Linear Programming Class 12 Notes - Class 12 Maths Ch 12

Download the Linear Programming Class 12 Notes as a free PDF. The this Class 12 page condense Class 12 Mathematics Chapter 12 Linear Programming into a compact, exam-ready text with theorems, proofs, formulae and solved examples set out in sequence. The notes cover graphical solution of linear programming problems.

• 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 plus a Miscellaneous set, so a tight revision sheet is enough to lock the the resource. Lose this question and you forfeit a full 5 marks from a topic with no derivation, no proof, and no surprise twist.

These Collegedunia notes are cross-verified against the 2026-27 NCERT Mathematics Part II textbook and the past five years of CBSE marking schemes for the graphical LPP question.

Also Check:

• Linear Programming Class 12 NCERT Solutions
• Linear Programming Class 12 Maths Formula Sheet
• Linear Programming Handwritten Notes PDF

Linear Programming Class 12 Notes: Topic-wise Weightage for CBSE Boards

Chapter 12 is single-question territory in the CBSE paper. The full 5 marks ride on one graphical LPP, so the weightage spread inside the chapter notes matters less than getting every sub-step of that one question right. The table below shows where examiners pull marks from inside the standard 5-marker.

Linear Programming Video Walkthrough

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 called decision variables. Inequalities of the form x ≥ 0, y ≥ 0 are called non-negative restrictions. Together, the objective function and the constraints form the LPP.

Three classical examples Class 12 students should recognise: a manufacturing problem (maximise profit per unit produced), a diet problem (minimise cost subject to nutritional minimums), and a transportation problem (minimise shipping cost across routes). The CBSE paper rotates between these three setups.

12.2 Graphical Method of Solving an LPP

Once the LPP is formulated, the graphical method follows five fixed steps. Plot each constraint as a line; shade the half-plane satisfying the inequality; the intersection of all shaded half-planes plus x ≥ 0, y ≥ 0 is the feasible region.

The feasible region is always a convex polygon (possibly unbounded). Identify its corner points (vertices) by solving pairs of constraint equations simultaneously.

The Corner-Point Theorem guarantees that the optimum of the objective function (if it exists) occurs at a corner point of the feasible region. So evaluate Z = ax + by at every corner, tabulate, and pick the maximum or minimum.

When the feasible region is unbounded, the optimum may not exist. Test by checking whether the open half-plane ax + by > M (for the candidate maximum M ) has any point in common with the feasible 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)

The chapter has no derivations, but six named definitions plus the Corner-Point Theorem account for every mark examiners deduct for "incomplete answer".

How will Collegedunia's Linear Programming Notes Help You?

These the PDF are built for the one-shot revision a Class 12 student needs the night before the Maths Board paper. The structure mirrors what examiners actually check on the 5-mark question.

• The full 5-step graphical workflow on a single page, in the exact order CBSE marking schemes follow.
• Every standard LPP type (manufacturing, diet, transportation, allocation) gets one solved template you can adapt to any worded variant.
• The Corner-Point Theorem boxed alongside the unbounded-region exception, so you handle both "maximum exists" and "no maximum exists" variants.
• Iso-profit and iso-cost line tips for visual cross-checking before you commit to the answer.

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 underlying mathematics is identical.

Full year-wise PYQ map: Linear Programming PYQ year map in NCERT Solutions.

Common Misconceptions in Linear Programming Class 12

• Forgetting non-negativity constraints: students plot the constraint lines but skip x ≥ 0, y ≥ 0 , turning a first-quadrant problem into a four-quadrant one. The feasible region then includes phantom corner points.
• Confusing maximisation with the topmost vertex: the optimum is whichever corner gives the largest Z value, not the highest y -coordinate. A high vertex with a small x coefficient can lose to a low vertex with a large x coefficient.
• Skipping the unbounded check: for unbounded feasible regions, simply tabulating Z at corners is not enough. The candidate optimum must be verified by checking the half-plane condition.
• Reading inequality direction wrong: 2x + 3y ≤ 12 shades the side below the line (toward origin if origin satisfies it). Shading the wrong side flips the feasible region and the answer.

Real-World Applications of Linear Programming

Linear Programming is the engine behind operations research. Modern uses students should be aware of:

• Airline crew scheduling: IndiGo and Air India use LP to assign pilots and cabin crew across daily flights while respecting duty-hour limits.
• Portfolio optimisation: mutual fund managers solve LPs to allocate capital across asset classes for a target return at minimum risk.
• Hospital diet planning: dietitians use the diet-problem formulation Class 12 covers, scaled up to 50+ food items and a dozen nutrient minimums.
• Logistics routing: Flipkart and Amazon India route delivery trucks using LP variants of the transportation problem.

NCERT Notes for Class 12 Maths: All Chapters

Use this index to jump to any other chapter's notes during your revision rotation.

Frequently Asked Questions on Linear Programming Class 12 Notes

this chapter: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.

Linear Programming Class 12 Notes - Quick Summary

• These notes cover every section of Class 12 Mathematics Chapter 12 Linear Programming, aligned to the 2026-27 NCERT print.
• The this Class 12 page include formal definitions, solved examples and end-of-section formula recap suitable for board and JEE Main preparation.
• The the resource are downloadable as a free PDF and follow the notation of the official NCERT textbook line for line.

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.

PDF Download Formats and Languages for the Linear Programming Chapter

The Linear Programming Class 12 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:

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 the resource 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: The linear programming class 12 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 the chapter notes 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 PDF 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:

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.

• this chapter 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 these notes 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:

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:

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 linear programming class 12 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 this Class 12 page all formulas - it lists every identity used in the chapter.
• State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board the resource 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:

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:

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.