# RD Sharma Class 12 Chapter 30 Solutions (Linear Programming)

RD Sharma Class 12 Maths Solutions for Chapter 30 ‘Linear Programming’ will teach you how to solve problems using the concepts of linear programming. This chapter includes definitions and mathematical formulation of linear programming problems. You will learn the graphical methods of solving linear programming problems, ISO-Profit or ISO Cost method, along with some exceptional cases.

RD Sharma Solutions for Class 12 Chapter 30 ‘Linear Programming’ contains a total of 3 exercises and 92 questions based on all the topics of the chapter. This chapter prepares you for attempting any and all questions of Linear Programming you can expect in competitive examinations like JEE and NEET as well as CBSE. RD Sharma Solutions has sufficient questions on linear programming to help you to fully acquire whatever is being taught in this chapter.

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## Some Topics Discussed in RD Sharma Class 12 Maths Solutions Chapter 30 Linear Programming

**General Linear Programming**

Let there be m linear inequalities or equations

Let there be n variables

Our purpose is to find non-negative values of n number of variables satisfying the m linear equalities or equations as well as minimizing or maximizing linear functions of variables.

Call these equations or inequalities as constraints. The function to be maximized or minimized is also known as the objective function that is either of maximization type or minimization type.

The general form of linear programming problem is to minimize (maximize)

Z = c1x1 + c2x2 +…+ cnxn …. (1)

Subjected to

a1x1 + a12x2 + ….. + a1nxn {≤, =, ≥}b1

a21x1 + a22x2 + ….+a2nxn {≤, =, ≥}b2

…. ….. …. …

am1x1 + am2x2+ … + amnxn. {≤, =, ≥}bm ……. (2)

and, x1, x2, x3,…. xn ≥ 0 … (3)

Where,

- X1, x2, x3,… xn be the variables whose value we will need to determine and can be known as decision variables
- Z is the linear function that should be either minimized or maximized and you may call it an objective function.
- The equations or inequalities in (2) can be called as constraints.
- The set of inequalities in (3) are the non-negativity restrictions set.
- bi where (i = 1, 2, …, m) is the availability or the requirement of the ith constraint.
- aij(i=1, 2,…,m; j = 1,2, …., n) are the technological coefficients or substitution coefficients.
- cj where j = 1, 2, …., n is the cost or profit to the objective function xj, jth variable as well as the row matrix C having the set of values {c1, c2, …., cn]. C is the profit matrix-vector.
- {≤, =, ≥} is an expression which means that only one sigh ≤, =, ≥ will be true for a particular constraint. However, the sign may vary from one constraint to the other.

**Fundamental Extreme Point Theorem**

An optimal solution of Linear Programming Problem, if it is there, will occur at one of the extreme corners (i.e. corner point) in the convex polygon of the feasible solutions set.

It is probable that the corner polygon’s two vertices will provide the optimal value if a given objective function. In this case, every point found on the line segment that joins these two vertices will provide the optimal values. This Linear Programming Problem will have infinitely many solutions. There may be a case where the Convex Polygon will be an empty set. In that case, Linear Programming Solution will have no solution. If you find the LPP region bounded i.e. the region is enclosed within a circle, you’ll see the objective function to have a maximum and a minimum value. These values will occur at a corner of the region.

### Discussion of Exercises in RD Sharma Class 12 Maths Solutions Chapter 30 Linear Programming

- Exercise 30.1 discusses questions on diet problems, transportation problems, optimal product line problems, etc.
- The exercise 30.2 discusses questions on graphical methods of solving linear programming problems, corner-point method, and ISO-profit or ISO cost method.
- Exercise 30.3 discusses questions on different types of linear programming problems.

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