Sequence of real numbers, convergence of sequences, bounded and monotone sequences, convergence criteria for sequences of real numbers, Cauchy sequences, subsequences, Bolzano-Weierstrass theorem. Series of real numbers, absolute convergence, tests of convergence for series of positive terms - comparison test, ratio test, root test; Leibniz test for convergence of alternating series.

Functions of One Real Variable:

Limit, continuity, intermediate value property, differentiation, Rolle's Theorem, mean value theorem, L'Hospital rule, Taylor's theorem, maxima and minima.

Functions of Two or Three Real Variables:

Limit, continuity, partial derivatives, differentiability, maxima and minima.

Vector Calculus:

Scalar and vector fields, gradient, divergence, curl, line integrals, surface integrals, Green, Stokes and Gauss theorems.

Group Theory:

Groups, subgroups, Abelian groups, non-Abelian groups, cyclic groups, permutation groups, normal subgroups, Lagrange's Theorem for finite groups, group homomorphisms and basic concepts of quotient groups.

Linear Algebra:

Finite dimensional vector spaces, linear independence of vectors, basis, dimension,linear transformations, matrix representation, range space, null space, rank-nullity theorem. Rank andinverse of a matrix, determinant, solutions of systems of linear equations, consistency conditions,eigen values and eigenvectors for matrices, Cayley-Hamilton theorem.

Real Analysis:

Interior points, limit points, open sets, closed sets, bounded sets, connected sets,compact sets, completeness of R. Power series (of real variable), Taylor's series, radius and intervalof convergence, term-wise differentiation and integration of power series.

## Syllabus - Mathematics (MA)

## Sequences and Series of Real Numbers:

Sequence of real numbers, convergence of sequences, bounded and monotone sequences, convergence criteria for sequences of real numbers, Cauchy sequences, subsequences, Bolzano-Weierstrass theorem. Series of real numbers, absolute convergence, tests of convergence for series of positive terms - comparison test, ratio test, root test; Leibniz test for convergence of alternating series.

## Functions of One Real Variable:

Limit, continuity, intermediate value property, differentiation, Rolle's Theorem, mean value theorem, L'Hospital rule, Taylor's theorem, maxima and minima.

## Functions of Two or Three Real Variables:

Limit, continuity, partial derivatives, differentiability, maxima and minima.

## Vector Calculus:

Scalar and vector fields, gradient, divergence, curl, line integrals, surface integrals, Green, Stokes and Gauss theorems.

## Group Theory:

Groups, subgroups, Abelian groups, non-Abelian groups, cyclic groups, permutation groups, normal subgroups, Lagrange's Theorem for finite groups, group homomorphisms and basic concepts of quotient groups.

## Linear Algebra:

Finite dimensional vector spaces, linear independence of vectors, basis, dimension,linear transformations, matrix representation, range space, null space, rank-nullity theorem. Rank andinverse of a matrix, determinant, solutions of systems of linear equations, consistency conditions,eigen values and eigenvectors for matrices, Cayley-Hamilton theorem.

## Real Analysis:

Interior points, limit points, open sets, closed sets, bounded sets, connected sets,compact sets, completeness of R. Power series (of real variable), Taylor's series, radius and intervalof convergence, term-wise differentiation and integration of power series.