Gate Syllabus
MA Mathematics
Calculus: Finite, countable and uncountable sets, Real number system as a complete ordered
field, Archimedean property; Sequences and series, convergence; Limits, continuity,
uniform continuity, differentiability, mean value theorems; Riemann integration, Improper
integrals; Functions of two or three variables, continuity, directional derivatives, partial
derivatives, total derivative, maxima and minima, saddle point, method of Lagrange's
multipliers; Double and Triple integrals and their applications; Line integrals and Surface
integrals, Green's theorem, Stokes' theorem, and Gauss divergence theorem.
Linear Algebra: Finite dimensional vector spaces over real or complex fields; Linear
transformations and their matrix representations, rank and nullity; systems of linear
equations, eigenvalues and eigenvectors, minimal polynomial, Cayley-Hamilton Theorem,
diagonalization, Jordan canonical form, symmetric, skew-symmetric, Hermitian, skew-
Hermitian, orthogonal and unitary matrices; Finite dimensional inner product spaces, Gram-
Schmidt orthonormalization process, definite forms.
Real Analysis: Metric spaces, connectedness, compactness, completeness; Sequences and
series of functions, uniform convergence; Weierstrass approximation theorem; Power series;
Functions of several variables: Differentiation, contraction mapping principle, Inverse and
Implicit function theorems; Lebesgue measure, measurable functions; Lebesgue integral,
Fatou's lemma, monotone convergence theorem, dominated convergence theorem.
Complex Analysis: Analytic functions, harmonic functions; Complex integration: Cauchy's
integral theorem and formula; Liouville's theorem, maximum modulus principle, Morera's
theorem; zeros and singularities; Power series, radius of convergence, Taylor's theorem and
Laurent's theorem; residue theorem and applications for evaluating real integrals; Rouche's
theorem, Argument principle, Schwarz lemma; conformal mappings, bilinear
transformations.
Ordinary Differential equations: First order ordinary differential equations, existence and
uniqueness theorems for initial value problems, linear ordinary differential equations of
higher order with constant coefficients; Second order linear ordinary differential equations
with variable coefficients; Cauchy-Euler equation, method of Laplace transforms for solving
ordinary differential equations, series solutions (power series, Frobenius method); Legendre
and Bessel functions and their orthogonal properties; Systems of linear first order ordinary
differential equations.
Algebra: Groups, subgroups, normal subgroups, quotient groups, homomorphisms,
automorphisms; cyclic groups, permutation groups, Sylow's theorems and their applications;
Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domains,
Principle ideal domains, Euclidean domains, polynomial rings and irreducibility criteria;
Fields, finite fields, field extensions.
Functional Analysis: Normed linear spaces, Banach spaces, Hahn-Banach theorem, open
mapping and closed graph theorems, principle of uniform boundedness; Inner-product
spaces, Hilbert spaces, orthonormal bases, Riesz representation theorem.
Numerical Analysis: Numerical solutions of algebraic and transcendental equations:
bisection, secant method, Newton-Raphson method, fixed point iteration; Interpolation:
error of polynomial interpolation, Lagrange and Newton interpolations; Numerical
differentiation; Numerical integration: Trapezoidal and Simpson's rules; Numerical solution
of a system of linear equations: direct methods (Gauss elimination, LU decomposition),
iterative methods (Jacobi and Gauss-Seidel); Numerical solution of initial value problems of
ODEs: Euler's method, Runge-Kutta methods of order 2.
Partial Differential Equations: Linear and quasi-linear first order partial differential
equations, method of characteristics; Second order linear equations in two variables and
their classification; Cauchy, Dirichlet and Neumann problems; Solutions of Laplace and
wave equations in two dimensional Cartesian coordinates, interior and exterior Dirichlet
problems in polar coordinates; Separation of variables method for solving wave and
diffusion equations in one space variable; Fourier series and Fourier transform and Laplace
transform methods of solutions for the equations mentioned above.
Topology: Basic concepts of topology, bases, subbases, subspace topology, order topology,
product topology, metric topology, connectedness, compactness, countability and separation
axioms, Urysohn's Lemma.
Linear Programming: Linear programming problem and its formulation, convex sets and
their properties, graphical method, basic feasible solution, simplex method, two phase
methods; infeasible and unbounded LPP's, alternate optima; Dual problem and duality
theorems; Balanced and unbalanced transportation problems, Vogel's approximation method
for solving transportation problems; Hungarian method for solving assignment problems.
XE-A (Compulsory for all XE candidates)
Engineering Mathematics
Section 1: Linear Algebra
Algebra of matrices; Inverse and rank of a matrix; System of linear equations; Symmetric,
skew-symmetric and orthogonal matrices; Determinants; Eigenvalues and eigenvectors;
Diagonalisation of matrices; Cayley-Hamilton Theorem.
Section 2: Calculus
Functions of single variable: Limit, continuity and differentiability; Mean value theorems;
Indeterminate forms and L'Hospital's rule; Maxima and minima; Taylor's theorem;
Fundamental theorem and mean value-theorems of integral calculus; Evaluation of
definite and improper integrals; Applications of definite integrals to evaluate areas and
volumes.
Functions of two variables: Limit, continuity and partial derivatives; Directional derivative;
Total derivative; Tangent plane and normal line; Maxima, minima and saddle points;
Method of Lagrange multipliers; Double and triple integrals, and their applications.
Sequence and series: Convergence of sequence and series; Tests for convergence;
Power series; Taylor's series; Fourier Series; Half range sine and cosine series.
Section 3: Vector Calculus
Gradient, divergence and curl; Line and surface integrals; Green's theorem, Stokes
theorem and Gauss divergence theorem (without proofs).
Section 4: Complex variables
Analytic functions; Cauchy-Riemann equations; Line integral, Cauchy's integral theorem
and integral formula (without proof); Taylor's series and Laurent series; Residue theorem
(without proof) and its applications.
Section 5: Ordinary Differential Equations
First order equations (linear and nonlinear); Higher order linear differential equations with
constant coefficients; Second order linear differential equations with variable
coefficients; Method of variation of parameters; Cauchy-Euler equation; Power series
solutions; Legendre polynomials, Bessel functions of the first kind and their properties.
Section 6: Partial Differential Equations
Classification of second order linear partial differential equations; Method of separation
of variables; Laplace equation; Solutions of one dimensional heat and wave equations.
Section 7: Probability and Statistics
Axioms of probability; Conditional probability; Bayes' Theorem; Discrete and continuous
random variables: Binomial, Poisson and normal distributions; Correlation and linear
regression.
Section 8: Numerical Methods
Solution of systems of linear equations using LU decomposition, Gauss elimination and
Gauss-Seidel methods; Lagrange and Newton's interpolations, Solution of polynomial and
transcendental equations by Newton-Raphson method; Numerical integration by
trapezoidal rule, Simpson's rule and Gaussian quadrature rule; Numerical solutions of first
order differential equations by Euler's method and 4th order Runge-Kutta method.
Section 1: Mathematical Physics
Linear vector space: basis, orthogonality and completeness; matrices; vector
calculus; linear differential equations; elements of complex analysis: Cauchy-
Riemann conditions, Cauchy’s theorems, singularities, residue theorem and
applications; Laplace transforms, Fourier analysis; elementary ideas about tensors:
covariant and contravariant tensor, Levi-Civita and Christoffel symbols.
Section 2: Classical Mechanics
D’Alembert’s principle, cyclic coordinates, variational principle, Lagrange’s
equation of motion, central force and scattering problems, rigid body motion;
small oscillations, Hamilton’s formalisms; Poisson bracket; special theory of
relativity: Lorentz transformations, relativistic kinematics, mass‐energy equivalence.
Section 3: Electromagnetic Theory
Solutions of electrostatic and magnetostatic problems including boundary value
problems; dielectrics and conductors; Maxwell’s equations; scalar and vector
potentials; Coulomb and Lorentz gauges; Electromagnetic waves and their
reflection, refraction, interference, diffraction and polarization; Poynting vector,
Poynting theorem, energy and momentum of electromagnetic waves; radiation
from a moving charge.
Section 4: Quantum Mechanics
Postulates of quantum mechanics; uncertainty principle; Schrodinger equation;
one-, two- and three-dimensional potential problems; particle in a box, transmission
through one dimensional potential barriers, harmonic oscillator, hydrogen atom;
linear vectors and operators in Hilbert space; angular momentum and spin;
addition of angular momenta; time independent perturbation theory; elementary
scattering theory.
Section 5: Thermodynamics and Statistical Physics
Laws of thermodynamics; macrostates and microstates; phase space; ensembles;
partition function, free energy, calculation of thermodynamic quantities; classical
and quantum statistics; degenerate Fermi gas; black body radiation and Planck’s
distribution law; Bose‐Einstein condensation; first and second order phase
transitions, phase equilibria, critical point.
Section 6: Atomic and Molecular Physics
Spectra of one‐ and many‐electron atoms; LS and jj coupling; hyperfine structure;
Zeeman and Stark effects; electric dipole transitions and selection rules; rotational
and vibrational spectra of diatomic molecules; electronic transition in diatomic
molecules, Franck‐Condon principle; Raman effect; NMR, ESR, X-ray spectra;
lasers: Einstein coefficients, population inversion, two and three level systems.
Section 7: Solid State Physics & Electronics
Elements of crystallography; diffraction methods for structure determination;
bonding in solids; lattice vibrations and thermal properties of solids; free electron
theory; band theory of solids: nearly free electron and tight binding models; metals,
semiconductors and insulators; conductivity, mobility and effective mass; optical,dielectric and magnetic properties of solids; elements of superconductivity: Type-I
and Type II superconductors, Meissner effect, London equation.
Semiconductor devices: diodes, Bipolar Junction Transistors, Field Effect Transistors;
operational amplifiers: negative feedback circuits, active filters and oscillators;
regulated power supplies; basic digital logic circuits, sequential circuits, flip‐flops,
counters, registers, A/D and D/A conversion.
Section 8: Nuclear and Particle Physics
Nuclear radii and charge distributions, nuclear binding energy, Electric and
magnetic moments; nuclear models, liquid drop model: semi‐empirical mass
formula, Fermi gas model of nucleus, nuclear shell model; nuclear force and two
nucleon problem; alpha decay, beta‐decay, electromagnetic transitions in nuclei;
Rutherford scattering, nuclear reactions, conservation laws; fission and fusion;
particle accelerators and detectors; elementary particles, photons, baryons,
mesons and leptons; quark model.
CY Chemistry
Section 1: Physical Chemistry
Structure: Postulates of quantum mechanics. Time dependent and time independent
Schrödinger equations. Born interpretation. Particle in a box. Harmonic oscillator. Rigid
rotor. Hydrogen atom: atomic orbitals. Multi-electron atoms: orbital approximation.
Variation and first order perturbation techniques. Chemical bonding: Valence bond
theory and LCAO-MO theory. Hybrid orbitals. Applications of LCAO-MOT to H2+, H2 and
other homonuclear diatomic molecules, heteronuclear diatomic molecules like HF, CO,
NO, and to simple delocalized π– electron systems. Hückel approximation and its
application to annular π – electron systems. Symmetry elements and operations. Point
groups and character tables. Origin of selection rules for rotational, vibrational, electronic
and Raman spectroscopy of diatomic and polyatomic molecules. Einstein coefficients.
Relationship of transition moment integral with molar extinction coefficient and oscillator
strength. Basic principles of nuclear magnetic resonance: nuclear g factor, chemical shift,
nuclear coupling.
Equilibrium: Laws of thermodynamics. Standard states. Thermochemistry. Thermodynamic
functions and their relationships: Gibbs-Helmholtz and Maxwell relations, van’t Hoff
equation. Criteria of spontaneity and equilibrium. Absolute entropy. Partial molar
quantities. Thermodynamics of mixing. Chemical potential. Fugacity, activity and activity
coefficients. Chemical equilibria. Dependence of equilibrium constant on temperature
and pressure. Non-ideal solutions. Ionic mobility and conductivity. Debye-Hückel limiting
law. Debye-Hückel-Onsager equation. Standard electrode potentials and electrochemical cells. Potentiometric and conductometric titrations. Phase rule. Clausius-
Clapeyron equation. Phase diagram of one component systems: CO2, H2O, S; two component systems: liquid-vapour, liquid-liquid and solid-liquid systems. Fractional
distillation. Azeotropes and eutectics. Statistical thermodynamics: microcanonical and
canonical ensembles, Boltzmann distribution, partition functions and thermodynamic
properties.
Kinetics: Transition state theory: Eyring equation, thermodynamic aspects. Potential
energy surfaces and classical trajectories. Elementary, parallel, opposing and consecutive
reactions. Steady state approximation. Mechanisms of complex reactions. Unimolecular
reactions. Kinetics of polymerization and enzyme catalysis. Fast reaction kinetics:
relaxation and flow methods. Kinetics of photochemical and photophysical processes.
Surfaces and Interfaces: Physisorption and chemisorption. Langmuir, Freundlich and BET
isotherms. Surface catalysis: Langmuir-Hinshelwood mechanism. Surface tension, viscosity.
Self-assembly. Physical chemistry of colloids, micelles and macromolecules.
Section 2: Inorganic Chemistry
Main Group Elements: Hydrides, halides, oxides, oxoacids, nitrides, sulfides – shapes and
reactivity. Structure and bonding of boranes, carboranes, silicones, silicates, boron nitride,
borazines and phosphazenes. Allotropes of carbon. Chemistry of noble gases,
pseudohalogens, and interhalogen compounds. Acid-base concepts.
Transition Elements: Coordination chemistry – structure and isomerism, theories of bonding
(VBT, CFT, and MOT). Energy level diagrams in various crystal fields, CFSE, applications of
CFT, Jahn-Teller distortion. Electronic spectra of transition metal complexes: spectroscopic
term symbols, selection rules, Orgel diagrams, charge-transfer spectra. Magnetic properties of transition metal complexes. Reaction mechanisms: kinetic and
thermodynamic stability, substitution and redox reactions.
Lanthanides and Actinides: Recovery. Periodic properties, spectra and magnetic
properties.
Organometallics: 18-Electron rule; metal-alkyl, metal-carbonyl, metal-olefin and metal-
carbene complexes and metallocenes. Fluxionality in organometallic complexes. Types of organometallic reactions. Homogeneous catalysis - Hydrogenation, hydroformylation, acetic acid synthesis, metathesis and olefin oxidation. Heterogeneous catalysis - Fischer-
Tropsch reaction, Ziegler-Natta polymerization.
Radioactivity: Decay processes, half-life of radioactive elements, fission and fusion
processes.
Bioinorganic Chemistry: Ion (Na+ and K+) transport, oxygen binding, transport and
utilization, electron transfer reactions, nitrogen fixation, metalloenzymes containing
magnesium, molybdenum, iron, cobalt, copper and zinc.
Solids: Crystal systems and lattices, Miller planes, crystal packing, crystal defects, Bragg’s
law, ionic crystals, structures of AX, AX2, ABX3 type compounds, spinels, band theory,
metals and semiconductors.
Instrumental Methods of Analysis: UV-visible spectrophotometry, NMR and ESR
spectroscopy, mass spectrometry. Chromatography including GC and HPLC.
Electroanalytical methods- polarography, cyclic voltammetry, ion-selective electrodes.
Thermoanalytical methods.
Section 3: Organic Chemistry
Stereochemistry: Chirality of organic molecules with or without chiral centres and
determination of their absolute configurations. Relative stereochemistry in compounds
having more than one stereogenic centre. Homotopic, enantiotopic and diastereotopic
atoms, groups and faces. Stereoselective and stereospecific synthesis. Conformational
analysis of acyclic and cyclic compounds. Geometrical isomerism. Configurational and
conformational effects, and neighbouring group participation on reactivity and
selectivity/specificity.
Reaction Mechanisms: Basic mechanistic concepts – kinetic versus thermodynamic
control, Hammond’s postulate and Curtin-Hammett principle. Methods of determining
reaction mechanisms through identification of products, intermediates and isotopic
labeling. Nucleophilic and electrophilic substitution reactions (both aromatic and
aliphatic). Addition reactions to carbon-carbon and carbon-heteroatom (N,O) multiple
bonds. Elimination reactions. Reactive intermediates – carbocations, carbanions,
carbenes, nitrenes, arynes and free radicals. Molecular rearrangements involving electron
deficient atoms.
Organic Synthesis: Synthesis, reactions, mechanisms and selectivity involving the following
classes of compounds – alkenes, alkynes, arenes, alcohols, phenols, aldehydes, ketones,
carboxylic acids, esters, nitriles, halides, nitro compounds, amines and amides. Uses of Mg,
Li, Cu, B, Zn and Si based reagents in organic synthesis. Carbon-carbon bond formation
through coupling reactions - Heck, Suzuki, Stille and Sonogoshira. Concepts of multistep synthesis - retrosynthetic analysis, strategic disconnections, synthons and synthetic
equivalents. Umpolung reactivity – formyl and acyl anion equivalents. Selectivity in
organic synthesis – chemo-, regio- and stereoselectivity. Protection and deprotection of
functional groups. Concepts of asymmetric synthesis – resolution (including enzymatic),
desymmetrization and use of chiral auxilliaries. Carbon-carbon bond forming reactions
through enolates (including boron enolates), enamines and silyl enol ethers. Michael
addition reaction. Stereoselective addition to C=O groups (Cram and Felkin-Anh models).
Pericyclic Reactions and Photochemistry: Electrocyclic, cycloaddition and sigmatropic
reactions. Orbital correlations - FMO and PMO treatments. Photochemistry of alkenes,
arenes and carbonyl compounds. Photooxidation and photoreduction. Di-π-methane
rearrangement, Barton reaction.
Heterocyclic Compounds: Structure, preparation, properties and reactions of furan,
pyrrole, thiophene, pyridine, indole, quinoline and isoquinoline.
Biomolecules: Structure, properties and reactions of mono- and di-saccharides,
physicochemical properties of amino acids, chemical synthesis of peptides, structural
features of proteins, nucleic acids, steroids, terpenoids, carotenoids, and alkaloids.
Spectroscopy: Applications of UV-visible, IR, NMR and Mass spectrometry in the structural
determination of organic molecules.
CS Computer Science and Information Technology
Section1: Engineering Mathematics
Discrete Mathematics: Propositional and first order logic. Sets, relations, functions, partial
orders and lattices. Groups. Graphs: connectivity, matching, coloring. Combinatorics:
counting, recurrence relations, generating functions.
Linear Algebra: Matrices, determinants, system of linear equations, eigenvalues and
eigenvectors, LU decomposition.
Calculus: Limits, continuity and differentiability. Maxima and minima. Mean value
theorem. Integration.
Probability: Random variables. Uniform, normal, exponential, poisson and binomial
distributions. Mean, median, mode and standard deviation. Conditional probability and
Bayes theorem.
Computer Science and Information Technology
Section 2: Digital Logic
Boolean algebra. Combinational and sequential circuits. Minimization. Number
representations and computer arithmetic (fixed and floating point).
Section 3: Computer Organization and Architecture
Machine instructions and addressing modes. ALU, data‐path and control unit. Instruction
pipelining. Memory hierarchy: cache, main memory and secondary storage; I/O
interface (interrupt and DMA mode).
Section 4: Programming and Data Structures
Programming in C. Recursion. Arrays, stacks, queues, linked lists, trees, binary search
trees, binary heaps, graphs.
Section 5: Algorithms
Searching, sorting, hashing. Asymptotic worst case time and space complexity.
Algorithm design techniques: greedy, dynamic programming and divide‐and‐conquer.
Graph search, minimum spanning trees, shortest paths.
Section 6: Theory of Computation
Regular expressions and finite automata. Context-free grammars and push-down
automata. Regular and contex-free languages, pumping lemma. Turing machines and
undecidability.
Section 7: Compiler Design
Lexical analysis, parsing, syntax-directed translation. Runtime environments. Intermediate
code generation.
Section 8: Operating System
Processes, threads, inter‐process communication, concurrency and synchronization.
Deadlock. CPU scheduling. Memory management and virtual memory. File systems.
Section 9: Databases
ER‐model. Relational model: relational algebra, tuple calculus, SQL. Integrity constraints,
normal forms. File organization, indexing (e.g., B and B+ trees). Transactions and
concurrency control.
Section 10: Computer Networks
Concept of layering. LAN technologies (Ethernet). Flow and error control techniques,
switching. IPv4/IPv6, routers and routing algorithms (distance vector, link state). TCP/UDP
and sockets, congestion control. Application layer protocols (DNS, SMTP, POP, FTP, HTTP).
Basics of Wi-Fi. Network security: authentication, basics of public key and private key
cryptography, digital signatures and certificates, firewalls.
ST Statistics
Calculus: Finite, countable and uncountable sets, Real number system as a complete ordered
field, Archimedean property; Sequences and series, convergence; Limits, continuity,
uniform continuity, differentiability, mean value theorems; Riemann integration, Improper
integrals; Functions of two or three variables, continuity, directional derivatives, partial
derivatives, total derivative, maxima and minima, saddle point, method of Lagrange's
multipliers; Double and Triple integrals and their applications; Line integrals and Surface
integrals, Green's theorem, Stokes' theorem, and Gauss divergence theorem.
Linear Algebra: Finite dimensional vector spaces over real or complex fields; Linear
transformations and their matrix representations, rank; systems of linear equations,
eigenvalues and eigenvectors, minimal polynomial, Cayley-Hamilton Theorem, diagonalization, Jordan canonical form, symmetric, skew-symmetric, Hermitian, skew-
Hermitian, orthogonal and unitary matrices; Finite dimensional inner product spaces, Gram-
Schmidt orthonormalization process, definite forms.
Probability: Classical, relative frequency and axiomatic definitions of probability,
conditional probability, Bayes' theorem, independent events; Random variables and
probability distributions, moments and moment generating functions, quantiles; Standard
discrete and continuous univariate distributions; Probability inequalities (Chebyshev,
Markov, Jensen); Function of a random variable; Jointly distributed random variables,
marginal and conditional distributions, product moments, joint moment generating
functions, independence of random variables; Transformations of random variables,
sampling distributions, distribution of order statistics and range; Characteristic functions;
Modes of convergence; Weak and strong laws of large numbers; Central limit theorem for
i.i.d. random variables with existence of higher order moments.
Stochastic Processes: 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.
Inference: Unbiasedness, consistency, sufficiency, completeness, uniformly minimum
variance unbiased estimation, method of moments and maximum likelihood estimations;
Confidence intervals; Tests of hypotheses, most powerful and uniformly most powerful tests, likelihood ratio tests, large sample test, Sign test, Wilcoxon signed rank test, Mann-
Whitney U test, test for independence and Chi-square test for goodness of fit.
Regression Analysis: Simple and multiple linear regression, polynomial regression,
estimation, confidence intervals and testing for regression coefficients; Partial and multiple
correlation coefficients.
Multivariate Analysis: Basic properties of multivariate normal distribution; Multinomial
distribution; Wishart distribution; Hotellings T2and related tests; Principal component analysis; Discriminant analysis; Clustering.
Design of Experiments: One and two-way ANOVA, CRD, RBD, LSD, 22 and 23
Factorial experiments.
EC Electronics and Communications Engineering
Section 1: Engineering Mathematics
Linear Algebra: Vector space, basis, linear dependence and independence, matrix
algebra, eigen values and eigen vectors, rank, solution of linear equations – existence
and uniqueness.
Calculus: Mean value theorems, theorems of integral calculus, evaluation of definite and
improper integrals, partial derivatives, maxima and minima, multiple integrals, line, surface
and volume integrals, Taylor series.
Differential Equations: First order equations (linear and nonlinear), higher order linear
differential equations, Cauchy's and Euler's equations, methods of solution using variation
of parameters, complementary function and particular integral, partial differential
equations, variable separable method, initial and boundary value problems.
Vector Analysis: Vectors in plane and space, vector operations, gradient, divergence and
curl, Gauss's, Green's and Stoke's theorems.
Complex Analysis: Analytic functions, Cauchy's integral theorem, Cauchy's integral
formula; Taylor's and Laurent's series, residue theorem.
Numerical Methods: Solution of nonlinear equations, single and multi-step methods for
differential equations, convergence criteria.
Probability and Statistics: Mean, median, mode and standard deviation; combinatorial
probability, probability distribution functions - binomial, Poisson, exponential and normal;
Joint and conditional probability; Correlation and regression analysis.
Section 2: Networks, Signals and Systems
Network solution methods: nodal and mesh analysis; Network theorems: superposition,
Thevenin and Norton’s, maximum power transfer; Wye‐Delta transformation; Steady state
sinusoidal analysis using phasors; Time domain analysis of simple linear circuits; Solution of
network equations using Laplace transform; Frequency domain analysis of RLC circuits;
Linear 2‐port network parameters: driving point and transfer functions; State equations for
networks.
Continuous-time signals: Fourier series and Fourier transform representations, sampling
theorem and applications; Discrete-time signals: discrete-time Fourier transform (DTFT),
DFT, FFT, Z-transform, interpolation of discrete-time signals; LTI systems: definition and
properties, causality, stability, impulse response, convolution, poles and zeros, parallel and
cascade structure, frequency response, group delay, phase delay, digital filter design
techniques.
Section 3: Electronic Devices
Energy bands in intrinsic and extrinsic silicon; Carrier transport: diffusion current, drift
current, mobility and resistivity; Generation and recombination of carriers; Poisson and
continuity equations; P-N junction, Zener diode, BJT, MOS capacitor, MOSFET, LED, photo
diode and solar cell; Integrated circuit fabrication process: oxidation, diffusion, ion
implantation, photolithography and twin-tub CMOS process.
Section 4: Analog Circuits
Small signal equivalent circuits of diodes, BJTs and MOSFETs; Simple diode circuits:
clipping, clamping and rectifiers; Single-stage BJT and MOSFET amplifiers: biasing, bias
stability, mid-frequency small signal analysis and frequency response; BJT and MOSFET
amplifiers: multi-stage, differential, feedback, power and operational; Simple op-amp circuits; Active filters; Sinusoidal oscillators: criterion for oscillation, single-transistor and op-
amp configurations; Function generators, wave-shaping circuits and 555 timers; Voltage reference circuits; Power supplies: ripple removal and regulation.
Section 5: Digital Circuits
Number systems; Combinatorial circuits: Boolean algebra, minimization of functions using
Boolean identities and Karnaugh map, logic gates and their static CMOS
implementations, arithmetic circuits, code converters, multiplexers, decoders and PLAs;
Sequential circuits: latches and flip‐flops, counters, shift‐registers and finite state machines;
Data converters: sample and hold circuits, ADCs and DACs; Semiconductor memories:
ROM, SRAM, DRAM; 8-bit microprocessor (8085): architecture, programming, memory and
I/O interfacing.
Section 6: Control Systems
Basic control system components; Feedback principle; Transfer function; Block diagram
representation; Signal flow graph; Transient and steady-state analysis of LTI systems;
Frequency response; Routh-Hurwitz and Nyquist stability criteria; Bode and root-locus plots;
Lag, lead and lag-lead compensation; State variable model and solution of state
equation of LTI systems.
Section 7: Communications
Random processes: autocorrelation and power spectral density, properties of white noise,
filtering of random signals through LTI systems; Analog communications: amplitude
modulation and demodulation, angle modulation and demodulation, spectra of AM and
FM, superheterodyne receivers, circuits for analog communications; Information theory:
entropy, mutual information and channel capacity theorem; Digital communications:
PCM, DPCM, digital modulation schemes, amplitude, phase and frequency shift keying
(ASK, PSK, FSK), QAM, MAP and ML decoding, matched filter receiver, calculation of
bandwidth, SNR and BER for digital modulation; Fundamentals of error correction,
Hamming codes; Timing and frequency synchronization, inter-symbol interference and its
mitigation; Basics of TDMA, FDMA and CDMA.
Section 8: Electromagnetics
Electrostatics; Maxwell’s equations: differential and integral forms and their interpretation,
boundary conditions, wave equation, Poynting vector; Plane waves and properties:
reflection and refraction, polarization, phase and group velocity, propagation through
various media, skin depth; Transmission lines: equations, characteristic impedance,
impedance matching, impedance transformation, S-parameters, Smith chart;
Waveguides: modes, boundary conditions, cut-off frequencies, dispersion relations;
Antennas: antenna types, radiation pattern, gain and directivity, return loss, antenna
arrays; Basics of radar; Light propagation in optical fibers.
MA Mathematics
Calculus: Finite, countable and uncountable sets, Real number system as a complete ordered
field, Archimedean property; Sequences and series, convergence; Limits, continuity,
uniform continuity, differentiability, mean value theorems; Riemann integration, Improper
integrals; Functions of two or three variables, continuity, directional derivatives, partial
derivatives, total derivative, maxima and minima, saddle point, method of Lagrange's
multipliers; Double and Triple integrals and their applications; Line integrals and Surface
integrals, Green's theorem, Stokes' theorem, and Gauss divergence theorem.
Linear Algebra: Finite dimensional vector spaces over real or complex fields; Linear
transformations and their matrix representations, rank and nullity; systems of linear
equations, eigenvalues and eigenvectors, minimal polynomial, Cayley-Hamilton Theorem,
diagonalization, Jordan canonical form, symmetric, skew-symmetric, Hermitian, skew-
Hermitian, orthogonal and unitary matrices; Finite dimensional inner product spaces, Gram-
Schmidt orthonormalization process, definite forms.
Real Analysis: Metric spaces, connectedness, compactness, completeness; Sequences and
series of functions, uniform convergence; Weierstrass approximation theorem; Power series;
Functions of several variables: Differentiation, contraction mapping principle, Inverse and
Implicit function theorems; Lebesgue measure, measurable functions; Lebesgue integral,
Fatou's lemma, monotone convergence theorem, dominated convergence theorem.
Complex Analysis: Analytic functions, harmonic functions; Complex integration: Cauchy's
integral theorem and formula; Liouville's theorem, maximum modulus principle, Morera's
theorem; zeros and singularities; Power series, radius of convergence, Taylor's theorem and
Laurent's theorem; residue theorem and applications for evaluating real integrals; Rouche's
theorem, Argument principle, Schwarz lemma; conformal mappings, bilinear
transformations.
Ordinary Differential equations: First order ordinary differential equations, existence and
uniqueness theorems for initial value problems, linear ordinary differential equations of
higher order with constant coefficients; Second order linear ordinary differential equations
with variable coefficients; Cauchy-Euler equation, method of Laplace transforms for solving
ordinary differential equations, series solutions (power series, Frobenius method); Legendre
and Bessel functions and their orthogonal properties; Systems of linear first order ordinary
differential equations.
Algebra: Groups, subgroups, normal subgroups, quotient groups, homomorphisms,
automorphisms; cyclic groups, permutation groups, Sylow's theorems and their applications;
Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domains,
Principle ideal domains, Euclidean domains, polynomial rings and irreducibility criteria;
Fields, finite fields, field extensions.
Functional Analysis: Normed linear spaces, Banach spaces, Hahn-Banach theorem, open
mapping and closed graph theorems, principle of uniform boundedness; Inner-product
spaces, Hilbert spaces, orthonormal bases, Riesz representation theorem.
Numerical Analysis: Numerical solutions of algebraic and transcendental equations:
bisection, secant method, Newton-Raphson method, fixed point iteration; Interpolation:
error of polynomial interpolation, Lagrange and Newton interpolations; Numerical
differentiation; Numerical integration: Trapezoidal and Simpson's rules; Numerical solution
of a system of linear equations: direct methods (Gauss elimination, LU decomposition),
iterative methods (Jacobi and Gauss-Seidel); Numerical solution of initial value problems of
ODEs: Euler's method, Runge-Kutta methods of order 2.
Partial Differential Equations: Linear and quasi-linear first order partial differential
equations, method of characteristics; Second order linear equations in two variables and
their classification; Cauchy, Dirichlet and Neumann problems; Solutions of Laplace and
wave equations in two dimensional Cartesian coordinates, interior and exterior Dirichlet
problems in polar coordinates; Separation of variables method for solving wave and
diffusion equations in one space variable; Fourier series and Fourier transform and Laplace
transform methods of solutions for the equations mentioned above.
Topology: Basic concepts of topology, bases, subbases, subspace topology, order topology,
product topology, metric topology, connectedness, compactness, countability and separation
axioms, Urysohn's Lemma.
Linear Programming: Linear programming problem and its formulation, convex sets and
their properties, graphical method, basic feasible solution, simplex method, two phase
methods; infeasible and unbounded LPP's, alternate optima; Dual problem and duality
theorems; Balanced and unbalanced transportation problems, Vogel's approximation method
for solving transportation problems; Hungarian method for solving assignment problems.
XE-A (Compulsory for all XE candidates)
Engineering Mathematics
Section 1: Linear Algebra
Algebra of matrices; Inverse and rank of a matrix; System of linear equations; Symmetric,
skew-symmetric and orthogonal matrices; Determinants; Eigenvalues and eigenvectors;
Diagonalisation of matrices; Cayley-Hamilton Theorem.
Section 2: Calculus
Functions of single variable: Limit, continuity and differentiability; Mean value theorems;
Indeterminate forms and L'Hospital's rule; Maxima and minima; Taylor's theorem;
Fundamental theorem and mean value-theorems of integral calculus; Evaluation of
definite and improper integrals; Applications of definite integrals to evaluate areas and
volumes.
Functions of two variables: Limit, continuity and partial derivatives; Directional derivative;
Total derivative; Tangent plane and normal line; Maxima, minima and saddle points;
Method of Lagrange multipliers; Double and triple integrals, and their applications.
Sequence and series: Convergence of sequence and series; Tests for convergence;
Power series; Taylor's series; Fourier Series; Half range sine and cosine series.
Section 3: Vector Calculus
Gradient, divergence and curl; Line and surface integrals; Green's theorem, Stokes
theorem and Gauss divergence theorem (without proofs).
Section 4: Complex variables
Analytic functions; Cauchy-Riemann equations; Line integral, Cauchy's integral theorem
and integral formula (without proof); Taylor's series and Laurent series; Residue theorem
(without proof) and its applications.
Section 5: Ordinary Differential Equations
First order equations (linear and nonlinear); Higher order linear differential equations with
constant coefficients; Second order linear differential equations with variable
coefficients; Method of variation of parameters; Cauchy-Euler equation; Power series
solutions; Legendre polynomials, Bessel functions of the first kind and their properties.
Section 6: Partial Differential Equations
Classification of second order linear partial differential equations; Method of separation
of variables; Laplace equation; Solutions of one dimensional heat and wave equations.
Section 7: Probability and Statistics
Axioms of probability; Conditional probability; Bayes' Theorem; Discrete and continuous
random variables: Binomial, Poisson and normal distributions; Correlation and linear
regression.
Section 8: Numerical Methods
Solution of systems of linear equations using LU decomposition, Gauss elimination and
Gauss-Seidel methods; Lagrange and Newton's interpolations, Solution of polynomial and
transcendental equations by Newton-Raphson method; Numerical integration by
trapezoidal rule, Simpson's rule and Gaussian quadrature rule; Numerical solutions of first
order differential equations by Euler's method and 4th order Runge-Kutta method.
PH PHYSICS
Section 1: Mathematical Physics
Linear vector space: basis, orthogonality and completeness; matrices; vector
calculus; linear differential equations; elements of complex analysis: Cauchy-
Riemann conditions, Cauchy’s theorems, singularities, residue theorem and
applications; Laplace transforms, Fourier analysis; elementary ideas about tensors:
covariant and contravariant tensor, Levi-Civita and Christoffel symbols.
Section 2: Classical Mechanics
D’Alembert’s principle, cyclic coordinates, variational principle, Lagrange’s
equation of motion, central force and scattering problems, rigid body motion;
small oscillations, Hamilton’s formalisms; Poisson bracket; special theory of
relativity: Lorentz transformations, relativistic kinematics, mass‐energy equivalence.
Section 3: Electromagnetic Theory
Solutions of electrostatic and magnetostatic problems including boundary value
problems; dielectrics and conductors; Maxwell’s equations; scalar and vector
potentials; Coulomb and Lorentz gauges; Electromagnetic waves and their
reflection, refraction, interference, diffraction and polarization; Poynting vector,
Poynting theorem, energy and momentum of electromagnetic waves; radiation
from a moving charge.
Section 4: Quantum Mechanics
Postulates of quantum mechanics; uncertainty principle; Schrodinger equation;
one-, two- and three-dimensional potential problems; particle in a box, transmission
through one dimensional potential barriers, harmonic oscillator, hydrogen atom;
linear vectors and operators in Hilbert space; angular momentum and spin;
addition of angular momenta; time independent perturbation theory; elementary
scattering theory.
Section 5: Thermodynamics and Statistical Physics
Laws of thermodynamics; macrostates and microstates; phase space; ensembles;
partition function, free energy, calculation of thermodynamic quantities; classical
and quantum statistics; degenerate Fermi gas; black body radiation and Planck’s
distribution law; Bose‐Einstein condensation; first and second order phase
transitions, phase equilibria, critical point.
Section 6: Atomic and Molecular Physics
Spectra of one‐ and many‐electron atoms; LS and jj coupling; hyperfine structure;
Zeeman and Stark effects; electric dipole transitions and selection rules; rotational
and vibrational spectra of diatomic molecules; electronic transition in diatomic
molecules, Franck‐Condon principle; Raman effect; NMR, ESR, X-ray spectra;
lasers: Einstein coefficients, population inversion, two and three level systems.
Section 7: Solid State Physics & Electronics
Elements of crystallography; diffraction methods for structure determination;
bonding in solids; lattice vibrations and thermal properties of solids; free electron
theory; band theory of solids: nearly free electron and tight binding models; metals,
semiconductors and insulators; conductivity, mobility and effective mass; optical,dielectric and magnetic properties of solids; elements of superconductivity: Type-I
and Type II superconductors, Meissner effect, London equation.
Semiconductor devices: diodes, Bipolar Junction Transistors, Field Effect Transistors;
operational amplifiers: negative feedback circuits, active filters and oscillators;
regulated power supplies; basic digital logic circuits, sequential circuits, flip‐flops,
counters, registers, A/D and D/A conversion.
Section 8: Nuclear and Particle Physics
Nuclear radii and charge distributions, nuclear binding energy, Electric and
magnetic moments; nuclear models, liquid drop model: semi‐empirical mass
formula, Fermi gas model of nucleus, nuclear shell model; nuclear force and two
nucleon problem; alpha decay, beta‐decay, electromagnetic transitions in nuclei;
Rutherford scattering, nuclear reactions, conservation laws; fission and fusion;
particle accelerators and detectors; elementary particles, photons, baryons,
mesons and leptons; quark model.
CY Chemistry
Section 1: Physical Chemistry
Structure: Postulates of quantum mechanics. Time dependent and time independent
Schrödinger equations. Born interpretation. Particle in a box. Harmonic oscillator. Rigid
rotor. Hydrogen atom: atomic orbitals. Multi-electron atoms: orbital approximation.
Variation and first order perturbation techniques. Chemical bonding: Valence bond
theory and LCAO-MO theory. Hybrid orbitals. Applications of LCAO-MOT to H2+, H2 and
other homonuclear diatomic molecules, heteronuclear diatomic molecules like HF, CO,
NO, and to simple delocalized π– electron systems. Hückel approximation and its
application to annular π – electron systems. Symmetry elements and operations. Point
groups and character tables. Origin of selection rules for rotational, vibrational, electronic
and Raman spectroscopy of diatomic and polyatomic molecules. Einstein coefficients.
Relationship of transition moment integral with molar extinction coefficient and oscillator
strength. Basic principles of nuclear magnetic resonance: nuclear g factor, chemical shift,
nuclear coupling.
Equilibrium: Laws of thermodynamics. Standard states. Thermochemistry. Thermodynamic
functions and their relationships: Gibbs-Helmholtz and Maxwell relations, van’t Hoff
equation. Criteria of spontaneity and equilibrium. Absolute entropy. Partial molar
quantities. Thermodynamics of mixing. Chemical potential. Fugacity, activity and activity
coefficients. Chemical equilibria. Dependence of equilibrium constant on temperature
and pressure. Non-ideal solutions. Ionic mobility and conductivity. Debye-Hückel limiting
law. Debye-Hückel-Onsager equation. Standard electrode potentials and electrochemical cells. Potentiometric and conductometric titrations. Phase rule. Clausius-
Clapeyron equation. Phase diagram of one component systems: CO2, H2O, S; two component systems: liquid-vapour, liquid-liquid and solid-liquid systems. Fractional
distillation. Azeotropes and eutectics. Statistical thermodynamics: microcanonical and
canonical ensembles, Boltzmann distribution, partition functions and thermodynamic
properties.
Kinetics: Transition state theory: Eyring equation, thermodynamic aspects. Potential
energy surfaces and classical trajectories. Elementary, parallel, opposing and consecutive
reactions. Steady state approximation. Mechanisms of complex reactions. Unimolecular
reactions. Kinetics of polymerization and enzyme catalysis. Fast reaction kinetics:
relaxation and flow methods. Kinetics of photochemical and photophysical processes.
Surfaces and Interfaces: Physisorption and chemisorption. Langmuir, Freundlich and BET
isotherms. Surface catalysis: Langmuir-Hinshelwood mechanism. Surface tension, viscosity.
Self-assembly. Physical chemistry of colloids, micelles and macromolecules.
Section 2: Inorganic Chemistry
Main Group Elements: Hydrides, halides, oxides, oxoacids, nitrides, sulfides – shapes and
reactivity. Structure and bonding of boranes, carboranes, silicones, silicates, boron nitride,
borazines and phosphazenes. Allotropes of carbon. Chemistry of noble gases,
pseudohalogens, and interhalogen compounds. Acid-base concepts.
Transition Elements: Coordination chemistry – structure and isomerism, theories of bonding
(VBT, CFT, and MOT). Energy level diagrams in various crystal fields, CFSE, applications of
CFT, Jahn-Teller distortion. Electronic spectra of transition metal complexes: spectroscopic
term symbols, selection rules, Orgel diagrams, charge-transfer spectra. Magnetic properties of transition metal complexes. Reaction mechanisms: kinetic and
thermodynamic stability, substitution and redox reactions.
Lanthanides and Actinides: Recovery. Periodic properties, spectra and magnetic
properties.
Organometallics: 18-Electron rule; metal-alkyl, metal-carbonyl, metal-olefin and metal-
carbene complexes and metallocenes. Fluxionality in organometallic complexes. Types of organometallic reactions. Homogeneous catalysis - Hydrogenation, hydroformylation, acetic acid synthesis, metathesis and olefin oxidation. Heterogeneous catalysis - Fischer-
Tropsch reaction, Ziegler-Natta polymerization.
Radioactivity: Decay processes, half-life of radioactive elements, fission and fusion
processes.
Bioinorganic Chemistry: Ion (Na+ and K+) transport, oxygen binding, transport and
utilization, electron transfer reactions, nitrogen fixation, metalloenzymes containing
magnesium, molybdenum, iron, cobalt, copper and zinc.
Solids: Crystal systems and lattices, Miller planes, crystal packing, crystal defects, Bragg’s
law, ionic crystals, structures of AX, AX2, ABX3 type compounds, spinels, band theory,
metals and semiconductors.
Instrumental Methods of Analysis: UV-visible spectrophotometry, NMR and ESR
spectroscopy, mass spectrometry. Chromatography including GC and HPLC.
Electroanalytical methods- polarography, cyclic voltammetry, ion-selective electrodes.
Thermoanalytical methods.
Section 3: Organic Chemistry
Stereochemistry: Chirality of organic molecules with or without chiral centres and
determination of their absolute configurations. Relative stereochemistry in compounds
having more than one stereogenic centre. Homotopic, enantiotopic and diastereotopic
atoms, groups and faces. Stereoselective and stereospecific synthesis. Conformational
analysis of acyclic and cyclic compounds. Geometrical isomerism. Configurational and
conformational effects, and neighbouring group participation on reactivity and
selectivity/specificity.
Reaction Mechanisms: Basic mechanistic concepts – kinetic versus thermodynamic
control, Hammond’s postulate and Curtin-Hammett principle. Methods of determining
reaction mechanisms through identification of products, intermediates and isotopic
labeling. Nucleophilic and electrophilic substitution reactions (both aromatic and
aliphatic). Addition reactions to carbon-carbon and carbon-heteroatom (N,O) multiple
bonds. Elimination reactions. Reactive intermediates – carbocations, carbanions,
carbenes, nitrenes, arynes and free radicals. Molecular rearrangements involving electron
deficient atoms.
Organic Synthesis: Synthesis, reactions, mechanisms and selectivity involving the following
classes of compounds – alkenes, alkynes, arenes, alcohols, phenols, aldehydes, ketones,
carboxylic acids, esters, nitriles, halides, nitro compounds, amines and amides. Uses of Mg,
Li, Cu, B, Zn and Si based reagents in organic synthesis. Carbon-carbon bond formation
through coupling reactions - Heck, Suzuki, Stille and Sonogoshira. Concepts of multistep synthesis - retrosynthetic analysis, strategic disconnections, synthons and synthetic
equivalents. Umpolung reactivity – formyl and acyl anion equivalents. Selectivity in
organic synthesis – chemo-, regio- and stereoselectivity. Protection and deprotection of
functional groups. Concepts of asymmetric synthesis – resolution (including enzymatic),
desymmetrization and use of chiral auxilliaries. Carbon-carbon bond forming reactions
through enolates (including boron enolates), enamines and silyl enol ethers. Michael
addition reaction. Stereoselective addition to C=O groups (Cram and Felkin-Anh models).
Pericyclic Reactions and Photochemistry: Electrocyclic, cycloaddition and sigmatropic
reactions. Orbital correlations - FMO and PMO treatments. Photochemistry of alkenes,
arenes and carbonyl compounds. Photooxidation and photoreduction. Di-π-methane
rearrangement, Barton reaction.
Heterocyclic Compounds: Structure, preparation, properties and reactions of furan,
pyrrole, thiophene, pyridine, indole, quinoline and isoquinoline.
Biomolecules: Structure, properties and reactions of mono- and di-saccharides,
physicochemical properties of amino acids, chemical synthesis of peptides, structural
features of proteins, nucleic acids, steroids, terpenoids, carotenoids, and alkaloids.
Spectroscopy: Applications of UV-visible, IR, NMR and Mass spectrometry in the structural
determination of organic molecules.
CS Computer Science and Information Technology
Section1: Engineering Mathematics
Discrete Mathematics: Propositional and first order logic. Sets, relations, functions, partial
orders and lattices. Groups. Graphs: connectivity, matching, coloring. Combinatorics:
counting, recurrence relations, generating functions.
Linear Algebra: Matrices, determinants, system of linear equations, eigenvalues and
eigenvectors, LU decomposition.
Calculus: Limits, continuity and differentiability. Maxima and minima. Mean value
theorem. Integration.
Probability: Random variables. Uniform, normal, exponential, poisson and binomial
distributions. Mean, median, mode and standard deviation. Conditional probability and
Bayes theorem.
Computer Science and Information Technology
Section 2: Digital Logic
Boolean algebra. Combinational and sequential circuits. Minimization. Number
representations and computer arithmetic (fixed and floating point).
Section 3: Computer Organization and Architecture
Machine instructions and addressing modes. ALU, data‐path and control unit. Instruction
pipelining. Memory hierarchy: cache, main memory and secondary storage; I/O
interface (interrupt and DMA mode).
Section 4: Programming and Data Structures
Programming in C. Recursion. Arrays, stacks, queues, linked lists, trees, binary search
trees, binary heaps, graphs.
Section 5: Algorithms
Searching, sorting, hashing. Asymptotic worst case time and space complexity.
Algorithm design techniques: greedy, dynamic programming and divide‐and‐conquer.
Graph search, minimum spanning trees, shortest paths.
Section 6: Theory of Computation
Regular expressions and finite automata. Context-free grammars and push-down
automata. Regular and contex-free languages, pumping lemma. Turing machines and
undecidability.
Section 7: Compiler Design
Lexical analysis, parsing, syntax-directed translation. Runtime environments. Intermediate
code generation.
Section 8: Operating System
Processes, threads, inter‐process communication, concurrency and synchronization.
Deadlock. CPU scheduling. Memory management and virtual memory. File systems.
Section 9: Databases
ER‐model. Relational model: relational algebra, tuple calculus, SQL. Integrity constraints,
normal forms. File organization, indexing (e.g., B and B+ trees). Transactions and
concurrency control.
Section 10: Computer Networks
Concept of layering. LAN technologies (Ethernet). Flow and error control techniques,
switching. IPv4/IPv6, routers and routing algorithms (distance vector, link state). TCP/UDP
and sockets, congestion control. Application layer protocols (DNS, SMTP, POP, FTP, HTTP).
Basics of Wi-Fi. Network security: authentication, basics of public key and private key
cryptography, digital signatures and certificates, firewalls.
ST Statistics
Calculus: Finite, countable and uncountable sets, Real number system as a complete ordered
field, Archimedean property; Sequences and series, convergence; Limits, continuity,
uniform continuity, differentiability, mean value theorems; Riemann integration, Improper
integrals; Functions of two or three variables, continuity, directional derivatives, partial
derivatives, total derivative, maxima and minima, saddle point, method of Lagrange's
multipliers; Double and Triple integrals and their applications; Line integrals and Surface
integrals, Green's theorem, Stokes' theorem, and Gauss divergence theorem.
Linear Algebra: Finite dimensional vector spaces over real or complex fields; Linear
transformations and their matrix representations, rank; systems of linear equations,
eigenvalues and eigenvectors, minimal polynomial, Cayley-Hamilton Theorem, diagonalization, Jordan canonical form, symmetric, skew-symmetric, Hermitian, skew-
Hermitian, orthogonal and unitary matrices; Finite dimensional inner product spaces, Gram-
Schmidt orthonormalization process, definite forms.
Probability: Classical, relative frequency and axiomatic definitions of probability,
conditional probability, Bayes' theorem, independent events; Random variables and
probability distributions, moments and moment generating functions, quantiles; Standard
discrete and continuous univariate distributions; Probability inequalities (Chebyshev,
Markov, Jensen); Function of a random variable; Jointly distributed random variables,
marginal and conditional distributions, product moments, joint moment generating
functions, independence of random variables; Transformations of random variables,
sampling distributions, distribution of order statistics and range; Characteristic functions;
Modes of convergence; Weak and strong laws of large numbers; Central limit theorem for
i.i.d. random variables with existence of higher order moments.
Stochastic Processes: 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.
Inference: Unbiasedness, consistency, sufficiency, completeness, uniformly minimum
variance unbiased estimation, method of moments and maximum likelihood estimations;
Confidence intervals; Tests of hypotheses, most powerful and uniformly most powerful tests, likelihood ratio tests, large sample test, Sign test, Wilcoxon signed rank test, Mann-
Whitney U test, test for independence and Chi-square test for goodness of fit.
Regression Analysis: Simple and multiple linear regression, polynomial regression,
estimation, confidence intervals and testing for regression coefficients; Partial and multiple
correlation coefficients.
Multivariate Analysis: Basic properties of multivariate normal distribution; Multinomial
distribution; Wishart distribution; Hotellings T2and related tests; Principal component analysis; Discriminant analysis; Clustering.
Design of Experiments: One and two-way ANOVA, CRD, RBD, LSD, 22 and 23
Factorial experiments.
EC Electronics and Communications Engineering
Section 1: Engineering Mathematics
Linear Algebra: Vector space, basis, linear dependence and independence, matrix
algebra, eigen values and eigen vectors, rank, solution of linear equations – existence
and uniqueness.
Calculus: Mean value theorems, theorems of integral calculus, evaluation of definite and
improper integrals, partial derivatives, maxima and minima, multiple integrals, line, surface
and volume integrals, Taylor series.
Differential Equations: First order equations (linear and nonlinear), higher order linear
differential equations, Cauchy's and Euler's equations, methods of solution using variation
of parameters, complementary function and particular integral, partial differential
equations, variable separable method, initial and boundary value problems.
Vector Analysis: Vectors in plane and space, vector operations, gradient, divergence and
curl, Gauss's, Green's and Stoke's theorems.
Complex Analysis: Analytic functions, Cauchy's integral theorem, Cauchy's integral
formula; Taylor's and Laurent's series, residue theorem.
Numerical Methods: Solution of nonlinear equations, single and multi-step methods for
differential equations, convergence criteria.
Probability and Statistics: Mean, median, mode and standard deviation; combinatorial
probability, probability distribution functions - binomial, Poisson, exponential and normal;
Joint and conditional probability; Correlation and regression analysis.
Section 2: Networks, Signals and Systems
Network solution methods: nodal and mesh analysis; Network theorems: superposition,
Thevenin and Norton’s, maximum power transfer; Wye‐Delta transformation; Steady state
sinusoidal analysis using phasors; Time domain analysis of simple linear circuits; Solution of
network equations using Laplace transform; Frequency domain analysis of RLC circuits;
Linear 2‐port network parameters: driving point and transfer functions; State equations for
networks.
Continuous-time signals: Fourier series and Fourier transform representations, sampling
theorem and applications; Discrete-time signals: discrete-time Fourier transform (DTFT),
DFT, FFT, Z-transform, interpolation of discrete-time signals; LTI systems: definition and
properties, causality, stability, impulse response, convolution, poles and zeros, parallel and
cascade structure, frequency response, group delay, phase delay, digital filter design
techniques.
Section 3: Electronic Devices
Energy bands in intrinsic and extrinsic silicon; Carrier transport: diffusion current, drift
current, mobility and resistivity; Generation and recombination of carriers; Poisson and
continuity equations; P-N junction, Zener diode, BJT, MOS capacitor, MOSFET, LED, photo
diode and solar cell; Integrated circuit fabrication process: oxidation, diffusion, ion
implantation, photolithography and twin-tub CMOS process.
Section 4: Analog Circuits
Small signal equivalent circuits of diodes, BJTs and MOSFETs; Simple diode circuits:
clipping, clamping and rectifiers; Single-stage BJT and MOSFET amplifiers: biasing, bias
stability, mid-frequency small signal analysis and frequency response; BJT and MOSFET
amplifiers: multi-stage, differential, feedback, power and operational; Simple op-amp circuits; Active filters; Sinusoidal oscillators: criterion for oscillation, single-transistor and op-
amp configurations; Function generators, wave-shaping circuits and 555 timers; Voltage reference circuits; Power supplies: ripple removal and regulation.
Section 5: Digital Circuits
Number systems; Combinatorial circuits: Boolean algebra, minimization of functions using
Boolean identities and Karnaugh map, logic gates and their static CMOS
implementations, arithmetic circuits, code converters, multiplexers, decoders and PLAs;
Sequential circuits: latches and flip‐flops, counters, shift‐registers and finite state machines;
Data converters: sample and hold circuits, ADCs and DACs; Semiconductor memories:
ROM, SRAM, DRAM; 8-bit microprocessor (8085): architecture, programming, memory and
I/O interfacing.
Section 6: Control Systems
Basic control system components; Feedback principle; Transfer function; Block diagram
representation; Signal flow graph; Transient and steady-state analysis of LTI systems;
Frequency response; Routh-Hurwitz and Nyquist stability criteria; Bode and root-locus plots;
Lag, lead and lag-lead compensation; State variable model and solution of state
equation of LTI systems.
Section 7: Communications
Random processes: autocorrelation and power spectral density, properties of white noise,
filtering of random signals through LTI systems; Analog communications: amplitude
modulation and demodulation, angle modulation and demodulation, spectra of AM and
FM, superheterodyne receivers, circuits for analog communications; Information theory:
entropy, mutual information and channel capacity theorem; Digital communications:
PCM, DPCM, digital modulation schemes, amplitude, phase and frequency shift keying
(ASK, PSK, FSK), QAM, MAP and ML decoding, matched filter receiver, calculation of
bandwidth, SNR and BER for digital modulation; Fundamentals of error correction,
Hamming codes; Timing and frequency synchronization, inter-symbol interference and its
mitigation; Basics of TDMA, FDMA and CDMA.
Section 8: Electromagnetics
Electrostatics; Maxwell’s equations: differential and integral forms and their interpretation,
boundary conditions, wave equation, Poynting vector; Plane waves and properties:
reflection and refraction, polarization, phase and group velocity, propagation through
various media, skin depth; Transmission lines: equations, characteristic impedance,
impedance matching, impedance transformation, S-parameters, Smith chart;
Waveguides: modes, boundary conditions, cut-off frequencies, dispersion relations;
Antennas: antenna types, radiation pattern, gain and directivity, return loss, antenna
arrays; Basics of radar; Light propagation in optical fibers.
Comments
Post a Comment