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Relations and Functions
Relations and Functions
Inverse Trigonometric Functions
Algebra
Matrices
- Concept of Matrices
- Types of Matrices
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- Multiplication of a Matrix by a Scalar
- Properties of Matrix Addition
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- Negative of Matrix
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- Proof of the Uniqueness of Inverse
- Elementary Transformations
Calculus
Vectors and Three-dimensional Geometry
Determinants
- Determinants of Matrix of Order One and Two
- Determinant of a Matrix of Order 3 × 3
- Minors and Co-factors
- Inverse of a Square Matrix by the Adjoint Method
- Applications of Determinants and Matrices
- Elementary Transformations
- Properties of Determinants
- Determinant of a Square Matrix
- Rule A=KB
- Overview of Determinants
- Geometric Interpretation of the Area of a Triangle
Linear Programming
Continuity and Differentiability
- Concept of Continuity
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivatives of Composite Functions - Chain Rule
- Derivatives of Implicit Functions
- Derivatives of Inverse Trigonometric Functions
- 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
- Continuous Function of Point
- Mean Value Theorem
- Overview of Continuity and Differentiability
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
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- 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 Fractions
- Methods of Integration: Integration by Parts
- Fundamental Theorem of 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
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- Differential Equations with Variables Separable Method
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- Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
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Vectors
- Vector
- Basic Concepts of Vector Algebra
- Direction Cosines
- Vector Operations>Addition and Subtraction of Vectors
- Properties of Vector Addition
- Vector Operations>Multiplication of a Vector by a Scalar
- Components of Vector
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- Section Formula
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- Scalar (Or Dot) Product of Two Vectors
- Projection of a Vector on a Line
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- Scalar Triple Product of Vectors
- 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
- Forms of the Equation of a Straight Line
- 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
- Distance of a Point from a Plane
- Plane Passing Through the Intersection of Two Given Planes
- Overview of Three Dimensional Geometry
Linear Programming
Probability
Notes
Let R, S and T be three non collinear points on the plane with position vectors `vec a` , `vec b` and `vec c` respectively in following fig.
The vectors `vec (RS)` and `vec (RT)` are in the given plane. Therefore , the vector `vec (RS) xx vec (RT)` is perpendicular to the plane containing points R,S and T. Let `vec r`be the position vector of any point P in the plane. Therefore, the equation of the plane passing through R and perpendicular to the vector `vec (RS) xx vec (RT)` is
`(vec r - vec a) . (vec (RS) xx vec (RT)) = 0`
or `(vec r - vec a) . [(vec b - vec a) xx (vec c - vec a)] = 0` ...(1)
If the three points were on the same line, then there will be many planes that will contain them Fig.
for example , These planes will resemble the pages of a book where the line containing the points R, S and T are members in the binding of the book.
Cartesian form:
Let `(x_1,y_1,z_1) , (x_2 , y_2 , z_2)` and `(x_3 , y_3 , z_3)` be the coordinates of the points R, S and T respectively. Let (x, y, z) be the coordinates of any point P on the plane with position vector `vec r`. Then
`vec (RP) = (x - x_1) hat i + ( y - y_1) hat j + (z - z_1) hat k`
`vec (RS) = (x_2 - x_1) hat i + ( y_2 - y_1) hat j + (z_2 - z_1) hat k`
`vec (RT) = (x_3 - x_1) hat i + ( y_3 - y_1) hat j + (z_3 - z_1) hat k`
Substituting these values in equation (1) of the vector form and expressing it in the form of a determinant, we have
`|(x - x_1 , y - y_1 , z - z_1),(x_2 - x_1 , y_2 - y_1, z_2 -z_1) ,(x_3 - x_1, y_3 - y_1, z_3 - z_1) | = 0`
which is the equation of the plane in Cartesian form passing through three non collinear points `(x_1, y_1, z_1), (x_2, y_2, z_2)` and `(x_3, y_3, z_3).`
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