Class 12 Maths - KARNATAKA
Linear Programming
The chapter 'Linear Programming' in Class 12 Mathematics bridges algebraic modeling and real-world decision-making. It deals with optimizing a linear objective function subject to a system of linear inequalities, known as constraints. Students will learn how to visually and analytically solve problems related to manufacturing, diet, and transportation to find maximum or minimum values. For the Karnataka (KSEEB) board exams, this chapter is high-scoring and straightforward, typically featuring a mandatory long-answer question that requires careful graphical plotting of feasible regions and correct identification of corner points.
Start Learning FreeKey Concepts
Linear Programming Problem (LPP)
A mathematical problem that aims to maximize or minimize a linear function subject to several linear constraints and non-negativity restrictions.
Objective Function
A linear function Z = ax + by whose maximum or minimum value is to be found under given conditions.
Constraints
Linear inequalities or equations representing limitations or restrictions on the decision variables x and y, usually including non-negativity constraints (x >= 0, y >= 0).
Feasible Region
The common region determined by all the given constraints, including the non-negativity restrictions, representing all acceptable solutions.
Corner Point Method
A method to solve LPPs graphically by evaluating the objective function at the corner points (vertices) of the feasible region.
Important Formulas
Board Exam Info
In the Karnataka (KSEEB) Class 12 Mathematics board examination, Linear Programming generally carries around 6 to 8 marks. The blueprint usually features one major 5-mark or 6-mark question requiring a complete graphical solution, and sometimes a 1-mark or 2-mark question on defining terms like 'feasible region' or 'objective function'.
Frequently Asked Questions
How do I know whether to shade towards the origin or away from it for an inequality?
Test the origin (0,0) in the inequality. If the resulting statement is true, shade the region containing the origin. If false, shade away from the origin.
Is the feasible region always bounded in a Linear Programming Problem?
No, the feasible region can be bounded or unbounded. However, for Class 12 board problems, most regions are bounded, though unbounded regions are tested using specific theorems.
What is the Corner Point Theorem?
It states that the optimal (maximum or minimum) value of the objective function always occurs at the corner points (vertices) of the feasible region.
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