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Combined Syllabus & Reference books For M.Sc & Ph.D Entrance Exam in Mathematics

This is useful for those preparing for M.Sc & Ph.D Entrance in Mathematics in IMSC, TIFR, IISC, ISI, CMI, IIT, IISER, NISER, UOHYD, RKMVU. This is duly compiled to cover all the entrances given above.

Classical Algebra:
De Moivre’s theorem, relation between roots and coefficient of nth degree equation, solution to cubic and biquadratic equation, transformation of equations. Arithmetic, geometric and harmonic progression. Continued fractions. Elementary combinatorics: Permutations and combinations, Binomial theorem. Theory of equations. Inequalities. Elementary set theory. Functions and relations. Elementary number theory: Divisibility, Congruences, Primality.
Reference Books:
1.Combinatorics - Krishnamurthy
3.Higher Algebra - Bernald•Child.

Abstract Algebra:
Groups, homomorphisms, cosets, Lagrange's Theorem, Sylow Theorems, symmetric group Sn, conjugacy class, rings, ideals, quotient by ideals, maximal and prime ideals, fields,
algebraic extensions, finite fields.
Reference Books:
1. Algebra by Artin
2. Topics in Algebra, I. N. Herstein, J. Wiley
3. Abstract Algebra, D. S. Dummit and R. M. Foote, J. Wiley

General :
Elementary Combinatorics, Binomial Theorem, Elementary Probability Theory, Logarithms, Progressions.
Reference Books:
1. An Introduction to Probability-Feller.

Linear Algebra: Vector Space, subspace and its properties, linear independence and dependence of vectors, matrices, rank of a matrix, reduction to normal forms, linear homogenous and non-homogenous equations,Rank, inverse of a matrix. systems of linear equations. Linear transformations, eigenvalues and eigenvectors. Cayley-Hamilton theorem, symmetric, skewsymmetric and orthogonal matrices.
Reference Books:
1.Linear Algebra - Rao•Bhimasankaram
2.Linear Algebra - Hoffman•Kunze
3.Higher Algebra - S.K.Mapa

Complex Analysis:
Holomorphic functions, Cauchy-Riemann equations, integration, zeroes of analytic functions, Cauchy formulas, maximum modulus theorem, open mapping theorem, Louville's theorem, poles and singularities, residues and contour integration, conformal maps, Rouche's theorem, Morera's theorem.
Reference Books:
Complex Analysis-L.V.Ahlfors 

Calculus and Real Analysis:
(a) Real Line: Limits, continuity, di_erentiablity, Reimann integration, sequences, series, lim-sup, liminf, pointwise and uniform convergence, uniform continuity, Taylor expansions,
(b) Multivariable: Limits, continuity, partial derivatives, chain rule, directional derivatives, total derivative, Jacobian, gradient, line integrals, surface integrals, vector _elds, curl, divergence, Stoke's theorem
(c) General: Metric spaces, Heine Borel theorem, Cauchy sequences, completeness, Weierstrass approximation.
Reference Books:
1. Mathematical Analysis, T. M. Apostol, Narosa.
2. Introduction To Real Analysis - Bartle•Sherbert
3. Principles of Mathematical Analysis - Rudin.
Topology:
General: Metric spaces, Heine Borel theorem, Cauchy sequences, completeness, Weierstrass approximation. Topological spaces, base of open sets, product topology, accumulation points, boundary, continuity, connectedness, path connectedness, compactness
Reference Books:
1. Topology of Metric Spaces by Kumaresan
2. Principles of Topology by Fred H. Croom
3. Topology-James Munkres

Differential Equations: Ordinary differential equations of the first order of the form y' = f(x,y). Linear differential equations of the second order with constant coefficients. Linear, homogenous, separable equations, first order higher degree equations, algebraic properties of solutions, linear homogenous equations with constant coefficients.
Reference Books:
Introduction to Differential Equation - Ghosh & Maity.

Coordinate geometry:
Straight lines, circles, parabolas, ellipses and hyperbolas. Three Dimensional Geometry.
Reference Books:
Co-ordinate Geometry - S.L.Loney