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Class XII Mathematics
Chapter 11: Linear Programming
Standard NCERT & CBSE aligned study curriculum. Master concepts, track accuracy, revise weak areas, and challenge yourself with 9 customized practice modes.
Syllabus Sections
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Chapter Overview
Welcome to Class XII Mathematics: Linear Programming. This chapter forms a core structural component of the math syllabus, designed to build analytical rigor and key formula models.
Use the detailed subtopic guide below to review standard definitions, key mathematical rules, and study guidelines.
Prerequisite Concepts
Linear InequalitiesLinear Equations in Two Variables
About This Chapter
This comprehensive study guide for Linear Programming is designed for Class XII students following the CBSE and NCERT Mathematics curriculum. It covers 4 key subtopics including Linear Programming problems mathematical formulation, Graphical method for solving LP in two variables, Feasible and infeasible boundary regions, and 1 more essential concepts. Whether you are preparing for school examinations, CBSE board exams, or competitive tests, this resource provides everything you need to build a strong conceptual foundation and achieve mastery.
The chapter includes 1 key formulas and equations, 1 fully worked step-by-step example problems, interactive practice exercises across 9 difficulty categories, timed mock quizzes, and downloadable worksheets. Each topic is explained with detailed concept definitions, mathematical representations, and expert study guidelines to help you understand not just the "how" but the "why" behind every formula and method.
Mathematics is a subject that rewards consistent practice and conceptual clarity over rote memorization. As you work through this chapter on Linear Programming, focus on understanding the underlying principles first, then gradually increase problem difficulty. Use the practice sections to identify and strengthen weak areas, and refer to the common mistakes section to avoid the pitfalls that most students encounter.
What You'll Learn in This Chapter
By the end of studying Linear Programming for Class XII, you will have developed proficiency in the following learning outcomes as outlined by the NCERT syllabus:
Formulate constraint systems for profit-maximization LP problems.
Graph constraint lines and shade feasible overlap zones.
Evaluate corner points of bounded regions to locate optimal objective targets.
Prerequisites for This Chapter
Before studying Linear Programming, make sure you are comfortable with the following prerequisite concepts. A strong foundation in these areas will help you understand new topics faster and solve problems more confidently:
Linear InequalitiesLinear Equations in Two Variables
If any of these prerequisites feel unfamiliar, consider reviewing them first using the Related Chapters section at the bottom of this page. Building a solid base ensures you can tackle Linear Programming with full confidence.
Real-World Applications of Linear Programming
Students often wonder “Where will I use Linear Programmingin real life?” The answer is: everywhere. The mathematical concepts you learn in this chapter have practical applications across science, engineering, technology, medicine, finance, and everyday problem-solving. Here are some notable examples:
Manufacturing Optimization
Factories maximize output or minimize costs by solving linear programming problems with constraints on labor, materials, and machine time.
Transportation & Logistics
Shipping companies minimize delivery costs by optimizing routes and vehicle loads using linear programming algorithms.
Diet & Nutrition Planning
Dietitians create minimum-cost meal plans that satisfy all nutritional requirements using LP constraint optimization.
Workforce Scheduling
Airlines and hospitals optimize staff schedules to ensure coverage while minimizing overtime costs using LP models.
Understanding the real-world relevance of Linear Programming not only makes learning more engaging but also helps you appreciate how mathematical thinking is a superpower that opens doors in virtually every career path — from engineering and medicine to finance and technology.
Study Tips for Linear Programming
Follow these expert study strategies to maximize your understanding and exam performance in this chapter. These tips are specifically tailored for the type of content covered in Linear Programming:
✏️
Practice Step-by-Step
Write out every intermediate step when solving problems. Skipping steps is the most common source of errors in calculation-heavy chapters. Build speed only after achieving consistent accuracy.
🔄
Verify by Back-Substitution
After finding your answer, substitute it back into the original equation to verify correctness. This simple habit catches most arithmetic and sign errors before they cost you marks.
📝
Maintain an Error Log
Keep a dedicated notebook of mistakes you make during practice. Review it weekly to identify patterns — you will notice the same types of errors recurring and can actively work to eliminate them.
Pro Tip: Consistency beats intensity. Studying Linear Programming for 30 minutes daily is far more effective than cramming for 5 hours before the exam. Use the practice sections below to build muscle memory through regular problem-solving.
Detailed Topic Breakdown
Detailed Subtopics Study Guide
Review detailed conceptual explanations, mathematical equations, and guidelines for each subtopic in this chapter:
1Linear Programming problems mathematical formulation
Concept Explanation
Formulating a Linear Programming Problem (LPP) involves defining decision variables, constructing a linear objective function to maximize or minimize, and setting up linear inequalities representing constraints.
Mathematical Representation
\text{Max/Min } Z = c_1 x + c_2 y \quad \text{subject to } a_{ij} x_j \le b_i \, \text{ and } \, x_j \ge 0
Study Guideline: Clearly define your decision variables (x and y) first, then write down the constraints from the problem's physical limitations.
2Graphical method for solving LP in two variables
Concept Explanation
The graphical method solves LPPs by plotting constraints on a coordinate grid, shading the feasible region, and finding the optimal vertex (corner point) using the Corner Point Theorem.
Mathematical Representation
\text{Optimal solution occurs at one of the corner points of the feasible region}
Study Guideline: The Corner Point Theorem guarantees that the maximum or minimum value of the objective function always lies at one of the vertices of the feasible region.
3Feasible and infeasible boundary regions
Concept Explanation
The feasible region is the common region determined by all constraints, including non-negativity constraints. If no common region satisfies all constraints simultaneously, the problem is infeasible.
Mathematical Representation
\text{Feasible Region } \neq \emptyset \implies \text{Solutions exist}; \quad \text{Feasible Region } = \emptyset \implies \text{Infeasible}
Study Guideline: An unbounded feasible region may have a minimum but might not have a maximum value.
4Optimal corner point solutions
Concept Explanation
To find the optimal solution, identify all vertices (corner points) of the feasible region, calculate the value of the objective function Z at each vertex, and select the maximum or minimum value.
Mathematical Representation
Z_i = c_1 x_i + c_2 y_i \implies \text{Choose max/min } Z_i
Study Guideline: Find the corner points by solving the intersecting boundary line equations simultaneously.
Learning Outcomes
By the end of this chapter, students will be able to master and solve questions on these outcomes:
Formulate constraint systems for profit-maximization LP problems.
Graph constraint lines and shade feasible overlap zones.
Evaluate corner points of bounded regions to locate optimal objective targets.
Sub-Topics Directory
Linear Programming problems mathematical formulationGraphical method for solving LP in two variablesFeasible and infeasible boundary regionsOptimal corner point solutions
Key Formulas & Equations
Click Copy to duplicateObjective Function
Z = ax + by
Target linear function optimized under boundary constraints.
Related Formulas from Adjacent Chapters
Line Vector Equation
\vec{r} = \vec{a} + \lambda \vec{b}
[From Three Dimensional Geometry] Vector form of line passing through a parallel to b.
Shortest Distance Skew Lines
d = \frac{|(\vec{b}_1 \times \vec{b}_2) \cdot (\vec{a}_2 - \vec{a}_1)|}{|\vec{b}_1 \times \vec{b}_2|}
[From Three Dimensional Geometry] Distance between skew lines.
Conditional Probability
P(A|B) = \frac{P(A \cap B)}{P(B)}
[From Probability] Probability of event A occurring given that event B has occurred.
Bayes Theorem
P(E_i|A) = \frac{P(E_i)P(A|E_i)}{\sum P(E_j)P(A|E_j)}
[From Probability] Inverse conditional probability updates.
Interactive Problem Solver
**Optimal corner point solver**
Evaluate Z = 4x + y at corner points of feasible region: (0,0), (0,30), (20,0). Find maximum Z.
Scientific Calculator Active: A custom solver for the specific subtopics of this chapter is currently in development. In the meantime, please use our comprehensive Scientific Functions Calculator below to verify your study equations.
Scientific Functions Calculator
Solve trigonometric, exponential, logarithmic, and root functions for standard algebraic values.
Adjust Inputs
45
Calculated Results
Function Output
0.71
Audit & Export Center
NEXPROTOOLS COMPUTATIONAL REPORT
Deterministic Mathematical Simulation Engine • Verified Calculations
Date: 9/1/2026
Time: 9:27:41 AM
Scientific Functions Calculator
Solve trigonometric, exponential, logarithmic, and root functions for standard algebraic values.
1. Input Parameters
| Parameter | Value | Unit |
|---|---|---|
| Value (X) | 45 | — |
| Function | sin | — |
2. Calculated Results
| Metric | Calculated Output |
|---|---|
| Function Output | 0.707 |
Generated by NexProTools (www.nexprotools.com)
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Student Solver & Visualizer Guide
Real-time calculationsStep-by-step solving
1. Collect input parameters:
- operand1 = 45
- operation = sin
2. Feed parameters through standard algorithmic calculation methods.
3. Compile calculated results:
- calcResult = 0.7071
Student-friendly explanations
"This math utility processes input parameters using standard algebraic algorithms. By substituting variables into target formulas, the engine solves equations and computes results instantly with high arithmetic precision!"
Visual explanations
MATH SOLVER RUNNING: [Inputs] ──► [Mathematical Formula] ──► [Outputs] Processed elements successfully.
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Easy Practice
Simple algebraic checks with hints and solutions enabled.
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Medium Practice
Standard curriculum queries matching board exam templates.
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Hard Practice
Complex math application questions with numeric inputs.
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Timed Chapter Quiz
5 mixed questions with a ticking clock to evaluate speed.
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Prerequisite & Mixed Revision
Combines prerequisite concepts and related chapter formulas.
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Previous Year style (Board Exam)
Original questions from previous year CBSE board exams.
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Olympiad & JEE Challenge
High-order logical problems matching Olympiad standards.
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Class XII Quiz Challenge
Test your math skills with this 5-question chapter mock exam. A live ticking clock will monitor your performance.
Class XII CBSE Board Practice Questions
Review and solve 10 real Board paper questions from the 16 mock sample sets matching Linear Programming.
Filter by Section:
Sample Paper 03 | Question 29Section C (3 Marks)
Minimize and maximize Z = 600x + 400y Subject to the constraints: x + 2y ≤ 12; 2x + y ≤ 12; 4x + 5y ≤ 20; x≥ 0; y ≥ 0 by graphical method
Sample Paper 06 | Question 12Section A (1 Mark)
Solution of LPP To maximise Z = 4x + 8y subject to constraints : 2x + y ≤ 30, x + 2y ≤ 24, x ≥ 3, y ≤ 9, y ≥ 0 is
Sample Paper 06 | Question 26Section C (3 Marks)
Solve the following problem graphically: Minimise and Maximise Z = 3x + 9y subject to the constraints: x + 3y ≤ 60; x + y ≥ 10; x ≤ y; x ≥ 0, y ≥ 0
Sample Paper 07 | Question 35Section D (5 Marks)
Maximise Z = 8x + 9y subject to the constraints given below : 2x + 3y ≤ 6; 3x - 2y ≤ 6; y ≤ 1; x, y ≥ 0
Sample Paper 08 | Question 29Section C (3 Marks)
The feasible region of a ∠PR is given as follows: (i) Write the constraints with respect to the above in terms of x and y. (ii) Find the coordinate of B and C and maximize, z = x + y.
Sample Paper 09 | Question 13Section A (1 Mark)
Feasible region is the set of points which satisfy
Sample Paper 12 | Question 10Section A (1 Mark)
Feasible region (shaded) for a LPP is shown in the given figure. The maximum value of the Z = 0.4x + y is
Sample Paper 13 | Question 31Section C (3 Marks)
Minimise Z = 13x - 15y subject to the constraints x + y ≤ 7, 2x - 3y + 6 ≥ 0, x ≥ 0 and y ≥ 0.
Sample Paper 14 | Question 33Section D (5 Marks)
Solve the following LPP graphically: Maximise Z = 3x + 4y Subject to x + y ≤ 4, x ≥ 0 and y ≥ 0.
Sample Paper 15 | Question 35Section D (5 Marks)
Minimize and maximize Z = 5x + 2y subject to the following constraints: x - 2y ≤ 2, 3x + 2y ≤ 12, -3x + 2y ≤ 3, x ≥ 0, y ≥ 0
Critical Mistakes to Avoid
Recognizing and eliminating common errors is a key step to scoring high marks in school and board exams:
!Order of Operations (BODMAS)
Solving additions or subtractions before evaluating parentheses, exponents, divisions or multiplications.
!Improper Algebraic Isolation
Doing addition/subtraction to only one side of the equation when rearranging terms.
!Decimal Place Errors
Misplacing the decimal dot when dividing by factors of ten.
Revision Summary Cards
1Formula Memorization
Commit core formulas to memory and write them down before solving problems.
2Units Consistency
Verify that all physical parameters use compatible measurement units.
3Calculator Check
Use our interactive calculator solvers to verify manually computed decimal answers.
4Prerequisites Checked
Ensure all prerequisite concepts are fully understood before working on advanced calculations.
5Step-by-Step Checking
Always check equations step-by-step to avoid simple sign and arithmetic errors.
Frequently Asked Questions
Resolve equations using alternative methods (e.g., check quadratic roots by plugging them back into ax² + bx + c), verify decimals using scientific solvers, and review step-by-step worked calculations.
Trigonometry is used in architecture, mechanics and navigation. Quadratic equations model rocket trajectories and profit boundaries. Basic arithmetic and fractions are applied in accounting, budgeting and lab chemistry.
Recommended Study Syllabus
Check out these related chapters and pre-requisites to strengthen your analytical foundation:
Three Dimensional Geometry
[Adjacent Chapter] Convert line equations between vector and Cartesian formats.
Read Guide
Probability
[Adjacent Chapter] Calculate conditional probabilities.
Read Guide
Relations and Functions
Prove if a relation is reflexive, symmetric, and transitive, classifying it as equivalence.
Read Guide
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Select any key syllabus topic below to open its detailed learning guide, worked models, and study guidelines.
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Print-Ready Math Worksheet
Generate and print a beautiful chapter worksheet complete with key equations, problems, and dedicated writing spaces.
Worksheet Details:
- • Title: Class XII Math: Linear Programming Printable Sheet
- • Format: Print Optimized CSS stylesheet
- • Sections: Formulas checklist, 5 assessment questions, worked keys
NexPro Education Worksheet
Class XII Mathematics: Linear Programming
Name: ______________________Date: __________________Score: ______ / 5
Section I: Key Formulas
Objective Function
Z = ax + by
Section II: Answer the Questions
1. Minimize and maximize Z = 600x + 400y Subject to the constraints: x + 2y ≤ 12; 2x + y ≤ 12; 4x + 5y ≤ 20; x≥ 0; y ≥ 0 by graphical method
Solve and write the step-by-step solution in the workspace below.
2. Solve the following problem graphically: Minimise and Maximise Z = 3x + 9y subject to the constraints: x + 3y ≤ 60; x + y ≥ 10; x ≤ y; x ≥ 0, y ≥ 0
Solve and write the step-by-step solution in the workspace below.
3. Maximise Z = 8x + 9y subject to the constraints given below : 2x + 3y ≤ 6; 3x - 2y ≤ 6; y ≤ 1; x, y ≥ 0
Solve and write the step-by-step solution in the workspace below.
4. The feasible region of a ∠PR is given as follows: (i) Write the constraints with respect to the above in terms of x and y. (ii) Find the coordinate of B and C and maximize, z = x + y.
Solve and write the step-by-step solution in the workspace below.
5. Minimise Z = 13x - 15y subject to the constraints x + y ≤ 7, 2x - 3y + 6 ≥ 0, x ≥ 0 and y ≥ 0.
Solve and write the step-by-step solution in the workspace below.