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)
Related QuestionsVIEW ALL [27]
Read the following passage and answer the questions given below.
 There are two antiaircraft guns, named as A and B. The probabilities that the shell fired from them hits an airplane are 0.3 and 0.2 respectively. Both of them fired one shell at an airplane at the same time. |
- What is the probability that the shell fired from exactly one of them hit the plane?
- If it is known that the shell fired from exactly one of them hit the plane, then what is the probability that it was fired from B?
