NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming
Linear Programming is the chapter where Class 12 Maths finally starts to look like something you might actually use outside an exam hall, deciding how a factory should allocate raw material, or how a farmer should split land between two crops to earn the most profit. Myclass24 has built these NCERT Solutions for Class 12 Maths Chapter 12 around that practical spirit, walking through every word problem the way you would actually think it through on paper: read the situation, define the variables, write the constraints, then solve. The aim is to make optimisation problems feel like logical puzzles rather than abstract algebra.
Summary of Chapter 12 Linear Programming
Linear Programming-Introduction- Solution of inequalities using the graphical approach, formation of LPP optional, Solution of LPP.
Find the PDF of NCERT Solutions for Class 12 Maths Chapter 12
Many students ask for a downloadable version of this chapter specifically because the graphical method involves drawing feasible regions, and having a clean reference copy to compare your own graph against is genuinely useful while practising. The Myclass24 PDF for this chapter includes properly labelled feasible region sketches alongside the corner point evaluations, so you can check not just your final answer but whether your shaded region itself was drawn correctly, which is usually where the actual mistake hides.
Class | 12 |
Subject | Mathematics |
Chapter Number | 12 |
Chapter Name | Linear Programming |
Board | CBSE / NCERT |
Total Exercises | 2 (including Miscellaneous Exercise) |
Key Topics | Linear programming problem and its formulation, objective function, constraints, feasible region, optimal solution, graphical method of solution, corner point method |
Weightage | Typically 4 to 6 marks in the board exam, often as one full long-answer question |
Difficulty Level | Easy to moderate, mostly about careful graph reading |
Prerequisite Chapter | Basic coordinate geometry and inequalities from Class 11 |
Understanding Chapter 12: Where Marks Are Actually Won or Lost
This chapter has a reputation for being one of the more scoring topics in the Class 12 syllabus, and that reputation is fair, but only if a student respects the graphical method instead of rushing it. The single biggest reason marks are lost here is a poorly drawn or mislabelled feasible region. Myclass24's solutions insist on plotting every constraint line carefully and shading the correct side based on the inequality sign, because an exam evaluator is checking your diagram as closely as your final number. A correct objective function value attached to a sloppy or incorrect graph rarely gets full marks.
The second thing students consistently underestimate is the corner point method itself. It is tempting to assume the optimal value lies somewhere in the middle of the feasible region, but the corner point theorem guarantees that the maximum or minimum of the objective function, if it exists, always occurs at a vertex of the feasible region, never inside it. Our solutions explicitly list every vertex and substitute each one into the objective function in a table format, so you build the habit of checking all corner points rather than guessing which one looks right.
A subtler issue appears in unbounded feasible regions. Some problems have a feasible region that extends infinitely in one direction, and a maximum may not exist even though a minimum does, or vice versa. Myclass24's solutions flag this explicitly wherever it occurs in the chapter, since this is exactly the kind of conceptual trap that NCERT and CBSE both like to test through application-based questions. Recognising an unbounded region from the graph itself, before you even start substituting values, saves a lot of wasted calculation.
Finally, the word problems in this NCERT Solutions for Class 12 chapter, the diet problem, the manufacturing problem, the transportation-style problem, all reward a student who writes the constraints in plain language first before converting them into inequalities. Our step-by-step format always begins by restating what each variable represents in the context of the question, because a wrong constraint built on a misread sentence cannot be fixed by correct graphing later. Getting the setup right is, in this chapter more than most, the entire battle.
NCERT Solutions for Class 12 Maths Chapter 12 – Linear Programming FAQs
Linear Programming is a mathematical technique used to find the best possible solution to a problem involving limited resources and specific constraints. In this chapter, students learn how to maximize or minimize an objective function while satisfying given linear inequalities. NCERT exercises introduce practical situations related to production, profit, cost, and resource allocation. Understanding Linear Programming is important because it demonstrates how mathematics can be applied to real-life decision-making problems. The concepts learned in this chapter are widely used in business, economics, engineering, and management. A strong understanding of the topic helps students solve application-based questions confidently in examinations.
The objective function is the mathematical expression that needs to be maximized or minimized in a Linear Programming Problem. It represents the goal of the problem, such as maximizing profit, minimizing cost, or achieving the highest output. Students must first identify the objective function correctly before solving the problem. NCERT exercises often provide real-life situations where the objective function must be formed using the given information. Once the feasible region is determined, the objective function is evaluated at the corner points to obtain the optimal solution. Understanding this concept is essential because it forms the core of every Linear Programming Problem.
The feasible region is the set of all points that satisfy the given constraints of a Linear Programming Problem. To determine it, students graph the linear inequalities on a coordinate plane and identify the common region that satisfies all conditions simultaneously. NCERT exercises frequently include questions where the feasible region must be shaded and analysed before finding the optimal solution. Understanding how to construct the feasible region accurately is crucial because the final answer depends on it. Careful graphing and verification of inequalities help students avoid mistakes and solve Linear Programming problems more efficiently.
Corner points play a vital role in solving Linear Programming Problems because the optimal value of the objective function occurs at one of these points. After identifying the feasible region, students calculate the coordinates of all corner points and evaluate the objective function at each one. The highest or lowest value obtained determines the required solution. NCERT exercises provide several examples that demonstrate this principle. Understanding the significance of corner points helps students apply the graphical method correctly. Regular practice of finding and testing corner points improves accuracy and confidence in solving optimisation-based questions.
Students often make mistakes while graphing inequalities, which can result in an incorrect feasible region. Another common error is failing to identify all corner points before evaluating the objective function. Some students also substitute coordinates incorrectly or overlook the conditions given in the problem. NCERT exercises encourage a step-by-step approach to minimise these errors. Drawing graphs neatly, verifying constraints carefully, and checking calculations can significantly improve accuracy. Consistent practice helps students understand the logic behind optimisation problems and develop confidence in solving Linear Programming questions effectively during board examinations.