IIT JAM Mathematics Syllabus

IIT JAM Chemistry Syllabus

IIT JAM Biotechnology Syllabus

IIT JAM Physics Syllabus

Topic

Weightage

Real Analysis

21%

Calculus of Single Variable

18%

Linear Algebra

14%

Calculus of Two Variables

14%

Vector Calculus

12%

Differential Equation

11%

Abstract Algebra

10%

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Sl.no

Topics

Sub- Topics

Unit - 1

Sequences and Series of Real Numbers

Real number sequences, sequence convergence, bounded and monotone sequences, real number sequence convergence criteria. There are also Cauchy sequences and sub-sequences, as well as the Bolzano-Weierstrass theorem. There are also series of real numbers, absolute convergence, tests of convergence for series of positive terms that include comparison tests, ratio tests, root tests, and the Leibniz test for alternating series convergence, among other things.

Unit - 2

Differential Equations

Bernoulli's equation, exact differential equations, integrating factors, orthogonal trajectories, homogeneous differential equations, variable separation method, linear differential equations of second order with constant coefficients, parameter variation method, Cauchy-Euler equation.

Unit - 3

Integral Calculus

Integration is defined as the inverse process of differentiation, as well as definite integrals and all the properties associated with the calculus fundamental theorem. Aside from that, there are double and triple integrals, information on changing the order of integration, the process of calculating surface areas and volumes using double integrals, and calculating volumes using triple integrals.

Unit - 4

Linear Algebra

Finite-dimensional vector spaces with linear vector independence, basis and dimension, and linear transformations, details on matrix representation and also range space with null space, concepts on rank-nullity theorem. It also covers Rank and inverse of a matrix, determinant and all the solutions of linear equation's systems. On the contrary there are consistency conditions along with eigenvalues, and also eigenvectors for matrices, Cayley-Hamilton theorem, etc in this section.

Unit - 5

Functions of Two or Three Real Variables

Limit, continuity, partial derivatives, total derivatives, maxima and minima are all included in this unit. 

Unit - 6

Finite-Dimensional Vector Spaces

Linear independence of vectors, basis, dimension, linear transformations, matrix representation, range space, null space, rank-nullity theorem are all covered in this unit.

Unit - 7

Matrices

Systems of linear equations, rank, nullity, rank-nullity theorem, inverse, determinant, eigenvalues, and eigenvectors are all covered in this unit.

Unit - 8

Groups

Cyclic groups, abelian groups, non-abelian groups, permutation groups, normal subgroups, quotient groups, Lagrange's theorem for finite groups, group homomorphisms.

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Units

Topics

Sub topics

Unit-1

Probability

Experiments at Random. Sample Space and Event Algebra (Event Space). Probability is defined using relative frequency and axiomatically. Probability function properties. Probability function addition theorem (inclusion-exclusion principle). Probability based on geometry. Boole's and Bonferroni's inequalities. Multiplication rule and conditional probability. Total probability theorem and Bayes' theorem. Events are pairwise and mutually independent.

Unit - 2

Univariate Distributions

Random variables are defined as follows. A random variable's cumulative distribution function (c.d.f.). Random variables, both discrete and continuous. A random variable's probability mass function (p.m.f.) and probability density function (p.d.f.). Distribution (c.d.f., p.m.f., p.d.f.) of a random variable function using variable transformation and Jacobian method. Moments and mathematical expectation. A probability distribution's mean, median, mode, variance, standard deviation, coefficient of variation, quantiles, quadriles, coefficient of variation, and measures of skewness and kurtosis. The properties and uniqueness of the moment generating function (m.g.f.). The applications of Markov and Chebyshev inequalities.

Unit - 3

Standard Univariate Distributions

Degenerate, Bernoulli, Binomial, Negative binomial, Geometric, Poisson, Hypergeometric, Uniform, Exponential, Double exponential, Gamma, Beta (of the first and second types), Normal, and Cauchy distributions, as well as their properties, interrelationships, and limiting (approximation) cases.

Unit - 4

Multivariate Distributions

• Random vector definition. A random vector's joint and marginal c.d.f.s. Random vectors of the discrete and continuous types. Joint and marginal p.m.f., joint and marginal p.d.f.. Conditional c.d.f., conditional p.m.f. and conditional p.d.f.. Independence of random variables. Variable transformation and the Jacobian method are used to distribute functions of random vectors.
• The mathematical expectation of random vector functions. 
• Covariance and correlation are examples of joint moments. The properties of the joint moment generating function. The distinctiveness of joint m.g.f. and its applications. Conditional expectations, conditional moments, and conditional variance. Binomial, Poisson, Negative Binomial, Gamma, and Normal Distributions' additive properties using their m.g.f.

Unit - 5

Standard Multivariate Distributions

The multinomial distribution and its properties (moments, correlation, marginal distributions, additive property) are a generalization of the binomial distribution and its properties. The bivariate normal distribution, as well as its marginal and conditional distributions, and their related properties.

Unit - 6

Limit Theorems

The interrelationships of probability and distribution convergence. The Weak Law of Large Numbers and the Central Limit Theorem are used in applications (i.i.d. case).

Unit - 7

Estimation

• Unbiasedness. Sufficiency of a statistic. Factorization Theorem. Complete statistic. Estimator consistency and efficiency are important considerations. Uniformly Minimum variance unbiased estimator (UMVUE). RaoTheorems of Blackwell and Lehmann-Scheffe and their applications. The Cramer-Rao inequity and UMVUEs.
• Estimation methods include the method of moments, the method of maximum likelihood, and the invariance of maximum likelihood estimators. Estimation of least squares and its applications in simple linear regression models. The confidence intervals and the confidence coefficient. Confidence intervals for univariate normal, two independent normal, and exponential distribution parameters.

Unit - 8

Testing of Hypotheses

Type-I and Type-II errors, null and alternative hypotheses (simple and composite). A critical region. The level of significance, the size and power of a test, and the p-value. Uniformly powerful critical regions and powerful (MP) tests. Uniformly most powerful (UMP) tests are performed. Pearson, Neyman Lemma (without proof) and its applications to the development of MP and UMP tests for single parameter parametric families. Likelihood ratio tests for univariate normal distribution parameters.

Unit - 9 

Sampling Distributions

Random sample, parameter, and statistic definitions. A statistic's sampling distribution.

• Order Statistics: Define and distribute the rth order statistic (d.f. and p.d.f. for i.i.d. case for continuous distributions). Distribution of smallest and largest order statistics (c.d.f., p.m.f., p.d.f) (i.i.d. case for discrete as well as continuous distributions).
• Central Chi-square distribution: Using m.g.f., define and derive the p.d.f. of the central 2 distribution with n degrees of freedom (d.f.). Properties of the central 2 distribution, including additive properties and the limiting form of the central 2 distribution.
• Central Student's t-distribution: Definition and derivation of the p.d.f. of the Central Student's t-distribution with n d.f., Properties and limiting form of the Central Student's t-distribution.

Definition and derivation of p.d.f. of Snedecor's Central F-distribution with (m, n) d.f. The properties of the Central F-distribution, as well as the distribution of the reciprocal of the F-distribution. The relationship between the t, F, and 2 distributions.

Sl.no

Topics

Recommended Books

1.

Books for Linear Algebra

• Chapter 0 of Serge Lang's Introduction Linear Algebra
• Introduction to Linear Algebra by Gilbert Strang
• Linear Algebra: A geometric Approach by S Kumaresan
• Linear Algebra by Hoffman & Kunze
• Algebra by Artin
• Topics in Algebra by Herstein
• Linear Algebra Done Right by Axler

2.

Books for Abstract Algebra

• Contemporary Abstract Algebra by Gallian
• SAGE Math (computation software)
• Abstract Algebra by Dummit and Foote

3.

Books for Real Analysis

• Introduction to Real Analysis by Bartle Sherbet
• Problems in Real Analysis by Kaczor
• A basic course in Real Analysis by Kumar & Kumaresan
• Analysis 1 & 2 by Terence Tao
• Principles of Mathematical Analysis by Rudin

4.

Books for Vector Calculus and Differential Equation

• Calculus, Early Transcendentals by James Stewart
• Basic Multivariable Calculus by Marsden, Tromba, and Weinstein.
• Calculus Vol 1 and 2 by Apostle
• Calculus on Manifolds by Spivak

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