NCERT Solutions for Class 12 Mathematics Chapter 12: Linear Programming – Free PDF Download

Chapter 12 of Class 12 Maths is Linear Programming. It covers formulating real-world optimization problems as linear programming problems (LPPs) and solving them graphically. These Class 12 Mathematics Chapter 12 solutions are also useful as quick revision notes before exams.

Linear Programming Problem (LPP)

An LPP involves optimizing (maximizing or minimizing) a linear objective function Z=ax+by, subject to a set of linear constraints (inequalities) and non-negativity restrictions (x≥0, y≥0).

Feasible Region

The feasible region is the common region satisfying all constraints simultaneously, represented graphically. It is bounded if enclosed within a finite area, or unbounded otherwise. Points within/on the boundary are feasible solutions.

Corner Point Method

The optimal value of Z (if it exists) occurs at a corner point (vertex) of the feasible region. Method: identify all corner points, evaluate Z at each, and select the maximum/minimum value.

Types of LPP

Common real-world applications: diet problems (minimize cost subject to nutritional constraints), manufacturing problems (maximize profit subject to resource constraints), transportation problems (minimize transportation cost).

Special Cases

If the feasible region is unbounded, a maximum may not exist even if Z appears large at corner points — must verify by checking if the constraint half-plane intersects the region. Multiple optimal solutions occur when Z is constant along an entire edge of the feasible region.

Frequently Asked Questions

What is the corner point method?
Evaluating the objective function at each vertex of the feasible region to find the optimal value.

What is a feasible region?
The region satisfying all the given linear constraints simultaneously.

Can an unbounded feasible region have a maximum?
Not always — it depends on whether the corresponding half-plane of Z=constant intersects the feasible region for large values.

Written by Satish

NCERTBooks.org is an independent educational resource run by a small team focused on making official NCERT textbooks easy to find, read, and download for students, parents, and teachers across India. We are not affiliated with NCERT or the Ministry of Education -- we organise publicly available NCERT content by class and subject, verify links against official sources, and build tools (like our in-browser reader) that make studying more convenient. Every guide we publish is written and reviewed by our team based on the actual NCERT curriculum and syllabus.