NOTE: Attempt any FIVE questions selecting at least TWO questions from each section.

Section-I (5/9)

Real Number System

•    Ordered sets, Fields, Completeness property of real numbers

•    The extended real number system, Euclidean spaces

Sequences and Series

•    Sequences, Subsequences, Convergent sequences, Cauchy sequences

•    Monotone and bounded sequences, Bolzano Weierstrass theorem

•    Series, Convergence of series, Series of non-negative terms, Cauchy condensation test

•    Partial sums, The root and ratio tests, Integral test, Comparison test

•    Absolute and conditional convergence

Limit and Continuity

•    The limit of a function, Continuous functions, Types of discontinuity

•    Uniform continuity, Monotone functions

Differentiation

•    The derivative of a function

•    Mean value theorem, Continuity of derivatives

•    Properties of differentiable functions.

Functions of Several Variables

•    Partial derivatives and differentiability, Derivatives and differentials of composite functions

•    Change in the order of partial derivative, Implicit functions, Inverse functions, Jacobians

•     Maxima and minima, Lagrange multipliers



Section-II (4/9)

The Riemann-Stieltjes Integrals

•    Definition and existence of integrals, Properties of integrals

•    Fundamental theorem of calculus and its applications

•    Change of variable theorem

•    Integration by parts

Functions of Bounded Variation

•    Definition and examples

•    Properties of functions of bounded variation

Improper Integrals

•    Types of improper integrals

•    Tests for convergence of improper integrals

•    Beta and gamma functions

•    Absolute and conditional convergence of improper integrals

Sequences and Series of Functions

•    Definition of point-wise and uniform convergence

•    Uniform convergence and continuity

•    Uniform convergence and integration

•    Uniform convergence and differentiation