Analytical Reasoning

Algebra

Arithmetic, geometric and harmonic progression. Continued fractions. Elementary combinatorics: Permutations and combinations, Binomial theorem. Theory of equations. Inequalities. Complex numbers and De Moivre’s theorem. Elementary set theory. Functions and relations. Elementary number theory: Divisibility, Congruences, Primality. Algebra of matrices. Determinant, rank and inverse of a matrix. Solutions of linear equations. Eigenvalues and eigenvectors of matrices. Simple properties of a group.

Coordinate Geometry

Straight lines, circles, parabolas, ellipses and hyperbolas

Calculus

Sequences and series: Power series, Taylor and Maclaurin series. Limits and continuity of functions of one variable. Differentiation and integration of functions of one variable with applications. Definite integrals. Maxima and minima. Functions of several variables - limits, continuity, differentiability. Double integrals and their applications. Ordinary linear differential equations.

Elementary discrete probability theory

Combinatorial probability, Conditional probability, Bayes theorem. Binomial and Poisson distributions.

S.No.

Topics

1

Analytical Reasoning

2

Elementary Euclidean geometry and trigonometry

3

Elements of set theory, Functions and relations, Permutations and combinations, Principle of inclusion and exclusion, Pigeon-hole principle

4

Theory of equations, Inequalities

5

Elementary number theory, divisibility, congruences, primality

6

Determinants, matrices, solutions of linear equations, vector spaces, linear independence, dimension, rank and inverse

7

Limits, continuity, sequences and series, differentiation and integration with applications, maxima-minima

8

Combinatorial probability, Conditional probability, Discrete random variables and expectation, Binomial distribution

Mathematics

1

Linear algebra

vector spaces and linear transformations, direct sum, matrices and systems of linear equations, characteristic roots and characteristic vectors, Cayley Hamilton theorem, diagonalization and triangular forms, quadratic forms.

2

Abstract algebra

Groups: subgroups, products, cosets, Lagranges theorem, group homomorphism, normal subgroups and quotient groups, permutation groups, Sylow theorems. Rings and integral domains: subrings, ring homomorphism, ideals and quotient rings, prime and maximal ideals, products, Chinese remainder theorem, prime and irreducible elements, principal ideal domain, unique factorization domains. Polynomial rings: division algorithm, roots of polynomials. Fields: characteristic of a field, field extensions, finite fields.

3

Calculus and real analysis

Real numbers, limits, continuity, uniform continuity of functions, differentiability of functions of one or more variables and applications, convergence of sequences and series; indefinite integral, fundamental theorem of Calculus, Riemann integration, improper integrals, double and multiple integrals and applications, sequences and series of functions, uniform convergence, solutions of ordinary differential equations.

4

Graph Theory

Connectedness, trees, vertex coloring, planar graphs, Eulerian graphs, Hamiltonian graphs, digraphs and tournaments.

5

Combinatorics

Counting principles, Ramsey theory, binomial coefficients, recurrence relations, divide-and-conquer recurrences, recurrences involving convolution and their use in counting, Fibonacci numbers, generating functions, solving recurrence relations using generating functions

Statistics

1

Notions of sample space and probability, combinatorial probability, conditional probability, Bayes’ theorem and independence

2

Random variable and expectation, moments, standard univariate discrete and continuous distributions, sampling distribution of statistics based on normal samples, central limit theorem, approximation of binomial to normal, Poisson law

3

Multinomial, bivariate normal and multivariate normal distributions.

4

Descriptive statistical measures, product-moment correlation, partial and multiple correlation.

5

Regression – simple and multiple.

6

Elementary theory and methods of estimation – unbiasedness, minimum variance, sufficiency, maximum likelihood method, method of moments, least squares methods.

7

Tests of hypotheses – basic concepts and simple applications of Neyman-Pearson lemma, confidence intervals.

8

Tests of regression, elements of non-parametric inference, contingency tables and Chi-square, ANOVA, basic designs (CRD/RBD/LSD) and their analyses, elements of factorial designs.

9

Conventional sampling techniques, ratio and regression methods of estimation.

Physics

1

Classical mechanics

Lagrangian and Hamiltonian formulation, symmetries and conservation laws, motion in central field of force, planetary motion, simple harmonic motion - damped, undamped and forced, special theory of relativity

2

Electrodynamics

electrostatics, magnetostatics, electromagnetic induction, self and mutual inductance, capacitance, Maxwell’s equation in free space.

3

Nonrelativistic quantum mechanics

Planck’s law, photoelectric effect, Compton effect, wave-particle duality, Heisenberg’s uncertainty principle, Schrodinger equation and applications.

4

Thermodynamics and statistical Physics

laws of thermodynamics and their consequences, thermodynamic potentials and Maxwell’s relations, chemical potential, phase equilibrium, phase space, microstates and macrostates, partition function, free energy, classical statistics.

5

Atomic and molecular physics

quantum states of an electron in an atom, Hydrogen atom spectrum, electron spin, spin-orbit coupling, fine structure, Zeeman effect.

6

Condensed matter physics

crystal classes, 2D and 3D lattice, reciprocal lattice, bonding, diffraction and structure factor, point defects and dislocations, lattice vibration, free electron theory, electron motion in periodic potential, energy bands in metals, insulators and semiconductors

7

Basic nuclear physics

nuclear properties, nuclear forces, nuclear structures, nuclear reactions, radioactive nuclear decay

8

Electronics

semiconductor physics; diodes - clipping, clamping, rectification; Zener regulated power supply, bipolar junction transistor - CC, CB, and CE configurations; transistor as a switch; amplifiers

9

Operational Amplifier and its applications

inverting & noninverting amplifiers, adder, integrator, differentiator, waveform generator, comparator, Schmidt trigger. Digital integrated circuits – NAND and NOR gates as building blocks, XOR gates, half and full adder.

Computer Science

1

Data structures

array, stack, queue, linked list, binary tree, heap, AVL tree, Btree.

2

Discrete Mathematics

recurrence relations, generating functions, graph theory - paths and cycles, connected components, trees, digraphs.

3

Programming languages

Fundamental concepts - abstract data types, procedure call and parameter passing, dynamic memory allocation, at least one of C, C++, Java and Python.

4

Design and analysis of algorithms

Asymptotic notation, searching, sorting, selection, graph traversal, minimum spanning tree.

5

Switching Theory and Logic Design

Boolean algebra, minimization of Boolean functions, combinational and sequential circuits - synthesis and design.

6

Computer organization and architecture

Number representation, computer arithmetic, memory organization, I/O organization, microprogramming, pipelining, instruction level parallelism.

7

Operating systems

Memory management, processor management, critical section problem, deadlocks, device management, file systems

8

Formal languages and automata theory

Finite automata and regular expressions, pushdown automata, context-free grammars, Turing machines, elements of undecidability

9

Database management systems

Relational model, relational algebra, relational calculus, functional dependency, normalization (up to 3rd normal form).

10

Computer networks

OSI, LAN technology - Bus/tree, Ring, Star; MAC protocols; WAN technology - circuit switching, packet switching; data communications - data encoding, routing, flow control, error detection/correction, Inter-networking, TCP/IP networking including IPv4

Engineering & Technology

1

At least one of C, C++, Java, Python.

2

Gravitation, moments of inertia, particle dynamics, elasticity, friction, strength of materials, surface tension and viscosity.

3

Laws of thermodynamics and heat engines.

4

Electrostatics, magnetostatics and electromagnetic induction

5

Laws of electrical circuits – transient and steady state responses of resistive and reactive circuits.

6

D.C. generators, D.C. motors, induction motors, alternators, transformers

7

Diode circuits, bipolar junction transistors & FET devices and circuits, oscillator, operational amplifier.

8

Boolean algebra, minimization of Boolean functions.

9

Combinatorial and sequential circuits – multiplexer, de-multiplexer, encoder, decoder, flip-flops, registers and counters, A/D and D/A converters.

S.No.

Topics

Part 1

Statistics (S1)

1

Descriptive statistics for univariate, bivariate and multivariate data.

2

Standard univariate probability distributions [Binomial, Poisson, Normal] and their fittings, properties of distributions. Sampling distributions.

3

Theory of estimation and tests of statistical hypotheses.

4

Simple and Multiple linear regression, linear statistical models, ANOVA.

5

Principles of experimental designs and basic designs [CRD, RBD & LSD], Full factorial design, Confounding and blocking in 2k factorial designs

6

Elements of non-parametric inference.

7

Elements of categorical data analysis.

8

Sample surveys – simple random sampling with and without replacement, stratified and cluster sampling.

Probability (S2)

1

Classical definition of probability and standard results on operations with events, conditional probability and independence.

2

Distributions of discrete type [Bernoulli, Binomial, Multinomial, Hypergeometric, Poisson, Geometric and Negative Binomial] and continuous type [Uniform, Exponential, Normal, Gamma, Beta] random variables and their moments.

3

Bivariate distributions (with special emphasis on bivariate normal), marginal and conditional distributions, correlation and regression.

4

Multivariate distributions, marginal and conditional distributions, regression, independence, partial and multiple correlations.

5

Order statistics [including distributions of extreme values and of sample range for uniform and exponential distributions].

6

Distributions of functions of random variables.

7

Multivariate normal distribution [density, marginal and conditional distributions, regression].

8

Weak law of large numbers, central limit theorem.

9

Basics of Markov chains and Poisson processes.

Part 2 “ Engineering Stream

Mathematics (E1)

1

Quadratic equations, Roots of polynomial, AP, GP, HP, Divisibility and Prime numbers, Binomial theorem

2

Inequalities, permutation and combination, complex numbers and De Moivre’s theorem.

3

Elementary set theory, functions and relations, matrices, determinants, solutions of linear equations.

4

Trigonometry [multiple and sub-multiple angles, inverse circular functions, identities, solutions of equations, properties of triangles].

5

Coordinate geometry (two dimensions) [straight line, circle, parabola, ellipse and hyperbola], plane geometry, Mensuration.

6

Sequences, series and their convergence and divergence, power series, limit and continuity of functions of one or more variables, differentiation and its applications, maxima and minima, integration, definite integrals areas using integrals, ordinary and partial differential equations (up to second order)

Engineering and Technology (E2)

Engineering Mechanics and Thermodynamics

1

Forces in plane and space, analysis of trusses, beams, columns, friction, principles of strength of materials, work-energy principle, moment of inertia, plane motion of rigid bodies, belt drivers, gearing.

2

Laws of thermodynamics, internal energy, work and heat changes, reversible changes, adiabatic changes, heat of formation, combustion, reaction, solution and dilution, entropy and free energy and maximum work function, reversible cycle and its efficiency, principles of internal combustion engines, principles of refrigeration.

Electrical and Electronics Engineering

1

DC circuits, AC circuits (1-φ), energy and power relationships, transformer, DC and AC machines, concepts of control theory and applications.

2

Network analysis, 2 port network, transmission lines, elementary electronics (including amplifiers, oscillators, op-amp circuits), analog and digital electronic circuits, basics of computer architecture.

Engineering Drawing

1

Concept of projection, point projection, line projection, plan, elevation, sectional view (1st angle / 3rd angle) of simple mechanical objects, isometric view, dimensioning, sketch of machine parts. (Use of Set Square, compass and diagonal scale should suffice).