Board exam focus — Linear Programming (CBSE & HBSE)

Linear Programming carries 5-6 marks in CBSE as one 5-mark question (or LA). Students must formulate the LPP, solve graphically, find the feasible region, and optimise the objective function using the corner point method.

Formulation and Graphical Method

Linear Programming Problem (LPP)

Key terms:

• Decision variables: The unknowns (x, y, etc.)
• Objective function: Z = ax + by (to be maximised or minimised)
• Constraints: Linear inequalities involving the decision variables
• Non-negativity conditions: x ≥ 0, y ≥ 0
• Feasible region: Set of all points satisfying ALL constraints
• Feasible solution: Any point in the feasible region
• Optimal solution: The feasible solution that maximises/minimises the objective function

Steps to Solve LPP Graphically

1. Convert each constraint to a line equation (replace inequality with =)
2. Plot each constraint line on the coordinate plane
3. Identify the half-plane for each constraint (check with (0,0) or another point)
4. Find the feasible region = intersection of all half-planes (including x≥0, y≥0)
5. Identify the corner points (vertices) of the feasible region
6. Evaluate Z at each corner point
7. Report the maximum/minimum value of Z

Fundamental Theorem of Linear Programming

If an optimal solution exists, it occurs at a corner point (vertex) of the feasible region.

Special cases:

• Bounded region: Both maximum and minimum exist (both at corner points)
• Unbounded region: Maximum/minimum may or may not exist
• Minimum exists if the feasibility region is unbounded towards the objective function direction
• Infeasible: No feasible region → no solution
• Multiple optimal solutions: Optimal value achieved on an edge (all points on that edge are optimal)

CBSE Tip: Always shade the feasible region clearly and mark corner points. Evaluate Z at EVERY corner point — do not guess. For unbounded regions, check whether Z can become larger/smaller than the optimal value found.

Applications and Special Cases of LPP

LPP Formulation — Common Word Problems

Manufacturing problem: Company makes two products. Each requires machine time/labor. Resources are limited. Maximise profit.

Diet problem: Mix foods to meet minimum nutritional requirements at minimum cost.

Transportation problem: Minimise cost of shipping goods from sources to destinations.

Formulation Template

1. Identify decision variables: Let x = (units of product A), y = (units of product B)
2. Write objective function: Z = p₁x + p₂y (profits) or Z = c₁x + c₂y (costs)
3. Write constraints: Machine hours, raw material, demand etc.
4. Add non-negativity: x ≥ 0, y ≥ 0

Working With an Unbounded Feasible Region

Check for maximum: If the objective function line Z = k can be moved beyond the feasible region for arbitrarily large k, then there is NO maximum (objective is unbounded above).

Worked Example: Maximise Z = 10x + 15y subject to:

• 2x + y ≤ 40 (machine hours)
• x + 2y ≤ 50 (labor hours)
• x ≥ 0, y ≥ 0

Corner points:

• (0,0): Z=0
• (20,0): Z=200
• (10,20): Z = 100+300 = 400 ← Maximum
• (0,25): Z=375

Maximum Z = 400 at x=10, y=20.

Summary of Solution Types

CBSE Tip: The LPP 5-mark question follows a standard structure: formulate (1 mark) + draw graph (1 mark) + shade feasible region (1 mark) + corner points (1 mark) + optimal value (1 mark). Don't skip any step.

Frequently asked questions

Are these Linear Programming notes free?

Yes — the Linear Programming notes for Mathematics (Class 12) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Linear Programming notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Linear Programming chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Linear Programming.