CBSE Class 12 Maths: Linear Programming — Complete Notes 2026

LPP Terminology

• Objective function: linear function Z = ax + by to be maximised or minimised
• Constraints: linear inequalities that restrict the values of variables; non-negativity: x ≥ 0, y ≥ 0
• Feasible region: set of all points satisfying all constraints simultaneously; shown as shaded region on graph

Corner Point (Vertex) Method

• Optimal value of Z occurs at one of the corner points of the feasible region (convex polygon)
• Steps: plot constraint lines; shade feasible region; find corner points (solve simultaneous equations); evaluate Z at each
• Multiple optimal solutions: if two adjacent corner points give same optimal Z, all points on that edge are optimal

Types of LPP Feasible Regions

• Bounded feasible region: finite area; both max and min exist
• Unbounded feasible region: extends to infinity; max may exist (if Z increasing toward boundary) or may not
• Empty feasible region: no point satisfies all constraints; infeasible problem; LPP has no solution

Standard CBSE LPP Problems

• Manufacturing: max profit given two products, limited machines and materials; variables = quantities produced
• Diet problem: min cost satisfying nutritional requirements; variables = servings of different food items
• Transportation: optimise cost or time of routing goods; variables = units transported on each route

Solving Graphically — Step by Step

• Convert inequality to equation; find x and y intercepts; draw line; shade correct side (test (0,0))
• Find intersection of constraint lines using simultaneous equations; these are the corner points
• Evaluate Z at each corner point; state optimal solution and optimal value clearly

Special Cases

• Multiple optimal solutions: objective function parallel to one edge of feasible region; infinitely many solutions
• Unbounded solution: if feasible region unbounded, maximisation may have no finite solution
• CBSE: state 'the feasible region is unbounded, hence maximum may not exist' and check if interior region satisfies −Z condition

CBSE Board Focus

• Linear Programming: 5–6 marks; one full question in Board exam; graph and corner point method
• Label graph clearly: constraint lines, feasible region (shade or label), corner points with coordinates
• Write final answer: 'Z is maximised/minimised at (x, y) with value Z = …'