JAM is an abbreviated form of (Joint Admission Test) for M.Sc. (JAM) conducted by the Indian Institute of Technology (IITs) for admission in Ph.D. Degree programs at IISc, Bangalore and M.Sc. (Two Years), Joint M. Sc. -Ph.D. Dual Degree, Joint M.Sc. –Ph.D., M. Sc. -M. tech, M.Sc. – M.S. (Research) /Ph.D. Dual Degree and other Post-Bachelor’s Degree Programs at IITs. The main objective of this exam is to consolidate Science as a career option for bright students across the country. The academic environment, interdisciplinary interaction and research infrastructure which are given to the student at these top Institutes like IISC Bangalore and IITs motivate the scholars to pursue their career not only in research and development of basic science but also in the interdisciplinary areas of Science and Technology.

The entrance exam is open to all Indians without any age limit. The applicant should have scored at least 55% marks (for general categories / OBC) and 50% marks for SC, ST and PD category candidates in the bachelor degree. The candidates who have appeared or are waiting for the final result are also eligible to participate in the entrance exam. The candidates who have cleared BSc through distance education or correspondence course have to appear for the additional interview for admission into M.SC. physics or M.SC. mathematics.

Good quality education is a useful resource for your bright future. The institutions like IITs and IISc provide you world quality infrastructure and exposure to globally accepted education. These institutions get more than 50% of the government budget for higher education. One should not miss the opportunity if he/she is wanting to pursue higher education.

**Sequences And Series Of Real Numbers:**

Sequence of real numbers, convergence of sequences, bounded and monotone sequences, convergence criteria for sequences of real numbers, Cauchy sequences, subsequences, Bolzano-Weierstrass theorem. Series of real numbers, absolute convergence, tests of convergence for series of positive terms - comparison test, ratio test, root test; Leibniz test for convergence of alternating series.

**Functions Of One Real Variable:**

Limit, continuity, intermediate value property, differentiation, Rolle's Theorem, mean value theorem, L'Hospital rule, Taylor's theorem, maxima and minima.

**Functions Of Two Or Three Real Variables:**

Limit, continuity, partial derivatives, differentiability, maxima and minima.

**Vector Calculus:**

Scalar and vector fields, gradient, divergence, curl, line integrals, surface integrals, Green, Stokes and Gauss theorems.

**Group Theory:**

Groups, subgroups, Abelian groups, non-Abelian groups, cyclic groups, permutation groups, normal subgroups, Lagrange's Theorem for finite groups, group homomorphisms and basic concepts of quotient groups.

**Linear Algebra:**

Finite dimensional vector spaces, linear independence of vectors, basis, dimension,linear transformations, matrix representation, range space, null space, rank-nullity theorem. Rank andinverse of a matrix, determinant, solutions of systems of linear equations, consistency conditions,eigen values and eigenvectors for matrices, Cayley-Hamilton theorem.

**Real Analysis:**

Interior points, limit points, open sets, closed sets, bounded sets, connected sets,compact sets, completeness of R. Power series (of real variable), Taylor's series, radius and intervalof convergence, term-wise differentiation and integration of power series.

The Mathematical Statistics (MS) test paper comprises of Mathematics (40% weightage) and Statistics (60% weightage).

**Sequences And Series:**

Convergence of sequences of real numbers, Comparison, root and ratiotests for convergence of series of real numbers.

**Differential Calculus:**

Limits, continuity and differentiability of functions of one and two variables.Rolle's theorem, mean value theorems, Taylor's theorem, indeterminate forms, maxima and minimaof functions of one and two variables.

**Integral Calculus:**

Fundamental theorems of integral calculus. Double and triple integrals, applicationsof definite integrals, arc lengths, areas and volumes.

**Matrices:**

Rank, inverse of a matrix. Systems of linear equations. Linear transformations, eigenvaluesand eigenvectors. Cayley-Hamilton theorem, symmetric, skew-symmetric and orthogonal matrices.

**Differential Equations:**

Ordinary differential equations of the first order of the form y'=f(x,y), Bernoulli's equation, exact differential equations, integrating factor, orthogonal trajectories, homogeneous differential equations, variable separable equations, linear differential equations of second order with constant coefficients, method of variation of parameters, Cauchy-Euler equation.

**Probability:**

Axiomatic definition of probability and properties, conditional probability, multiplication rule. Theorem of total probability. Bayes' theorem and independence of events.

**Random Variables:**

Probability mass function, probability density function and cumulative distributionfunctions, distribution of a function of a random variable. Mathematical expectation, moments andmoment generating function. Chebyshev's inequality.

**Standard Distributions:**

Binomial, negative binomial, geometric, Poisson, hypergeometric, uniform, exponential, gamma, beta and normal distributions. Poisson and normal approximations of a binomial distribution.

**Sampling Distributions**

Chi-square, t and F distributions, and their properties.

**Limit Theorems:**

Weak law of large numbers. Central limit theorem (i.i.d.with finite variance case only).

**Estimation:**

Unbiasedness, consistency and efficiency of estimators, method of moments and method of maximum likelihood. Sufficiency, factorization theorem. Completeness, Rao-Blackwell and Lehmann-Scheffe theorems, uniformly minimum variance unbiased estimators. Rao-Cramer inequality. Confidenceintervals for the parameters of univariate normal, two independent normal, and one parameter exponentialdistributions.

**Testing Of Hypotheses:**

Basic concepts, applications of Neyman-Pearson Lemma for testing simpleand composite hypotheses. Likelihood ratio tests for parameters of univariate normal distribution.