Topics
Relations and Functions
Mathematics
Applied Mathematics
Inverse Trigonometric Functions
Matrices
- Concept of Matrices
- Types of Matrices
- Equality of Matrices
- Operations on Matrices>Scalar Multiplication
- Operations on Matrices> Addition and Subtraction of Matrices
- Operations on Matrices>Scalar Multiplication
- Operations on Matrices> Matrix Multiplication
- Negative of Matrix
- Transpose of a Matrix
- Symmetric and Skew Symmetric Matrices
- Symmetric and Skew Symmetric Matrices
- Invertible Matrices
Determinants
- Determinant Method (Cramer’s Rule)
- Determinant of a Matrix
- Determinant of a Matrix
- Properties of Determinants
- Application of Determinants
- Area of a Triangle Using Determinants
- Minors and Co-factors
- Adjoint & Inverse of Matrix
- Operations on Matrices> Matrix Multiplication
- Applications of Determinants and Matrices
Continuity and Differentiability
- Concept of Differentiability
- Continuous and Discontinuous Functions
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivatives 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
- Mean Value Theorem
Applications of Derivatives
Integrals
- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Geometrical Interpretation of Indefinite Integrals
- Properties of Indefinite Integral
- Comparison Between Differentiation and Integration
- Methods of Integration> Integration by Substitution
- Methods of Integration> Integration Using Partial Fraction
- Integrals of Some Particular Functions
- Methods of Integration> Integration by Parts
- Methods of Integration>Integration Using Trigonometric Identities
- Definite Integrals
- Definite Integral as the Limit of a Sum
- Fundamental Theorem of Integral 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 Bounded by Two Curves
Differential Equations
- Basic Concepts of 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
- 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
- Solutions of Linear Differential Equation
Vectors
- Basic Concepts of Vector Algebra
- Vector
- Vector Operations>Addition and Subtraction of Vectors
- Algebra of Vector Addition
- Components of Vector in Algebra
- Vector Joining Two Points in Algebra
- Section Formula in Coordinate Geometry
- Product of Two Vectors
- Projection of a Vector on a Line
- Magnitude and Direction of a Vector
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Scalar Triple Product
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
- Equations of Line in Different Forms
- 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
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
- Derivative 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
Notes
(a) If the given family `F_1` of curves depends on only one parameter then it is represented by an equation of the form
`F_1` (x, y, a) = 0 ... (1)
For example, the family of parabolas `y^2` = ax can be represented by an equation of the form f (x, y, a) : `y^2` = ax.
Differentiating equation (1) with respect to x, we get an equation involving y′, y, x, and a, i.e.,
g (x, y, y′, a) = 0 ... (2)
The required differential equation is then obtained by eliminating a from equations (1) and (2) as
F(x, y, y′) = 0 ...(3)
b) If the given family `F_2` of curves depends on the parameters a, b (say) then it is represented by an equation of the from
`F_2` (x, y, a, b) = 0 ... (4)
Differentiating equation (4) with respect to x, we get an equation
involving y′, x, y, a, b, i.e.,
g (x, y, y′, a, b) = 0 ... (5)
But it is not possible to eliminate two parameters a and b from the two equations and so, we need a third equation. This equation is obtained by differentiating equation (5), with respect to x, to obtain a relation of the form h (x, y, y′, y″, a, b) = 0 ... (6)
The required differential equation is then obtained by eliminating a and b from equations (4), (5) and (6) as
F (x, y, y′, y″) = 0 ... (7)
