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MATHEMATICS
Paper-I
Section A
Linear Algebra: Vector space, Linear dependance and, independence, Sub-speces, Bases, Dimensions, Finite dimensional vector spaces.
Matrices, Cayley-Hamilton theorem, Eigenvalues and Eigenvectors, Matrix of linear transformation, Row and column reduction, Echelon form, Equivalence, Congruence and Similrity, Reduction to Canonical form, Rank, Orthogonal, Symmetrical, Skew Symmetrical, Unitary, Hermitian Skew-Hermitian forms their eigenvalues. Orthogonal and Unitary reduction of quadratic and Hermition forms, Positive definite quadratic forms, Simultaneous reduction, Sylvester's law of inertia.
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Calculus: Real numbers, Limits, Continuity, Differentiability, Mean-value Theorems, Taylor's theorem with remainders, indeterminate forms, Maxima and Minima, Asymptotes, Functions of several variables, Continuity, Differentiability, Partial derivatives, Maxima and Minima, Lagrange's method of Multipliers,
Jacobian, Riemann's definition of definite integrals; Indefinite integrals, infinite and improper integral, Double and triple integrals (techniques only). Repeated integrals, Beta and Gamma functions. Areas, Surface and Volumes, Centre of Gravity.
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Geometry: Cartesian and Polar coordinates in two and three dimension; Second degree equations in two and three dimensions, Reduction to Cannonical forms; Straight lines, Plane, Sphere, Cone, Cylinder, Paraboloid, Ellipsoid, Hyperboloid of one and two sheets and their properties. Shortest distance between two skew lines; Curves in space, Curvature and torsion. Serret-Frenet's formulae.
Section-B
Ordinary Differential Equations: Formation of differential equations. Order and Degree. Equations of first order and first degree. Integrating factor. Equations of first order but not of first degree. Clairaut's equation, singular solution. Higher order linear equations with constant coefficients. Complementary function and particular integral. General solution. Euler-Cauchy equation.
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Second order linear equations with variable coefficients. Determination of complete solution when one solution is known. Method of variation of parameters.
Statics: Equilibrium of a system of particles, work and potential energy. Friction. Common Catenary. Principle of Virtual work. Stability of Equilibrium. Equilibrium of forces in three Dimensions.
Dynamics: Degree of freedom an constraints. Rectilinear motion. Simple Harmonic motion. Motion in a plane. Projectiles. Constrained Motion. Work and energy, Conservation of energy. Motion under Impulsive forces. Kepler's laws. Orbits under Central forces. Motion of varying mass. Motion under resistance.
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Hydrostatics: Pressure of heavy fluids. Equilibrium of fluids under given system of forces. Centre of pressure. Thrust on curved surfaces. Equilibrium of floating bodies. Stability of equilibrium, Metacentre, Pressure of gases, problems relating to atmosphere.
Vector Analysis: Scalar and vector fields, triple products Differentiation of Vector function of a scalar variable, Gradient, Divergence and Curl in Cartesian, Cylindrical and Spherical coordinates and their physical interpretation. Higher order derivatives, Vector Identities and Vector Equations, Application to Geometry, Gauss and Stoke's Theorems, Green's identities.
Tensor Analysis: Definition of a Tensor, Transformation of coordinates, contravariant and covariant tensors. Addition and multiplication of tensors, contraction of tensors. Inner product, fundamental tensor, Christoffel symbols, Covariant differentiation. Gradient, Curl and Divergence in tensor notation.
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Special Theory-of Relativity: Lorentz transformation, Addition of velocities, Fitzgerald contraction Time dl1ation, Variation of mass with velocity, Energy momentum vector, Mass-energy relationship.
Paper-II
Section-A
Algebra: Group, Subgroups, Normal subgroups, Homomorphism of groups, Quotient Groups. Basic Isomorphism Theorems. Sylow's Group. Permutation Groups. Cayley-Hamilton Theorem. Rings and Ideals. Principal Ideal Domains, unique Factorisation Domains and Euclidean Domains. Field Extension. Finite Fields.
Real Analysis: Real number system, Ordered sets, Bounds, Ordered Field, Real number system as an Ordered Field with least Upper Bound, Cauchy Sequence, Completeness. Completion, Continuous Functions, Uniform Continuity, Properties of continuous functions on compact sets. Riemann integral, Improper integrals. Differentiation of functions of several variables, Change in the order of Partial derivatives. Implicit function theorem, Maxima and Minima, Absolute and Conditional Convergence of series of real and Complex terms, Rearrangement of series, Uniform convergence, Infinite Products. Continuity, differentiability and integrability for series, Multiple integrals.
Complex Analysis: Analytic function, Cauchy-Riemann equations. Cauchy's Theorem, Cauchy's intera1 formula, Power series. Taylor's series, Laurent's series, Singularities, Cauchy's Residue Theorem Contour integration. Conformal mapping,' Bilinear Transformations.
Numerical Analysis and Computer Programming Numerical methods: Solution of algebraic and transcendental equations of one variable by bisection, Regula-falsi and Newton Raphson methods. Solution of system of linear equations by' Gaussian elimination and Gauss-Jordan (direct) methods.. Gauss Seid (interactive) method.
Interpolation: Newton's (forward and backward) and Lagrange's method.
Numerical integration: Simpson's one-third rule, Trapezoidal rule, Gaussian quadrature formula.
Numerical solution of Ordinary Differential Equations: Euler and Runge Kutta-methods.
Computer Programming: Storage of number in Computers: Bits, Bytes and Words. Binary system. Arithmetic and Logical operations on numbers. Bitwise operations. AND, OR, XOR, NOT and shift/rotate operators. Octal and Hexadecimal Systems. Conversion to and from. Decimal Systems. Representation of unsigne integers, Signed Integers and Reals. Double precision reals and Long integers.
Algorithms and Flow charts for solving numerical analysis problems.
Developing simple programs in BASIC for problems involving techniques covered in the numerical analysis.
Section-B
Partial differential equations: Curves and surfaces in three dimensions; Formation of Partial differential equations; Solutions of equations of type dx/p=dy/Q=dz/R; Orthogonal Trajectories, Plaffian Differential Equational; Partial Differential Equations of the First order; Solution by Cauchy's method of Characteristics; Charpit's method of solution; Linear Partial Differential Equational; Partial Differential Equations of the First order; Solution by Cauchy's method of Characteristics; Charpit's method of solution; Linear Partial Different Equations of the second order with constant coefficients; Equations of vibrating string; Heat equation; Laplace equation.
Mechanics: Generalised Coordinates, Constraints, Holonomic and Non-Holonomic Systems. D'Alembert's Principle and Lagrange's equations, Hamilton equations, Moment of Inertia, Motion of rigid bodies in two dimensions.
Hydrodynamics: Equation of continuity, Euler's equation of motion for inviseid Flow, Stream-liner, Path of a Particle, Potential flow, Two-dimensional and axisymmetric motion, Sources and Sink, Vortex motion, Flow past a Cylinder and a Sphere, Method of Images, Navier-Stokes Equation for a viscous fluid and its limitations.
Probability and Statistics-Probability: Sample space, Events, Algebra of events, Probability-Classical, Statistical and Axiomatic Approaches. Combinatorial Problems. Geometric Problems. Conditional Probability and Baye's Theorem. Random Variables and Probability. Distributions-Discrete and Continuous. Mathematical Expectations. Binomial, Poisson and Normal Distributions. Joint Distribution of Random Variables, Independence. Central limit theorem in i.i.d. cases.
Statistics: Concepts of Population, Sample, Variable, Attribute, Parameter and Statistic. Measures of Location and Dispersion. Moments, Skewness and Kurtosis. Correlation. Simple Random sampling and Sampling Distribution of Sample Means and Sample Proportions.
Operations Research: Linear programming problems. Basic, Basic Feasible and Optimal solution. Graphical method and Simple method of solution. Duality.
Transporatat on assignment problems. Travelling salesman problems.
Game Theory: Rectangular games, pure strategy and mixed strategy, Saddle point, value of the game and optimal strategy. Principle of dominance. Algebrac method Relations between game theory and LPP.
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