ML Aggarwal Linear Programming ISC Class-12 Maths Solutions Chapter-3 of Section-C. Step by step Solutions of ML Aggarwal ISCClass 12 Mathematics for Exercise Questions with Chapter Test. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

## ML Aggarwal Linear Programming ISC Class-12 Maths Solutions Chapter-3 of Section-C

Board | ISC |

Class | 12 |

Subject | Mathematics |

Cha-3 Sec-C | Linear Programming |

Session | 2024-25 |

Topics | Solutions of ML Aggarwal |

### Linear Programming

ML Aggarwal Linear Programming ISC Class-12 Maths Solutions Chapter-3 of Section-C

**Linear Programming Problem:** A linear programming problem is one in which we have to find optimal value (maximum or minimum) of a linear function of several variables (called objective function) subject to certain conditions that the variables are non-negative and satisfying by a set of linear inequalities with variables, are sometimes called division variables.

**Terms related to Linear Programming**

**Objective Function:** A linear function z = px + qy (p and q are constants) which has to be maximized or minimised, is called an objective function.

**Constraints:** The linear inequalities or equations or restrictions on the variables of the linear programming problem are called constraints. The conditions x ≥ 0, y ≥ 0 are called non-negative restrictions.

**Optimal Value:** The maximum or minimum value of an objective function is known as its optimal value.

**Optimization Problem:** A problem, which seeks to maximize or minimize a linear function subject to certain constraints as determined by a set of linear inequalities, is called an optimization problem.

**Feasible Region:** The common region determined by all the constraints including non-negative constraints x,y>0 of a linear programming problem is called the feasible region for the problem. The region other than the feasible region is called an infeasible region. The feasible region is always a convex polygon.

**Feasible Solutions:** Points within and on the boundary of the feasible region represent feasible solutions of the constraints. Any point outside the feasible region is called an infeasible solution.

**Optimal Feasible Solution:** Any point in the feasible region that gives the optimal value of the objective function is called the optimal feasible solution.

**Bounded and Unbounded Region:** A feasible region of a system of linear inequalities is said to be bounded, if it can be enclosed within a circle. Otherwise, it is called unbounded.

**Fundamental Theorems for Solving Linear Programming**

**Theorem 1:**Let R be the feasible region for a linear programming problem and let z = ax + by be the objective function. When z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities. This optimal value must occur at a corner point (vertex) of the feasible region.

Note: A corner point of a feasible region is a point in the region which is the intersection of two boundary lines.**Theorem 2:**Let R be the feasible region for a linear programming problem and let z = ax + by be the objective function. If R is bounded, then z has both a maximum and a minimum value on R and each of these recurs at a corner point of JR.

Note: Maximum or a minimum may not exist,- if the feasible region is unbounded.

**Steps for Applying Corner Point Method**

Find the feasible region of the linear programming problem and determine its corner points either by inspection or by solving the two equations of the lines intersecting at that point.

Evaluate the objective function z = ax + by at each corner point. Let M and m be, respectively denote the largest and the smallest values of these points.

### Exercise –3.1

ML Aggarwal Linear Programming ISC Class-12 Maths Solutions Chapter-3 of Section-C

### Exercise –3.2

ML Aggarwal Linear Programming ISC Class-12 Maths Solutions Chapter-3 of Section-C

### Chapter Test

ML Aggarwal Linear Programming ISC Class-12 Maths Solutions Chapter-3 of Section-C

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