Questions
3 questions per paper
Difficulty
Easy
Importance
High yield — guaranteed marks
Overview
Linear Programming (LPP) is the optimization of a linear objective function subject to a set of linear equality or inequality constraints. It is a high-scoring section in the Class 12 mathematics curriculum, primarily testing your ability to translate word problems into mathematical models and solve them graphically. Mastery involves identifying the feasible region correctly to determine the optimal solution.
Formulation of LPP
Formulation involves converting real-world constraints into mathematical inequalities. You must identify decision variables, the objective function (Z = ax + by), and the subject-to constraints.
- Objective function represents cost, profit, or resources
- Constraints are linear inequalities like ax + by ≤ c
- Non-negativity constraints: x ≥ 0, y ≥ 0
- Always express constraints in standard form
Graphical Method
This method is used for problems with two variables by plotting constraint lines on a Cartesian plane. The intersection of these half-planes defines the solution space.
- Use test point (0,0) to determine inequality shading direction
- Convert inequalities to equations to find intercepts
- Intersection points of lines are potential vertices
- Solution space must satisfy all constraints simultaneously
Feasible Region and Corner Point Theorem
The feasible region is the common shaded area satisfying all constraints. The fundamental theorem states that the optimal solution occurs at a corner point of the bounded feasible region.
- Feasible region must be a convex polygon
- Corner Point Theorem: Optimal values are at vertices
- Bounded region yields unique max/min values
- Unbounded region may result in no solution
Formula Sheet
Z = ax + by (Objective Function)
ax + by ≤ c or ax + by ≥ c (Constraints)
x ≥ 0, y ≥ 0 (Non-negativity)
Exam Tip
Always prepare a neat table listing every corner point and its corresponding value of Z to avoid calculation errors during the final evaluation step.
Common Mistakes
- Failing to shade the region correctly by ignoring non-negativity constraints (x≥0, y≥0).
- Miscalculating coordinates of intersection points of the constraint lines.
- Assuming the optimal value always exists if the feasible region is unbounded.
More Revision Notes
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