Chapter Hub
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
Work through the concepts in order. Each one has an explanation, a worked example, a self-check and an exam answer.
All Concepts
All 4 concepts of the chapter, in the order the NCERT chapter teaches them.
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
The traps that cost marks in this chapter, and how its main concepts turn up in the paper.
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.
How CBSE Asks This Chapter
- Mathematical Formulation of an LPP
Usually asked as a short-answer or long-answer first part: define variables, write the objective function, list constraints, and mention non-negativity restrictions.
- Graphical Method and Corner Point Evaluation
Often asked as a complete long-answer problem requiring graph, feasible region, corner point table, optimum value, and final interpretation.
- Bounded and Unbounded Feasible Regions
Asked through short-answer reasoning, assertion-reason items, or as the final check in a graphical LPP where the region is open.
- Application-Based Linear Programming Problems
Usually appears as a long-answer or case-study problem where students must form the LPP, draw the feasible region, evaluate corner points, and write the final real-world decision.
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
Mathematical Formulation of an LPP
Remember
General form for Class 12 graphical LPP: maximize or minimize Z = ax + by, subject to constraints a1x + b1y ≤, ≥, or = c1; a2x + b2y ≤, ≥, or = c2; and usually x ≥ 0, y ≥ 0.Trap to avoid
Students often write the objective function correctly but reverse a constraint inequality, such as using ≥ for an upper limit like available time or raw material.
Graphical Method and Corner Point Evaluation
Remember
Corner point principle: For a bounded feasible region, maximum or minimum of Z = ax + by occurs at one or more vertices of the feasible region.Trap to avoid
A common error is shading the wrong side of a line after plotting it correctly, which changes the feasible region completely.
Bounded and Unbounded Feasible Regions
Remember
Condition for bounded region: feasible region is closed and enclosed by constraint boundaries.Trap to avoid
Students sometimes assume every unbounded feasible region has no solution.
Application-Based Linear Programming Problems
Remember
Application modelling pattern: decision variables x, y; objective function Z = cost/profit expression; constraints from resource limits or minimum requirements; non-negativity x ≥ 0, y ≥ 0.Trap to avoid
Students often ignore units and mix quantities, such as adding kilograms of material directly to hours of labour in one constraint.
| Concept | Remember | Trap to avoid |
|---|---|---|
| Mathematical Formulation of an LPP | General form for Class 12 graphical LPP: maximize or minimize Z = ax + by, subject to constraints a1x + b1y ≤, ≥, or = c1; a2x + b2y ≤, ≥, or = c2; and usually x ≥ 0, y ≥ 0. | Students often write the objective function correctly but reverse a constraint inequality, such as using ≥ for an upper limit like available time or raw material. |
| Graphical Method and Corner Point Evaluation | Corner point principle: For a bounded feasible region, maximum or minimum of Z = ax + by occurs at one or more vertices of the feasible region. | A common error is shading the wrong side of a line after plotting it correctly, which changes the feasible region completely. |
| Bounded and Unbounded Feasible Regions | Condition for bounded region: feasible region is closed and enclosed by constraint boundaries. | Students sometimes assume every unbounded feasible region has no solution. |
| Application-Based Linear Programming Problems | Application modelling pattern: decision variables x, y; objective function Z = cost/profit expression; constraints from resource limits or minimum requirements; non-negativity x ≥ 0, y ≥ 0. | Students often ignore units and mix quantities, such as adding kilograms of material directly to hours of labour in one constraint. |
Practice
Use short concept checks first, then move into the full chapter test.
Free Chapter MCQ Quiz
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