#### Topics

##### Relations and Functions

##### Relations and Functions

##### Inverse Trigonometric Functions

##### Algebra

##### 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 Transformations
- 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

##### Calculus

##### Vectors and Three-dimensional Geometry

##### 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

##### Linear Programming

##### 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

##### Probability

##### 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

##### Sets

##### 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

##### 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

##### Vectors

- 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
- Vectors and Their Types
- 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

##### 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

##### Linear Programming

##### Probability

- Variance of a Random Variable
- Probability Examples and Solutions
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Bayes’ 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

The scalar product of two nonzero vectors `vec a` and `vec b`, denoted by `vec a .vec b` , is defined as `vec a . vec b = |vec a| |vec b| cos theta,`

where , θ is the angle between Fig.

If either `vec a = 0` or `vec b = 0` then θ is not defined, and in this case , we define `vec a . vec b = 0`

Observations:

1) `vec a .vec b` is a real number.

2) Let `vec a` and `vec b` be two nonzero vectors, then `vec a . vec b = 0` if and only if `vec a` and `vec b` are perpendicular to each other . i.e.

`vec a . vec b = 0 <=> vec a ⊥ vec b`

3) If θ = 0, then `vec a .vec b = |vec a| |vec b|`

In particular , `vec a .vec a = |vec a|^2 , as θ in this case is 0.`

4) If θ = π, then `vec a . vec b = - |vec a||vec b|`

In particular, `vec a . vec b = - |vec a||vec b|`, as θ in this case is π.

5) In view of the Observations 2 and 3, for mutually perpendicular unit vectors

`hat i , hat j "and" hat k` we have

`hat i . hat i = hat j . hat j = hat k. hat k = 1,`

`hat i . hat j = hat j . hat k = hat k. hat i = 0`

6) The angle between two nonzero vectors `vec a` and `vec b` is given by

`cos theta = (vec a .vec b)/(|vec a||vec b|),` or `theta = cos ^(-1) ((vec a . vec b)/(|vec a||vec b|))`

7) The scalar product is commutative. i.e.

`vec a . vec b` = `vec b . vec a`

#### text

**Two important properties of scalar product :****Property:** (Distributivity of scalar product over addition)

Let `vec a, vec b` and `vec c` be any three vectors , then `vec a (vec b + vec c) = vec a . vec b + vec a. vec c`

**Let `vec a` and `vec b` be any two vectors, and l be any scalar. Then**

Property:

Property:

`(lambda vec a). vec b = (lambda vec a).vec b = lambda (vec a . vec b) = vec a . (lambda vec b)`

If two vectors `vec a` and `vec b` are given in component form as

`a_1 hat i + a_2hat j + a_3 hat k` and `b_1 hat i + b_2hat j + b_3 hat k .`, then their scalar product is given as

`vec a . vec b` = `(a_1hat i + a_2 hat j + a_3 hat k) . (b_1 hat i + b_2 hat j + b_3 hat k)`

= `a_1hat i . (b_1 hat i + b_2 hat j + b_3 hat k) + a_2 hat j . (b_1 hat i + b_2 hat j + b_3 hat k) + a_3 hat k . (b_1 hat i + b_2 hat j + b_3 hat k)`

= `a_1 b_1 (hat i . hat i) + a_1b_2 (hat i . hat j) +a_1b_3 (hat i . hat k)+a_2b_1 (hat j . hat i) + a_2 b_2 (hat j . hat j) + a_2b_3 (hat j . hat k) + a_3b_1 (hat k . hat i) + a_3b_2 (hat k . hat j) + a_3b_3 (hat k . hat k)` (Using the above Properties 1 and 2)

= `a_1b_1 + a_2b_2 + a_3b_3 ` (Using Observation 5)

Thus `vec a . vec b = a_1b_1 + a_2b_2 + a_3b_3`