DSSSB Syllabus - DSSSB TGT Mathematics

Topic: Undergraduate (UG)

Calculus:

Unit 1: Derivatives for Graphing and Applications

The first-derivative test for relative extrema Video
Concavity and inflection points Video
Second derivative test for relative extrema Video
Curve sketching using first and second derivative tests Video
Limits to infinity and infinite limits Video
Graphs with asymptotesVideo
L’Hôpital’s rule Video
Applications in business, economics and life sciences
Higher order derivatives Video
Leibniz rule Video

Unit 2: Sketching and Tracing of Curves

Parametric representation of curves and tracing of parametric curves (except lines in \(\mathbb{R}^3\)) Video
Polar coordinates and tracing of curves in polar coordinates Video
Techniques of sketching conics Video
Reflection properties of conics
Rotation of axes and second-degree equations
Classification into conics using the discriminant

Unit 3: Volume and Area of Surfaces

Volumes by slicing disks and method of washers
Volumes by cylindrical shells
Arc length
Arc length of parametric curves
Area of surface of revolution
Hyperbolic functions
Reduction formulae

Unit 4: Vector Calculus and its Applications

Introduction to vector functions and their graphs
Operations with vector functions
Limits and continuity of vector functions
Differentiation and integration of vector functions
Modeling ballistics and planetary motion
Kepler’s second law
Unit tangent, Normal and binormal vectors
Curvature

Algebra

Unit 1: Theory of Equations and Complex numbers

Polynomials
The remainder and factor theorem
Synthetic division
Factored form of a polynomial
Fundamental theorem of algebra
Relations between the roots and the coefficients of polynomial equations
Theorems on imaginary, integral and rational roots
Polar representation of complex numbers
De Moivre’s theorem for integer and rational indices and their applications
The nth roots of unity

Unit 2: Equivalence Relations and Functions

Equivalence relations
Functions
Composition of functions
Invertibility and inverse of functions
One-to-one correspondence and the cardinality of a set

Unit 3: Basic number Theory

Well ordering principle
The division algorithm in ℤ
Divisibility and the Euclidean algorithm
Fundamental theorem of arithmetic
Modular arithmetic and basic properties of congruences
Principle of mathematical induction

Unit 4: Row Echelon Form of Matrices and Applications

Systems of linear equations
Row reduction and echelon forms
Vector equations
The matrix equation \(Ax = b\)
Solution sets of linear systems
The inverse of a matrix
Subspaces
Linear independence
Basis and dimension
The rank of a matrix and applications
Introduction to linear transformations
The matrix of a linear transformation
Applications to computer graphics
Eigenvalues and eigenvectors
The characteristic equation
Cayley−Hamilton theorem

Real Analysis:

Unit 1: Real number System \(\mathbb{R}\)

Algebraic and order properties of \(\mathbb{R}\)
Absolute value of a real number
Bounded above and bounded below sets
Supremum and infimum of a nonempty subset of \(\mathbb{R}\)

Unit 2: Properties of \(\mathbb{R}\)

The completeness property of \(\mathbb{R}\)
Archimedean property
Density of rational numbers in \(\mathbb{R}\)
Definition and types of intervals
Nested intervals property
Neighborhood of a point in \(\mathbb{R}\)
Open and closed sets in \(\mathbb{R}\)

Unit 3: Sequences in \(\mathbb{R}\)

Convergent sequence
Limit of a sequence
Bounded sequence
Limit theorems
Monotone sequences
Monotone convergence theorem
Subsequences
Bolzano−Weierstrass theorem for sequences
Limit superior and limit inferior for bounded sequence
Cauchy sequence
Cauchy’s convergence criterion

Unit 4: Infinite Series

Convergence and divergence of infinite series of real numbers
Necessary condition for convergence
Cauchy criterion for convergence
Tests for convergence of positive term series: Integral test, Basic comparison test, Limit comparison test, D’Alembert’s ratio test, Cauchy’s nth root test
Alternating series
Leibniz test
Absolute and conditional convergence

Differential Equation:

Unit 1: Differential Equations and Mathematical Modeling

Differential equations and mathematical models
Order and degree of a differential equation
Exact differential equations and integrating factors of first order differential equations
Reducible second order differential equations
Applications of first order differential equations to acceleration-velocity model
Growth and decay model

Unit 2: Population Growth Models

Introduction to compartmental models
Lake pollution model (with case study of Lake Burley Griffin)
Drug assimilation into the blood (case of a single cold pill, case of a course of cold pills, case study of alcohol in the bloodstream)
Exponential growth of population
Limited growth of population
Limited growth with harvesting

Unit 3: Second and Higher Order Differential Equations

General solution of homogeneous equation of second order
Principle of superposition for a homogeneous equation
Wronskian, its properties and applications
Linear homogeneous and non-homogeneous equations of higher order with constant coefficients
Euler’s equation
Method of undetermined coefficients
Method of variation of parameters
Applications of second order differential equations to mechanical vibrations

Unit 4: Analysis of Mathematical Models

Interacting population models
Epidemic model of influenza and its analysis
Predator-prey model and its analysis
Equilibrium points
Interpretation of the phase plane
Battle model and its analysis

Theory of Real Function:

Unit 1: Limits of Functions

Limits of functions (ε δ − approach)
Sequential criterion for limits
Divergence criteria
Limit theorems
One-sided limits
Infinite limits and limits at infinity

Unit 2: Continuous Functions and their Properties

Continuous functions
Sequential criterion for continuity and discontinuity
Algebra of continuous functions
Properties of continuous functions on closed and bounded intervals
Uniform continuity
Non-uniform continuity criteria
Uniform continuity theorem

Unit 3: Derivability and its Applications

Differentiability of a function
Algebra of differentiable functions
Carathéodory’s theorem
Chain rule
Relative extrema
Interior extremum theorem
Rolle’s theorem
Mean-value theorem and applications
Intermediate value property of derivatives
Darboux’s theorem

Unit 4: Taylor’s Theorem and its Applications

Taylor polynomial
Taylor’s theorem with Lagrange form of remainder
Application of Taylor’s theorem in error estimation
Relative extrema
Criterion for convexity
Taylor’s series expansions of e^x, sin x, and cos x

Multivariate Calculus:

Unit 1: Calculus of Functions of Several Variables

Functions of several variables
Level curves and surfaces
Limits and continuity
Partial differentiation
Higher order partial derivative
Tangent planes
Total differential and differentiability
Chain rule
Directional derivatives
The gradient
Maximal and normal property of the gradient
Tangent planes and normal lines

Unit 2: Extrema of Functions of Two Variables and Properties of Vector Field

Extrema of functions of two variables
Method of Lagrange multipliers
Constrained optimization problems
Definition of vector field
Divergence and curl

Unit 3: Double and Triple Integrals

Double integration over rectangular and nonrectangular regions
Double integrals in polar coordinates
Triple integral over a parallelopiped and solid regions
Volume by triple integrals
Triple integration in cylindrical and spherical coordinates
Change of variables in double and triple integrals

Unit 4: Green’s, Stokes’ and Gauss Divergence Theorem

Line integrals
Applications of line integrals: Mass and Work
Fundamental theorem for line integrals
Conservative vector fields
Green’s theorem
Area as a line integral
Surface integrals
Stokes’ theorem
Gauss divergence theorem

Partial Differential Equations:

Unit 1: First Order PDE and Method of Characteristics

Introduction
Classification
Construction and geometrical interpretation of first order partial differential equations (PDE)
Method of characteristic and general solution of first order PDE
Canonical form of first order PDE
Method of separation of variables for first order PDE

Unit 2: Mathematical Models and Classification of 2nd Order Linear PDE

Gravitational potential
Conservation laws and Burger’s equations
Classification of second order PDE
Reduction to canonical forms
Equations with constant coefficients
General solution

Unit 3: The Cauchy Problem and Wave Equations

Mathematical modeling of vibrating string and vibrating membrane
Cauchy problem for second order PDE
Homogeneous wave equation
Initial boundary value problems
Nonhomogeneous boundary conditions
Finite strings with fixed ends
Non-homogeneous wave equation
Goursat problem

Unit 4: Method of Separation of Variables

Method of separation of variables for second order PDE
Vibrating string problem
Existence and uniqueness of solution of vibrating string problem
Heat conduction problem
Existence and uniqueness of solution of heat conduction problem
Non-homogeneous problem

Riemann Integration and Series of Functions

Unit 1: Riemann Integration

Definition of Riemann integration
Inequalities for upper and lower Darboux sums
Necessary and sufficient conditions for Riemann integrability
Definition of Riemann integration by Riemann sum and equivalence of the two definitions
Riemann integrability of monotone functions and continuous functions
Properties of Riemann integrable functions
Definitions of piecewise continuous and piecewise monotone functions and their Riemann integrability
Intermediate value theorem for integrals
Fundamental theorems (I and II) of calculus
Integration by parts

Unit 2: Improper Integral

Improper integrals of Type-I, Type-II and mixed type
Convergence of beta and gamma functions, and their properties

Unit 3: Sequence and Series of Functions

Pointwise and uniform convergence of sequence of functions
Theorem on the continuity of the limit function of a sequence of functions
Theorems on the interchange of the limit and derivative, and the interchange of the limit and integrability of a sequence of functions
Pointwise and uniform convergence of series of functions
Theorems on the continuity, derivability and integrability of the sum function of a series of functions
Cauchy criterion and the Weierstrass M-test for uniform convergence

Unit 4: Power Series

Definition of a power series
Radius of convergence
Absolute convergence (Cauchy−Hadamard theorem)
Uniform convergence
Differentiation and integration of power series
Abel’s theorem

Ring Theory and Linear Algebra-I

Unit 1: Introduction of Rings

Definition and examples of rings
Properties of rings
Subrings
Integral domains and fields
Characteristic of a ring
Ideals, Ideal generated by a subset of a ring
Factor rings
Operations on ideals
Prime and maximal ideals

Unit 2: Ring Homomorphisms

Ring homomorphisms
Properties of ring homomorphisms
First, Second and Third Isomorphism theorems for rings
The Field of quotients

Unit 3: Introduction of Vector Spaces

Vector spaces
Subspaces
Algebra of subspaces
Quotient spaces
Linear combination of vectors
Linear span
Linear independence
Basis and dimension
Dimension of subspaces

Unit 4: Linear Transformations

Linear transformations
Null space
Range
Rank and nullity of a linear transformation
Matrix representation of a linear transformation
Algebra of linear transformations
Isomorphisms
Isomorphism theorems
Invertibility and the change of coordinate matrix

Metric Spaces

Unit 1: Basic Concepts

Definition and examples of metric spaces
Sequences in metric spaces
Cauchy sequences
Complete metric space

Unit 2: Topology of Metric Spaces

Open and closed ball
Neighborhood
Open set
Interior of a set
Limit point of a set
Derived set
Closed set
Closure of a set
Diameter of a set
Cantor’s theorem
Subspaces
Dense set

Unit 3: Continuity & Uniform Continuity

Continuous mappings
Sequential criterion and other characterizations of continuity
Uniform continuity
Homeomorphism
Contraction mapping
Banach fixed point theorem

Unit 4: Connectedness and Compactness

Connectedness
Connected subsets of ℝ
Connectedness and continuous mappings
Compactness
Compactness and boundedness
Continuous functions on compact spaces

Group Theory-II

Unit 1: Automorphisms and Properties

Automorphism
Inner automorphism
Automorphism groups
Automorphism groups of finite and infinite cyclic groups
Characteristic subgroups
Commutator subgroup and its properties
Applications of factor groups to automorphism groups

Unit 2: External and Internal Direct Products of Groups

External direct products of groups and its properties
The group of units modulo ( n ) as an external direct product
Applications to data security and electric circuits
Internal direct products
Classification of groups of order ( p^n ), where ( p ) is a prime
Fundamental theorem of finite abelian groups and its isomorphism classes

Unit 3: Group Action

Group actions and permutation representations
Stabilizers and kernels of group actions
Groups acting on themselves by left multiplication and consequences
Conjugacy in ( S_n )

Unit 4: Sylow Theorems and Applications

Conjugacy classes
Class equation
( p )-groups
Sylow theorems and consequences
Applications of Sylow theorems
Finite simple groups
Nonsimplicity tests
Generalized Cayley’s theorem
Index theorem
Embedding theorem and applications
Simplicity of ( A_n )

Complex Analysis

Unit 1: Analytic Functions and Cauchy-Riemann Equations

Functions of complex variable
Mappings by the exponential function
Limits
Theorems on limits
Limits involving the point at infinity
Continuity
Derivatives
Differentiation formulae
Cauchy-Riemann equations
Sufficient conditions for differentiability
Analytic functions and their examples

Unit 2: Elementary Functions and Integrals

Exponential function
Logarithmic function
Branches and derivatives of logarithms
Trigonometric function
Derivatives of functions
Definite integrals of functions
Contours
Contour integrals and its examples
Upper bounds for moduli of contour integrals

Unit 3: Cauchy’s Theorems and Fundamental Theorem of Algebra

Antiderivatives
Proof of antiderivative theorem
Cauchy-Goursat theorem
Cauchy integral formula
An extension of Cauchy integral formula
Consequences of Cauchy integral formula
Liouville’s theorem
Fundamental theorem of algebra

Unit 4: Series and Residues

Convergence of sequences and series
Taylor series and its examples
Laurent series and its examples
Absolute and uniform convergence of power series
Uniqueness of series representations of power series
Isolated singular points
Residues
Cauchy’s residue theorem
Residue at infinity
Types of isolated singular points
Residues at poles and its examples

Group Theory & Linear Algebra-II

Unit 1: Polynomial Rings and Unique Factorization Domain (UFD)

Polynomial rings over commutative rings
Division algorithm and consequences
Principal ideal domains
Factorization of polynomials
Reducibility tests
Irreducibility tests
Eisenstein’s criterion
Unique factorization in ( \mathbb{Z}[x] )
Divisibility in integral domains
Irreducibles
Primes
Unique factorization domains
Euclidean domains

Unit 2: Dual Spaces and Diagonalizable Operators

Dual spaces
Double dual
Dual basis
Transpose of a linear transformation and its matrix in the dual basis
Annihilators
Eigenvalues
Eigenvectors
Eigenspaces and characteristic polynomial of a linear operator
Diagonalizability
Invariant subspaces
Cayley-Hamilton theorem
Minimal polynomial for a linear operator

Unit 3: Inner Product Spaces

Inner product spaces and norms
Orthonormal basis
Gram-Schmidt orthogonalization process
Orthogonal complements
Bessel’s inequality

Unit 4: Adjoint Operators and Their Properties

Adjoint of a linear operator
Least squares approximation
Minimal solutions to systems of linear equations
Normal, self-adjoint, unitary and orthogonal operators and their properties