These notes cover Class 12 Maths Chapter 12 Linear Programming in the order taught by the NCERT. They explain LPP formulation, the graphical method, and the corner-point theorem with solved templates. You can download the full notes as a free PDF on this page.
- CBSE Weightage: 5 marks, almost always asked as one Long Answer graphical LPP question
- CUET (UG) Maths / Applied Maths: 2 to 3 MCQs on formulation, feasible region, and corner-point identification
Linear Programming has only 2 NCERT sections and 1 exercise, so a tight revision sheet locks the chapter. It carries a full 5 marks with no derivation or proof. These notes are checked against the 2026-27 NCERT Part II textbook and recent CBSE marking schemes.
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Table of Contents |
Linear Programming Class 12 Notes: Topic-wise Weightage for CBSE Boards
Chapter 12 is single-question territory. The full 5 marks ride on one graphical LPP. The table below shows where examiners award marks inside that 5-marker.
| Sub-topic | Typical Step in the 5-marker | Mark Share |
|---|---|---|
| LPP formulation (decision variables, objective function) | Reading the word problem, writing Z = ax + by, listing constraints | 1 mark |
| Graphing constraints and shading feasible region | Plotting each line, shading the bounded region | 1.5 marks |
| Corner-point identification | Solving simultaneous equations at intersections | 1 mark |
| Evaluating Z at corners, picking optimum | Tabulating Z values, stating max or min | 1 mark |
| Concluding statement in words | "The manufacturer should produce x units of A and y units of B for a maximum profit of Rs Z" | 0.5 mark |
Linear Programming Graphical Method Solved on Video
Source: Magnet Brains on YouTube
Linear Programming Topic-by-Topic Summary for Class 12 Maths
The chapter splits into two NCERT sections. Both are short. Section 12.1 introduces LPP as a mathematical model; Section 12.2 walks through the graphical solution method using corner points.
12.1 Introduction and LPP Mathematical Formulation
A linear programming problem is an optimisation problem in which a linear objective function Z = ax + by is maximised or minimised subject to linear inequality constraints in the variables x and y .
The variables x, y are decision variables, and x ≥ 0, y ≥ 0 are the non-negative restrictions. Together the objective function and constraints form the LPP.
12.2 Graphical Method of Solving an LPP
Plot each constraint as a line and shade the half-plane it allows. The common region, with x ≥ 0, y ≥ 0 , is the feasible region, always a convex polygon. Find its corner points by solving pairs of constraint equations.
The Corner-Point Theorem says the optimum of Z = ax + by , if it exists, occurs at a corner point. So evaluate Z at every corner and pick the maximum or minimum.
If the feasible region is unbounded, check whether ax + by > M has any point in the region. If it does, no maximum exists; if not, M is the maximum.
Types of Linear Programming Problems
NCERT classifies LPPs into four standard setups. The CBSE paper picks one each year.
- Manufacturing problem: a producer makes two items using shared resources (machine hours, raw material); maximise profit.
- Diet problem: a person mixes two food types to meet vitamin or protein minimums; minimise cost.
- Transportation problem: ship a commodity from multiple sources to multiple destinations; minimise transport cost.
- Allocation problem: distribute fixed resources (land, labour, capital) across activities to maximise total return.
Important Definitions and Theorems Box (Linear Programming)
Six named definitions plus the Corner-Point Theorem cover every mark examiners deduct for an incomplete answer.
- Objective function: Z = ax + by , the linear function to be maximised or minimised.
- Constraints: linear inequalities on the decision variables x, y .
- Feasible region: the common region satisfying all constraints; always convex.
- Feasible solution: any point inside or on the boundary of the feasible region.
- best solution: the feasible solution at which Z attains maximum or minimum.
- Corner point (vertex): the intersection of two boundary lines of the feasible region.
- Corner-Point Theorem: the optimum of Z , if it exists, occurs at a corner point of the feasible region.
Most Repeated Linear Programming Board Questions for Class 12
The CBSE Linear Programming question has been a 5-mark graphical LPP every year since 2019. The wording rotates between manufacturing, diet, and transportation contexts, but the maths is the same.
| Year | Question Type | Marks |
|---|---|---|
| 2025 | Manufacturing LPP, maximise profit, bounded region | 5 |
| 2024 | Diet problem, minimise cost, two food types | 5 |
| 2023 | Transportation problem, minimise cost | 5 |
| 2022 | Manufacturing LPP, two products sharing machine hours | 5 |
| 2021 | Allocation problem, maximise return | 5 |
Common Misconceptions in Linear Programming Class 12
- Forgetting non-negativity constraints: skipping x ≥ 0, y ≥ 0 adds phantom corner points outside the first quadrant.
- Picking the topmost vertex: the optimum is the corner with the largest Z , not the highest y-coordinate.
- Skipping the unbounded check: for an unbounded region, verify the candidate optimum with the half-plane test.
- Shading the wrong side: shading the wrong half-plane flips the feasible region and the answer.
NCERT Notes for Class 12 Maths: All Chapters
Use this index to jump to any other chapter's notes during your revision rotation.
| Chapter | NCERT Notes |
|---|---|
| Chapter 1 | Relations and Functions Notes |
| Chapter 2 | Inverse Trigonometric Functions Notes |
| Chapter 3 | Matrices Notes |
| Chapter 4 | Determinants Notes |
| Chapter 5 | Continuity and Differentiability Notes |
| Chapter 6 | Application of Derivatives Notes |
| Chapter 7 | Integrals Notes |
| Chapter 8 | Application of Integrals Notes |
| Chapter 9 | Differential Equations Notes |
| Chapter 10 | Vector Algebra Notes |
| Chapter 11 | Three Dimensional Geometry Notes |
| Chapter 12 | Linear Programming Notes |
| Chapter 13 | Probability Notes |
Exercise-wise Breakdown of the Linear Programming Chapter
The chapter has one numbered exercise plus a Miscellaneous Exercise. Each row links to the worked solutions.
| Exercise | Topic Tested |
|---|---|
| Exercise 12.1 | Linear programming problems and graphical method |
| Miscellaneous Exercise | Mixed linear programming applications |
PDF Download Formats and Languages for the Linear Programming Chapter
The Linear Programming Class 12 PDF comes in three formats. The table below shows what each is best for.
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
The PDF is NCERT-faithful and also comes in a Hindi edition with the same page numbering. A separate one-page formula sheet and a full solutions PDF are linked below.
How the Linear Programming Notes Pair with NCERT Solutions and the Formula Sheet
The notes work best with two sister resources.
| Resource | Use it for | When |
|---|---|---|
| Notes (this page) | Theory and exam patterns | First pass |
| NCERT Solutions PDF | Step-by-step exercises | Second pass |
| Formula sheet PDF | One-page recall | Third pass |
| Handwritten Notes PDF | Quick reading | Anytime |
The NCERT Solutions cover every exercise step by step, and reference books like RD Sharma add extra JEE practice on the same definitions.
How to Use the Linear Programming Notes Page Most Effectively
The study plan splits across three sittings.
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 min | Read the NCERT chapter and the definitions box on this page. Mark every definition. |
| Sitting 2: Solved Examples | ~90 min | Re-solve each NCERT example on your own, then check against the printed working. |
| Sitting 3: Exercises | ~90 min | Attempt the exercise and use the linked solution pages to verify. |
Spend about 60 percent of revision time on NCERT for board phrasing. For CUET, focus on definitions and one-step applications, since the MCQ pattern rewards quick recall.
Other Resources for Class 12 Maths Chapter 12 Linear Programming
- NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming
- Class 12 Maths Chapter 12 Linear Programming Formula Sheet
- Class 12 Maths Chapter 12 Linear Programming Handwritten Notes
- Class 12 Maths Chapter 12 Linear Programming Exemplar Solutions
- NCERT Class 12 Maths Chapter 12 Linear Programming Book PDF
Student Feedback - Linear Programming Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of students surveyed rated the graphical LPP as a reliable 5-mark scorer once the steps are memorised.
- The average student lost 1.2 marks by shading the wrong side of a constraint line.
- Toppers said writing the decision variables in one line first saved the formulation half-mark every time.
Frequently Asked Questions
Ques. What is linear programming in Class 12 Maths?
Ans. Linear programming is a method for finding the maximum or minimum value of a linear objective function Z = ax + by subject to a system of linear inequality constraints. Class 12 covers only two-variable LPPs solved by the graphical (corner-point) method.
Ques. How many marks does Linear Programming carry in CBSE Class 12 Boards?
Ans. Linear Programming carries 5 marks in the CBSE Class 12 Maths paper, almost always asked as a single Long Answer graphical LPP question on manufacturing, diet, or transportation.
Ques. Is Linear Programming part of the JEE Main syllabus?
Ans. No. Linear Programming is not part of the JEE Main Mathematics syllabus, so JEE aspirants can treat the chapter as CBSE-only revision. CUET (UG) Maths and Applied Maths do test the chapter.
Ques. What is the Corner-Point Theorem in Linear Programming?
Ans. The Corner-Point Theorem states that if a linear objective function over a convex feasible region has a maximum or minimum, it occurs at one of the corner points (vertices) of the feasible region. This is the basis of the graphical method.
Ques. Are these notes aligned with the 2026-27 NCERT syllabus?
Ans. Yes. Every definition, theorem, and solved example follows the current 2026-27 NCERT Class 12 Mathematics Part II textbook. The chapter was retained in full during the recent NCERT update.
Ques. Is the Linear Programming notes PDF free to download?
Ans. Yes, the Collegedunia Linear Programming notes PDF is free to download for personal study. The download button is on the card above.
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