Paper-I
Differential and Integral Calculus
Partial Differentiation
Euler's Theorem for homogeneous functions
Total Differentiation, Maxima and Minima of two and three variables
Lagrange's Multipliers Method
Curvature
Asymptotes
Envelopes and Evolutes, Singular Points
Rectification, Multiple Integral
volume and surface of revolution of curves
Beta and Gamma functions.
Two Dimensional Coordinate Geometry (Catesian and Polar coordinates)
Polar equation of conics
Polar equation of tangent, normal, asymptotes and chord of contact
Auxiliary and Director circle
Second degree equation of General Conic
Centre, Asymptotes, eccentricity, foci, directrix axes and latus rectum of a conic, Co-ordinate of center, equation of conic referred to center as origin, lengths and position of axes of a standard conic
Three Dimensional Coordinate Geometry
Straight Line, Sphere, Cylinder, Cone and their properties (Rectangular Coordinates only)
Central Conicoids and their properties (Referred to principal axes only)
Vector Calculus
Differentiation of Vectors
Del operator, Gradient, divergent, Curl and directional derivative, their identities and related theorems
Integration of Vectors, line, Surface and Volume integration of vectors
Gauss Divergence, Stokes and Green theorem.
Ordinary Differential Equations
First order non-linear differential equation
singular solutions and extraneous Loci
Second order linear differential equation with constant and variable coefficients
imultaneous and Total Differential Equations
Partial Differential Equations
Linear and Non-linear Partial differential equation of first order
Liner Partial Differential Equations of Second Order
Solution of Partial Differential Equations by Lagrange's, Charpit's and Monge's Method
Mechanics
Equilibrium of coplanar forces, Moments, Friction, Catenary
Simple harmonic motion
Rectilinear motion under variable laws
Motion in resisting medium
Projectile
Abstract Algebra
Groups- Normal Sub-groups, Quotient groups, Homomorphism, Isomorphism of groups
Classification of finite groups
Cauchy's Theorem for finite abelian groups, Permutation groups, Solvable groups and their properties
Rings, Morphism, Principal Ideal domain
Euclidean Rings, Polynomial Rings
Irreducibility criteria, Fields, Finite fields, Field extensions
Integral domain
Linear Algebra
Vector Spaces, Linear dependence and independence
Bases, Dimensions, Linear transformations
Matrix representation of Linear transformations, Change of bases
Inner product spaces, Orthonormal basis, Quadratic forms, reduction and classification of quadratic forms
Algebra of Matrices, Eigenvalues and Eigenvectors, Cayley-Hamilton theorem
Canonical, Diagonal, Triangular and Jordan forms, Rank of Matrix
Complex Analysis
Analytic Functions, Cauchy's Theorem, Cauchy's Integral Formulae
Power Series, Laurent's Series
Singularities, Theory of Residues
Complex Transformations, Contour Integration
Paper-II
Special Functions
Beta and Gamma Functions
Hypergeometric
Functions
Bessel Functions
Legendre Function of first kind
Hermite
Polynomials
Laguerre Polynomials.
Integral Transforms
Laplace transform
Inverse Laplace transform
convolution theorem
Fourier transform,
Inverse Fourier transform
Parseval
theorem
Hankel transform
Mellin transform.
Differential and Integral Equations
Classification of second order Partial
Differential Equations, Green's Functions, Sturm-Liouville Boundary Value
Problems, Cauchy's problems and Characteristics, Calculus of variation,
Euler-Lagrange equation
Integral Equations of first and second kind of Fredholm and Volterra type
Solution by successive substitutions and successive approximations.
Metric spaces and Topology
Metric spaces, compactness,
connectedness, Topological spaces, closed sets, closure, Dense set,
Neighbourhood
Interior, exterior and boundary points, Accumulation points
and derived sets
Bases and sub-bases
First and second countable spaces,
separable spaces, Separation axioms, compactness, continuous functions
and compact sets, connected spaces.
Differential Geometry
Curves in space
Osculating plane, Normal plane,
rectifying plane, Serret-Frenet formulae, curvature torsion, circle of
curvature, Sphere of curvature
envelopes
curves on sufaces.
Tensors
Covariant
Contravariant and mixed tensors
Invariants
Addition
Subtraction and Multiplication of tensors
Contraction of tensors
Quotient law of tensors
Fundamental tensors
Associated tensors
Christoffel symbols
Covariant differentiation of tensors
Law of covariant differentiation.
Mechanics
D'Alembert's Principle, Moment and product of inertia,
Motion in two-dimensions
Lagrange's equations of motion, Euler's
Equations of motion, motion of a top.
Numerical Analysis
Interpolation, Difference schemes, Lagrange
interpolation, Numerical differentiation and integration, Bisection, Secant,
Regula-Faisi and Newton's Methods, Roots of polynominal
Linear
Equation - Direct Methods (Jacobi, Gauss and Siedal Method).
Operations Research
Simplex methods, Duality, Degeneracy, Revised
Simplex method, Integer Programming Problems, Assignment problems,
Transportation Problems
Game Theory - Two person zero sum game.
Mathematical Statistics
Probability, conditional Probability,
Addition and multiplication theorems of probability, Baye's Theorem,
Expectations, Moment Generating Function
Probability
Distributions : Binomial, Poisson, Uniform and Normal, Correlation and
Regression.