Topics
Relations and Functions
Relations and Functions
Inverse Trigonometric Functions
- Basics of Inverse Trigonometric Functions
- Domain, Range & Principal Value
- Graphs of Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions
- Overview of Inverse Trigonometric Functions
Algebra
Calculus
Matrices
Determinants
- Determinant of a Matrix
- Expansion of Determinant
- Area of Triangle using Determinant
- Minors and Co-factors
- Adjoint & Inverse of Matrix
- Applications of Determinants and Matrices
- Overview of Determinants
Vectors and Three-dimensional Geometry
Continuity and Differentiability
- Continuous and Discontinuous Functions
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivative of Composite Functions
- Derivative of Implicit Functions
- Derivative of Inverse Function
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Overview of Continuity and Differentiability
Linear Programming
Applications of Derivatives
Probability
Integrals
- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Properties of Indefinite Integral
- Methods of Integration> Integration by Substitution
- Methods of Integration>Integration Using Trigonometric Identities
- Methods of Integration> Integration Using Partial Fraction
- Methods of Integration> Integration by Parts
- Integrals of Some Particular Functions
- Definite Integrals
- Fundamental Theorem of Integral Calculus
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Overview of Integrals
Sets
Applications of the Integrals
Differential Equations
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Methods of Solving Differential Equations> Variable Separable Differential Equations
- Methods of Solving Differential Equations> Homogeneous Differential Equations
- Methods of Solving Differential Equations>Linear Differential Equations
- Overview of Differential Equations
Vectors
- Vector
- Basic Concepts of Vector Algebra
- Direction Ratios, Direction Cosine & Direction Angles
- Types of Vectors in Algebra
- Vector Operations>Addition and Subtraction of Vectors
- Algebra of Vector Addition
- Multiplication in Vector Algebra
- Components of Vector in Algebra
- Vector Joining Two Points in Algebra
- Section Formula in Coordinate Geometry
- Product of Two Vectors
- Product of Two Vectors
- Projection of a Vector on a Line
- Geometrical Interpretation of Scalar
- Scalar Triple Product
- Vector Joining Two Points in Algebra
- Section Formula in Vector Algebra
- Product of Two Vectors
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Magnitude and Direction of a Vector
- Vectors Examples and Solutions
- Product of Two Vectors
- Overview of Vectors
Three - Dimensional Geometry
- Introduction of Three Dimensional Geometry
- Direction Cosines and Direction Ratios of a Line
- Relation Between Direction Ratio and Direction Cosines
- Equation of a Line in Space
- Angle Between Two Lines
- Shortest Distance Between Two Lines
- Three - Dimensional Geometry Examples and Solutions
- Equation of a Plane Passing Through Three Non Collinear Points
- Equations of Line in Different Forms
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Angle Between Line and a Plane
- Angle Between Two Planes
- Vector and Cartesian Equation of a Plane
- 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
- Overview of Three Dimensional Geometry
Linear Programming
Probability
Text
Property:- For any two vectors `vec a`and `vec b`,
`vec a + vec b` = `vec b + vec a` (Commutative property)
Proof: Consider the parallelogram ABCD
Let `vec (AB) = vec a` and `vec (BC) = vec b`, then using the triangle law, from triangle ABC , we have `vec (AC) = vec a + vec b`
Now, since the opposite sides of a parallelogram are equal and parallel, from abov fig.
we have , `vec (AD) = vec (BC) = vec b` and `vec (DC) = vec (AB) = vec a`
Again using triangle law, from triangle ADC, we have `vec (AC) = vec (AD) + vec (DC) = vec b + vec a`
Hence `vec a + vec b = vec b + vec a`
Property:- For any three vectors `vec a , vec b and vec c`
`(vec a + vec b) + vec c = vec a + (vec b + vec c)`
(Associative property)
Proof: Let the vectors `vec a , vec b and vec c` be represented by `vec (PQ) , vec (QR)` and `vec (RS)`, respectively , as shown in following fig.
Then `vec a + vec b = vec (PQ) + vec (QR) = vec (PR)`
and `vec b + vec c = vec (QR) + vec (RS) = vec (QS)`
So, `(vec a + vec b) + vec c = vec (PR) + vec (RS) = vec (PS)`
and `vec a + (vec b+vec c) = vec (PQ) + vec (QS) = vec (PS)`
Hence `(vec a + vec b) + vec c = vec a + (vec b+vec c)`
Remark: Any vector `vec a`, we have
`vec a + vec 0 =vec 0 + vec a = vec a`
Here, the zero vector `vec 0` is called the additive identity for the vector addition.
