IIT JAM Syllabus 2026 - Download PDF Subject wise & Check Important Topics

Unit- 1: 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, Complex numbers, Error analysis of Experimental Data: Significant digits and rounding of numbers, Types of errors, mean, median, standard deviation.

Unit- 2: Mechanics and General Properties of Matter Velocity

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.

Unit- 3: 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 oscillations, 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..

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. 

Unit- 4: Electricity and Magnetism

Coulomb’s law, Electric field and potential, Gauss’s law, Electrostatic boundary conditions, Solution of Laplace’s equation for simple cases – upto two dimensions.

Conductors, capacitors, Linear dielectrics, dielectric polarization, volume and surface bound 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 vector, Poynting’s theorem, Energy of Electromagnetic fields. 

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.

Unit- 5: 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.

Unit - 6: 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, Schr¨odinger equation and its solution for one, two and three dimensional boxes.

Solution of Schr¨odinger 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.

Unit - 7: 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.

Unit - 1: Atomic and Molecular Structure

Planck’s black body radiation, Photoelectric effect, Bohr’s theory, de Broglie postulate, Heisenberg’s Uncertainty Principle; Schr¨odinger’s wave equation (including mathematical treatment),

postulates of quantum mechanics, normalized and orthogonal wave functions, its complex conjugate (idea of complex numbers) and significance of Ψ2 ;

Operators; Particle in one- dimension box, radial and angular wave functions for hydrogen atom, radial probability distribution; Finding maxima of distribution functions (idea of maxima and minima), energy spectrum of hydrogen atom; Shapes of s, p, d and f orbitals; Pauli’s Exclusion Principle; Hund’s rule of maximum multiplicity.

Physical properties of Liquid, vapour pressure, surface tension and co-efficient of viscosity and their applications; effect of concentration of solutes on surface tension and viscosity; effect of temperature on viscosity of liquids.

Unit - 3: Gaseous State

Kinetic molecular model of a gas: collision frequency; collision diameter; mean free path and viscosity of gases;

Maxwell-Boltzmann distribution: molecular velocities, law of equipartition of energy, molecular basis of heat capacities; Ideal gases, and deviations from ideal gas behaviour,

van der Waals equation of state; critical state, law of corresponding states.

Unit - 4: Solid State

Unit Cells, Miller indices, crystal systems and Bravais Lattices, elementary applications of vectors to crystal systems; X-ray diffraction, Bragg’s Law, Structure of NaCl, CsCl, and KCl,

diamond, and graphite; Close packing in metals and metal compounds, semiconductors, insulators; Defects in crystals, lattice energy; isomorphism; heat capacity of solids.

Unit - 5: Chemical Thermodynamics

Mathematical treatment: Exact and in-exact differentials, partial derivatives, Euler’s reciprocity, cyclic rule; Reversible and irreversible processes; Laws of thermodynamics, thermochemistry, thermodynamic functions, such as enthalpy, entropy, and Gibbs free energy, their properties and applications;

Partial molar quantities, dependence of thermodynamic parameters on composition, Gibbs Duhem equation, chemical potential and its applications.

Unit - 6: Chemical and Phase Equilibria

Law of mass action; Kp, Kc, Kx and 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.

Unit - 7: Electrochemistry

Conductivity, equivalent and molar conductivity and their properties; Kohlrausch law; DebyeH¨uckel-Onsager equation; Ionic velocities, mobilities,
 

transference numbers; Applications of conductance measurement; Quantitative aspects of Faraday’s laws of electrolysis, applications of electrolysis in metallurgy and industry;

Electromotive force of a cell, Nernst equation; Standard electrode potential, Electrochemical series; Concentration cells with and without transference; Applications of EMF measurements including potentiometric titrations.

Unit - 8: Chemical Kinetics

Order and molecularity of a reaction, differential and integrated form 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, specificity and selectivity, mechanisms of catalyzed reactions at solid surfaces; Enzyme catalysis (Michaelis-Menten mechanism, Double reciprocal plot), Acid-base catalysis.

Unit - 9: Adsorption

Gibbs adsorption equation; adsorption isotherm; types of adsorption; surface area of adsorbents; surface films on liquids.

Unit - 10: Spectroscopy

Beer-Lambert’s law; fundamental concepts of rotational, vibrational, electronic and magnetic resonance spectroscopy.

Unit - 1: Basic Concepts in Organic Chemistry and Stereochemistry

Electronic effects (resonance, inductive, hyperconjugation) and steric effects and its applications (acid/base property); optical isomerism in compounds with and without any stereocenters (allenes, biphenyls);

Conformation of acyclic systems (substituted ethane/n-propane/n-butane) and cyclic systems, substituted cyclohexanes, and polycyclic (cis and trans decalins) systems.

Unit - 2: Organic Reaction Mechanism and Synthetic Applications

Chemistry of reactive intermediates (carbocations, carbanions, free radicals, carbenes, nitrenes, benzynes); nucleophilic substitution, elimination reactions and mechanisms;
 

Hofmann-Curtius- Lossen rearrangement, Wolff rearrangement, Simmons-Smith reaction, Reimer-Tiemann reaction, Michael reaction, Darzens reaction, Wittig reaction and McMurry reaction;

Wittig reaction and McMurry reaction;Pinacolpinacolone, Favorskii, benzilic acid rearrangement, Baeyer-Villeger reaction; oxidation and reduction reactions in organic chemistry;
 

Organometallic reagents in organic synthesis (Grignard, organolithium , organocopper and organozinc (Reformatsky only); Diels-Alder, electrocyclic and sigmatropic reactions; functional group inter-conversions and structural problems using chemical reactions.

Unit - 3: Qualitative Organic Analysis Identification

Identification of functional groups by chemical tests; elementary UV, IR and 1H NMR spectroscopic techniques as tools for structural elucidation of simple organic molecules.

Unit - 4: Natural Products Chemistry

Chemistry of alkaloids, steroids, terpenes, carbohydrates, amino acids, peptides and nucleic acids.
 

Unit - 5: Aromatic and Heterocyclic Chemistry

Monocyclic, bicyclic and tricyclic aromatic hydrocarbons, and monocyclic compounds with one hetero atom: synthesis, reactivity and properties, aromaticity; Electrophilic and nucleophilic aromatic substitution reactions.

Unit - 1: Periodic Table

Periodic classification of elements, Aufbau’s principle, periodicity; Variations of orbital energy, effective nuclear charge, atomic, covalent, and ionic radii ionization enthalpy

electron gain enthalpy, and electronegativity with atomic number, electronic configuration of diatomic molecules (first and second row elements).

General methods of isolation and purification of elements; Principles and applications of Ellingham diagram.
 

Unit - 3: Chemical Bonding and Shapes of Molecules

Ionic bond: Packing of ions in crystals, radius ratio rule, Born-Land´e equation, Kapustinskii expression, Madelung constant, Born-Haber cycle, solvation energy,

Polarizing power and polarizability; Fajan’s rules; Covalent bond: Lewis structure, valence bond theory. Hybridization, molecular orbital theory, molecular orbital diagrams of diatomic and simple polyatomic molecules and ions;

Multiple bonding (σ and π bond approach) and bond lengths; van der Waals forces, ion-dipole forces, dipole-dipole interactions, induced dipole interactions, instantaneous dipole- induced dipole interactions, hydrogen bonding; Effect of intermolecular forces on melting and boiling points,

solubility energetics of dissolution process; Bond dipole, dipole moment, and molecular polarizabilities; VSEPR theory and shapes of molecules; ionic solids.

Unit - 4: Main Group Elements (s and p blocks)

Reactions of alkali and alkaline earth metals with oxygen, hydrogen and water; Alkali and alkaline earth metals in liquid ammonia; Gradation in properties of main group element in a group;

Inert pair effect; Synthesis, structure and properties of diborane, ammonia, silane, phosphine and hydrogen sulphide;

Allotropes of carbon; Oxides of nitrogen, phosphorus and sulphur; Oxoacids of phosphorus, sulphur and chlorine; Halides of silicon and phosphorus; Synthesis and properties of borazine, silicone and phosphazene; Synthesis and reactions of xenon fluorides.

Unit - 5: Transition Metals (d block)

Characteristics of d-block elements; oxide, hydroxide and salts of first row metals; coordination complexes: structure, isomerism, reaction mechanism and electronic spectra;

VB, MO and crystal field theoretical approaches for structure, color and magnetic properties of metal complexes;

Organometallic compounds with metal-ligand single and multiple bonds (such as metal carbonyls, metal nitrosyls and metallocenes); Homogenous catalysis involving Wilkinson’s catalyst.

Unit - 6: Bioinorganic Chemistry

Essentials and trace elements of life; basic reactions in the biological systems and the role of metal ions, especially F e2+, and Zn2+; structure and function of myoglobin, hemoglobin and carbonic anhydrase. 

Unit - 7: Instrumental Methods of Analysis

Basic principles; instrumentations and simple applications of conductometry, potentiometry and UV-vis spectrophotometry; analyses of water, air and soil samples.

Unit - 8: Analytical Chemistry

Principles of qualitative and quantitative analysis; Acidbase, oxidation- reduction and complexometric titrations using EDTA; Precipitation reactions;

Use and types of indicators; Use of organic reagents in inorganic analysis; Radioactivity, nuclear reactions, applications of isotopes; Mathematical treatment in error analysis, elementary statistics and probability theory.

Unit - 1: 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.

Unit - 2: 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.

Unit - 3: Functions of Two or Three Real Variables

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

Unit - 4: Integral Calculus

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

Unit - 5: 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.

Unit - 6: Matrices

Systems of linear equations, rank, nullity, rank-nullity theorem, inverse, determinant, eigenvalues, eigenvectors.

Unit - 7: Cyclic

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

Unit - 8: Finite-Dimensional Vector Spaces

Linear independence of vectors, basis, dimension, linear transformations, matrix representation, range space, null space, rank-nullity theorem.

Unit - 1: General Biology (Cell Biology)

Structure of prokaryotic and eukaryotic cells; Membrane structure and function; Organelles and internal organization of the eukaryotic cell; Cell communication – signalling pathways: endocrine and paracrine signalling; Extracellular matrix and apoptosis; Cell cycle – stages of mitosis and meiosis.
 

Unit - 2: Biochemistry:

Structure and function of biological macromolecules; Enzymes – basic mechanisms of enzyme catalysis and regulation, Hill coefficient, Michaelis-Menten kinetics, enzyme inhibition, vitamins as coenzymes;
 

Bioenergetics – free-energy change, high-energy compounds, biological oxidation-reduction reactions and reduction potential;

Metabolism – glycolysis, TCA cycle and their regulation, oxidative phosphorylation, photosynthesis, nitrogen fixation, urea cycle.

Unit - 3: Genetics

Mendelian inheritance; Exception to Mendelian law of independent assortment; Genetic interactions, linkage, recombination and chromosome mapping, Chromosomal mutations; Genetic disorders; Population Genetics.

Unit - 4: Molecular Biology:​​​​​​​

Landmark experiments that established DNA is the genetic material; DNA replication; Proof-reading and repair of DNA; DNA recombination; Transcription;

RNA processing; Translation; Regulation of gene expression including operons and differential gene expression in multicellular eukaryotes.

Unit - 5: Evolution and Ecology:

Darwinian view – natural selection, fossil record and descent with modification; Different types of speciation; Phylogenetic classification; Origin of life – abiotic synthesis of biological macromolecules, protocell, dating fossils and origin of multicellularity;


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

Unit - 6: Microbiology

Microbial genetics - transformation, conjugation and transduction; Structural features of viruses, bacteria, fungi and protozoa; Pathogenic microorganisms; Nutritionbased classification of microbes; Microbial metabolism; Isolation and Cultivation of microorganisms;


Growth kinetics; Microbial control and sterilization; Microbial fermentation – batch, fed-batch and continuous; Bioreactor and its components; Introduction to downstream processing - product recovery and purification; Effluent treatment.

Types of tissues and organs; Primary and secondary growth; Morphogenesis; Transport in vascular plants; Plant nutrition; Development of flowering plants – gametophytic and sporophytic generations;


Plant growth regulators; Photobiology; Plant Tissue Culture – Cellular totipotency and microporopagation; Transgenic plants; Plant response to biotic and abiotic stresses.

Digestive, circulatory, respiratory, excretory, nervous, reproductive and endocrine systems; Basics of immunology – Innate and adaptive immunity, Immune cells, immunoglobulins and major histocompatibility complexes;


Animal development – Fertilization, embryonic pattern formation, cleavage, gastrulation, cellular differentiation and morphogenesis; Mammalian cell culture, animal cloning; Transgenic animals.

Chromatography; Centrifugation; Electrophoresis; ELISA, Western blotting and immunostaining; Principles of light, fluorescence and electron microscopy.

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

Bioinformatics; Sequence and structure databases; DNA, RNA and protein sequence analysis; Secondary structure and 3D structure prediction; Biochemical databases.

Unit - 11: Instrumental Techniques

Spectroscopy: fundamentals of molecular spectroscopy, emission and absorption spectroscopy, UV-Vis, circular dichroism, FTIR and 1-D proton NMR spectroscopy, basics of mass spectrometry; Basics of calorimetry; Basic concepts of crystallography; Flowcytometry.

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); and 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; and 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.

Sets; Relations and Functions; 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; 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.

Probability & Statistics: Mean, median, mode and standard deviation; Random variables; Poisson, normal and binomial distributions; Correlation and regression analysis

: 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, and Young’s experiment: Dual nature of radiation and matter; Atoms, nuclei and nuclear physics; Semiconductor materials, devices and simple circuits.

Unit - 1: The Planet Earth

Origin of the Solar System and the Earth; Internal structure, composition, and age of the Earth; Pressure-temperature-density variations within the Earth; Principles of radiometric dating (Rb-Sr, Sm-Nd, 14C);

Volcanism and volcanic landforms; Earthquakes; Earth’s magnetism and gravity, Isostasy; Concepts of Plate Tectonics and orogeny.

Unit - 2: Geomorphology

Weathering, erosion, deposition; Endogenous and exogenous processes of landform development; Soil formation; River and drainage basin/drainage pattern, network characteristics; Fluvial, aeolian, marine, glacial and karst landforms.
 

Unit - 3: Structural Geology

Concept of dip, strike, rake and plunge; Contour lines; Rule of ‘V’s and outcrop patterns; Interpretation of geological maps; Cross-section construction; Classification and origin of folds, faults, joints, unconformities, foliations and lineations;

Stereographic and equal-area projections of planes and lines; Quantitative interpretation of structures, outcrops, and bore-hole data.

Unit - 4: Palaeontology

Major stages in the evolution of life forms; Fossils and their mode of preservation; Application of macrofossils in age determination and paleoenvironmental interpretations;


Morphology, major evolutionary trends and ages of important groups of invertebrates – Brachiopoda, Mollusca, Trilobita, Echinodermata; Gondwana plant fossils; Vertebrate fossils (Equidae, Proboscidea) in India.

Unit - 5: Stratigraphy

Principles of stratigraphy; Litho-, chrono- and bio-stratigraphic 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, Cenozonic sequences of Assam, Bengal and Siwaliks.

Unit - 6: Mineralogy

Symmetry and forms in common crystal classes; Miller indices; Twinning and twinning laws; Isomorphism, polymorphism, solid solution and exsolution;

Elements of Optical Mineralogy; Classification, structure, chemistry, physical, and optical properties of common rock-forming minerals.

Unit - 7: Petrology

Igneous rocks – classification and texture; Forms of igneous bodies; Evolution and diversification of magma; Use of binary systems to understand melting and crystallization behaviour of rocks and magmas; Genesis of common igneous rocks and associations.

Sedimentary rocks – classification, texture, and structure; Petrology of sandstone and limestone; Basics 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.

Unit - 8: Economic Geology

Physical properties of common economic minerals; Processes of for mations of ore mineral deposits-magmatic concentration, hydrothermal processes, oxidation and supergene sulphide enrichment, residual and mechanical concentration; Mode of occurrence and distribution of metallic and non-metallic mineral deposits in India;

Ore grade and reserve estimation; Coal and hydrocarbon geology and their Indian occurrences.

Unit - 9 - Applied Geology

Basics of groundwater geology; Types of aquifers, porosity and permeability; Groundwater flow; Principles of engineering geology; Geological considerations in construction of dams and tunnels; Basics of remote sensing.

Unit - 1: 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.

Limit superior and limit inferior of 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 n th 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.

Unit - 2: 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, Lebnitz’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. Lebnitz’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).

Unit - 3: Integral Calculus

Fundamental theorems of integral calculus (single integral). Lebnitz’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.

Unit - 4: Matrices and Determinants

Rn and C n as vector spaces over real field. Span of a set. Linear dependence and independence. Dimension and basis. Null space. Algebra of matrices. Standard matrices (Symmetric and Skew Symmetric matrices,

Hermitian and Skew Hermitian matrices, Orthogonal and Unitary matrices, Idempotent and Nilpotent matrices). 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 and nullity 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. Consistent and inconsistent systems of linear equations. Properties of solutions of system of linear equations. Use of determinants in solving the system of linear equations. Cramer’s rule. Characteristic roots and Characteristic vectors.

Properties of characteristic roots and vectors. Cayley-Hamilton theorem. Quadratic forms, positive definite, positive semi-definite, negative definite,and negative semi-definite matrices, and their simple properties.

Unit - 1: Descriptive Statistics & Probability

Concepts of sample and population. Different types of data. Tabular and graphical representation of data. Measures of central tendency (arithmetic mean, geometric mean, harmonic mean, median, mode).Measures of dispersion (range, inter quartile range, mean deviation about a point, standard deviation, variance, coefficient of variation). Moments, central moments, skewness and kurtosis. Bivariate data: Scatter diagram, covariance, simple, partial and multiple correlations (3 variables only), Spearman’s rank correlation.

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.

Unit - 2: 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, 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.

Unit - 3: 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.

Unit - 4: 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.
 

Unit - 5: 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.

Unit - 6: Limit Theorems

Convergence in probability, convergence in mean square, almost sure convergence, convergence in distribution, and their inter-relations. Weak law of large numbers, Strong law of large numbers, and Central Limit Theorem (i.i.d. and finite variance case).

Unit - 7: Estimation&Nonparametric Methods

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.

Unit - 8: 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 parameter of one-parameter parametric families. Likelihood ratio tests for parameters of univariate normal distribution.

Definitions of random sample, parameter and statistic. Sampling distribution of a statistic. Order Statistics: Definition and distribution of the rth 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 x 2 distribution with n degrees of freedom (d.f.) using m.g.f. Properties of central x 2 distribution, additive property and limiting form of central x 2 distribution.


Central t - distribution: Definition and derivation of p.d.f. of Central t-distribution with n d.f., Properties and limiting form of central t-distribution. Central F-distribution: Definition and derivation of p.d.f. of Central F- distribution with (m,n) d.f. Properties of Central F-distribution, distribution of the reciprocal of F-distribution. Relationship between t, F and 2 distributions.

Discrete time Markov chain: transition probability matrix, higher order transition probabilities, Markov chain as a graph, Chapman-Kolmogorov equation, classification of states and chains, stability of Markov chain (stationary and limiting distributions). Poisson process and its properties, interarrival and waiting times.

Unit - 1: Microeconomics

Consumer theory:Preference, utility and representation theorem, budget constraint, choice, demand (ordinary and compensated), Slutsky equation, revealed preference axioms.

Theory of production and cost: Production technology, isoquants, production function with one and more inputs, returns to scale, short run and long run costs, cost curves in the short run and long run.

General equilibrium and welfare:Equilibrium and efficiency under pure exchange and production, welfare economics, theorems of welfare economics.

Market structure: Perfect competition, monopoly, pricing with market power, price discrimination (first, second and third), monopolistic competition and oligopoly.

Game theory: Strategic form games, iterated elimination of dominated strategies, Nash equilibrium, mixed extension and mixed strategy Nash equilibrium, examples: Cournot, Bertrand duopolies, Prisoner’s dilemma.

Public goods and market failure: Externalities, public goods and markets with asymmetric information (adverse selection and moral hazard)

Unit - 2: Macroeconomics

National income accounting: Structure, key concepts, measurements, and circular flow of income - for closed and open economy, money, fiscal and foreign sector variables - concepts and measurements.

Behavioural and technological functions: Consumption functions - absolute income hypothesis, life-cycle and permanent income hypothesis, random walk model of consumption, investment functions - Keynesian, money demand and supply functions, production function.

Business cycles and economic models (closed economy): Business cycles-facts and features, the Classical model of the business cycle, the Keynesian model of the business cycle, simple Keynesian cross model of income and employment determination and the multiplier (in a closed economy), IS-LM Model, Hicks’ IS-LM synthesis, role of monetary and fiscal policies.

Business cycles and economic models (open economy): Open economy, MundellFleming model, Keynesian flexible price (aggregate demand and aggregate supply) model, role of monetary and fiscal policies.

Inflation and unemployment: Inflation - theories, measurement, causes, and effects, unemployment - types, measurement, causes, and effects.

Growth models: Harrod-Domar, Solow and Neo-classical growth models (AK model, Romer model and Schumpeterian growth model).

Unit - 3: Statistics for Economics

Probability theory: Sample space and events, axioms of probability and their properties, conditional probability and Bayes’ rule, independent events, random variables and probability distributions, expectation, variance and higher order moments, functions of random variables,

 

properties of commonly used discrete and continuous distributions, density and distribution functions for jointly distributed random variables, mean and variance of jointly distributed random variables, covariance and correlation coefficients.

Mathematical statistics:Random sampling, types of sampling, point and interval estimation, estimation of population parameters using methods of moments and maximum likelihood procedures, properties of estimators, sampling distribution, confidence intervals, central limit theorem, law of large number.

Hypothesis testing: distributions of test statistics, testing hypotheses related to population parameters, Type I and Type II errors, the power of a test, tests for comparing parameters from two samples.

Correlation and regression: Correlation and types of correlation, the nature of regression analysis, method of Ordinary Least Squares (OLS), CLRM assumptions, properties of OLS, goodness of fit, variance and covariance of OLS estimator.

Unit - 4: Indian Economy

Indian economy before 1950: Transfer of tribute, deindustrialization of India.

Planning and Indian development: Planning models, relation between agricultural and industrial growth, challenges faced by Indian planning.

Indian economy after 1991: Balance of payments crisis in 1991, major aspects of economic reforms in India after 1991, reforms in trade and foreign investment.

Banking, finance and macroeconomic policies: aspects of banking in India, CRR and SLR, financial sector reforms in India, fiscal and monetary policy, savings and investment rates in India. 

Inequalities in social development: India’s achievements in health, education and other social sectors, disparities between Indian States in human development.

Poverty: Methodology of poverty estimation, Issues in poverty estimation in India.

India’s labour market: unemployment, labour force participation rates.

Unit - 5: Mathematics for Economics

Preliminaries and functions: Set theory and number theory, elementary functions: quadratic, polynomial, power, exponential, logarithmic, functions of several variables, graphs and level curves, convex set, concavity and quasiconcavity of function, convexity and quasiconvexity of functions, sequences and series: convergence, algebraic properties and applications, complex numbers and its geometrical representation, De Moivre’s theorem and its application

Differential calculus:Limits, continuity and differentiability, mean value theorems, Taylor’s theorem, partial differentiation, gradient, chain rule, second and higher order derivatives:
 

properties and applications, implicit function theorem, and application to comparative statics problems, homogeneous and homothetic functions: characterisations and applications

Integral calculus: Definite integrals, fundamental theorems, indefinite integrals and applications.

Differential equations, and difference equations: First order difference equations, first order differential equations and applications.

Linear algebra: Matrix representations and elementary operations, systems of linear equations: properties of their solution, linear independence and dependence, rank, determinants, eigenvectors and eigenvalues of square matrices, symmetric matrices and quadratic forms, definiteness and semidefiniteness of quadratic forms.

Optimization: Local and global optima: geometric and calculus-based characterisations, and applications, multivariate optimization, constrained optimization and method of Lagrange multiplier, second order condition of optima, definiteness and optimality,

properties of value function: envelope theorem and applications, linear programming: graphical solution, matrix formulation, duality, economic interpretation.