CSIR-UGC NET/JRF EXAM SYLLABUS FOR ENGINEERING SCIENCE

SUBJECT-ENGINEERING SCIENCE

CSIR-UGC (NET) EXAM FOR AWARD OF JUNIOR RESEARCH
FELLOWSHIP AND ELIGIBILITY FOR LECTURERSHIP
EXAM SCHEME FOR SINGLE PAPER CSIR-UGC NET in Engineering Sciences
The pattern for the Single Paper MCQ test in Engineering Sciences shall be as given below:-
The MCQ test paper in Engineering Science shall carry a maximum of 200 marks. The
duration of exam shall be three hours. The question paper shall be divided in three parts
Part ‘A’. This part shall carry 20 questions of General Aptitude (Logical reasoning,
graphical analysis, analytical and numerical ability, quantitative comparisons, series
formation, puzzles, etc). Candidates shall be required to answer any 15 questions. Each
question shall be of 2 marks. Total marks allocated to this section shall be 30 out of 200.

Part ‘B’: This part shall contain 25 questions related to Mathematics and Engineering
Aptitude. Candidates shall be required to answer any 20 questions. Each question shall be of
3.5 marks. Total marks allocated to this section shall be 70 out of 200.

Part ‘C’ shall contain subject related questions of the following 7 subject areas :
1. Computer Science & Information Technology
2 Electrical Science
3. Electronics
4. Materials Science
5. Fluid Mechanics
6. Solid Mechanics
7. Thermodynamics
Each subject area will have 10 questions. Candidates shall be required to answer any 20
questions out of a total of 70 questions. Each question shall be of 5 marks. The total marks
allocated to this part shall be 100 out of 200.
Negative marking for wrong answers shall be @ 25%
NB: The actual number of questions in each Part and Section to be asked and attempted may
vary from exam to exam.
SYLLABUS PART A
General aptitude with emphasis on logical reasoning, graphical analysis, analytical and
numerical ability, quantitative comparisons, series formation, puzzles, etc.
SYLLABUS PART B
Mathematics And Engineering Aptitude
Linear Algebra
Calculus
Complex variables
Vector Calculus
Ordinary Differential
Algebra of matrices, inverse, rank, system of linear equations,
symmetric, skew-symmetric and orthogonal matrices. Hermitian,
skew-Hermitian and unitary matrices. eigenvalues and
eigenvectors, diagonalisation of matrices.
Functions of single variable, limit, continuity and differentiability,
Mean value theorems, Indeterminate forms and L'Hospital rule,
Maxima and minima, Taylor's series, Newton’s method for finding
roots of polynomials. Fundamental and mean value-theorems of
integral calculus. Numerical integration by trapezoidal and
Simpson’s rule. Evaluation of definite and improper integrals,
Beta and Gamma functions, Functions of two variables, limit,
continuity, partial derivatives, Euler's theorem for homogeneous
functions, total derivatives, maxima and minima, Lagrange method
of multipliers, double integrals and their applications, sequence and
series, tests for convergence, power series, Fourier Series, Half
range sine and cosine series.
Analytic functions, Cauchy-Riemann equations, Line integral,
Cauchy's integral theorem and integral formula Taylor’s and
Laurent' series, Residue theorem and its applications.
Gradient, divergence and curl, vector identities, directional
derivatives, line, surface and volume integrals, Stokes, Gauss and
Green's theorems and their applications.
First order equation (linear and nonlinear), Second order linear
differential equations with variable coefficients, Variation of
Equations
Probability
parameters method, higher order linear differential equations with
constant coefficients, Cauchy-Euler's equations, power series
solutions, Legendre polynomials and Bessel's functions of the first
kind and their properties. Numerical solutions of first order
ordinary differential equations by Euler’s and Runge-Kutta
methods.
Definitions of probability and simple theorems, conditional
probability, Bayes Theorem.
Solid Body Motion and
Fluid Motion:
Energetics:
Electron Transport:
Electromagnetics:
Materials:
Particle dynamics; Projectiles; Rigid Body Dynamics; Lagrangian
formulation; Eularian formulation; Bernoulli’s Equation;
Continuity equation; Surface tension; Viscosity; Brownian Motion.
Laws of Thermodynamics; Concept of Free energy; Enthalpy, and
Entropy; Equation of State; Thermodynamics relations.
Structure of atoms, Concept of energy level, Bond Theory;
Definition of conduction, Semiconductor and Insulators; Diode;
Half wave & Full wave rectification; Amplifiers & Oscillators;
Truth Table.
Theory of Electric and Magnetic potential & field; Biot & Savart’s
Law; Theory of Dipole; Theory of Oscillation of electron;
Maxwell’s equations; Transmission theory; Amplitude &
Frequency Modulation.
Periodic table; Properties of elements; Reaction of materials;
Metals and non-Metals (Inorganic materials), Elementary
knowledge of monomeric and polymeric compounds;
Organometallic compounds; Crystal structure and symmetry,
Structure-property correlation-metals, ceramics, and polymers.
SYLLABUS PART C
1. COMPUTER SCIENCE AND INFORMATION TECHNOLOGY
Basic Discrete Mathematics: Counting principles, linear recurrence, mathematical induction,
equation sets, relations and function, predicate and propositional logic.
Digital Logic:
Logic functions, Minimization, Design and synthesis of combinational and sequential circuits;
Number representation and computer arithmetic (fixed and floating point).
Computer Organization and Architecture:
Machine instructions and addressing modes, ALU and data-path, CPU control design, Memory
interface, I/O interface (Interrupt and DMA mode), Instruction pipelining, Cache and main
memory, Secondary storage.
Programming and Data Structures:
Programming in C; Functions, Recursion, Parameter passing, Scope, Binding; Abstract data
types, Arrays, Stacks, Queues, Linked Lists, Trees, Binary search trees, Binary heaps.
Algorithms:
Analysis, Asymptotic notation, Notions of space and time complexity, Worst and average case
analysis; Design: Greedy approach, Dynamic programming, Divide-and conquer; Tree and graph
traversals, Connected components, Spanning trees, Shortest paths; Hashing, Sorting, Searching.
Asymptotic analysis (best, worst, average cases) of time and space, upper and lower bounds,
Basic concepts of complexity classes P, NP, NP-hard, NP-complete.
Operating System:
Processes, Threads, Inter-process communication, Concurrency, Synchronization, Deadlock,
CPU scheduling, Memory management and virtual memory, File systems.
Databases:
ER-model, Relational model (relational algebra, tuple calculus), Database design (integrity
constraints, normal forms), Query languages (SQL), File structures (sequential files, indexing, B
and B+ trees), Transactions and concurrency control.
Information Systems and Software Engineering:
information gathering, requirement and feasibility analysis, data flow diagrams, process
specifications, input/output design, process life cycle, planning and managing the project, design,
coding, testing, implementation, maintenance.
2. ELECTRICAL SCIENCES
Electric Circuits and Fields:
Node and mesh analysis, transient response of dc and ac networks, sinusoidal steady-state
analysis, resonance, basic filter concepts, ideal current and voltage sources, Thevenin’s,
Norton’s and Superposition and Maximum Power Transfer theorems, two port networks, three
phase circuits, measurement of power in three phase circuits, Gauss Theorem, electric field and
potential due to point, line, plane and spherical charge distributions, Ampere’s and Biot-Savart’s
laws, inductance, dielectrics , capacitance.
Electrical Machines: Magnetic circuits
Magnetic circuits, Single phase transformer- equivalent circuit, phasor diagram, tests, regulation
and efficiency, Three phase transformers- connections, parallel operation, auto-transformer;
energy conversion principles, DC Machines- types , starting and speed control of dc motors,
Three phase induction motors- principles, types, performance characteristics, starting and speed
control , Single phase induction motors, synchronous machines performance, regulation and
parallel operation of synchronous machine operating as generators, starting and speed control of
synchronous motors and its applications, servo and stepper motors.
Power Systems:
Basic power generation concepts, transmission line models and performance, cable performance,
insulation, corona and radio interference , Distribution systems, per-unit quantities, bus
impedance and admittance matrices, load flow, voltage and frequency control, power factor
correction; unbalanced analysis, symmetrical components, basic concepts of protection and
stability; Introduction to HVDC systems.
Control Systems:
Principles of feedback control, transfer function, block diagrams, steady state errors, Routh and
Nyquist techniques, Bode plots, Root loci, Lag , Lead and Lead-lag compensation; proportional,
PI, PID controllers, state space model , state transition matrix, controllability and observability.
Power Electronics and Drives:
Semiconductor Power devices - power diodes, power transistors, thyristors, triacs, GTOs,
MOSFETs, IGBTs – their characteristics and basic triggering circuits; diode rectifiers, thyristor
based line commutated ac to dc converters, dc to dc converters – buck, boost, buck-boost, c`uk,
flyback, forward, push-pull converters, single phase and three phase dc to ac inverters and
related pulse width modulation techniques, stability of electric drives; speed control issues of dc
motors, induction motors and synchronous motors.
3. ELECTRONICS
Analog Circuits and Systems:
Electronic devices: characteristics and small-signal equivalent circuits of diodes, BJTs and
MOSFETs. Diode circuits: clipping, clamping and rectifier. Biasing and bias stability of BJT and
FET amplifiers. Amplifiers: single-and multi-stage, differential and operational, feedback, and
power. Frequency response of amplifiers. Op-amp circuits: voltage-to-current and current-tovoltage
converters, active filters, sinusoidal oscillators, wave-shaping circuits, effect of practical
parameters (input bias current, input offset voltage, open loop gain, input resistance, CMRR).
Electronic measurements: voltage, current, impedance, time, phase, frequency measurements,
oscilloscope.
Digital Circuits and Systems:
Boolean algebra and minimization of Boolean functions. Logic gates, TTL and CMOS IC
families. Combinatorial circuits: arithmetic circuits, code converters, multiplexers and decoders.
Sequential circuits: latches and flip-flops, counters and shift-registers. Sample-and-hold
circuits,ADCs, DACs. Microprocessors and microcontrollers: number systems, 8085 and 8051
architecture, memory, I/O interfacing, Serial and parallel communication.
Signals and Systems:
Linear time invariant systems: impulse response, transfer function and frequency response of
first- and second order systems, convolution. Random signals and noise: probability, random
variables, probability density function, autocorrelation, power spectral density. Sampling
theorem, Discrete-time systems: impulse and frequency response, IIR and FIR filters.
Communications:
Amplitude and angle modulation and demodulation, frequency and time division multiplexing.
Pulse code modulation, amplitude shift keying, frequency shift keying and pulse shift keying for
digital modulation. Bandwidth and SNR calculations. Information theory and channel capacity.
4. MATERIALS SCIENCE
Structure:
Atomic structure and bonding in materials. Crystal structure of materials, crystal systems, unit
cells and space lattices, miller indices of planes and directions, packing geometry in metallic,
ionic and covalent solids. Concept of amorphous, single and polycrystalline structures and their
effect on properties of materials. Imperfections in crystalline solids and their role in influencing
various properties.
Diffusion: Fick's laws and application of diffusion.
Metals and Alloys:
Solid solutions, solubility limit, phase rule, binary phase diagrams, intermediate phases,
intermetallic compounds, iron-iron carbide phase diagram, heat treatment of steels, cold, hot
working of metals, recovery, recrystallization and grain growth. Microstrcture, properties and
applications of ferrous and non-ferrous alloys.
Ceramics, Polymers, & Composites:
Structure, properties, processing and applications of ceramics. Classification, polymerization,
structure and properties, processing and applications. Properties and applications of various
composites.
Materials Characterization Tools:
X-ray diffraction, optical microscopy, scanning electron microscopy and transmission electron
microscopy, differential thermal analysis, differential scanning calorimetry.
Materials Properties:
Stress-strain diagrams of metallic, ceramic and polymeric materials, modulus of elasticity, yield
strength, tensile strength, toughness, elongation, plastic deformation, viscoelasticity, hardness,
impact strength, creep, fatigue, ductile and brittle fracture.
Heat capacity, thermal conductivity, thermal expansion of materials. Concept of energy band
diagram for materials - conductors, semiconductors and insulators, intrinsic and extrinsic
semiconductors, dielectric properties. Origin of magnetism in metallic and ceramic materials,
paramagnetism, diamagnetism, antiferro magnetism, ferromagnetism, ferrimagnetism, magnetic
hysterisis.
Environmental Degradation:
Corrosion and oxidation of materials, prevention.
5. FLUID MECHANICS
Fluid Properties:
Relation between stress and strain rate for Newtonian fluids; Buoyancy, manometry, forces on
submerged bodies.
Kinematics
Eulerian and Lagrangian description of fluid motion, strain rate and vorticity; concept of local
and convective accelerations, steady and unsteady flows
Control Volume Based Analysis
Control volume analysis for mass, momentum and energy.
Differential equations of mass and momentum (Euler equation), Bernoulli's equation and its
applications, Concept of fluid rotation.
Potential flow:
Vorticity, Stream function and Velocity potential function; Elementary flow fields and principles
of superposition, potential flow past a circular cylinder.
Dimensional analysis:
Concept of geometric, kinematic and dynamic similarity, Non-dimensional numbers and their
usage.
Viscous Flows
Navier-Stokes Equations; Exact Solutions; Couette Flow, Fully-developed pipe flow,
Hydrodynamic lubrication, Basic ideas of Laminar and Turbulent flows, Prandtl-mixing length,
Friction factor, Darcy-Weisbach relation, Simple pipe networks.
Boundary Layer
Qualitative ideas of boundary layer, Boundary Layer Equation; Separation, Streamlined and
bluff bodies, drag and lift forces.
Measurements
Basic ideas of flow measurement using venturimeter, pitot-static tube and orifice plate.
6. SOLID MECHANICS
Equivalent force systems; free-body diagrams; equilibrium equations; analysis of determinate
trusses and frames; friction; simple particle dynamics; plane kinematics and kinetics; workenergy
and impulse-momentum principles;
Stresses and strains; principal stresses and strains; Mohr's circle; generalized Hooke's Law;
thermal strain.
Axial, shear and bending moment diagrams; axial, shear and bending stresses; deflection of
beams (symmetric bending); Torsion in circular shafts; thin walled pressure vessels. Energy
methods (Catigliano’s theorems) for analysis.
Combined axial, bending and torsional action; Theories of failure.
Buckling of columns.
Free vibration of single degree of freedom systems.
7. THERMODYNAMICS
Basic Concepts:
Continuum, macroscopic approach, thermodynamic system (closed and open or control volume);
thermodynamic properties and equilibrium; state of a system, state diagram, path and process;
different modes of work; Zeroth law of thermodynamics; concept of temperature; heat.
First Law of Thermodynamics:
Energy, enthalpy, specific heats, first law applied to closed systems and open systems (control
volumes), steady and unsteady flow analysis.
Second Law of Thermodynamics:
Kelvin-Planck and Clausius statements, reversible and irreversible processes, Carnot theorems,
thermodynamic temperature scale, Clausius inequality and concept of entropy, principle of
increase of entropy, entropy balance for closed and open systems, exergy (availability) and
irreversibility, non-flow and flow exergy.
Properties of Pure Substances:
Thermodynamic properties of pure substances in solid, liquid and vapor phases, P-V-T behaviour
of simple compressible substances, phase rule, thermodynamic property tables and charts, ideal
and real gases, equations of state, compressibility chart.
Thermodynamic Relations:
T-ds relations, Maxwell equations, Joule-Thomson coefficient, coefficient of volume expansion,
adiabatic and isothermal compressibilities, Clapeyron equation.
Thermodynamic cycles:
Carnot vapour power cycle; simple Rankine cycle, reheat and regenerative Rankine cycle; Air
standard cycles: Otto cycle, Diesel cycle, simple Brayton cycle, Brayton cycle with regeneration,
reheat and intercooling; vapour-compression refrigeration cycle.
Ideal Gas Mixtures:
Dalton's and Amagat's laws, calculations of properties (internal energy, enthalpy, entropy), airwater
vapour mixtures and simple thermodynamic processes involving them.

CSIR-UGC NET/JRF SYLLABUS FOR PHYSICAL SCIENCE

CSIR-UGC National Eligibility Test (NET) for JRF and Lecturer-ship

 SUBJECT- PHYSICAL SCIENCES

PART ‘A’ CORE

I. Mathematical Methods of Physics
Dimensional analysis. Vector algebra and vector calculus. Linear algebra, matrices, Cayley-Hamilton Theorem. Eigenvalues and eigenvectors. Linear ordinary differential equations of first & second order, Special functions (Hermite, Bessel, Laguerre and Legendre functions). Fourier series, Fourier and Laplace transforms. Elements of complex analysis, analytic functions; Taylor & Laurent series; poles, residues and evaluation of integrals. Elementary probability theory, random variables, binomial, Poisson and normal distributions. Central limit theorem.
II. Classical Mechanics
Newton’s laws. Dynamical systems, Phase space dynamics, stability analysis. Central force motions. Two body Collisions - scattering in laboratory and Centre of mass frames. Rigid body dynamics- moment of inertia tensor. Non-inertial frames and pseudoforces. Variational principle. Generalized coordinates. Lagrangian and Hamiltonian formalism and equations of motion. Conservation laws and cyclic coordinates. Periodic motion: small oscillations, normal modes. Special theory of relativity- Lorentz transformations, relativistic kinematics and mass–energy equivalence.
III. Electromagnetic Theory
Electrostatics: Gauss’s law and its applications, Laplace and Poisson equations, boundary value problems. Magnetostatics: Biot-Savart law, Ampere's theorem. Electromagnetic induction. Maxwell's equations in free space and linear isotropic media; boundary conditions on the fields at interfaces. Scalar and vector potentials, gauge invariance. Electromagnetic waves in free space. Dielectrics and conductors. Reflection and refraction, polarization, Fresnel’s law, interference, coherence, and diffraction. Dynamics of charged particles in static and uniform electromagnetic fields.
IV. Quantum Mechanics
Wave-particle duality. Schrödinger equation (time-dependent and time-independent). Eigenvalue problems (particle in a box, harmonic oscillator, etc.). Tunneling through a barrier. Wave-function in coordinate and momentum representations. Commutators and Heisenberg uncertainty principle. Dirac notation for state vectors. Motion in a central potential: orbital angular momentum, angular momentum algebra, spin, addition of angular momenta; Hydrogen atom. Stern-Gerlach experiment. Time-independent perturbation theory and applications. Variational method. Time dependent perturbation theory and Fermi's golden rule, selection rules. Identical particles, Pauli exclusion principle, spin-statistics connection.
V. Thermodynamic and Statistical Physics
Laws of thermodynamics and their consequences. Thermodynamic potentials, Maxwell relations, chemical potential, phase equilibria. Phase space, micro- and macro-states. Micro-canonical, canonical
and grand-canonical ensembles and partition functions. Free energy and its connection with thermodynamic quantities. Classical and quantum statistics. Ideal Bose and Fermi gases. Principle of detailed balance. Blackbody radiation and Planck's distribution law.
VI. Electronics and Experimental Methods
Semiconductor devices (diodes, junctions, transistors, field effect devices, homo- and hetero-junction devices), device structure, device characteristics, frequency dependence and applications. Opto-electronic devices (solar cells, photo-detectors, LEDs). Operational amplifiers and their applications. Digital techniques and applications (registers, counters, comparators and similar circuits). A/D and D/A converters. Microprocessor and microcontroller basics.
Data interpretation and analysis. Precision and accuracy. Error analysis, propagation of errors. Least squares fitting,

PART ‘B’ ADVANCED

I. Mathematical Methods of Physics
Green’s function. Partial differential equations (Laplace, wave and heat equations in two and three dimensions). Elements of computational techniques: root of functions, interpolation, extrapolation, integration by trapezoid and Simpson’s rule, Solution of first order differential equation using Runge-Kutta method. Finite difference methods. Tensors. Introductory group theory: SU(2), O(3).
II. Classical Mechanics
Dynamical systems, Phase space dynamics, stability analysis. Poisson brackets and canonical transformations. Symmetry, invariance and Noether’s theorem. Hamilton-Jacobi theory.
III. Electromagnetic Theory
Dispersion relations in plasma. Lorentz invariance of Maxwell’s equation. Transmission lines and wave guides. Radiation- from moving charges and dipoles and retarded potentials.
IV. Quantum Mechanics
Spin-orbit coupling, fine structure. WKB approximation. Elementary theory of scattering: phase shifts, partial waves, Born approximation. Relativistic quantum mechanics: Klein-Gordon and Dirac equations. Semi-classical theory of radiation.
V. Thermodynamic and Statistical Physics
First- and second-order phase transitions. Diamagnetism, paramagnetism, and ferromagnetism. Ising model. Bose-Einstein condensation. Diffusion equation. Random walk and Brownian motion. Introduction to nonequilibrium processes.
VI. Electronics and Experimental Methods
Linear and nonlinear curve fitting, chi-square test. Transducers (temperature, pressure/vacuum, magnetic fields, vibration, optical, and particle detectors). Measurement and control. Signal conditioning and recovery. Impedance matching, amplification (Op-amp based, instrumentation amp, feedback), filtering
and noise reduction, shielding and grounding. Fourier transforms, lock-in detector, box-car integrator, modulation techniques.
High frequency devices (including generators and detectors).
VII. Atomic & Molecular Physics
Quantum states of an electron in an atom. Electron spin. Spectrum of helium and alkali atom. Relativistic corrections for energy levels of hydrogen atom, hyperfine structure and isotopic shift, width of spectrum lines, LS & JJ couplings. Zeeman, Paschen-Bach & Stark effects. Electron spin resonance. Nuclear magnetic resonance, chemical shift. Frank-Condon principle. Born-Oppenheimer approximation. Electronic, rotational, vibrational and Raman spectra of diatomic molecules, selection rules. Lasers: spontaneous and stimulated emission, Einstein A & B coefficients. Optical pumping, population inversion, rate equation. Modes of resonators and coherence length.
VIII. Condensed Matter Physics
Bravais lattices. Reciprocal lattice. Diffraction and the structure factor. Bonding of solids. Elastic properties, phonons, lattice specific heat. Free electron theory and electronic specific heat. Response and relaxation phenomena. Drude model of electrical and thermal conductivity. Hall effect and thermoelectric power. Electron motion in a periodic potential, band theory of solids: metals, insulators and semiconductors. Superconductivity: type-I and type-II superconductors. Josephson junctions. Superfluidity. Defects and dislocations. Ordered phases of matter: translational and orientational order, kinds of liquid crystalline order. Quasi crystals.
IX. Nuclear and Particle Physics
Basic nuclear properties: size, shape and charge distribution, spin and parity. Binding energy, semi-empirical mass formula, liquid drop model. Nature of the nuclear force, form of nucleon-nucleon potential, charge-independence and charge-symmetry of nuclear forces. Deuteron problem. Evidence of shell structure, single-particle shell model, its validity and limitations. Rotational spectra. Elementary ideas of alpha, beta and gamma decays and their selection rules. Fission and fusion. Nuclear reactions, reaction mechanism, compound nuclei and direct reactions.
Classification of fundamental forces. Elementary particles and their quantum numbers (charge, spin, parity, isospin, strangeness, etc.). Gellmann-Nishijima formula. Quark model, baryons and mesons. C, P, and T invariance. Application of symmetry arguments to particle reactions. Parity non-conservation in weak interaction. Relativistic kinematics.

CSIR-UGC NET/JRF SYLLABUS FOR MATHEMATICAL SCIENCES

CSIR-UGC National Eligibility Test (NET) for JRF & LECTURERSHIP

MATHEMATICAL SCIENCES

UNIT – 1
Analysis: Elementary set theory, finite, countable and uncountable sets, Real number system as a
complete ordered field, Archimedean property, supremum, infimum.
Sequences and series, convergence, limsup, liminf.
Bolzano Weierstrass theorem, Heine Borel theorem.
Continuity, uniform continuity, differentiability, mean value theorem.
Sequences and series of functions, uniform convergence.
Riemann sums and Riemann integral, Improper Integrals.
Monotonic functions, types of discontinuity, functions of bounded variation, Lebesgue measure,
Lebesgue integral.
Functions of several variables, directional derivative, partial derivative, derivative as a linear
transformation, inverse and implicit function theorems.
Metric spaces, compactness, connectedness. Normed linear Spaces. Spaces of continuous functions
as examples.
Linear Algebra: Vector spaces, subspaces, linear dependence, basis, dimension, algebra of linear
transformations.
Algebra of matrices, rank and determinant of matrices, linear equations.
Eigenvalues and eigenvectors, Cayley-Hamilton theorem.
Matrix representation of linear transformations. Change of basis, canonical forms, diagonal forms,
triangular forms, Jordan forms.
Inner product spaces, orthonormal basis.
Quadratic forms, reduction and classification of quadratic forms

UNIT – 2
Complex Analysis: Algebra of complex numbers, the complex plane, polynomials, power series,
transcendental functions such as exponential, trigonometric and hyperbolic functions.
Analytic functions, Cauchy-Riemann equations.
Contour integral, Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, Maximum
modulus principle, Schwarz lemma, Open mapping theorem.
Taylor series, Laurent series, calculus of residues.
Conformal mappings, Mobius transformations.
Algebra: Permutations, combinations, pigeon-hole principle, inclusion-exclusion principle,
derangements.
Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem,
Euler’s Ø- function, primitive roots.
Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation
groups, Cayley’s theorem, class equations, Sylow theorems.
Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal
domain, Euclidean domain.
Polynomial rings and irreducibility criteria.
Fields, finite fields, field extensions, Galois Theory.
Topology: basis, dense sets, subspace and product topology, separation axioms, connectedness and
compactness.

UNIT – 3
Ordinary Differential Equations (ODEs):
Existence and uniqueness of solutions of initial value problems for first order ordinary differential
equations, singular solutions of first order ODEs, system of first order ODEs.
General theory of homogenous and non-homogeneous linear ODEs, variation of parameters,
Sturm-Liouville boundary value problem, Green’s function.
Partial Differential Equations (PDEs):
Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for first order PDEs.
Classification of second order PDEs, General solution of higher order PDEs with constant
coefficients, Method of separation of variables for Laplace, Heat and Wave equations.
Numerical Analysis :
Numerical solutions of algebraic equations, Method of iteration and Newton-Raphson method, Rate
of convergence, Solution of systems of linear algebraic equations using Gauss elimination and
Gauss-Seidel methods, Finite differences, Lagrange, Hermite and spline interpolation, Numerical
differentiation and integration, Numerical solutions of ODEs using Picard, Euler, modified Euler and
Runge-Kutta methods.
Calculus of Variations:
Variation of a functional, Euler-Lagrange equation, Necessary and sufficient conditions for extrema.
Variational methods for boundary value problems in ordinary and partial differential equations.
Linear Integral Equations:
Linear integral equation of the first and second kind of Fredholm and Volterra type, Solutions with
separable kernels. Characteristic numbers and eigenfunctions, resolvent kernel.
Classical Mechanics:
Generalized coordinates, Lagrange’s equations, Hamilton’s canonical equations, Hamilton’s
principle and principle of least action, Two-dimensional motion of rigid bodies, Euler’s dynamical
equations for the motion of a rigid body about an axis, theory of small oscillations.

UNIT – 4
Descriptive statistics, exploratory data analysis
Sample space, discrete probability, independent events, Bayes theorem. Random variables and
distribution functions (univariate and multivariate); expectation and moments. Independent random
variables, marginal and conditional distributions. Characteristic functions. Probability inequalities
(Tchebyshef, Markov, Jensen). Modes of convergence, weak and strong laws of large numbers, Central
Limit theorems (i.i.d. case).
Markov chains with finite and countable state space, classification of states, limiting behaviour of n-step
transition probabilities, stationary distribution, Poisson and birth-and-death processes.
Standard discrete and continuous univariate distributions. sampling distributions, standard errors and
asymptotic distributions, distribution of order statistics and range.
Methods of estimation, properties of estimators, confidence intervals. Tests of hypotheses: most powerful
and uniformly most powerful tests, likelihood ratio tests. Analysis of discrete data and chi-square test of
goodness of fit. Large sample tests.
Simple nonparametric tests for one and two sample problems, rank correlation and test for independence.
Elementary Bayesian inference.
Gauss-Markov models, estimability of parameters, best linear unbiased estimators, confidence intervals,
tests for linear hypotheses. Analysis of variance and covariance. Fixed, random and mixed effects models.
Simple and multiple linear regression. Elementary regression diagnostics. Logistic regression.
Multivariate normal distribution, Wishart distribution and their properties. Distribution of quadratic
forms. Inference for parameters, partial and multiple correlation coefficients and related tests. Data
reduction techniques: Principle component analysis, Discriminant analysis, Cluster analysis, Canonical
correlation.
Simple random sampling, stratified sampling and systematic sampling. Probability proportional to size
sampling. Ratio and regression methods.
Completely randomized designs, randomized block designs and Latin-square designs. Connectedness and
orthogonality of block designs, BIBD. 2K factorial experiments: confounding and construction.
Hazard function and failure rates, censoring and life testing, series and parallel systems.
Linear programming problem, simplex methods, duality. Elementary queuing and inventory models.
Steady-state solutions of Markovian queuing models: M/M/1, M/M/1 with limited waiting space, M/M/C,
M/M/C with limited waiting space, M/G/1.
All students are expected to answer questions from Unit I. Students in mathematics
are expected to answer additional question from Unit II and III. Students with in
statistics are expected to answer additional question from Unit IV.

CSIR NET/JRF SYLLABUS

CSIR NET/JRF SYLLABUS

 PAPER-1 GENERAL APTITUDE



 PAPER-2  & 3 SUBJECT WISE(COMMON SYLLABUS)

 1. CHEMICAL SCIENCE

 2. PHYSICAL SCIENCE

 3. MATHEMATICAL SCIENCE

 4. LIFE SCIENCE

 5. EARTH SCIENCE

 6. ENGINEERING SCIENCE

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