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
Consider a plane whose perpendicular distance from the origin is d (d ≠ 0). in following fig. 
If `vec (ON)` is the normal from the origin to the plane, and `hat n ` is the unit normal vector along `vec (ON)`. Then `vec (ON)` = d . `hat n` . Let P be any point on the plane. Therefore , `vec (NP)` is perpendicular to `vec (ON)`.
Therefore, `vec (NP) . vec (ON) = 0` ...(1)
Let `vec r` be the position vector of the point P, then `vec (NP) = vec r - d . hat n` (as `vec (ON) + vec (NP) = vec (OP)`)
Therefore, (1) becomes
`(vec r - d . hat n) . d hat n = 0`
or `(vec r - d.hat n). hat n =0` (d ≠ 0)
or `vec r . hat n - d hat n . hat n = 0`
i.e., `vec r . hat n = d` (as `hat n . hat n = 1`) ...(2)
This is the vector form of the equation of the plane.
Cartesian form
Equation (2) gives the vector equation of a plane, where `hat n` is the unit vector normal to the plane. Let P(x, y, z) be any point on the plane. Then
`vec (OP) = vec r = x hat i + y hat j + z hat k`
Let l, m, n be the direction cosines of `hat n` . Then
`hat n = l hat i + m hat j + n hat k `
Therefore, (2) gives
`(x hat i + y hat j + z hat k) . (l hat i + m hat j + n hat k) = d`
i.e. lx + my + nz = d ... (3)
