Linear Programming

part 1

Introduction to LPP

• Introduction to linear programming problems
• Objective function, constraints, feasible region
• Corner point theorem

part 2

Graphical Method

• Graphical method of solving LPP
• Bounded and unbounded feasible regions
• Multiple optimal solutions

part 3

Applications

• Diet problems
• Manufacturing problems
• Transportation problems

Corner point theorem

Optimal Z always occurs at a corner (vertex) of feasible region

★ board favourite

Objective function

Maximise/Minimise $Z = ax + by$

★ frequently asked

Feasible region satisfies all constraints

Feasible region = intersection of all constraint half-planes

★ frequently asked

Unbounded region: check if optimal exists

Draw half-plane $ax+by > M$ (or < m); if no common point, optimal exists

★ frequently asked

Minimise Z = 3x + 5y subject to the constraints x + 3y ≥ 3, x + y ≥ 2, x ≥ 0, y ≥ 0. Solve graphically and state whether the minimum value exists.

The corner points of the feasible region of an LPP are (0, 0), (0, 8), (5, 4) and (6, 0). If Z = 4x + 3y, find the point at which Z is maximum and its maximum value.

Define the term 'feasible region' in a linear programming problem.

A manufacturer produces two products A and B. Each unit of A requires 2 hours on machine I and 1 hour on machine II; each unit of B requires 1 hour on machine I and 3 hours on machine II. Machine I is available for 40 hours and machine II for 60 hours per week. Profits are ₹30 per unit of A and ₹20 per unit of B. Formulate the LPP and solve graphically to maximise profit.

Solve the following LPP graphically: Maximise Z = 3x + 9y subject to the constraints x + 3y ≤ 60, x + y ≥ 10, x ≤ y, x ≥ 0, y ≥ 0.

Define the term 'objective function' in a linear programming problem.

Maximise Z = 3x + 4y subject to x + y ≤ 4, x ≥ 0, y ≥ 0. Solve graphically and state whether the feasible region is bounded.

Maximise Z = 5x + 3y subject to the constraints 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0. Solve graphically.

In a linear programming problem, what is meant by the term 'constraints'?

How many marks is Linear Programming worth in the Class 12 board exam?

Around 5 marks of the 80-mark CBSE Maths theory paper, based on the official unit-wise weightage. Graph the feasible region neatly — corner point method.

Is Linear Programming easy or hard?

It's rated easy — one of the easier chapters in Class 12 Maths. Most students need about 2 hours to cover it well.

What are the important topics in Linear Programming?

The chapter covers 7 NCERT topics in 3 areas: Introduction to LPP; Graphical Method; Applications.

What questions come from Linear Programming in board exams?

Between 2019–2024, CBSE board papers asked questions from this chapter on Minimisation LPP, Corner Point Method, Feasible Region, Word Problem — 1- to 5-mark questions.