#### Topics

##### Relations and Functions

##### Algebra

##### Calculus

##### Vectors and Three-dimensional Geometry

##### Linear Programming

##### Probability

##### Sets

##### Inverse Trigonometric Functions

##### Relations and Functions

##### Determinants

- Applications of Determinants and Matrices
- Elementary Transformations
- Inverse of a Square Matrix by the Adjoint Method
- Properties of Determinants
- Determinant of a Square Matrix
- Determinants of Matrix of Order One and Two
- Introduction of Determinant
- Area of a Triangle
- Minors and Co-factors
- Determinant of a Matrix of Order 3 × 3
- Rule A=KB

##### Matrices

- Introduction of Operations on Matrices
- Inverse of a Nonsingular Matrix by Elementary Transformation
- Multiplication of Two Matrices
- Negative of Matrix
- Properties of Matrix Addition
- Concept of Transpose of a Matrix
- Subtraction of Matrices
- Addition of Matrices
- Symmetric and Skew Symmetric Matrices
- Types of Matrices
- Proof of the Uniqueness of Inverse
- Invertible Matrices
- Elementary Operation (Transformation) of a Matrix
- Multiplication of Matrices
- Properties of Multiplication of Matrices
- Equality of Matrices
- Order of a Matrix
- Matrices Notation
- Introduction of Matrices
- Multiplication of a Matrix by a Scalar
- Properties of Scalar Multiplication of a Matrix
- Properties of Transpose of the Matrices

##### Continuity and Differentiability

- Derivative - Exponential and Log
- Concept of Differentiability
- Proof Derivative X^n Sin Cos Tan
- Infinite Series
- Higher Order Derivative
- Algebra of Continuous Functions
- Continuous Function of Point
- Mean Value Theorem
- Second Order Derivative
- Derivatives of Functions in Parametric Forms
- Logarithmic Differentiation
- Exponential and Logarithmic Functions
- Derivatives of Implicit Functions
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Composite Functions - Chain Rule
- Concept of Continuity

##### Applications of Derivatives

- Maximum and Minimum Values of a Function in a Closed Interval
- Maxima and Minima
- Simple Problems on Applications of Derivatives
- Graph of Maxima and Minima
- Approximations
- Tangents and Normals
- Increasing and Decreasing Functions
- Rate of Change of Bodies Or Quantities
- Introduction to Applications of Derivatives

##### Applications of the Integrals

##### Differential Equations

- Linear Differential Equations
- Solutions of Linear Differential Equation
- Homogeneous Differential Equations
- Differential Equations with Variables Separable Method
- Formation of a Differential Equation Whose General Solution is Given
- General and Particular Solutions of a Differential Equation
- Order and Degree of a Differential Equation
- Basic Concepts of Differential Equation
- Procedure to Form a Differential Equation that Will Represent a Given Family of Curves

##### Integrals

- Definite Integrals Problems
- Indefinite Integral Problems
- Comparison Between Differentiation and Integration
- Geometrical Interpretation of Indefinite Integral
- Integrals of Some Particular Functions
- Indefinite Integral by Inspection
- Properties of Indefinite Integral
- Integration Using Trigonometric Identities
- Introduction of Integrals
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Fundamental Theorem of Calculus
- Definite Integral as the Limit of a Sum
- Evaluation of Simple Integrals of the Following Types and Problems
- Methods of Integration - Integration by Parts
- Methods of Integration - Integration Using Partial Fractions
- Methods of Integration - Integration by Substitution
- Integration as an Inverse Process of Differentiation

##### Three - Dimensional Geometry

- Three - Dimensional Geometry Examples and Solutions
- Introduction of Three Dimensional Geometry
- Equation of a Plane Passing Through Three Non Collinear Points
- Relation Between Direction Ratio and Direction Cosines
- Intercept Form of the Equation of a Plane
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Angle Between Line and a Plane
- Angle Between Two Planes
- Angle Between Two Lines
- Vector and Cartesian Equation of a Plane
- Shortest Distance Between Two Lines
- Equation of a Line in Space
- Direction Cosines and Direction Ratios of a Line
- Equation of a Plane in Normal Form
- Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
- Plane Passing Through the Intersection of Two Given Planes

##### Vectors

- Concept of Direction Cosines
- Properties of Vector Addition
- Geometrical Interpretation of Scalar
- Scalar Triple Product of Vectors
- Vector (Or Cross) Product of Two Vectors
- Scalar (Or Dot) Product of Two Vectors
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Multiplication of a Vector by a Scalar
- Addition of Vectors
- Introduction of Vector
- Magnitude and Direction of a Vector
- Basic Concepts of Vector Algebra
- Types of Vectors
- Components of a Vector
- Section formula
- Vector Joining Two Points
- Vectors Examples and Solutions
- Projection of a Vector on a Line
- Introduction of Product of Two Vectors

##### Linear Programming

##### Probability

- Variance of a Random Variable
- Probability Examples and Solutions
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Baye'S Theorem
- Random Variables and Its Probability Distributions
- Mean of a Random Variable
- Bernoulli Trials and Binomial Distribution
- Introduction of Probability
- Properties of Conditional Probability

#### definition

**Defination:** The adjoint of a square matrix A = `[a_(ij)]_(n × n)` is defined as the transpose of the matrix `[A_(ij)]_(n × n)`, where `A_(ij)` is the cofactor of the element `a_(ij)`. Adjoint of the matrix A is denoted by adj A.

Let A = `[(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)]`

Then adj A = Transpose of `[(A_11,A_12,A_13),(A_21,A_22,A_23),(A_31,A_32,A_33)] = [(A_11,A_21,A_31),(A_12,A_22,A_32),(A_13,A_23,A_33)]`

#### theorem

If A be any given square matrix of order n, then

A(adj A) = (adj A) A = |A| I ,

where I is the identity matrix of order n

Verification

Let A =`[(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)]` , then adj A =` [(A_11,A_21,A_31),(A_12,A_22,A_32),(A_13,A_23,A_33)]`

Since sum of product of elements of a row (or a column) with corresponding cofactors is equal to |A| and otherwise zero, we have

A (adj A) = `|(|A|,0,0),(0,|A|,0),(0,0,|A|)| = |A| [(1,0,0),(0,1,0),(0,0,1)]`

Similarly, we can show (adj A) A = A I

Hence A (adj A) = (adj A) A = A I

#### notes

**Definition :** A square matrix A is said to be singular if A = 0.

For example, the determinant of matrix A = `[(1,2),(4,8)]` is zero

Hence A is a singular matrix.

**Definition :** A square matrix A is said to be non-singular if A ≠ 0

Let A = `[(1,4),(3,2)]`. Then |A| = `|(1,4)(3,2)|` = 4 - 6 = 2 ≠ 0

Hence A is a nonsingular matrix.

We state the following theorems without proof.

**Theorem :** If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.**Theorem:** The determinant of the product of matrices is equal to product of their respective determinants, that is, |AB| = |A| |B| , where A and B are square matrices of the same order**Remark:** We know that (adj A) A = |A| I = `[(|A|,0,0),(0,|A|,0),(0,0,|A|)]` , |A| ≠ 0

Writing determinants of matrices on both sides, we have

`|(adj A)A| = |(|A|,0,0),(0,|A|,0),(0,0,|A|)|`

i.e,. `|(adjA)| |A| = |A|^3 |(1,0,0),(0,1,0),(0,0,1)|`

i.e. `|(adj A)| |A| = |A|3 (1) i.e. |(adj A)| = |A|2`

i.e. `|(adj A)| = |A|^2`

In general, if A is a square matrix of order n, then |adj(A)| = |A|^(n – 1).

**Theorem:** A square matrix A is invertible if and only if A is nonsingular matrix.

**Proof:** Let A be invertible matrix of order n and I be the identity matrix of order n. Then, there exists a square matrix B of order n such that

AB = BA = I

Now AB = I. So |AB| = I or |A| |B| = 1 (since |I| =1, |AB|=|A||B|)

This gives |A| ≠ 0.

Hence A is nonsingular.

Conversely, let A be nonsingular. Then A ≠ 0

Now A (adj A) = (adj A) A = |A| I (Theorem 1)

or `A(1/|A| adj A) = (1/|A| adj A) A = I`

or `AB = BA = I` ,where `B = 1/|A| adj A`

Thus, A is invertible and `A^-1 = 1/ |A|adj A`