Topics
Relations and Functions
Mathematics
Inverse Trigonometric Functions
- Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions
- Inverse Trigonometric Functions
- Graphs of Inverse Trigonometric Functions
- Inverse Trigonometric Functions - Principal Value Branch
Applied Mathematics
Matrices
- Concept of Matrices
- Types of Matrices
- Equality of Matrices
- Operation on Matrices
- Properties of Matrix Addition
- Multiplication of a Matrix by a Scalar
- Properties of Matrix Multiplication
- Negative of Matrix
- Transpose of a Matrix
- Symmetric and Skew Symmetric Matrices
- Elementary Transformations of a Matrix
- Invertible Matrices
Determinants
- Determinant Method (Cramer’s Rule)
- Determinants
- Determinants of Matrix of Order One and Two
- Determinant of a Matrix of Order 3 × 3
- Properties of Determinants
- Application of Determinants
- Area of a Triangle Using Determinants
- Minors and Co-factors
- Adjoint of a Matrix
- Properties of Matrix Multiplication
- Applications of Determinants and Matrices
Continuity and Differentiability
- Introduction of 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
- Mean Value Theorem
Applications of Derivatives
- Introduction to Applications of Derivatives
- Rate of Change of Bodies or Quantities
- Increasing and Decreasing Functions
- Tangents and Normals
- Approximations
- Maxima and Minima
- Maximum and Minimum Values of a Function in a Closed Interval
- Graph of Maxima and Minima
- Simple Problems on Applications of Derivatives
Integrals
- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Geometrical Interpretation of Indefinite Integrals
- Some Properties of Indefinite Integral
- Comparison Between Differentiation and Integration
- Methods of Integration: Integration by Substitution
- Methods of Integration: Integration Using Partial Fractions
- Integrals of Some Particular Functions
- Methods of Integration: Integration by Parts
- Integration Using Trigonometric Identities
- Definite Integrals
- Definite Integral as the Limit of a Sum
- Fundamental Theorem of Calculus
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
Applications of the Integrals
- Introduction of Applications of the Integrals
- Area Under Simple Curves
- Area of the Region Bounded by a Curve and a Line
- Area Between Two Curves
Differential Equations
- Introduction of Differential Equations
- Differential Equations
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Formation of a Differential Equation Whose General Solution is Given
- Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
- Methods of Solving First Order, First Degree Differential Equations
- Differential Equations with Variables Separable Method
- Homogeneous Differential Equations
- Linear Differential Equations
- Solutions of Linear Differential Equation
Vectors
- Basic Concepts of Vector Algebra
- Vector
- 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
- Product of Two Vectors
- Scalar (Or Dot) Product of Two Vectors
- Projection of a Vector on a Line
- Vector (Or Cross) Product of Two Vectors
- Magnitude and Direction of a Vector
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Scalar Triple Product of Vectors
Three-dimensional Geometry
- Introduction of Three Dimensional Geometry
- Direction Cosines and Direction Ratios of a Line
- Equation of a Line in Space
- Angle Between Two Lines
- Shortest Distance Between Two Lines
- Equation of a Plane in Normal Form
- Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
- Equation of a Plane Passing Through Three Non Collinear Points
- Forms of the Equation of a Straight Line
- Plane Passing Through the Intersection of Two Given Planes
- Coplanarity of Two Lines
- Angle Between Two Planes
- Distance of a Point from a Plane
- Angle Between Line and a Plane
- Vector and Cartesian Equation of a Plane
- Vector and Cartesian Equations of a Line
Linear Programming
Probability
- Concept of Probability
- Conditional Probability
- Properties of Conditional Probability
- Multiplication Theorem on Probability
- Bayes’ Theorem
- Random Variables and Its Probability Distributions
- Mean of Grouped Data
- Variance of a Random Variable
- Bernoulli Trials and Binomial Distribution
- Independent Events
- Binomial Distribution
Numbers, Quantification and Numerical Applications
- Modulo Arithmetic
- Apply Arithmetic Operations Using Modular Arithmetic Rules
- Apply the Definition of Congruence Modulo in Various Problems
- Allegation and Mixture
- Rule of Allegation to Produce a Mixture at a Given Price
- Determine the Mean Price of Amixture
- Apply Rule of Allegation
- Solve Real Life Problems Mathematically
- Boats and Streams (Entrance Exam)
- Express the Boats and Streams Problem in the Form of an Equation
- Pipes and Cisterns (Entrance Exam)
- Races and Games
- Differentiate Between Active Partner and Sleeping Partner
- Determination of Partner's Ratio
- Surface Area of a Combination of Solids
- Numerical Inequalities
Algebra
Calculus
- Second Order Derivative
- Higher Order Derivative
- Derivatives of Functions in Parametric Forms
- Derivatives of Implicit Functions
- Dependent and Independent Variables
- Marginal Cost and Marginal Revenue Using Derivatives
- Maxima and Minima
Probability Distributions
Index Numbers and Time Based Data
- Index Numbers
- Test of Adequacy of Index Numbers
- Population and Sample
- Differentiate Between Population and Sample
- Representative Sample from a Population
- Parameter
- Concepts of Statistics
- Relation Between Parameter and Statistic
- Limitations of Statistics to Generalize the Estimation for Population
- Statistical Significance and Statistical Inferences
- Central Limit Theorem
- Relation Between Population, Sampling Distribution, and Sample
- Time Series Analysis
- Components of a Time Series
- Time Series Analysis for Uni-variate Data
Financial Mathematics
- Perpetuity Fund
- Sinking Fund
- Calculate Perpetuity
- Differentiate Between Sinking Fund and Saving Account
- Valuation of Bond
- Calculate Value of Bond Using present Value Approach
- Concept of EMI
- Calculation of EMI
- Fixed Instalment Method
- Interpretation Cost, Residual Value and Useful Life of an Asset
Linear Programming
Text
Let E and F be events of a sample space S of an experiment, then we have
Property : P(S|F) = P(F|F) = 1
We know that
P(S|F) = `(P(S ∩ F))/(P(F)) = (P(F))/(P(F)) = 1`
Also P(F|F) = `(P(F ∩ F))/(P(F)) = (P(F))/(P(F)) = 1`
Thus P(S|F) = P(F|F) = 1
Property : If A and B are any two events of a sample space S and F is an event of S such that P(F) ≠ 0, then
P((A ∪ B)|F) = P(A|F) + P(B|F) – P((A ∩ B)|F)
In particular, if A and B are disjoint events, then
P((A∪B)|F) = P(A|F) + P(B|F)
We have
P((A∪B)|F) = `(P[(A ∪ B) ∩ F]) /(P(F))`
= `(P[(A ∩ F )∪ (B ∩ F)])/ (P(F))`
(by distributive law of union of sets over intersection)
`= (P(A ∩ F) + P (B ∩ F) - P(A ∩ B ∩ F))/(P(F))`
`= (P(A ∩ F))/(P(F)) + (P (B ∩ F)) / (P(F)) - (P[(A ∩ B ∩ F)]) /(P(F))`
= P(A|F) + P(B|F) – P((A∩B)|F)
When A and B are disjoint events, then
P((A ∩ B)|F) = 0
⇒ P((A ∪ B)|F) = P(A|F) + P(B|F)
Property : P(E′|F) = 1 − P(E|F)
From first Property , we know that P(S|F) = 1
⇒ P(E ∪ E′|F) = 1 since S = E ∪ E′
⇒ P(E|F) + P (E′|F) = 1 since E and E′ are disjoint events
Thus, P(E′|F) = 1 − P(E|F)
