical Representation
\text{Feasible Region } \neq \emptyset \implies \text{Solutions exist}; \quad \text{Feasible Region } = \emptyset \implies \text{Infeasible}
Study Guideline: An unbounded feasible region may have a minimum but might not have a maximum value.

4Optimal corner point solutions

Concept Explanation

To find the optimal solution, identify all vertices (corner points) of the feasible region, calculate the value of the objective function Z at each vertex, and select the maximum or minimum value.

Mathematical Representation
Z_i = c_1 x_i + c_2 y_i \implies \text{Choose max/min } Z_i
Study Guideline: Find the corner points by solving the intersecting boundary line equations simultaneously.