IIT JAM Syllabus 2026, Download Subject-Wise Syllabus PDF, Topics

Topic

Subtopics

Mathematical Methods

Calculus of single and multiple variables; partial derivatives; Jacobian; imperfect and perfect differentials; Taylor expansion; Fourier series.

Vector algebra; Vector Calculus; Multiple integrals; Divergence theorem; Green’s theorem; Stokes’ theorem.

First order equations and linear second-order differential equations with constant coefficients.

Matrices and determinants; Algebra of complex numbers.

Mechanics and General Properties of Matter

Newton’s laws of motion and applications; Velocity and acceleration in Cartesian, polar and cylindrical coordinate systems; uniformly rotating frame; centrifugal and Coriolis forces; Motion under a central force; Kepler’s laws; Gravitational Law and field; Conservative and non-conservative forces.

System of particles; Center of mass; equation of motion of the CM; conservation of linear and angular momentum; conservation of energy; variable mass systems.

Elastic and inelastic collisions.

Rigid body motion; fixed axis rotations; rotation and translation; moments of Inertia and products of Inertia; parallel and perpendicular axes theorem.

Principal moments and axes.

Kinematics of moving fluids; equation of continuity; Euler’s equation; Bernoulli’s theorem.

Oscillations, Waves and Optics

Differential equation for simple harmonic oscillator and its general solution; Superposition of two or more simple harmonic oscillators; Lissajous figures.

Damped and forced oscillators; resonance.

Wave equation; traveling and standing waves in one dimension.

Energy density and energy transmission in waves.

Group velocity and phase velocity.

Sound waves in media; Doppler Effect.

Fermat’s Principle; General theory of image formation; Thick lens, thin lens and lens combinations.

Interference of light; optical path retardation.

Fraunhofer diffraction; Rayleigh criterion and resolving power; Diffraction gratings.

Polarization: linear, circular, and elliptic polarization; Double refraction and optical rotation.

Electricity and Magnetism

Coulomb’s law; Gauss’s law; Electric field and potential; Electrostatic boundary conditions; Solution of Laplace’s equation for simple cases.

Conductors, capacitors, dielectrics; dielectric polarization; volume and surface charges; electrostatic energy.

Biot-Savart law; Ampere’s law; Faraday’s law of electromagnetic induction; Self and mutual inductance.

Alternating currents; Simple DC and AC circuits with R, L, and C components.

Displacement current; Maxwell’s equations and plane electromagnetic waves; Poynting’s theorem; reflection and refraction at a dielectric interface; transmission and reflection coefficients (normal incidence only).

Lorentz Force and motion of charged particles in electric and magnetic fields.

Kinetic Theory, Thermodynamics

Elements of Kinetic theory of gases; Velocity distribution and Equipartition of energy; Specific heat of Mono-, di- and tri-atomic gases; Ideal gas, van-der-Waals gas and equation of state.

Mean free path; Laws of thermodynamics: Zeroth law and concept of thermal equilibrium; First law and its consequences; Isothermal and adiabatic processes; Reversible, irreversible and quasi-static processes.

Second law and entropy; Carnot cycle; Maxwell’s thermodynamic relations and simple applications; Thermodynamic potentials and their applications.

Phase transitions and Clausius-Clapeyron equation; Ideas of ensembles, Maxwell Boltzmann, Fermi-Dirac and Bose-Einstein distributions.

Modern Physics

Inertial frames and Galilean invariance; Postulates of special relativity; Lorentz transformations; Length contraction, time dilation; Relativistic velocity addition theorem; mass energy equivalence.

Blackbody radiation, photoelectric effect, Compton effect, Bohr’s atomic model, X-rays.

Wave-particle duality, Uncertainty principle, the superposition principle, calculation of expectation values, Schrödinger equation and its solution for one, two and three-dimensional boxes.

Solution of Schrödinger equation for the one-dimensional harmonic oscillator; Reflection and transmission at a step potential, Pauli exclusion principle.

Structure of atomic nucleus; mass and binding energy; Radioactivity and its applications; Laws of radioactive decay.

Solid State Physics, Devices and Electronics

Crystal structure, Bravais lattices and basis; Miller indices; X-ray diffraction and Bragg's law.

Intrinsic and extrinsic semiconductors, variation of resistivity with temperature; Fermi level.

p-n junction diode, I-V characteristics, Zener diode and its applications.

BJT: characteristics in CB, CE, CC modes.

Single stage amplifier, two-stage R-C coupled amplifiers.

Simple Oscillators: Barkhausen condition, sinusoidal oscillators.

OPAMP and applications: Inverting and non-inverting amplifier.

Boolean algebra: Binary number systems; conversion from one system to another system; binary addition and subtraction.

Logic Gates AND, OR, NOT, NAND, NOR, exclusive OR; Truth tables; combination of gates; de Morgan’s theorem.

Topic

Subtopics

Transition Metals (d block)

Characteristics of d-block Elements, Coordination Complexes, Isomerism, Reaction Mechanism, VB, MO, and Crystal Field Theoretical Approaches, Organometallic Compounds, Homogeneous Catalysis

Spectroscopy

Beer-Lambert’s Law, Rotational Spectroscopy, Vibrational Spectroscopy, Electronic Spectroscopy, Magnetic Resonance Spectroscopy

Solid State

Unit Cells, Miller Indices, Crystal Systems, Bravais Lattices, Applications of Vectors to Crystal Systems, X-ray Diffraction, Bragg’s Law, Structure of NaCl, CsCl, KCl, Diamond, Graphite, Close Packing in Metals and Metal Compounds, Semiconductors, Insulators, Defects in Crystals, Lattice Energy, Isomorphism, Heat Capacity of Solids

Qualitative Organic Analysis

Identification of Functional Groups by Chemical Tests, Elementary UV, IR, and 1H NMR Spectroscopic Techniques

Periodic Table

Periodic Classification of Elements, Aufbau’s Principle, Periodicity, Orbital Energy, Effective Nuclear Charge, Atomic/Covalent/Ionic Radii, Ionization Enthalpy, Electron Gain Enthalpy, Electronegativity, Electronic Configuration of Diatomic Molecules

Organic Reaction Mechanism and Synthetic Applications

Chemistry of Reactive Intermediates (Carbocations, Carbanions, Free Radicals, Carbenes, Nitrenes, Benzynes), Nucleophilic Substitution, Elimination Reactions, Hofmann-Curtius-Lossen Rearrangement, Wolff Rearrangement, Simmons-Smith Reaction, Reimer-Tiemann Reaction, Michael Reaction, Darzens Reaction, Wittig Reaction, McMurry Reaction, Pinacol-Pinacolone Rearrangement, Oxidation and Reduction Reactions, Organometallic Reagents in Organic Synthesis, Diels-Alder Reactions, Functional Group Inter-conversions

Natural Products Chemistry

Chemistry of Alkaloids, Steroids, Terpenes, Carbohydrates, Amino Acids, Peptides, Nucleic Acids

Main Group Elements (s and p blocks)

Reactions of Alkali and Alkaline Earth Metals, Inert Pair Effect, Diborane, Ammonia, Silane, Phosphine, Oxides of Nitrogen, Phosphorus, Sulphur, Allotropes of Carbon, Oxoacids of Phosphorus/Sulphur/Chlorine, Xenon Fluorides, Borazine, Silicone, Phosphazene

Liquid State

Physical Properties of Liquids, Vapour Pressure, Surface Tension, Coefficient of Viscosity and Applications, Effect of Concentration of Solutes on Surface Tension and Viscosity, Effect of Temperature on Viscosity of Liquids

Instrumental Methods of Analysis

Principles, Instrumentation, Applications of Conductometry, Potentiometry, UV-Vis Spectrophotometry

Gaseous State

Kinetic Molecular Model of a Gas, Collision Frequency, Collision Diameter, Mean Free Path, Viscosity of Gases, Maxwell-Boltzmann Distribution, Molecular Velocities, Law of Equipartition of Energy, Molecular Basis of Heat Capacities, Ideal Gases, Deviations from Ideal Gas Behavior, Van der Waals Equation of State, Critical State, Law of Corresponding States

Extractions of Metals

General Methods of Isolation and Purification of Elements, Ellingham Diagram

Electrochemistry

Conductivity, Equivalent and Molar Conductivity, Kohlrausch Law, Debye-Hückel-Onsager Equation, Ionic Velocities, Mobilities, Transference Numbers, Applications of Conductance Measurement, Faraday’s Laws of Electrolysis, Electrolysis in Metallurgy and Industry, Electromotive Force of a Cell, Nernst Equation, Standard Electrode Potential, Electrochemical Series, Concentration Cells with and without Transference, EMF Measurements Including Potentiometric Titrations

Chemical Thermodynamics

Exact and Inexact Differentials, Partial Derivatives, Euler’s Reciprocity, Cyclic Rule, Reversible and Irreversible Processes, Laws of Thermodynamics, Thermochemistry, Thermodynamic Functions (Enthalpy, Entropy, Gibbs Free Energy), Partial Molar Quantities, Dependence of Thermodynamic Parameters on Composition, Gibbs-Duhem Equation, Chemical Potential and Its Application

Chemical Kinetics

Order and Molecularity of a Reaction, Differential and Integrated Forms of Rate Expressions, Kinetics of Opposing, Parallel, and Consecutive Reactions, Steady State Approximation in Reaction Mechanisms, Chain Reactions, Uni-Molecular Reaction (Lindemann Mechanism), Temperature Dependence of Reaction Rates, Arrhenius Equation, Activation Energy, Collision Theory of Reaction Rates, Types of Catalysts, Enzyme Catalysis, Acid-Base Catalysis

Chemical Bonding and Shapes of Molecules

Ionic Bond, Covalent Bond (Lewis Structure, Valence Bond Theory, Hybridization, Molecular Orbital Theory), Multiple Bonding, Intermolecular Forces, VSEPR Theory, Bond Dipole and Dipole Moment, Effect on Melting and Boiling Points, Solubility Energetics

Chemical and Phase Equilibria

Law of Mass Action, Kp, Kc, Kx, Kn, Effect of Temperature on K, Le-Chatelier Principle, Ionic Equilibria in Solutions, pH and Buffer Solutions, Salt Hydrolysis, Solubility and Solubility Product, Acid-Base Titration Curves, Indicators, Dilute Solutions, Raoult’s and Henry’s Laws and Their Applications, Colligative Properties, Gibbs Phase Rule, Phase Equilibria, Single and Two Component Phase Diagrams

Bioinorganic Chemistry

Essentials and Trace Elements of Life, Role of Metal Ions (Fe²⁺, Zn²⁺), Structure and Function of Myoglobin, Hemoglobin, Carbonic Anhydrase

Basic Concepts in Organic Chemistry and Stereochemistry

Electronic Effects (Resonance, Inductive, Hyperconjugation), Steric Effects, Optical Isomerism, Conformation of Acyclic and Cyclic Systems, Substituted Ethane/Propane/Butane, Substituted Cyclohexanes, Polycyclic Systems

Atomic and Molecular Structure

Planck’s Black Body Radiation, Photoelectric Effect, Bohr’s Theory, de Broglie Postulate, Heisenberg’s Uncertainty Principle, Schrödinger’s Wave Equation, Postulates of Quantum Mechanics, Normalized and Orthogonal Wave Functions, Complex Conjugate of Wave Function, Significance of Ѱ², Operators, Particle in One-Dimension Box, Radial and Angular Wave Functions for Hydrogen Atom, Radial Probability Distribution, Finding Maxima of Distribution Functions, Energy Spectrum of Hydrogen Atom, Shapes of s, p, d, and f Orbitals, Pauli’s Exclusion Principle, Hund’s Rule of Maximum Multiplicity

Aromatic and Heterocyclic Chemistry

Monocyclic, Bicyclic, and Tricyclic Aromatic Hydrocarbons, Monocyclic Compounds with One Heteroatom, Synthesis, Reactivity, Aromaticity, Electrophilic and Nucleophilic Aromatic Substitution

Analytical Chemistry

Qualitative and Quantitative Analysis, Acid-Base, Oxidation-Reduction, Complexometric Titrations, Precipitation Reactions, Indicators, Organic Reagents in Inorganic Analysis, Radioactivity, Error Analysis, Elementary Statistics and Probability Theory

Topic

Subtopics

Cell Biology

Structure of prokaryotic and eukaryotic cells, Membrane structure and function, Organelles and internal organization of eukaryotic cells, Protein trafficking in a eukaryotic cell, Cell communication – signalling pathways (endocrine, paracrine signalling), Extracellular matrix and apoptosis, Cell cycle – stages of mitosis and meiosis, control of cell division cycle

Biochemistry

Structure and function of biological macromolecules, Allostery, Enzymes – basic mechanisms of enzyme catalysis, Michaelis-Menten kinetics, enzyme inhibition, vitamins as coenzymes and regulation, Bioenergetics – free-energy change, high-energy compounds, biological oxidation-reduction reactions and reduction potential, Metabolism – glycolysis, TCA cycle, oxidative phosphorylation, photosynthesis, nitrogen fixation, urea cycle, regulation of glycolysis and TCA cycle

Genetics

Mendel’s laws, Inheritance patterns of polygenic traits, Mendelian inheritance patterns of human disorders, Pedigree analysis, Chromosomal basis of inheritance, Genetic recombination, Mapping genes on chromosomes based on linkage analysis, Plant breeding

Molecular Biology

Landmark experiments that established DNA as the genetic material, DNA replication, Proof-reading and repair of DNA, DNA recombination, Transcription, RNA processing, Translation, Regulation of gene expression (operons in bacteria, differential gene expression in multicellular eukaryotes)

Evolution

Darwinian view – natural selection, fossil record, descent with modification, Population genetics – sources of genetic variation, gene pools and allele frequencies, Hardy-Weinberg equation, genetic drift, gene flow, adaptive evolution, Types of speciation, Phylogenetic classification, Origin of life – abiotic synthesis of biological macromolecules, protocell, dating fossils, origin of multicellularity

Microbiology

Isolation, Cultivation, Structural features of viruses, bacteria, fungi, protozoa, Pathogenic microorganisms, Nutrition-based classification of microbes, Microbial metabolism, Growth kinetics, Submerged fermentation techniques, Microbial genetics

Plant Biology

Types of tissues and organs, Primary and secondary growth, Morphogenesis, Transport in vascular plants, Plant nutrition, Development of flowering plants – gametophytic and sporophytic generations, developmental phases, genetic control of flowering, gametogenesis, incompatibility, embryogenesis, dormancy, germination, environmental influence, Plant hormones, Photobiology, Plant response to biotic and abiotic stress

Animal Biology

Digestive, circulatory, respiratory, excretory, nervous, reproductive, and endocrine systems, Basics of immunology – Innate and adaptive immunity, Immune cells and Immunoglobulins, Animal development – Fertilization, embryonic pattern formation, cleavage, gastrulation, cellular differentiation, and morphogenesis

Ecology

Climate patterns, Terrestrial and aquatic biomes, Environmental constraints on species distribution, Factors affecting population density, Interactions among communities, Ecosystems, Ecological remediation

Biotechnology

Plant tissue culture, Cloning of animals through somatic cell nuclear transfer, Applications of recombinant DNA technology in medicine, agriculture, and forensic science

Methods in Biology

Cell Biology: Microscopy (light microscopy, electron microscopy), Staining proteins with antibodies, Visualizations using the GFP reporter; Biochemical Techniques: UV spectrophotometry, Biomolecular chromatography, Cell fractionation by centrifugation, Electrophoresis, Western blotting; Molecular Biology Techniques: DNA cloning (plasmid vectors, restriction enzymes), Polymerase Chain Reaction, Expression of cloned eukaryotic genes in bacteria, Hybridization techniques, DNA sequencing

Topic

Subtopics

Structure and Properties of Atoms

Bohr's Theory, Periodicity in Properties

Bonding in Molecules

Chemical Bonding, Complex Formation, Physical and Chemical Basis of Molecular Interactions

Chemical Kinetics, Thermodynamics, and Equilibrium

Chemical Equilibrium, Chemical Thermodynamics (First and Second Law), Chemical Kinetics (Zero and First Order Reactions)

Physical and Chemical Properties of Compounds

Chemical Catalysis, Acid-Base Concepts, Concepts of pH and Buffer, Conjugative Effects and Resonance, Inductive Effects, Electromeric Effects, Photochemistry, Electrochemistry

Chemistry of Organic Compounds

Hydrocarbons, Alkyl Halides, Alcohols, Aldehydes, Ketones, Carboxylic Acids, Amines and Their Derivatives, Aromatic Hydrocarbons, Halides, Nitro and Amino Compounds, Phenols, Diazonium Salts, Carboxylic and Sulphonic Acids, Soaps and Detergents, Stereochemistry of Carbon Compounds

Instrumental Techniques - Spectroscopy

Fundamentals of Molecular Spectroscopy, Emission and Absorption Spectroscopy, UV-Vis, IR and 1-D Proton NMR Spectroscopy, Basics of Mass Spectrometry, Basics of Calorimetry, Basic Concepts of Crystallography

Topic

Subtopics

Sets; Relations and Functions

Mathematical Induction; Logarithms; Complex numbers; Linear and Quadratic equations; Sequences and Series; Trigonometry; Cartesian System of Rectangular Coordinates; Straight lines and Family; Three Dimensional Geometry; Permutations and Combinations; Binomial Theorem; Vectors; Matrices and Determinants; Boolean Algebra; Functions; Limits and Continuity; Differentiation; Ordinary Differential Equations; Application of Derivatives; Integration as inverse process of differentiation; Definite and indefinite integrals; Methods of Integration; Integration by parts.

Statistics

Measures of dispersion; Mean Deviation for grouped and ungrouped data; Variance and Standard Deviation; and Analysis of Frequency Distribution.

Statistics

Random Experiments; Event; Axiomatic Approach to Probability; Conditional Probability and its properties; Multiplication Theorem on Probability; Independent Events; Bayes’ Theorem; Random Variables and its Probability Distributions; Bernoulli Trails and Binomial Distributions.

Topic

Subtopics

Units and measurements

Motion in one and two dimensions; Laws of motion; Work and kinetic energy; Conservation of energy; System of particles and rotational motion; Mechanical properties of solids and fluids; Thermal properties of matter; Heat and laws of thermodynamics; Kinetic theory of gases; Electric charge and field; Electric potential and capacitance; Current, resistance and simple circuits; Moving charges and magnetic field; Magnetism and matter; Electromagnetic induction; Electromagnetic waves; Alternating currents; Optics: Geometrical Optics – Reflection by spherical mirrors, Refraction at spherical surfaces and lenses, Total internal reflection and Optical instruments; Wave optics – Reflection and refraction of plane waves, Interference, Diffraction, Polarization

Wave optics

Reflection and refraction of plane waves, Interference, Diffraction, Polarization, and Young’s experiment: Dual nature of radiation and matter; Atoms, nuclei and nuclear physics; Semiconductor materials, devices and simple circuits.

Topic

Subtopics

The Planet Earth

Origin of the Solar System and the Earth; Geosphere and the composition of the Earth; Shape and size of the Earth; Earth-Moon system; Dating rocks and age of the Earth; Volcanism and volcanic landforms; Interior of the Earth; Earthquakes; Earth’s magnetism and gravity; Isostasy; Basic elements of Plate Tectonics; Orogenic cycles.

Geomorphology

Weathering and erosion; Soil formation; Transportation and deposition by wind, ice, river, sea, and resulting landforms.

Structural Geology

Orientation of planes and lines in space – concept of dip, strike, rake, and plunge. Contour lines; Rule of ‘V’s and outcrop patterns; Interpretation of geological maps and cross-section construction; Classification and origin of folds, faults, joints, unconformities, foliations, and lineations; Stereographic and equal-area projections of planes and lines; Numerical problems related to outcrop and bore-hole data.

Paleontology

Major steps in the evolution of life forms; Fossils, their mode of preservation and utility in age determination and paleoenvironmental interpretations; Morphology, major evolutionary trends, and ages of important groups of animals – Brachiopoda, Mollusca, Trilobita, Graptolitoidea, Anthozoa, Echinodermata; Gondwana plant fossils; Elementary idea of vertebrate fossils in India.

Stratigraphy

Principles of stratigraphy; Litho-, Chrono- and biostratigraphic classification; Stratigraphic correlation techniques; Archaean cratons of Peninsular India (Dharwar, Singhbhum and Aravalli); Proterozoic mobile belts; Stratigraphy of Cuddapah and Vindhyan basins; Stratigraphy of Paleozoic – Mesozoic of Spiti and Kashmir, Gondwana Supergroup, Jurassic of Kutch, Cretaceous of Trichinopoly, Tertiary and Quaternary sequences of Assam, Bengal, and Siwaliks.

Mineralogy

Symmetry and forms in common crystal classes; Physical properties of minerals; Isomorphism, polymorphism, solid solution and exsolution; Classification of minerals; Structure of silicates; Mineralogy of common rock-forming minerals; Elements of Optical Mineralogy, Optical properties of common rock-forming minerals.

Petrology

Definition and classification of rocks; Igneous rocks – forms of igneous bodies; Processes of evolution and diversification of magma; Classification, association, and genesis of common igneous rocks. Sedimentary rocks – classification, texture, and structure; Petrology of sandstone and limestone; Elements of sedimentary environments and facies. Metamorphic rocks – classification and texture; Types of metamorphism; Controls on metamorphism – pressure, temperature, and fluids; Concept of projections – ACF, AKF, and AFM diagrams; Phase Rule and its applications; Concepts of zones and facies, Characteristic mineral assemblages of pelites in the Barrovian zones and mafic rocks in common facies.

Economic Geology

Physical properties of common economic minerals; General processes of formation of mineral deposits; Mode of occurrence and distribution of metallic and non-metallic mineral deposits in India; Fundamentals of reserve calculation; Elements of coal and hydrocarbon geology, Coal and hydrocarbon occurrences in India.

Applied Geology

Groundwater and hydrological cycle, Types of aquifers, porosity, and permeability; Principles of engineering geology; Geological considerations in construction of dams and tunnels.

Real Analysis

Topic

Subtopics

Sequences and Series of Real Numbers

Convergence of sequences, bounded and monotone sequences, Cauchy sequences, Bolzano-Weierstrass theorem, absolute convergence, tests of convergence for series (comparison test, ratio test, root test), Power series (of one real variable), radius and interval of convergence, term-wise differentiation and integration of power series

Functions of One Real Variable

Limit, continuity, intermediate value property, differentiation, Rolle’s Theorem, mean value theorem, L'Hospital rule, Taylor's theorem, Taylor’s series, maxima and minima, Riemann integration (definite integrals and their properties), fundamental theorem of calculus

Multivariable Calculus and Differential Equations

Functions of Two or Three Real Variables

Limit, continuity, partial derivatives, total derivative, maxima and minima

Integral Calculus

Double and triple integrals, change of order of integration, calculating surface areas and volumes using double integrals, calculating volumes using triple integrals

Differential Equations

Bernoulli’s equation, exact differential equations, integrating factors, orthogonal trajectories, homogeneous differential equations, method of separation of variables, linear differential equations of second order with constant coefficients, method of variation of parameters, Cauchy-Euler equation

Topic

Subtopics

Sequences and Series of Real Numbers

Sequences of real numbers, their convergence, and limits.

Cauchy sequences and their convergence.

Monotonic sequences and their limits.

Limits of standard sequences.

Infinite series and its convergence, and divergence.

Convergence of series with non-negative terms.

Tests for convergence and divergence of a series: Comparison test, limit comparison test, D'Alembert’s ratio test, Cauchy’s root test, Cauchy’s condensation test, and integral test.

Absolute convergence of series.

Leibnitz’s test for the convergence of alternating series.

Conditional convergence.

Convergence of power series and radius of convergence.

Differential Calculus of One and Two Real Variables

Limits of functions of one real variable.

Continuity and differentiability of functions of one real variable.

Properties of continuous and differentiable functions of one real variable.

Rolle's theorem and Lagrange's mean value theorems.

Higher order derivatives, Leibnitz's rule and its applications.

Taylor's theorem with Lagrange's and Cauchy's form of remainders.

Taylor's and Maclaurin's series of standard functions.

Indeterminate forms and L'Hospital's rule.

Maxima and minima of functions of one real variable, critical points, local maxima and minima, global maxima and minima, and point of inflection.

Limits of functions of two real variables.

Continuity and differentiability of functions of two real variables.

Properties of continuous and differentiable functions of two real variables.

Partial differentiation and total differentiation.

Leibnitz's rule for successive differentiation.

Maxima and minima of functions of two real variables.

Critical points, Hessian matrix, and saddle points.

Constrained optimization techniques (with Lagrange multiplier).

Integral Calculus

Fundamental theorems of integral calculus (single integral).

Leibnitz's rule and its applications.

Differentiation under integral sign.

Improper integrals.

Beta and Gamma integrals: properties and relationship between them.

Double integrals.

Change of order of integration.

Transformation of variables.

Applications of definite integrals: Arc lengths, areas, and volumes.

Matrices and Determinants

Vector spaces with real field.

Subspaces and sum of subspaces.

Span of a set.

Linear dependence and independence.

Dimension and basis.

Algebra of matrices.

Standard matrices (Symmetric, Skew Symmetric, Hermitian, Skew Hermitian, Orthogonal, Unitary, Idempotent, Nilpotent).

Definition, properties, and applications of determinants.

Evaluation of determinants using transformations.

Determinant of product of matrices.

Singular and non-singular matrices and their properties.

Trace of a matrix.

Adjoint and inverse of a matrix and related properties.

Rank of a matrix, row-rank, column-rank, standard theorems on ranks, rank of the sum and the product of two matrices.

Row reduction and echelon forms.

Partitioning of matrices and simple properties.

Consistent and inconsistent system of linear equations.

Properties of solutions of system of linear equations.

Use of determinants in solution to the system of linear equations.

Cramer’s rule.

Characteristic roots and Characteristic vectors.

Properties of characteristic roots and vectors.

Cayley Hamilton theorem.

Probability

Random Experiments; Sample Space and Algebra of Events (Event space).

Relative frequency and Axiomatic definitions of probability.

Properties of probability function.

Addition theorem of probability function (inclusion exclusion principle).

Geometric probability.

Boole's and Bonferroni's inequalities.

Conditional probability and Multiplication rule.

Theorem of total probability and Bayes’ theorem.

Pairwise and mutual independence of events.

Univariate Distributions

Definition of random variables.

Cumulative distribution function (c.d.f.) of a random variable.

Discrete and Continuous random variables.

Probability mass function (p.m.f.) and Probability density function (p.d.f.) of a random variable.

Distribution (c.d.f., p.m.f., p.d.f.) of a function of a random variable using transformation of variable and Jacobian method.

Mathematical expectation and moments.

Mean, Median, Mode, Variance, Standard deviation, Coefficient of variation, Quantiles, Quartiles, Coefficient of Variation, and measures of Skewness and Kurtosis of a probability distribution.

Moment generating function (m.g.f.), its properties and uniqueness.

Markov and Chebyshev inequalities and their applications.

Standard Univariate Distributions

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

Multivariate Distributions

Definition of random vectors.

Joint and marginal c.d.f.s of a random vector.

Discrete and continuous type random vectors.

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.

Distribution of functions of random vectors using transformation of variables and Jacobian method.

Mathematical expectation of functions of random vectors.

Joint moments, Covariance and Correlation.

Joint moment generating function and its properties.

Uniqueness of joint m.g.f. and its applications.

Conditional moments, conditional expectations and conditional variance.

Additive properties of Binomial, Poisson, Negative Binomial, Gamma, and Normal Distributions using their m.g.f..

Standard Multivariate Distributions

Multinomial distribution as a generalization of binomial distribution and its properties (moments, correlation, marginal distributions, additive property).

Bivariate normal distribution, its marginal and conditional distributions and related properties.

Limit Theorems

Convergence in probability, convergence in distribution and their interrelations.

Weak law of large numbers and Central Limit Theorem (i.i.d. case) and their applications.

Sampling Distributions

Definitions of random sample, parameter and statistic.

Sampling distribution of a statistic.

Order Statistics: Definition and distribution of the i-th order statistic (d.f. and p.d.f. for i.i.d. case for continuous distributions).

Distribution (c.d.f., p.m.f., p.d.f.) of smallest and largest order statistics (i.i.d. case for discrete as well as continuous distributions).

Central Chi-square distribution: Definition and derivation of p.d.f. of central χ2 distribution with k degrees of freedom (d.f.) using m.g.f..

Properties of central χ2 distribution, additive property and limiting form of central χ2 distribution.

Central Student's t-distribution: Definition and derivation of p.d.f. of Central Student's t-distribution with ν d.f., Properties and limiting form of central t-distribution.

Snedecor's Central F-distribution: Definition and derivation of p.d.f. of Snedecor's Central F-distribution with (ν1,ν2) d.f..

Properties of Central F-distribution, distribution of the reciprocal of F-distribution.

Relationship between t,χ2,F distributions.

Estimation

Unbiasedness.

Sufficiency of a statistic.

Factorization theorem.

Complete statistic.

Consistency and relative efficiency of estimators.

Uniformly Minimum variance unbiased estimator (UMVUE).

Rao-Blackwell and Lehmann-Scheffe theorems and their applications.

Cramer-Rao inequality and UMVUEs.

Methods of Estimation: Method of moments, method of maximum likelihood, invariance of maximum likelihood estimators.

Least squares estimation and its applications in simple linear regression models.

Confidence intervals and confidence coefficient.

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

Testing of Hypotheses

Null and alternative hypotheses (simple and composite), Type-I and Type-II errors.

Critical region.

Level of significance, size and power of a test, p-value.

Most powerful critical regions and most powerful (MP) tests.

Uniformly most powerful (UMP) tests.

Neyman Pearson Lemma (without proof) and its applications to construction of MP and UMP tests for parameters of single parameter parametric families.

Likelihood ratio tests for parameters of univariate normal distribution.

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