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Linear Programming
Linear Programming deals with optimizing a linear expression under a system of linear inequalities. In Class 12, the focus is on two-variable problems that can be represented graphically. A typical problem begins with identifying decision variables, forming an objective function, writing constraints from the given conditions, and adding non-negativity restrictions when quantities cannot be negative. The graphical method depends on the feasible region. The optimum value of the objective function, when it exists, is tested at the corner points of the feasible region. Exam questions often combine algebra, graph interpretation, and application contexts such as manufacturing, diet planning, transport, and resource allocation. Marks are commonly lost when constraints are written with the wrong inequality sign or when corner points are not checked systematically.
Difficulty
Medium
Study time
70-90 min
Plan by time
Pick the window that matches what you have right now.
If you have 15 min
Last-pass revision
Skim the Quick Revision table — definitions, formulas, and the traps board examiners reuse.
Open Quick RevisionIf you have 45 min
Targeted practice
Read the high-priority concepts, then take the chapter MCQ quiz to find weak spots.
Start MCQ QuizIf you have 70 min
First full pass
Walk every concept in chapter order, then revise and quiz. Best for the first time you study this chapter.
Open Key ConceptsChapter Learning Map
Start with one of the buckets below, then open the full map when you want the complete concept roadmap.
Key Concepts
Concepts grouped the way the chapter is taught — open the bucket that matches what you want to revise.
Core Concepts
high priorityOpen the chapter concepts in a clean revision order.
Mathematical Formulation of an LPP
A linear programming problem is formulated by choosing decision variables, writing a linear objective function such as Z = ax + by, and expressing all given conditions as linear constraints.
Graphical Method and Corner Point Evaluation
The graphical method solves a two-variable LPP by drawing the constraint lines, shading the common feasible region, finding its corner points, and evaluating the objective function at those points.
Bounded and Unbounded Feasible Regions
A feasible region is bounded if it can be enclosed within a finite part of the plane; it is unbounded if it extends indefinitely in at least one direction.
Application-Based Linear Programming Problems
Application word problems use linear programming to model real situations such as diet planning, manufacturing, transport, and allocation of limited resources.
Exam Intelligence
Use this section to decide what deserves the most revision time.
High Probability Topics
- Mathematical Formulation of an LPP
- Graphical Method and Corner Point Evaluation
- Bounded and Unbounded Feasible Regions
- Application-Based Linear Programming Problems
Common Traps
- Reversing <= and >= while translating at most or at least conditions.
- Forgetting non-negativity restrictions.
- Using points that are not vertices of the feasible region in the final table.
- Assuming unbounded feasible region always means no optimum.
- Not interpreting the optimum point in the context of the word problem.
- Making arithmetic errors while solving intersection points of two boundary lines.
Likely Question Types
- MCQ: concept checks, applications, and common mistakes
- Very short answer: definitions, formulas, conditions, or terms
- Short answer: process, diagram, reasoning, or worked method
- Case-based: chapter scenario with linked subparts
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
- Linear Programming optimizes a linear objective function under linear constraints.
- Formulation is the foundation: variables, objective function, constraints, and non-negativity conditions.
- The graphical method is used for two-variable LPPs in Class 12.
- In a bounded feasible region, the optimum occurs at a corner point.
- An unbounded feasible region needs an additional direction check before deciding whether a finite optimum exists.
- Application problems require final answers in context, not only algebraic values.
- Mathematical Formulation of an LPP: A linear programming problem is formulated by choosing decision variables, writing a linear objective function such as Z = ax + by, and exp…
- Graphical Method and Corner Point Evaluation: The graphical method solves a two-variable LPP by drawing the constraint lines, shading the common feasible region, finding its corner poin…
Practice
Use short concept checks first, then move into the full chapter test.
Free Chapter MCQ Quiz
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