About the Event

• Venue: Indian Statistical Institute, 7, S. J. S. Sansanwal Marg, NRPC Colony, Block B, Qutab Institutional Area, New Delhi, Delhi 110016.


• Date: 23rd March 2024, Saturday


• Reporting Time: 9:30 AM

Instructions

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Syllabus

1. Sequences and Series of real numbers

• Sequences of real numbers, their convergence, and limits, Cauchy sequences and their convergence. Monotonic sequences and their limits, Limits of standard sequences, Infinite series and its convergence, and divergence, Convergence of series with non-negative terms, Tests for convergence and divergence of a series, Comparison test, limit comparison test, D’Alembert’s ratio test, Cauchy’s nth root test, Cauchy’s condensation test and integral test, Absolute convergence of series, Leibnitz’s test for the convergence of alternating series, Conditional convergence, Convergence of power series and radius of convergence.

2. Calculus

• Limit of a function, Continuity and Differentiability, Elementary Integration and Antiderivatives, Fundamental theorems of integral calculus, single integral, Leibnitz’s rule and its applications. Differentiation under integral sign, Improper integrals, Beta and Gamma integrals: properties and relationship between them, Double integrals, Change of order of integration, Transformation of variables, Applications of definite integrals, Arc lengths, areas and volumes.

3. Linear Algebra

• Vector spaces with real field, Subspaces and sum of subspaces, Span of a set, Linear dependence and independence. Dimension and basis, Algebra of matrices, Standard matrices (Symmetric and Skew Symmetric matrices, Hermitian and Skew Hermitian matrices, Orthogonal and Unitary matrices, Idempotent and Nilpotent matrices). Definition, properties and applications of determinants, Evaluation of determinants using transformations, Determinant of product of matrices. Singular and non-singular matrices and their properties, Trace of a matrix, Adjoint and inverse of a matrix and related properties, Rank of a matrix, row-rank, column-rank, standard theorems on ranks, rank of the sum and the product of two matrices, Row reduction and echelon forms, Partitioning of matrices and simple properties, Consistent and inconsistent system of linear equations, Properties of solutions of system of linear equations, Use of determinants in solution to the system of linear equations, Cramer’s rule, Characteristic roots and Characteristic vectors, Properties of characteristic roots and vectors, Cayley Hamilton theorem.

4. Probability and Statistics

• Sample Space and Algebra of Events (Event space), Relative frequency, Addition theorem of probability function (Inclusion Exclusion Principle), Conditional probability, Multiplication rule, Theorem of total probability, Pairwise, mutual independence of events and Bayes theorem, Random variables, Random Experiments, Geometric probability, Boole’s and Bonferroni’s inequalities, density functions, idea of distribution functions, Expectation and variance.

5. School Level Mathematics

• Number Theory : Basic Number Theory, Prime numbers and divisibility, GCD, LCM, Modular arithmetic, Congruences, Logarithms, Bezout’s Theorem.


• Combinatorics : Sets and Relations, Combinations and Permutations, Elementary counting techniques, Pigeonhole principle, Binomial theorem, Recurrences, Invariance Monovariance and Extremal Principles Basic Geometry.


• Algebra and Miscellaneous : Complex Number and De Moivre’s Theorem, Basic Set Theory and Set Operations, Principle of Mathematical Induction, Puzzles, Elementary Functional Equations, Basic Laws of Inequalities, AM-GM-HM inequality, Cauchy-Schwarz inequality, Rearrangement and Chebyshev Inequalities, Jensen Inequality, Basic Operations on Polynomials, Polynomial Division and GCD, Remainder and Factor Theorems, Fundamental Theorem of Algebra, Quadratic Polynomials and Equations, Vieta’s Relations, Integer Polynomials.


• Geometry and Trigonometry : Congruence of Triangles, Properties of triangles and properties of circles, properties of trigonometric functions and its inverse functions.

Question Papers

Question papers for both the subjective and objective parts of StoChastico '24 can be found in the following two links.

Selection List

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