Guidelines for students shortlisted for Ph.D. entrance test  

 

  Please report in room 114 of the Mathematics Department by 9 am on May 8, 2015 for a written test. The written test will be followed by an interview for the candidates who secure more than or equal to 40% of marks in the written test.

Here you may find some recent question papers for the written test:

December 2013,   May 2014,   December 2014.

The syllabus of the written test is as follows:

Linear Algebra (40% weightage): Vector spaces over fields, subspaces, bases and dimension. Systems of linear equations, matrices, rank, Gaussian elimination. Linear transformations, representation of linear transformations by matrices, rank-nullity theorem, duality and transpose. Determinants, Laplace expansions, cofactors, adjoint, Cramer's Rule. Eigenvalues and eigenvectors, characteristic polynomials, minimal polynomials, Cayley-Hamilton theorem, triangulation, diagonalization, rational canonical form, Jordan canonical form. Inner product spaces, Gram-Schmidt ortho-normalization, orthogonal projections, linear functionals and adjoints, Hermitian, self-adjoint, unitary and normal operators, spectral Theorem for normal operators. Bilinear forms, symmetric and skew-symmetric bilinear forms, real quadratic forms, Sylvester's law of inertia, positive definiteness.

Real Analysis (30% weightage): Archimedean property and completeness of real numbers. Metric spaces, compactness, connectedness (with emphasis on R^n). Continuity and uniform continuity. Monotonic functions, functions of bounded variation; absolutely continuous functions. Derivatives of functions and Taylor's theorem. Riemann integral and its properties, characterization of Riemann integrable functions. Improper integrals, Gamma functions. Sequences and series of functions, uniform convergence and its relation to continuity, differentiation and integration. Fourier series, pointwise convergence, Fejer's theorem, Weierstrass approximation theorem.

Probability (30% weightage): Axioms of probability, conditional probability and independence, random variables and distribution functions, random vectors and joint distributions, functions of random vectors. Expectation, moment generating functions and characteristic functions, conditional expectation and distribution. Modes of convergence, weak and strong laws of large numbers, central limit theorem.