Topics
Relations and Functions (Section A)
- Introduction of Relations and Functions
- Types of Relations
- Types of Relations - Identity Relation
- Types of Functions
- Composition of Functions and Invertible Function
- Inverse of a Function
- Concept of Binary Operations
- All Axioms and Properties
- Conditions of Invertibility
- Basic Concepts of Inverse Trigonometric Functions
- Inverse Trigonometric Functions - Principal Value Branch
- Graphs of Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions
Algebra (Section A)
Matrices and Determinants
- Introduction of Matrices
- Matrices Notation
- Order of a Matrix
- Equality of Matrices
- Types of Matrices
- Symmetric and Skew Symmetric Matrices
- Transpose of a Matrix
- Addition of Matrices
- Multiplication of Two Matrices
- Elementary Transformations
- Multiplication of Matrices
- Invertible Matrices
- Proof of the Uniqueness of Inverse
- Introduction of Determinant
- Determinants of Matrix of Order One and Two
- Determinant of a Square Matrix
- Determinant of a Matrix of Order 3 × 3
- Properties of Determinants
- Minors and Co-factors
- Area of a Triangle
- Inverse of a Square Matrix by the Adjoint Method
- Applications of Determinants and Matrices
- Martin’S Rule
Calculus (Section A)
Continuity, Differentiability and Differentiation
- Concept of Continuity
- Continuous Function of Point
- Algebra of Continuous Functions
- Exponential and Logarithmic Functions
- Derivatives of Composite Functions - Chain Rule
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Implicit Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Mean Value Theorem
- Second Order Derivative
- L' Hospital'S Theorem
Applications of Derivatives
Integrals
- Introduction of Integrals
- Indefinite Integral
- Integrals of the Type
- Integration as an Inverse Process of Differentiation
- Methods of Integration: Integration by Substitution
- Methods of Integration: Integration Using Partial Fractions
- Methods of Integration: Integration by Parts
- Some Properties of Indefinite Integral
- Anti-derivatives of Polynomials and Functions
- Evaluation of Simple Integrals of the Following Types and Problems
- Definite Integral as the Limit of a Sum
- Fundamental Theorem of Calculus
- Definite Integrals
- Properties of Definite Integrals
- Evaluation of Definite Integrals by Substitution
Differential Equations
- Basic Concepts of Differential Equation
- 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
- Formation of Differential Equation by Eliminating Arbitary Constant
- Differential Equations with Variables Separable Method
- Homogeneous Differential Equations
- Solutions of Linear Differential Equation
- Application on Growth and Decay
- Solve Problems on Velocity, Acceleration, Distance and Time
- Solve Population Based Problems on Application of Differential Equations
- Application on Coordinate Geometry
Probability (Section A)
- Introduction of Probability
- Dependent Events
- Conditional Event
- Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Bayes’ Theorem
- Addition Theorem of Probability
- Random Variables and Its Probability Distributions
- Mean of a Random Variable
- Bernoulli Trials and Binomial Distribution
- Laws of Probability
- Probability Distribution Function
- Mean of Binomial Distribution (P.M.F.)
- Variance of Binomial Distribution (P.M.F.)
Vectors (Section B)
- Magnitude and Direction of a Vector
- Vectors and Their Types
- Basic Concepts of Vector Algebra
- Components of a Vector
- Addition of Vectors
- Operations - Sum and Difference of Vectors
- Multiplication of a Vector by a Scalar
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Geometrical Interpretation of Scalar
- Scalar (Or Dot) Product of Two Vectors
- Vector (Or Cross) Product of Two Vectors
- Scalar Triple Product of Vectors
- Section Formula
Three - Dimensional Geometry (Section B)
- Direction Cosines and Direction Ratios of a Line
- Equation of a Line in Space
- Shortest Distance Between Two Lines
- Vector and Cartesian Equation of a Plane
- Angle Between Two Lines
- Angle Between Two Planes
- Angle Between Line and a Plane
- Intercept Form of the Equation of a Plane
- Distance of a Point from a Plane
- Direction Ratios of the Normal to the Plane.
- Intersection of the Line and Plane
- Equation of Plane Passing Through the Intersection of Two Given Planes
- Equation of Line Passing Through Given Point and Parallel to Given Vector
- One Point Form
- Normal Form
Application of Integrals (Section B)
- Area Under Simple Curves
- Area of the Region Bounded by a Curve and a Line
- Area Between Two Curves
- Application of Integrals - Polynomial Functions
- Application of Integrals - Modulus Function
- Application of Integrals - Trigonometric Function
- Application of Integrals - Exponential Functions
- Application of Integrals - Logarithmic Functions
Application of Calculus (Section C)
- Application of Calculus in Commerce and Economics in the Cost Function
- Application of Calculus in Commerce and Economics in the Average Cost
- Application of Calculus in Commerce and Economics in the Marginal Cost and Its Interpretation
- Application of Calculus in Commerce and Economics in the Demand Function
- Application of Calculus in Commerce and Economics in the Revenue Function
- Application of Calculus in Commerce and Economics in the Marginal Revenue Function and Its Interpretation
- Application of Calculus in Commerce and Economics in the Profit Function and Breakeven Point
- Rough Sketching
Linear Regression (Section C)
- Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
- Scatter Diagram
- The Method of Least Squares
- Lines of Best Fit
- Regression Coefficient of X on Y and Y on X
- Identification of Regression Equations
- Angle Between Regression Line and Properties of Regression Lines
- Estimation of the Value of One Variable Using the Value of Other Variable from Appropriate Line of Regression
Linear Programming (Section C)
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- Rough sketching of the following curves:- AR, MR, R, C, AC, MC and their mathematical interpretation using the concept of maxima & minima and increasing- decreasing functions.
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