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
Matrices
Calculus
Determinants
- Determinant of a Matrix
- Expansion of Determinant
- Area of Triangle using Determinant
- Minors and Co-factors
- Adjoint of a Matrix
- Inverse of a 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
- Derivatives of Implicit Functions
- Derivative of Inverse Function
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Derivative - Exponential and Log
- Proof Derivative X^n Sin Cos Tan
- Infinite Series
- Higher Order Derivative
- Mean Value Theorem
- Overview of Continuity and Differentiability
Linear Programming
Applications of Derivatives
- Introduction to Applications of Derivatives
- Rate of Change of Bodies or Quantities
- Increasing and Decreasing Functions
- Maxima and Minima
- Maximum and Minimum Values of a Function in a Closed Interval
- Simple Problems on Applications of Derivatives
- Graph of Maxima and Minima
- Approximations
- Tangents and Normals
- Overview of Applications of Derivatives
Probability
Sets
Integrals
- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Some Properties of Indefinite Integral
- Methods of Integration> Integration by Substitution
- Integration Using Trigonometric Identities
- Integrals of Some Particular Functions
- Methods of Integration> Integration Using Partial Fraction
- Methods of Integration> Integration by Parts
- Fundamental Theorem of Integral Calculus
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Definite Integrals
- Indefinite Integral Problems
- Comparison Between Differentiation and Integration
- Geometrical Interpretation of Indefinite Integrals
- Indefinite Integral by Inspection
- Definite Integral as the Limit of a Sum
- Evaluation of Simple Integrals of the Following Types and Problems
- Overview of Integrals
Applications of the Integrals
Differential Equations
- Differential Equations
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Linear Differential Equations
- Homogeneous Differential Equations
- Solutions of Linear Differential Equation
- Equations in Variable Separable Form
- Formation of a Differential Equation Whose General Solution is Given
- Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
- Overview of Differential Equations
Vectors
- Vector
- Basic Concepts of Vector Algebra
- Direction Ratios, Direction Cosine & Direction Angles
- Vector Operations>Addition and Subtraction of Vectors
- Properties of Vector Addition
- Vector Operations>Multiplication of a Vector by a Scalar
- Components of Vector
- Vector Joining Two Points
- Section Formula in Coordinate Geometry
- Vector (Or Cross) Product of Two Vectors
- Scalar (Or Dot) Product of Two Vectors
- Projection of a Vector on a Line
- Geometrical Interpretation of Scalar
- Scalar Triple Product
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Magnitude and Direction of a Vector
- Vectors Examples and Solutions
- Introduction of 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
Notes
For example , `3 xx [(2,4),(-5,1)]`
To get the result of matrix each element of the matrix multiply by 3.
The order of the above matrix is same as the order of the original matrix. The original matrix is of the order `2 xx 2` so the resulting matrix
will also be of the same order.
`3 xx [(2,4),(-5,1)] = [(3 xx 2,3 xx 4),(3 xx-5 ,3 xx 1)]`
=`[(6 , 12),(-15 , 3)]`
Thus `k xx [(a,b),(c,d)]` = `[(ka , kb),(kc , kd)]`
Multiplying a matrix to a number is called scalar Multiplication.
Negative of matrix :
Z = `[(1,3),(-5,2)]` -Z = ?
-Z = `-1 xx Z
=`-1 xx [(1,3),(-5,2)]`
=` [(1*-1 , 3*-1),(-5*-1, 2*-1)]`
=`[(-1 ,-3),(5, -2)] `
The negative of a matrix is denoted by –Z. We define –Z = (–1) Z.
Difference of matrices:
If A = `[a_(ij)]`, B = `[b_(ij)]` are two matrices of the same order, say m × n, then difference A – B is defined as a matrix D = `[d_(ij)]`, where `d_(ij) = a_(ij) – b_(ij)`, for all value of i and j. In other words, D = A – B = A + (–1) B, that is sum of the matrix A and the matrix – B.
Video link : https://youtu.be/4lHyTQH1iS8
Notes
The matrix is the sum of the two matrices. The sum of two matrices is a matrix obtained by adding the corresponding elements of the matrices.
Thus, if A = `[(a_11,a_12,a_13),(a_21,a_22,a_23)]` is a 2 x 3 matrix and
B = `[(b_11,b_12,b_13),(b_21,b_22,b_23)]` is another 2 x 3 matrix .
Then we define
A +B = `[(a_11 + b_11,a_12 + b_12,a_13 + b_13), (a_21 + b_21,a_22 + b_22,a_23 + b_23)]`
In general, if A = `[a_(ij)]` and B = `[b_(ij)]` are two matrices of the same order, say m × n. Then, the sum of the two matrices A and B is defined as a matrix C = `[c_(ij)]_(m × n)`, where `c_(ij)` = `a_(ij)` + `b_(ij)`, for all possible values of i and j.
Video link : https://youtu.be/ZCmVpGv6_1g
Notes
The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B. Let A = `[a_(ij)]` be an m × n matrix and B = `[b_(jk)]` be an n × p matrix. Then the product of the matrices A and B is the matrix C of order m × p. To get the `(i, k)th`element cik of the matrix C, we take the ith row of A and kth column of B, multiply them elementwise and take the sum of all these products. In other words, if A = `[a_(ij)]_(m × n)`, B = `[b_(jk)]_(n × p)`, then the ith row of A is `[a_(i1) a_(i2) ... a_("in")]` and the `k^(th)` column of
B is `[(b_(1k)) , (b_(2k)), (b_(nk))]`
, then `c_(ik) = a_(i1) b_(1k) + a_(i2) b_(2k) + a_(i3) b_(3k) + ... + a_("in") b_(nk)`
=\[\displaystyle\sum_{j=1}^{n} a_{ij} b_{jk}\].
Non-commutativity of multiplication of matrices:
The below example that even if AB and BA are both defined, it is not necessary that AB = BA.
If A = `[(1,0),(0,-1)]` and B =`[(0,1),(1,0)]` ,
then AB `[(0,1),(-1,0)]`
and BA = `[(0,-1),(1,0)]`.
Clearly AB ≠ BA.
Thus matrix multiplication is not commutative.
Zero matrix as the product of two non zero matrices:
The real numbers a, b if ab = 0, then either a = 0 or b = 0. This need not be true for matrices, we will observe this through an example.
find AB ,if A = `[(0,-1),(0,2)]` and B = `[(3,5),(0,0)]`
We have AB = `[(0,-1),(0,2)][(3,5),(0,0)]`
=`[(0,0),(0,0)]`
Thus, if the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix.
