Class 12 Mathematics — Chapter 6: LINEAR PROGRAMMING
Chapter 6: LINEAR PROGRAMMING is a chapter in Class 12 Mathematics (Part 2), part of the CBSE NCERT curriculum followed by over 25 million students across India. This chapter covers 7 topics including Introduction to Linear Programming Problems (LPP), Mathematical Formulation of LPP, Objective Function and Constraints. BrainWeave provides free AI-powered explanations — by voice or text, in Hindi or English — with no signup required.
What you'll learn
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▸Introduction to Linear Programming Problems (LPP)optimisation problemlinear programmingmaximiseminimisereal-life problems
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▸Mathematical Formulation of LPPCore conceptformulationword problemdecision variablesobjective functionconstraints
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▸Objective Function and ConstraintsCore conceptobjective functionlinear constraintsnon-negative restrictionsZ = ax + byinequalities
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▸Graphical Method: Feasible RegionCore conceptgraphical methodfeasible regionsolution regionplotting inequalitiescommon region
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▸Feasible and Infeasible Solutionsfeasible solutioninfeasible solutionboundaryfeasible regionconstraints
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▸Optimal Solution and the Corner Point MethodCore conceptoptimal solutioncorner point methodvertexmaximum valueminimum value
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▸Bounded and Unbounded Feasible Regionsbounded regionunbounded regionconvex polygonoptimal valuecorner point
Chapter Summary
Understand the concept of an optimisation problem, specifically a Linear Programming Problem (LPP), which involves maximizing or minimizing a linear objective function subject to linear constraints. Recognize real-world scenarios where LPP can be applied, such as maximizing profit or minimizing cost.
Learn to translate a real-world problem into a mathematical model by identifying decision variables, formulating a linear objective function (to be maximized or minimized), and establishing a set of linear inequalities as constraints, including non-negative constraints.
Define and identify the key components of an LPP. The objective function (Z = ax + by) is the linear function to be optimized. Constraints are the linear inequalities that limit the decision variables, including non-negative restrictions (x ≥ 0, y ≥ 0).
Learn the graphical method for solving LPPs by plotting the linear inequalities (constraints) on a graph. Identify the common shaded area, known as the feasible region, which represents all possible valid solutions to the problem.
Differentiate between a feasible solution, which is any point within or on the boundary of the feasible region, and an infeasible solution, which is any point outside this region. Understand that the optimal solution must be a feasible solution.
Understand the fundamental theorem stating that the optimal value (maximum or minimum) of the objective function for a bounded feasible region will always occur at one of its corner points (vertices). Learn to find the coordinates of all corner points and evaluate the objective function at each to determine the optimal solution.
Distinguish between a bounded feasible region, which is enclosed, and an unbounded one, which is not. Understand that for a bounded region, both a maximum and minimum value exist at corner points, while for an unbounded region, an optimal value may or may not exist.
Practice Questions from this Chapter
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- Show real-world optimization problems. Get Solution →
- Explain linear programming simply. Get Solution →
- List daily examples of constraints. Get Solution →
- What is the general class of problems that seek to maximise profit or minimise cost called? Get Solution →
- What is the linear function Z = ax + by, which has to be maximised or minimized, called? Get Solution →
- In a linear programming problem, what are the linear inequalities or restrictions on the variables called? Get Solution →
- The conditions x ≥ 0, y ≥ 0 in a linear programming problem are known as what? Get Solution →
- What is the common region determined by all the constraints, including non-negative constraints, of a linear programming problem called? Get Solution →
Did you know?
- 💡 Bees build hexagonal honeycomb cells for optimal storage of honey and pollen.
- 💡 Linear programming was secretly developed during World War II for military logistics.
- 💡 Your brain constantly optimizes decisions, even when choosing the fastest checkout line.
- 💡 Ancient Egyptians optimized pyramid construction by using basic geometry and resource planning.
- 💡 Traffic lights use optimization algorithms to minimize wait times and improve city flow.
Frequently Asked Questions
How many topics are covered in this chapter?
This chapter covers 7 key topics: Introduction to Linear Programming Problems (LPP), Mathematical Formulation of LPP, Objective Function and Constraints, Graphical Method: Feasible Region, Feasible and Infeasible Solutions, and more. The BrainWeave AI tutor explains each one with examples.
Is Chapter 6: LINEAR PROGRAMMING important for board exams?
Yes — Class 12 is a CBSE board exam year, and every NCERT chapter is part of the syllabus. Use BrainWeave's AI tutor to master this chapter, then practice with the auto-generated quizzes and mind maps.
Can I get NCERT solutions for this chapter in Hindi?
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