Subjects carrying not more than 200 marks can be opted.
Total Marks – 200
PAPER – I
• Candidates will he asked to attempt three questions from Section A and two questions from section B.
• Groups, subgroups, Languages Theorem, cyclic groups, normal sub groups, quotient groups, Fundamental theorem of homomorphism, Isomorphism theorems of groups, Inner automorphisms, Conjugate elements, conjugate subgroups, Commutator subgroups.
• Rings, Subrings, Integral domains, Quotient fields, Isomorphism theorems, Field extension and finite fields.
• Vector spaces, Linear independence, Bases, Dimension of a finitely generated space, Linear transformations, Matrices and their algebra, Reduction of matrices to their echelon form, Rank and nullity of a linear transformation.
• Solution of a system of homogeneous and non-homogeneous linear equations, Properties of determinants, Cayley-Hamilton theorem, Eigenvalues and eigenvectors, Reduction to canonical forms, specially diagonalisation.
• Conic sections in Cartesian coordinates, Plane polar coordinates and their use to represent the straight line and conic sections, (artesian and spherical polar coordinates in three dimensions, The plane, the sphere, the ellipsoid, the paraboloid and the hyperbiloid in Cartesian and spherical polar coordinates.
• Vector equations for Plane and for space-curves. The arc length. The osculating plane. The tangent, normal and binormal, Curvature and torsion, Serre-Frenet’s formulae, Vector equations for surfaces, The first and second fundamental forms, Normal, principal, Gaussian and mean curvatures,
Candidates will be asked to attempt any three questions from Section A and two questions from Section B.SECTION A
Calculus and Real Analysis
• Real Numbers, Limits, Continuity, Differentiabiliry, Indefinite integration, Mean value theorems, Taylor’s theorem, Indeterminate forms, Asymptotes. Curve tracing, Definite integrals, Functions of several variables, Partial derivatives. Maxima and minima Jacobians, Double and triple integration (techniques only). Applications of Beta and Gamma func tions. Areas and Volumes. Riemann-Stieltje’s integral, Improper integrals and their conditions of existences, Implicit function theorem, Absolute and conditional convergence of series of real terms, Rearrangement of series, Uniform convergence of series,
• Metric spaces, Open and closed spheres, Closure, Interior and Exterior of a set. Sequences in metric space, Cauchy sequence convergence of sequences, Examples, Complete metric spaces, Continuity in metric spaces, Properties of continuous functions,
SECTION – B
Function of a complex variable: Demoiver’s theorem and its applications, Analytic functions, Cauchy’s theorem, Cauchy’s integral formula, Taylor’s and Laurent’s series, Singularities, Cauchy residue theorem and contour integration, Fourier series and Fourier transforms, Analytic continuation.
Applied Mathematics Total Marks– 200
PAPER-I (Marks – 100)
Candidates will be asked to attempt any two questions from Section A and any three questions from Section B.
• Vector Analysis
Vector algebra, scalar and vector product of two or more vectors, Function of a scalar variable, Gradient, divergence and curl, Expansion formulae, curvilinear coordinates, Expansions for gradient, divergence and curl in orthogonal curvilinear coordinates, Line, surface and volume integrals, Green’s, Stoke’s and Gauss’s theorms
Composition and resolution of forces, Parallel forces, and couples, Equilibrium of a system of coplanar forces, Centre of mass and centre of gravity of a system of particles and rigid bodies, Friction, Principle of virtual work and its applications, equilibrium of forces in three dimensions.
Tangential, normal, radial and transverse components of velocity and acceleration, Rectilinear motion with constant and variable acceleration, Simple harmonic motion, Work, Power and Energy, Conservative forces and principles of energy, Principles of linear and angular momentum, Motion of a projectile, Ranges on horizontal and inclined planes, Parabola of’ safety. Motion under central forces, Apse and apsidal distances, Planetary orbits, Kepler’s laws, Moments and products of inertia of particles and rigid bodies, Kinetic energy and angular momentum of a rigid body, Motion of rigid bodies, Compound pendulum, Impulsive motion, collision of two spheres and coefficient of restitution.
PAPER – II (Marks – 100)
Candidates will be asked to attempt any two questions from Section A. one question from Section B and two questions from Section C.
SECTION – A
• Differential Equations
Linear differential equations with constant and variable coefficients, the power series method.
Formation of partial differential equations. Types of integrals of partial differential equations. Partial differential equations of first order Partial differential equations with constant coefficients. Monge’s method. Classification of partial differential equations of second order, Laplace’s equation and its boundary value problems. Standard solutions of wave equation and equation of heat induction.
SECTION – B
Definition of tensors as invariant quantities. Coordinate transformations. Contravariant and covariant laws of transformation of the components of tensors. Addition and multiplication of tensors, Contraction and inner product of tensors The Kronecker delta and Levi-Civita symbol. The metric tensor in Cartesian, polar and other coordinates, covariant derivatives and the Christoffel symbols. The gradient. divergence and curl operators in tensor notation.
• Elements of Numerical Analysis
Solution of non-linear equations, Use of x = g (x) form, Newton Raphson method, Solution of system of linear equations, Jacobi and Gauss Seidel Method, Numerical Integration, Trapezoidal and Simpson’s rule. Regula falsi and interactive method for solving non-linear equation with convergence. Linear and Lagrange interpolation. Graphical solution of linear programming problems.
Computer Science (Total Marks – 100)
Candidates will be asked to attempt total five questions including one compulsory objective type question. They will attempt atleast one question from each section. Each question will carry 20 marks.
• Computer Architecture
Introduction to modern machine Architecture, Storage Hierarchy Main/ Virtual/ Cache/ Secondary Memory, CPU, ALU, Peripheral communication, Designing of Instruction set, Stored program concept. Introduction to parallel computing; SIMD/MIMD.
• Operating System
Functions/Types of operating systems, Processes, Interprocess Communication/ Synchronization/Co-ordination, Process Scheduling Policies, Virtual Memory Management Techniques: Paging/Segmentation, File Management Systems.
• Computer Networks
LAN/WAN/MAN, Communication channels, Internetworking, Internet, Network layer structure, ISO Internet Protocol, OSI/TCP/IP reference model.
SECTION – B
• Structured and Object Oriented Programming
Basics of C/C++ environment, memory concepts. operators, control structures, selection structures, Array & functions/methods, classes & data Abstractions, inheritance and polymorphism.
• Data Structures and Algorithms
Pseudo language, Functions, Iteration, Recursion, Time/Complexity Analysis, Stacks Queue, hashing. linked list, Searching; Sequential. Binary, Soiling Algorithms, Graphs Algorithms, Tree Algorithms, Trees, ADTs, Implementation using Structured/object oriented languages.
• Software Engineering
Introduction to Software Engineering, Software life cycle, Software Design Methodologies: Structured/Object oriented, Software documentation and Management, Introduction to CASE tools.
SECTION – C
• Data Base Management
Data Models, E-R Models, Relational Database concepts, SQL, Normalization, Database Design.
• Web Programming
HTML, CGI, PERL, JAVA: Applet/Script, WWW, Web based unit face Design.
• Computer Graphics
Fundamentals of input, display and hard copy devices, scan conversion of geometric primitives. 2D and 3D geometric transformations, clipping and windowing, scene modeling and animation, algorithms for visible and surface determination.
Total Marks – 100
1. Basic Probability Axiomatic definition of probability, random variable, distribution function, probability density function, mathematical expectation; conditional probability, jointly distributed random variables, marginal and conditional distributions, conditional expectation, stochastic independence.
2. Some Special Distributions : Binomial, poisson. negative binomial, hypergeometric, normal distributions with their derivation of their mean and variance; Definition and Application of chisquare, “T” and ‘F’ distributions.
3. Statistical Inference: Maximum likelihood estimation of the mean and the variance of a normal population; confidence interval for mean, difference of means and for variance: testing hypothesis for the equality of two means (paired and unpaired observations); testing of equality of sever al means (ANOVA) and testing of variance and equality of two variance.
4. Correlation and regression: Simple linear regression model point and interval estimation of parameters, Simple Partial, Multiple Correlation and testing of these correlations.
5. Sampling, Simple random, stratified, systematic and cluster sampling, estimates of mean and total and their precision.
6. Applications of Statistics in social, economic and political problems public health, crimes, Law, social innovations economic development, socio-political inequality.