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# CBSE - Class 12 - Mathematics - Linear Programming - NCERT Solutions

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# NCERT solutions for class 12 Mathematics

## NCERT Class 12 Mathematics Chapter-wise Solutions

### Part I

• Chapter 1: Relations and Functions
• Chapter 2: Inverse Trigonometric Functions
• Chapter 3: Matrices
• Chapter 4: Determinants
• Chapter 5: Continuity and Differentiability

### Part II

• Chapter 6: Application of Derivatives
• Chapter 7: Integrals
• Chapter 8: Application of Integrals
• Chapter 9: Differential Equations
• Chapter 10: Vector Algebra
• Chapter 11: Three-Dimensional Geometry
• Chapter 12: Linear Programming
• Chapter 13: Probability

CBSE class 12th Mathematics have two books. Each book has chapters and topics.

1. NCERT Mathematics Book Class 12 Part-1
2. NCERT Mathematics Book Class 12 Part-2

Here is the list of topics covered under each chapter of class 12 Mathematics NCERT text book.

### 1. Relations and Functions

• 1.1 Introduction
• 1.2 Types of Relations
• 1.3 Types of Functions
• 1.4 Composition of Functions and Invertible Function
• 1.5 Binary Operations

### 2. Inverse Trigonometric Functions

• 2.1 Introduction
• 2.2 Basic Concepts
• 2.3 Properties of Inverse Trigonometric Functions

### 3. Matrices

• 3.1 Introduction
• 3.2 Matrix
• 3.3 Types of Matrices
• 3.4 Operations on Matrices
• 3.5 Transpose of a Matrix
• 3.6 Symmetric and Skew Symmetric Matrices
• 3.7 Elementary Operation (Transformation) of a Matrix
• 3.8 Invertible Matrices

### 4. Determinants

• 4.1 Introduction
• 4.2 Determinant
• 4.3 Properties of Determinants
• 4.4 Area of a Triangle
• 4.5 Minors and Cofactors
• 4.6 Adjoint and Inverse of a Matrix
• 4.7 Applications of Determinants and Matrices

### 5. Continuity and Differentiability

• 5.1 Introduction
• 5.2 Continuity
• 5.3 Differentiability
• 5.4 Exponential and Logarithmic Functions
• 5.5 Logarithmic Differentiation
• 5.6 Derivatives of Functions in Parametric Forms
• 5.7 Second Order Derivative
• 5.8 Mean Value Theorem

### 6. Application of Derivatives

• 6.1 Introduction
• 6.2 Rate of Change of Quantities
• 6.3 Increasing and Decreasing Functions
• 6.4 Tangents and Normal
• 6.5 Approximations
• 6.6 Maxima and Minima

### 7. Integrals

• 7.1 Introduction
• 7.2 Integration as an Inverse Process of Differentiation
• 7.3 Methods of Integration
• 7.4 Integrals of some Particular Functions
• 7.5 Integration by Partial Fractions
• 7.6 Integration by Parts
• 7.7 Definite Integral
• 7.8 Fundamental Theorem of Calculus
• 7.9 Evaluation of Definite Integrals by Substitution
• 7.10 Some Properties of Definite Integrals

### 8. Application of Integrals

• 8.1 Introduction
• 8.2 Area under Simple Curves
• 8.3 Area between Two Curves

### 9. Differential Equations

• 9.1 Introduction
• 9.2 Basic Concepts
• 9.3 General and Particular Solutions of a Differential Equation
• 9.4 Formation of a Differential Equation whose General Solution is given
• 9.5 Methods of Solving First order, First Degree Differential Equations

### 10. Vector Algebra

• 10.1 Introduction
• 10.2 Some Basic Concepts
• 10.3 Types of Vectors
• 10.5 Multiplication of a Vector by a Scalar
• 10.6 Product of Two Vectors

### 11. Three Dimensional Geometry

• 11.1 Introduction
• 11.2 Direction Cosines and Direction Ratios of a Line
• 11.3 Equation of a Line in Space
• 11.4 Angle between Two Lines
• 11.5 Shortest Distance between Two Lines
• 11.6 Plane
• 11.7 Coplanarity of Two Lines
• 11.8 Angle between Two Planes
• 11.9 Distance of a Point from a Plane
• 11.10 Angle between a Line and a Plane

12. Linear Programming

• 12.1 Introduction
• 12.2 Linear Programming Problem and its Mathematical Formulation
• 12.3 Different Types of Linear Programming Problems

13. Probability

• 13.1 Introduction
• 13.2 Conditional Probability
• 13.3 Multiplication Theorem on Probability
• 13.4 Independent Events
• 13.5 Bayes' Theorem
• 13.6 Random Variables and its Probability Distributions
• 13.7 Bernoulli Trials and Binomial Distribution

## NCERT Solution Chapter 12: Linear Programming

In earlier classes, we have discussed systems of linear equations and their applications in day to day problems. In Class XI, we have studied linear inequalities and systems of linear inequalities in two variables and their solutions by graphical method. Many applications in mathematics involve systems of inequalities/equations. In this chapter, we shall apply the systems of linear inequalities/equations to solve some real life problems of the type as given below: A furniture dealer deals in only two items–tables and chairs. He has Rs 50,000 to invest and has storage space of at most 60 pieces. A table costs Rs 2500 and a chair Rs 500. He estimates that from the sale of one table, he can make a profit of Rs 250 and that from the sale of one chair a profit of Rs 75. He wants to know how many tables and chairs he should buy from the available money so as to maximise his total profit, assuming that he can sell all the items which he buys. Such type of problems which seek to maximise (or, minimise) profit (or, cost) form a general class of problems called optimisation problems. Thus, an optimisation problem may involve finding maximum profit, minimum cost, or minimum use of resources etc. A special but a very important class of optimisation problems is linear programming problem. The above stated optimisation problem is an example of linear programming problem. Linear programming problems are of much interest because of their wide applicability in industry, commerce, management science etc. In this chapter, we shall study some linear programming problems and their solutions by graphical method only, though there are many other methods also to solve such problems.

## NCERT Solutions for Class 12th Mathematics

NCERT Solutions Class 12 Mathematics PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Mathematics includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Mathematics have total 13 chapters. Class 12 Mathematics ncert Solutions in pdf for free Download are given in this website. Ncert Mathematics class 12 solutions PDF and Mathematics ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide.

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