Download the Linear Programming Formula below as a free PDF. The sheet contains every formula tested by CBSE in the 5-mark question block on Class 12 Mathematics Chapter 12 Linear Programming, plus the JEE Main extensions used in the same area. The Linear Programming Formula are structured for quick lookup.
| Snapshot | Value |
|---|---|
| CBSE Weightage | 5 marks, one long answer |
| Concepts to recall | Standard LPP form, corner-point method, iso-profit line, region rules |
| Sheet length | 5 to 7 pages, single-column Collegedunia format |
The chapter is procedure-heavy rather than formula-heavy, so this Linear Programming Formula is built around two boxed procedures, three classification rules, and one inline objective-function template. The full reference appears in the sections below.
This Formula Sheet is curated by Class 12 Maths experts at Collegedunia, mapped to the 2026-27 NCERT edition, and refined against the last five years of CBSE Board papers.
Class 12 Maths NCERT Formulas PDF Download Chapter 12 Linear Programming: Why a Procedure Sheet, Not a Formula List
The Linear Programming Formula address this in the same order as the NCERT textbook.
Unlike Integrals or Vector Algebra, Chapter 12 does not test memorised algebraic identities. Examiners check whether you can set up the LPP, plot the feasible region, and read the optimum off the corner points. The sheet therefore groups content into procedures and decision rules. More than 80% of board marks on the Linear Programming Formula come from the corner-point evaluation step.
Linear Programming Video Walkthrough
Source: Magnet Brains on YouTube
How the Linear Programming Formula on the Linear Programming Formula Help You
The Linear Programming Formula address this in the same order as the NCERT textbook.
This Collegedunia sheet replaces 20 pages of NCERT prose with a printable procedural reference. Every entry is mapped to a standard board-paper sub-task.
- Procedure-first layout orders content by task, not by formula: setup, plot, evaluate, conclude.
- Bounded vs unbounded region rules are isolated, since this is where most marks are lost.
- LPP type table classifies manufacturing, diet, and transportation problems with the constraint pattern each one uses.
- 2026-27 syllabus aligned with the current NCERT print; the dropped sub-topics from earlier editions are not included.
Bounded vs Unbounded Feasible Region: The Optimum-Existence Rule
The feasible region in a Class 12 LPP is either bounded (encloses a finite area) or unbounded (extends to infinity in some direction). The rule for declaring an optimum is different in each case, and CBSE has tested this distinction in 4 of the last 6 board papers.
| Region type | Maximum of Z | Minimum of Z |
|---|---|---|
| Bounded | Always exists, at one of the corner points. | Always exists, at one of the corner points. |
| Unbounded | Exists only if the open half-plane ax + by > M (where M is the largest corner value) has no point in common with the feasible region. | Exists only if the open half-plane ax + by < m (where m is the smallest corner value) has no point in common with the feasible region. |
For unbounded regions, skipping the open-half-plane check is the single most common reason a 5-mark LPP answer is cut to 3 marks.
Iso-Profit and Iso-Cost Line Method (Alternative to Corner-Point)
The iso-profit (or iso-cost) line method is a graphical alternative to the corner-point method. CBSE accepts either approach for full marks, but the corner-point method is faster and less error-prone in exam conditions.
| Step | Action |
|---|---|
| 1 | Draw the line ax + by = k for any convenient value of k. This is an iso-profit line (for maximisation) or an iso-cost line (for minimisation). |
| 2 | Move the line parallel to itself in the direction of increasing Z (for max) or decreasing Z (for min). |
| 3 | The last corner point the moving line touches before leaving the feasible region gives the optimum. |
Types of Linear Programming Problems in NCERT Class 12 Maths
NCERT Chapter 12 illustrates the corner-point procedure on three canonical problem types. The constraint pattern differs in each case, so recognising the type quickly tells you what the inequalities will look like.
| LPP type | Objective | Constraint pattern |
|---|---|---|
| Manufacturing | Maximise profit Z = p1 x + p2 y | Resource constraints of the form ai x + bi y ≤ ci (machine hours, raw material limits) |
| Diet | Minimise cost Z = c1 x + c2 y | Nutrient constraints of the form ai x + bi y ≥ ni (minimum protein, vitamin, calorie content) |
| Transportation | Minimise cost Z = ∑ cij xij | Supply and demand equalities of the form j xij = si and i xij = dj |
Recognition shortcut: if the stem mentions "profit", you are in a manufacturing LPP with ≤ constraints. If it mentions "cost" with minimum nutrient requirements, it is a diet LPP with ≥ constraints.
Class 12 Maths Chapter 12 PYQ-Linked Procedures
The table below tags each PYQ year with the procedure it tested. Use it to prioritise drill order before the board paper.
| Year | Marks | Procedure tested |
|---|---|---|
| CBSE 2025 | 5 | Manufacturing LPP, corner-point method, bounded region |
| CBSE 2024 | 5 | Diet LPP, corner-point method with unbounded-region check |
| CBSE 2023 | 5 | Manufacturing LPP, two-product profit maximisation |
| CBSE 2022 | 5 | Diet LPP with three constraints, minimum cost |
| CBSE 2021 | - | Term-format paper, LPP omitted from the syllabus subset |
Full year-wise PYQ map: NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming.
Common Slips Examiners Penalise in Class 12 LPP Answers
- Forgetting to write the non-negativity restrictions x ≥ 0, y ≥ 0 in the constraint list.
- Plotting the line but shading the wrong half-plane; always test the origin (0, 0) unless it lies on the line.
- Reading corner-point coordinates from the graph by eye, instead of solving the simultaneous equations exactly.
- Declaring the maximum of Z on an unbounded region without the open-half-plane check.
- Mixing up the inequality direction: profit maximisation uses ≤ constraints, diet minimisation uses ≥.
- Writing the corner-point table without labels, so examiners cannot tell which Z value belongs to which vertex.
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 Notes
- Class 12 Maths Chapter 12 Linear Programming Handwritten Notes
- Class 12 Maths Chapter 12 Linear Programming Exemplar Solutions
NCERT Formula Sheets for Class 12 Maths: All Chapters
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| Chapter 12 | Linear Programming Formula Sheet |
| Chapter | Formula Sheet |
|---|---|
| Chapter 1 | Relations and Functions Formula Sheet |
| Chapter 2 | Inverse Trigonometric Functions Formula Sheet |
| Chapter 3 | Matrices Formula Sheet |
| Chapter 4 | Determinants Formula Sheet |
| Chapter 5 | Continuity and Differentiability Formula Sheet |
| Chapter 6 | Application of Derivatives Formula Sheet |
| Chapter 7 | Integrals Formula Sheet |
| Chapter 8 | Application of Integrals Formula Sheet |
| Chapter 9 | Differential Equations Formula Sheet |
| Chapter 10 | Vector Algebra Formula Sheet |
| Chapter 11 | Three Dimensional Geometry Formula Sheet |
| Chapter 13 | Probability Formula Sheet |
Linear Programming Formula: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Student Feedback - 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.
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