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IIT – JAM

Syllabus

Mathematics (MA)

1. 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.

2. 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.

3. Functions of Two or Three Real Variables:

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

4. Integral Calculus:

Integration as the inverse process of differentiation, definite integrals and their properties, fundamental theorem of calculus. Double and triple integrals, change of order of integration, calculating surface areas and volumes using double integrals, calculating volumes using triple integrals.

5. 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.

6. Vector Calculus:

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

7. 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.

8. Linear Algebra:

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

Statistics (MS)

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

Part – A : Maths

1. Applied Analysis

Sequences and Series: Convergence of sequences of real numbers, Comparison, root and ratio tests for convergence of series of real numbers.

2. Advance 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 minima of functions of one and two variables.

3. Advance Integral Calculus

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

4. Matrices

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

5. 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.

Part – B : Statistics

1. Probability

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

2. Random Variables

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

3. Standard Distributions

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

4. Joint Distributions

Joint, marginal and conditional distributions. Distribution of functions of random variables. Product moments, correlation, simple linear regression. Independence of random variables.

5. Sampling distributions

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

6. Limit Theorems

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

7. 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. Confidence intervals for the parameters of univariate normal, two independent normal, and one parameter exponential distributions.

8. Testing of Hypotheses

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

Mathematical Aptitude Test Areas (+2 Level):

Logarithms, Inequalities, Matrices and Determinants, Progressions, Binomial Expansion, Permutation and Combination, Equations (upto degree 2), Function and Relation, Complex Arithmetic, 2-D Coordinate Geometry, Basics of Calculus, Basic Concepts of Probability. Elementary set theory, Finite, countable and uncountable sets, Real numbersystem as a complete ordered field, Archimedean property, supremum, infimum. Sequence and series, Convergence, limsup, liminf. Bolzano Weierstrass theorem, Heine Borel theorem. Continuity, Uniform continuity, Intermediate value theorem, Differentiability,

Mean value theorem, Maclaurin’s theorem and series, Taylor's series.

Sequences and series of functions, Uniform convergence. Riemann sums and Riemann integral, Improper integrals. Monotonic functions, Types of discontinuity. Functions of several variables, Directional derivative, Partial derivative. Metric spaces, Completeness, Total boundedness, Separability, Compactness, Connectedness.

M.SC MATH’S WITH COMPUTER SCEINCE JMIU

• Calculus and Differential Equations
• Algebra of limits
• Continuous functions and classification of discontinuous functions
• Differentiability
• Successive differentiation
• Leibnitz theorem
• Rolle’s Theorem
• Mean Value theorems
• Taylor’s theorem with Lagrange’s and Cauchy’s form of remainder
• Taylor’s and Maclaurins’s series of elementary functions
• Test for concavity and convexity
• Points of inflexion
• Multiple points
• Tracing of curves in Cartesian and polar coordinates
• Reduction formulae
• Quadrature
• Rectification
• Intrinsic equation
• Volumes and surfaces of solids of revolution
• Order and degree of a differential equation
• Equations of first order and first degree
• Equations in which the variables are separable
• Homogeneous equation
• Linear equations and equations reducible to linear form
• Homogeneous linear differential equations
• Geometry of Two and Three Dimensions
• Conic sections
• General equation of second degree
• Pair of lines
• Lines joining the origin to the points of intersection of a curve and a line
• Equations of ellipse and hyperbola in standard and parametric forms
• Tangent Normal
• Pole and polar and their elementary properties
• Polar Equation of a conic
• Equation of plane
• Equation of sphere
• Tangent plane, plane of contact and polar plane
• Intersection of two spheres
• Equation cylinder
• Enveloping and right circular cylinders
• Equations of central conicoids
• Tangent plane
• Functions of several variables
• Domains and Range
• Functional notation
• Level curves and level surfaces
• Limits and continuity
• Partial derivatives
• Implicit functions
• Inverse functions
• Curvilinear co-ordinates
• Geometrical Applications
• The directional derivatives
• Partial derivatives of higher order
• Higher derivatives of composite functions
• Vector fields and scalar fields
• The gradient field
• The divergence of a vector field
• The curl of a vector field
• Combined operations
• Irrotational fields and Solenoidal fields
• Line integrals in the plane
• Integrals with respect to arc length
• Basic properties of line integrals
• Line integrals as integrals of vectors
• Green’s Theorem
• Independence of path
• Simply connected domains
• Extension of results to multiply connected domains
• De Moivre’s theorem and its applications
• Expansion of sinn¬, cosn¬ and tann¬
• Separation into real and imaginary parts
• Summation of series based on C+iS method
• Abstract Algebra
• Sets
• Relations
• Functions
• Binary operations
• Groups and its elementary properties
• Subgroups and its properties
• Group homomorphism
• Isomorphism
• The isomorphism theorems
• Permutation groups
• Even and odd permutations
• Homomorphism of rings and its properties
• The isomorphism theorems
• Rings of Polynomials
• Some properties of R[X]
• Euclidean domain
• Principal ideal domain
• Real Analysis
• Real Sequences
• Riemann Integration
• Uniform Convergence
• Improper Integrals
• Linear Systems and Gaussian Elimination
• Vector Spaces
• Linear Transformations
• Orthogonality in Vector Spaces
• Eigenvalues and Eigenvectors
• Canonical Forms
• Mechanics
• Metric Spaces, Complex Analysis and Differential Equations
• Mathematical Statistics and Operational Research
• Operational Research
• Linear Programming
• Theory of Games
• Introduction to Computer
• C Programming
• Algebraic and Transcendental Equations
• Systems of Linear and Nonlinear Equations
• Interpolation
• Numerical Differentiation, Numerical Integration and Ordinary Differential Equations

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