Maths 12th Std (TN 12th Maths English Medium)

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Maths 12th Std

12th Maths



Volume I

Chapter 1: Applications of Matrices and Determinants


Introduction
Inverse of a Non-Singular Square Matrix
Adjoint of a Square Matrix
Definition of inverse matrix of a square matrix
Properties of inverses of matrices
Application of matrices to Geometry
Application of matrices to Cryptography
Exercise 1.1: Inverse of a Non-Singular Square Matrix
Inverse of a Non-Singular Square Matrix: Solved Example Problems
Elementary Transformations of a Matrix
Row Echelon form
Rank of a Matrix
Gauss Jordan Method
Exercise 1.2: Elementary Transformations of a Matrix
Elementary Transformations of a Matrix: Solved Example Problems
Applications of Matrices: Solving System of Linear Equations
Formation of a System of Linear Equations
System of Linear Equations in Matrix Form
Solution to a System of Linear equations
Matrix Inversion Method
Exercise 1.3: Matrices linear equations by matrix inversion method
Matrices: Cramerís Rule
Exercise 1.4: Matrices: Cramerís Rule
Matrix: Gaussian Elimination Method
Exercise 1.5: Matrix: Gaussian Elimination Method
Solved Example Problems on Applications of Matrices: Solving System of Linear Equations
Matrix: Non-homogeneous Linear Equations
Exercise 1.6: Matrix: Non-homogeneous Linear Equations
Matrix: Homogeneous system of linear equations
Exercise 1.7: Matrix: Homogeneous system of linear equations
Choose the Correct Answers
Summary

Chapter 2: Complex Numbers


Introduction
Introduction to Complex Numbers
Exercise 2.1: Introduction to Complex Numbers
Complex Numbers
Exercise 2.2: Complex Numbers
Basic Algebraic Properties of Complex Numbers
Exercise 2.3: Properties of Complex Numbers
Conjugate of a Complex Number
Exercise 2.4: Conjugate of a Complex Number
Modulus of a Complex Number
Properties of Modulus of a complex number
Modulus of a Complex Number: Solved Example Problems
Square roots of a complex number
Exercise 2.5: Modulus of a Complex Number
Geometry and Locus of Complex Numbers
Exercise 2.6: Geometry and Locus of Complex Numbers
Polar form of a complex number
Eulerís Form of the complex number
Exercise 2.7: Polar and Euler form of a Complex Number
de Moivreís Theorem and its Applications
de Moivre's Theorem
Finding nth roots of a complex number
The nth roots of unity
Exercise 2.8: de Moivreís Theorem and its Applications
Solved Example Problems on de Moivreís Theorem
Choose the correct Answers
Summary

Chapter 3: Theory of Equations


Introduction
Basics and types of Polynomial Equations
Vietaís formula for Quadratic Equations
Vietaís formula for Polynomial Equations
Exercise 3.1: Vietaís Formulae and Formation of Polynomial Equations
Nature of Roots and Nature of Coefficients of Polynomial Equations
Imaginary Roots
Irrational Roots
Rational Roots
Applications of Polynomial Equation in Geometry
Exercise 3.2: Polynomial Equation in Geometry
Roots of Higher Degree Polynomial Equations
Polynomials with Additional Information
Exercise 3.3: Polynomials with Additional Information
Partly Factored Polynomial
Exercise 3.4: Partly Factored Polynomial
Rational Root Theorem
Reciprocal Equations
Non-polynomial Equations
Exercise 3.5: Polynomial Equations with no Additional Information
Descartes Rule
Exercise 3.6: Descartes Rule
Choose the correct Answers
Summary

Chapter 4: Inverse Trigonometric Functions


Introduction
Some Fundamental Concepts of Inverse Trigonometric Functions
Sine Function and Inverse Sine Function
Exercise 4.1: Sine Function and Inverse Sine Function
The Cosine Function and Inverse Cosine Function
Exercise 4.2: Cosine Function and Inverse Cosine Function
The Tangent Function and the Inverse Tangent Function
Exercise 4.3: Tangent Function and the Inverse Tangent Function
Cosecant Function and the Inverse Cosecant Function
The Secant Function and Inverse Secant Function
The Cotangent Function and the Inverse Cotangent Function
Principal Value of Inverse Trigonometric Functions
Exercise 4.4: Principal Value of Inverse Trigonometric Functions
Properties of Inverse Trigonometric Functions
Solved Example Problems on Inverse Trigonometric Functions
Exercise 4.5: Inverse Trigonometric Functions
Choose the correct answers
Summary

Chapter 5: Two Dimensional Analytical Geometry II


Introduction
Circle
Equation of a circle in standard form
Equations of tangent and normal at a point P on a given circle
Condition for the line y = mx + c to be a tangent to the circle and finding the point of contact
Circle: Solved Example Problems
Exercise 5.1: Circle
Conics
The general equation of a Conic
Parabola
Ellipse
Hyperbola
Exercise 5.2: Conics
Conic Sections
Exercise 5.3: Conic Sections
Parametric form of Conics
Tangents and Normals to Conics
Exercise 5.4: Tangents and Normals to Conics
Real life Applications of Conics
Exercise 5.5: Real life Applications of Conics
Solved Example Problems on Real life Applications of Conics
Choose the correct Answer
Summary

Chapter 6: Applications of Vector Algebra


Introduction
Geometric introduction to vectors
Scalar Product and Vector Product
Exercise 6.1: Scalar Product and Vector Product
Scalar triple product
Exercise 6.2: Scalar triple product
Vector triple product
Jacobiís Identity and Lagrangeís Identity
Exercise 6.3: Vector triple product, Jacobiís Identity and Lagrangeís Identity
Application of Vectors to 3-Dimensional Geometry
Exercise 6.4: Application of Vectors to 3-Dimensional Geometry
A point on the straight line and the direction of the straight line
Straight Line passing through two given points
Angle between two straight lines
Point of intersection of two straight lines
Shortest distance between two straight lines
Exercise 6.5: Point of intersection of two straight lines
Different forms of Equation of a plane
Equation of a plane when a normal to the plane
Equation of a plane perpendicular to a vector and passing through a given point
Intercept form of the equation of a plane
Exercise 6.6: Equation of a plane
Equation of a plane passing through three given non-collinear points
Equation of a plane passing through a given point and parallel to two given non-parallel vectors
Equation of a plane passing through two given distinct points and is parallel to a non-zero vector
Exercise 6.7: Equation of a plane
Condition for a line to lie in a plane
Condition for coplanarity of two lines
Equation of plane containing two non-parallel coplanar lines
Exercise 6.8: Equation of a plane
Angle between two planes
Angle between a line and a plane
Distance of a point from a plane
Distance between two parallel planes
Equation of line of intersection of two planes
Equation of a plane passing through the line of intersection of two given planes
Image of a Point in a Plane
Meeting Point of a Line and a Plane
Exercise 6.9: Equation of intersection of the planes
Choose the correct answer
Summary

Volume II

Chapter 7: Applications of Differential Calculus


Applications of Differential Calculus
Derivative as slope
Derivative as rate of change
Related rates
Exercise 7.1 : Meaning of Derivatives
Equations of Tangent and Normal
Angle between two curves
Exercise 7.2 : Equations of Tangent and Normal, Angle between two curves
Mean Value Theorem
Rolleís Theorem
Lagrangeís Mean Value Theorem
Applications - Mean Value Theorem
Exercise 7.3: Mean Value Theorem
Series Expansions: Maclaurinís and Taylorís Series
Exercise 7.4 : Series Expansions: Maclaurinís and Taylorís Series
Indeterminate Forms
Exercise 7.5 : Indeterminate Forms
Applications of First Derivative
Exercise 7.6 : Applications of First Derivative
Applications of Second Derivative
Exercise 7.7: Applications of Second Derivative
Applications in Optimization
Exercise 7.8: Applications in Optimization
Symmetry and Asymptotes
Sketching of Curves
Exercise 7.9: Symmetry and Asymptotes, Sketching of Curves
Exercise 7.10: Choose the correct or the most suitable answer
Summary

Chapter 8: Differentials and Partial Derivatives


Differentials and Partial Derivatives
Linear Approximation
Errors: Absolute Error, Relative Error, and Percentage Error
Exercise 8.1: Linear Approximation
Differentials
Exercise 8.2: Differentials
Functions of Several Variables
Recall of Limit and Continuity of Functions of One Variable
Limit and Continuity of Functions of Two Variables
Exercise 8.3: Functions of Several Variables
Partial Derivatives
Exercise 8.4: Partial Derivatives
Linear Approximation and Differential of a function of several variables
Exercise 8.5: Linear Approximation and Differential of a function of several variables
Function of Function Rule
Exercise 8.6: Function of Function Rule
Homogeneous Functions and Eulerís Theorem
Exercise 8.7: Homogeneous Functions and Eulerís Theorem
Exercise 8.8: Choose the correct answer
Summary

Chapter 9: Applications of Integration


Applications of Integration
Definite Integral as the Limit of a Sum
Exercise 9.1: Definite Integral as the Limit of a Sum
Limit Formula to Evaluate Definite Integral as the Limit of a Sum
Exercise 9.2: Limit Formula to Evaluate Definite Integral as the Limit of a Sum
Fundamental Theorems of Integral Calculus and their Applications
Exercise 9.3: Fundamental Theorems of Integral Calculus and their Applications
Bernoulliís Formula
Exercise 9.4: Bernoulliís Formula
Improper Integrals
Exercise 9.5: Improper Integrals
Reduction Formulae
Exercise 9.6: Reduction Formulae
Gamma Integral
Exercise 9.7: Gamma Integral
Evaluation of a Bounded Plane Area by Integration
Exercise 9.8: Evaluation of a Bounded Plane Area by Integration
Volume of a solid obtained by revolving area about an axis
Exercise 9.9: Volume of a solid obtained by revolving area about an axis
Exercise 9.10: Choose the correct answer
Summary

Chapter 10: Ordinary Differential Equations


Ordinary Differential Equations
Differential Equation, Order, and Degree
Exercise 10.1: Differential Equation, Order, and Degree
Classification of Differential Equations
Formation of Differential equations from Physical Situations
Exercise 10.2: Formation of Differential equations from Physical Situations
Formation of Differential Equations from Geometrical Problems
Exercise 10.3: Formation of Differential Equations from Geometrical Problems
Solution of Ordinary Differential Equations
Exercise 10.4: Solution of Ordinary Differential Equations
Variables Separable Method
Substitution Method
Exercise 10.5: Variables Separable Method, Substitution Method
Homogeneous Form or Homogeneous Differential Equation
Exercise 10.6: Homogeneous Form or Homogeneous Differential Equation
First Order Linear Differential Equations
Exercise 10.7: First Order Linear Differential Equations
Applications of First Order Ordinary Differential Equations
Exercise 10.8: Applications of First Order Ordinary Differential Equations
Exercise 10.9: Choose the correct answer
Summary

Chapter 11: Probability Distributions


Probability Distributions
Random Variable
Exercise 11.1: Random Variable
Types of Random Variable
Discrete random variables
Probability Mass Function
Cumulative Distribution Function or Distribution Function
Cumulative Distribution Function from Probability Mass function
Probability Mass Function from Cumulative Distribution Function
Exercise 11.2: Types of Random Variable
Continuous Distributions
Exercise 11.3: Continuous Distributions
Mathematical Expectation
Properties of Mathematical expectation and variance
Exercise 11.4: Mathematical Expectation
Theoretical Distributions: Some Special Discrete Distributions
Exercise 11.5: Theoretical Distributions: Some Special Discrete Distributions
Exercise 11.6: Choose the Correct answer
Summary

Chapter 12: Discrete Mathematics


Discrete Mathematics
Definitions of Binary Operations
Some more properties of a binary operation
Some binary operations on Boolean Matrices
Binary operations: Modular Arithmetic
Exercise 12.1: Binary operations
Mathematical Logic
Mathematical Logic: Statement and its truth value
Mathematical Logic: Compound Statements, Logical Connectives, and Truth Tables
Mathematical Logic: Logical Connectives and their Truth Tables
Mathematical Logic: Tautology, Contradiction, and Contingency
Mathematical Logic: Duality
Some Laws of Logical Equivalence
Exercise 12.2: Mathematical Logic
Exercise 12.3: Choose the correct answer
Summary



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