**Paper-I**

**Section – A**

**1. Abstract Algebra :**

(i) Integers, Congruences.

(ii) Groups, Subgroup, Normal Subgroups, Permutation groups, Homomorphism, Isomorphism, Counting Principles, Sylow’s Theorem, Caley’s Group.

(iii) Rings, Integral Domain, Field, Subring, Homomorphism, Ideal, Principal Ideal Ring, Maximal Ideal, Polynomial rings, Unique Factorization Theorem.

** 2. Linear Algebra :**

(i) Vector space, Linear dependence, Independence, Subspaces, Basis, Dimension, Finite Dimensional Vector space, Linear Transformation, Rank-nullity Theorem.

(ii) Matrices, Determinants, Igenvalue, Igenvectors, Row-column reduction, Echelon form, Orthogonal, Symmetrical, Skew-symmetrical, Unitary, Hermitian Matrices.

** 3. Analytic Geometry :**

(i) 2-D Geometry : Straight lines, Pairs of lines, Circle, System of Circles, Conic sections.

(ii) 3-D Geometry : Planes, Lines, Skew-lines, Sphere, Intersection of Plane and sphere, Cone, Cylinder, Conicoids, Tangent plane to conicoids.

**Section – B**

**1. Real and Complex Analysis :**

(i) Real Analysis : Real number system, Order relation, Bounds, l.u.b. g.l.b., Cauchy sequence, Completeness, Compactness, Continuity, Uniform Continuity of functions, Riemann-Theory of Integration, Fundamental Theorem of calculus, Convergence of sequence and series, Uniform convergence.

(ii) Complex Analysis : Analytic function, Cauchy Riemann Equation, Cauchy Integral Formula, Taylor, Laurent’s series, Singuralities, Poles, residues, Contour Integral.

** 2. Calculus :**

(i) Functions of one variable : Limit, Continuity, Differentiability, Meanvalue theorem, Maxima, Minima.

(ii) Asymptotes and Curvatures : Rectification, Area , Volume and Surface area of revolution (Equations in Cartesian and Parametric forms only)

(iii) Functions of several variables : Limit, Continuity, Differentiability, Jacobians, Euler’s theorem.

(iv) Improper integrals : Convergence, Gamma and Beta functions.

(v) Multiple integrals : Double and Triple integrals and their Evaluations.

** 3. Vector Analysis :**

(i) Dot and Vector products, Vector and scalar Triple Products.

(ii) Differentiation of Vector functions, Divergence, Gradient, Curl of Vectors (in Cartesian forms only).

(iii) Green, Gauss and Stokes theorems and applications.

(iv) Tangent, normal and binormal of curves in space, serret-frenet formulas.

**Paper – II**

**Section – A**

**1. Numerical Analysis :**

(i) Interpolation : Lagrange, Newton divided difference forms, Forward and back ward interpolation polynomials.

(ii) Approximations : Least squares approximations and curve fitting.

(iii) Numerical solution of non-linear equations : Bisection, Secant, NewtonRaphson and fixed point iteration techniques.

(iv) Numerical differentiation and integration: Differentiation formulas involving differences, Newton-Cotes rules, Compound rules, Gauss- Legendre 2 and 3 point rules.

(v) Numerical solution. of I.V.P.: Euler method, Taylor’s method, RungeKutta Method of order two

** 2. Graph Theory :**

Simple graphs, Regular, Complete graphs, Bipartite graphs, Matrix representation of graphs, Connected graphs, Isomorphic graphs, Trees, Planar graph, Hamiltonian and Eulerian graphs, Vertex colouring of graphs and Chromatic number.

** 3. Ordinary and Partial differential equations:**

(i) Linear first order O.D.E.

(ii) Higher order linear differential equations with constant and variable coefficients.

(iii) Series solution of O.D.E.

(iv) Solution of O.D.E. by Laplace transformation techniques.

(v) Solution of equations Pdx + Qdy + Rdz=O and dx/P = dy/Q = dz/R

(vi) Char pits method for partial differential equations.

(vii) Linear second order P.D.E. and solutions.

**Section – B**

**1. Computer programming :**

(i) Flow charting and algorithms.

(ii) Basics of Fortran language, arithmetic and logical operations, Arithmetc and Logical Statements.

(iii) GO TO and Computed GO TO Statements, Arithmetic and Logical IF, IF… THEN….ELSE Statements, DO Loops.

(iv) Arrays and subscripted variables.

(v) Functions, Subprograms and Subroutines.

(vi) Programme writing in Fortran.

** 2. Mechanics and Hydrodynamics:**

(i) Statics : Law of parallelogram of forces, Equilibrium of forces, Couple and Moments, Frictions.

(ii) Dynamics : Laws of motion, D’ Alemberts principle, Motion of a particle I in a plane, Projectiles, Motion of rigid bodies, Moment of inertia.

(iii) Hydrodynamics : Equation of continuity, Euler equation of motion (in Cartesian forms) Stream lines, Path Line, Potential flow, Stream functions and Potential functions, Sources, Sinks and Image system with respect to Plane and Circle.

** 3. Operations Research :**

(i) Formulation of L.P.P., Graphical solution.

(ii) Simplex method and Duality.

(iii) Transportation and Assignment problems.