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