The Linear Programming Class 12 NCERT Book page compiles NCERT Class 12 Mathematics Chapter 12 into a single download-ready resource, aligned to the 2026-27 NCERT syllabus. The page covers definitions, solved examples, exam-weightage data and common mistakes, with every formula matched to the CBSE marking scheme used in recent board papers.
- CBSE Weightage 2026-27: 5 to 6 marks (Unit V: Linear Programming, weight 5)
The chapter contains 2 exercises (Exercise 12.1 and the Miscellaneous Exercise) totalling about 21 exercise questions, plus roughly 10 solved examples drawn from real allocation, diet, manufacturing and transport problems.
Student Pulse - Linear Programming Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
- 74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
- Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
- Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
Sourced directly from the official NCERT 2026-27 print, so every page number, figure label and exercise number matches the copy your school uses.
Also Check:
- Linear Programming Class 12 NCERT Solutions
- Linear Programming Class 12 Maths Notes
- Linear Programming Formula Sheet Class 12
What is Inside the Linear Programming PDF Class 12 NCERT Chapter
Chapter 12 is part of Unit V: Linear Programming and is the only chapter in that unit. The 2026-27 print keeps the Linear Programming PDF Class 12 compact, with three numbered sections, two exercises and a short summary box at the end. The full structure is shown below.
| Section | Title | What it covers |
|---|---|---|
| 12.1 | Introduction | Real situations where a quantity (profit, cost, distance, calories) must be optimised under given limits. Introduces the idea of an objective function and constraints. |
| 12.2 | Linear Programming Problem and its Mathematical Formulation | Formal definition of a linear programming problem, decision variables, objective function Z = ax + by , linear constraints, non-negative restrictions, the feasible region, feasible and infeasible solutions, best solution, and the corner-point method with its two underlying theorems. |
| 12.3 | Different Types of Linear Programming Problems | The four standard families: manufacturing problems, diet problems, transportation problems and allocation problems. Each is illustrated with a fully solved example. |
| - | Exercise 12.1 | 10 questions on graphical solution of two-variable LPPs (max and min of Z over a feasible region). |
| - | Miscellaneous Exercise | 11 questions mixing all four problem types, plus bounded and unbounded feasible regions. |
| - | Summary | One-page recap of every definition and the corner-point rule. |
The chapter has roughly 10 solved examples in the running text, and the Miscellaneous Exercise pulls in word problems whose feasible region is bounded by 3 to 5 linear inequalities.
Linear Programming Video Chapter Walkthrough
Source: Magnet Brains on YouTube
Linear Programming Class 12 Chapter Snapshot
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| Item | Value |
|---|---|
| Class | 12 |
| Subject | Mathematics (Part-II) |
| Chapter | 12: Linear Programming |
| Unit | V: Linear Programming (weight 5) |
| Pages in PDF | about 15 |
| Numbered sections | 3 (12.1, 12.2, 12.3) |
| Exercises | 2 (Exercise 12.1, Miscellaneous Exercise) |
| Total exercise questions | about 21 |
| Solved examples | about 10 |
| CBSE Board weightage 2026-27 | 5 to 6 marks |
| Status in 2026-27 syllabus | Retained, no topic dropped |
The snapshot makes clear why students who plan their preparation by chapter length often save Linear Programming for the final revision week: it is short, self-contained and yields a near-guaranteed 5 marks if the corner-point method is practised on at least 8 to 10 word problems.
How will Collegedunia's NCERT Class 12 Maths Chapter 12 Linear Programming PDF Help You?
The Collegedunia mirror gives you the official chapter file with three practical advantages:
- Direct, single-click download of the 2026-27 print, no NCERT site work through
- The page number on the Linear Programming PDF Class 12 matches the page number teachers quote in class
- Paired with our matching Solutions, Notes and Formula Sheet pages so you can switch resources without re-searching
For students who study from a printed school copy, the Linear Programming PDF Class 12 is the cleanest digital reference for cross-checking question numbers, since CBSE marking schemes always quote the NCERT exercise number.
Linear Programming PDF Class 12: Key Definitions to Pull from the Book
While reading the Linear Programming PDF Class 12 for the first time, mark these definitions, all of which appear inside Section 12.2 and are tested verbatim in board papers.
| Term | Definition (NCERT phrasing) |
|---|---|
| Linear Programming Problem | A problem that seeks to optimise (maximise or minimise) a linear function of two variables subject to certain conditions in the form of linear inequalities. |
| Objective function | The linear function Z = ax + by , where a, b are constants, that has to be maximised or minimised. |
| Constraints | The system of linear inequalities (and the non-negative restrictions x ≥ 0, y ≥ 0 ) that limit the values of the variables. |
| Feasible region | The common region determined by all the constraints including the non-negative restrictions. Points inside or on the boundary are feasible solutions. |
| best solution | A feasible solution that gives the maximum or minimum value of the objective function. |
| Corner-point method | Theorem-based procedure: the best value of Z occurs at a corner (vertex) of the feasible region. |
Every CBSE long-answer LPP question follows the same five-step routine: identify variables, write the objective, write the constraints, draw the feasible region, evaluate Z at each corner.
NCERT Class 12 Maths Chapter 12 Question Types You Will Find
The 2026-27 print covers four families of word problems. Each appears at least once in the Miscellaneous Exercise.
- Manufacturing problems: two products, limited machine hours, profit per unit, find production mix.
- Diet problems: two foods, minimum daily nutrients, minimise total cost of the diet.
- Transportation problems: ship goods from factories to godowns, minimise the total shipping cost.
- Allocation problems: split available resources (land, labour, capital) between two activities to maximise return.
All four families use the same corner-point method, so once one solved example is internalised, the remaining word problems become a translation exercise.
NCERT Class 12 Maths PDF Linear Programming: Suggested Reading Order
Open the Linear Programming PDF Class 12 and follow this sequence for the cleanest first pass.
- Read Section 12.1 (Introduction) and one of the everyday examples it opens with.
- Read Section 12.2 fully, paying attention to the two theorems that justify the corner-point method.
- Work through the first three solved examples in Section 12.3 (one manufacturing, one diet, one transportation).
- Attempt Exercise 12.1 questions 1 to 6 on paper, then check against Collegedunia's Linear Programming Class 12 NCERT Solutions.
- Move to the Miscellaneous Exercise. Solve at least 8 of the 11 questions before the next test.
- Finish with the Summary box and the Linear Programming Formula Sheet for last-minute revision.
Full topic notes: Class 12 Maths Linear Programming Notes
Linear Programming Class 12 PDF: CBSE Board Exam Relevance
The chapter contributes a single long-answer question to every CBSE Class 12 Maths paper, usually worth 5 marks (occasionally 6 in years where Section E carries a longer LPP).
| Year | Question stem | Marks |
|---|---|---|
| 2025 | Manufacturing LPP (two products, machine and labour constraints) | 5 |
| 2024 | Diet problem with two food items | 5 |
| 2023 | Transportation problem from two depots to three retail outlets | 5 |
| 2022 | Allocation problem with bounded feasible region | 5 |
The pattern is stable: a single word problem solved by graphing the feasible region and applying the corner-point method. Practising 8 to 10 such problems from the Miscellaneous Exercise covers the full question bank.
NCERT Class 12 Maths: All Chapters NCERT Book PDF
Use the full set of Class 12 Maths chapter PDFs alongside Chapter 12.
| Chapter | NCERT Book PDF |
|---|---|
| Chapter 1 | Relations and Functions NCERT Book PDF |
| Chapter 2 | Inverse Trigonometric Functions NCERT Book PDF |
| Chapter 3 | Matrices NCERT Book PDF |
| Chapter 4 | Determinants NCERT Book PDF |
| Chapter 5 | Continuity and Differentiability NCERT Book PDF |
| Chapter 6 | Application of Derivatives NCERT Book PDF |
| Chapter 7 | Integrals NCERT Book PDF |
| Chapter 8 | Application of Integrals NCERT Book PDF |
| Chapter 9 | Differential Equations NCERT Book PDF |
| Chapter 10 | Vector Algebra NCERT Book PDF |
| Chapter 11 | Three Dimensional Geometry NCERT Book PDF |
| Chapter 13 | Probability NCERT Book PDF |
Linear Programming PDF Class 12 - Frequently Asked Questions
Linear Programming PDF Class 12: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Exercise-wise Breakdown of the Linear Programming Chapter
The Linear Programming chapter splits into 1 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into 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 the PDF PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format 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 linear programming class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
- NCERT-faithful: Every definition, theorem and exercise on the linear programming class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The this chapter pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The linear programming class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: These notes solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this this Class 12 page pdf - only the exercise numbers differ.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the linear programming class 12 formulas pdf), and the full the resource pdf on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Linear Programming Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Linear Programming chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
Walking through one example of each template before the exam covers most of the predictable linear programming class 12 important questions you will see on board day.
- the chapter notes previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The linear programming class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching the PDF extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
The table below maps the dominant question type asked from the Linear Programming chapter across recent CBSE Class 12 Maths boards:
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
The full linear programming class 12 important questions with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.
How the Linear Programming Notes Pair with NCERT Solutions and the Formula Sheet
The Linear Programming Class 12 notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
| Resource | Use it for | When |
|---|---|---|
| Linear Programming Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| linear programming class 12 ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| linear programming class 12 formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
- The linear programming class 12 ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The this chapter solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The linear programming class 12 formulas reference sheet is the same A4 file students sometimes refer to as these notes all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board this Class 12 page textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 linear programming solutions, class 12 linear programming ncert solutions, ncert class 12 linear programming solutions - the same files cover the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:
| Reference | How it maps to the resource |
|---|---|
| RD Sharma Class 12 Linear Programming | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Linear Programming | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo the chapter notes | Free online walkthroughs; useful for video-style learning. |
| Shaalaa the PDF solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board this chapter textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Linear Programming | Advanced problems for JEE Main/JEE Advanced preparation. |
How to Use the Linear Programming Notes Page Most Effectively
The recommended study plan for these notes chapter splits across three sittings. The table below outlines what to do in each:
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use the linear programming class 12 ncert solutions PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
For students preparing for both CBSE board and JEE Main:
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The linear programming class 12 important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
Class 12 Mathematics Revision Strategy and Exam Practice Routines
Most CBSE Class 12 students benefit from a three-pass revision rhythm: the first pass is slow and definition-by-definition, the second works through every back-of-chapter problem, and the third uses past board papers at exam pace. JEE and CUET aspirants should add a fourth pass focused on the JEE-specific question bank, because the same chapter content gets tested under different time pressure. Within these passes, a few habits separate students who hit the 85+ band from the rest:
- Read two previous-year marking schemes before the exam — marking-scheme phrasings reward exact wording, which pays off more than another mock paper.
- Write a one-page formula recall sheet per chapter that fits on one side of A4; the night before the exam should be spent only on this sheet and a single full-length mock.
- Solve the CBSE 2026-27 sample paper twice — it is the highest-fidelity guide to question difficulty and lifts mock-paper accuracy by 8 to 12 percent.
- Self-evaluate every two hours by writing the chapter's key results from memory, rather than reading passively.
- Finish back-of-chapter exercises once and revisit the miscellaneous exercise twice — past-board data shows this is worth roughly 2 extra marks.
Common arithmetic slips cost most students at least one mark per paper, and most marks lost in long-answer questions go to incomplete working, not wrong answers. Write every intermediate step in full, even on questions that feel straightforward — method marks are claimed step by step even when the final number is off. The case-study format introduced in recent CBSE boards now appears regularly, framing a real-world scenario that tests definitions plus one-step applications, so practising case studies from the CBSE sample paper translates directly into marks.
Time allocation in the last fortnight matters most. Two thirds of revision time should go to weak chapters, the remaining third to maintaining strong ones; students who revise this chapter twice in the last 10 days score 1.5 to 2 marks higher on past boards. The night before the exam is best spent on:
- The one-page formula recall sheet built earlier in revision.
- A single full-length mock paper at exam timing.
- Avoid learning any new material the night before — sleep matters more.
Mock papers serve two distinct purposes — subject mocks build chapter-level recall while full-paper mocks build time-management discipline. Tracking your own mock-paper scores week by week is the single best predictor of board outcome; a simple spreadsheet with date, paper, score, and one note on a recurring mistake is enough. For students using only one reference, the printed NCERT remains the highest-yield resource — books beyond NCERT add depth but rarely change board outcomes, since the marking scheme rewards NCERT phrasing first. Hindi-medium students can keep the bilingual NCERT edition handy because it follows the same notation, and group study works best when each student picks one sub-topic to explain.
Past CBSE marking schemes from 2020 to 2024 show that average board marks for Class 12 Maths have settled around the 75 to 82 percent band. Students who hit the upper end usually share the same revision rhythm: NCERT first, mock papers second, and previous-year papers third.
Frequently Asked Questions
Ques. What is the Linear Programming PDF Class 12 from NCERT?
Ans. It is the Chapter 12 file from the official NCERT Class 12 Mathematics Part-II textbook for session 2026-27. It covers the introduction, the mathematical formulation of a linear programming problem, the corner-point method, four standard problem families, Exercise 12.1 and the Miscellaneous Exercise.
Ques. How many pages and exercises are there in Class 12 Maths Chapter 12 Linear Programming?
Ans. The chapter runs across about 15 pages, making it the shortest chapter in Class 12 Maths Part-II. It contains 3 numbered sections, 2 exercises (Exercise 12.1 with 10 questions and a Miscellaneous Exercise with 11 questions) and roughly 10 solved examples.
Ques. Is Linear Programming Chapter 12 in the 2026-27 NCERT syllabus?
Ans. Yes. The chapter is fully retained in the current 2026-27 syllabus and carries 5 marks under Unit V: Linear Programming. No topic has been dropped from this chapter in the latest edition.
Ques. What is a feasible region in this chapter?
Ans. The feasible region is the common region in the plane determined by all the linear constraints and the non-negative restrictions x ≥ 0, y ≥ 0 .
Every point inside or on the boundary of this region is a feasible solution to the LPP, and the best value of the objective function Z = ax + by occurs at one of its corners.
Ques. How many marks does Linear Programming carry in CBSE Class 12 Board exam?
Ans. Under the 2026-27 CBSE rationalised pattern, Unit V: Linear Programming carries 5 marks. The board paper carries a single long-answer LPP word problem (5 to 6 marks) every year.
Ques. Where can I download these notes PDF for free?
Ans. The official NCERT chapter PDF is hosted free on the NCERT site, and Collegedunia mirrors it on the download card above so you can grab the 2026-27 print in one click without work through the NCERT portal.
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