IIT JAM Syllabus 2027: Subject-Wise Topics, Topic-Wise Weightage and PDF Download

Convergence of sequences. Bounded and monotone sequences. Convergence criteria for real sequences. Cauchy sequences. Subsequences. Bolzano-Weierstrass theorem. Series of positive terms: comparison test, ratio test, root test. Leibniz’s test for alternating series. Absolute and conditional convergence.

Limit, continuity, and intermediate value property. Differentiation. Rolle’s theorem and mean value theorem. L’Hospital’s rule. Taylor’s theorem. Maxima and minima.

Limit, continuity, and partial derivatives. Differentiability. Maxima and minima. Method of Lagrange multipliers. Homogeneous functions and Euler’s theorem.

Integration as the inverse of differentiation. Definite integrals and their properties. Fundamental theorem of integral calculus. Double and triple integrals. Change of order of integration. Surface areas and volumes using double integrals. Volumes using triple integrals.

Ordinary differential equations of the first order. Bernoulli’s equation. Exact differential equations and integrating factors. Orthogonal trajectories. Homogeneous differential equations. Variable separable equations. Linear differential equations of second and higher order with constant coefficients. Euler-Cauchy equation. Method of variation of parameters.

Scalar and vector fields. Gradient, divergence, curl, and Laplacian. Scalar and vector line integrals. Scalar and vector surface integrals. Green’s theorem. Stokes’ theorem. Gauss’ divergence theorem and their applications.

Groups, subgroups, Abelian groups, non-Abelian groups, cyclic groups, permutation groups. Normal subgroups. Lagrange’s theorem for finite groups. Group homomorphisms. Basic concepts of quotient groups.

Finite-dimensional vector spaces over real or complex fields. Linear independence of vectors. Basis and dimension. Linear transformations and their matrix representations. Range space and null space. Rank-nullity theorem. Rank and inverse of a matrix. Determinants. Eigenvalues and eigenvectors. Diagonalization of matrices. Cayley-Hamilton theorem. Symmetric, skew-symmetric, and orthogonal matrices.

Interior points and limit points. Open and closed sets. Bounded, connected, and compact sets. Heine-Borel theorem. Sequences and series of functions. Pointwise and uniform convergence. M-test. Properties of uniformly convergent sequences and series.

Calculus of single and multiple variables. Partial derivatives. Jacobian. Imperfect and perfect differentials. Taylor expansion. Fourier series. Vector algebra and vector calculus. Multiple integrals. Divergence theorem. Green’s theorem. Stokes’ theorem. First-order equations and linear second-order ODEs with constant coefficients. Special functions: Hermite, Bessel, Laguerre, and Legendre functions.

Newton’s laws of motion and their applications. Projectile and circular motion. Galilean invariance. Non-inertial frames and pseudo forces. Work-energy theorem. Conservative and non-conservative forces. Potential energy. System of particles — centre of mass and equations of motion. Conservation of linear and angular momenta. Rigid body motion. Moment of inertia tensor. Euler’s equations. Elastic and inelastic collisions. Special theory of relativity: Lorentz transformations, time dilation, length contraction, relativistic velocity addition, mass-energy equivalence.

Differential equation of SHM. Damped and forced simple harmonic oscillators. Energy of SHM and resonance. Wave equation. Travelling and standing waves in 1D. Energy density and energy transmission. Group velocity and phase velocity. Sound waves and Doppler effect. Fermat’s principle and image formation. Optical instruments. Interference: Young’s double-slit experiment, Newton’s rings. Diffraction: Fresnel and Fraunhofer types. Polarization: plane, circular, and elliptic polarization.

Coulomb’s law and Gauss’s law. Electric field and electric potential. Electrostatic boundary conditions. Solution of Laplace’s equation for simple cases. Capacitors. Dielectrics. Lorentz force. Biot-Savart’s law. Ampere’s law. Faraday’s law of electromagnetic induction. Self and mutual inductance. Maxwell’s equations. Electromagnetic waves in free space. Poynting’s theorem and energy transport.

Elements of kinetic theory of gases. Velocity distribution and equipartition of energy. Specific heat of mono-, di-, and tri-atomic gases. Ideal gas and van der Waals gas. Maxwell-Boltzmann distribution. Transport phenomena: viscosity and thermal conductivity. Zeroth, first, second, and third laws of thermodynamics. Thermodynamic potentials. Maxwell’s thermodynamic relations. Phase transitions and phase diagrams. Clausius-Clapeyron equation. Heat engines. Carnot engine and entropy.

Blackbody radiation. Photoelectric effect. De Broglie wavelength. Bohr’s model of the hydrogen atom. X-rays. Nuclear models: liquid drop model and nuclear shell model. Radioactivity — alpha, beta, and gamma decay. Fission and fusion. Nuclear reactions and Q-value. Elementary particles. Quarks and the Standard Model of particle physics.

Crystal structure and X-ray diffraction in solids. Bonding in solids. Lattice vibrations and phonons. Free electron theory. Band theory of solids. Semiconductor devices: p-n junction diodes and transistors. Logic circuits: AND, OR, NOT, NAND, NOR. Operational amplifiers and their basic applications.

Convergent sequences, bounded and monotone sequences. Convergence criteria. Cauchy sequences and subsequences. Series of real numbers: absolute convergence, ratio test, root test, and Leibniz test. Functions of one variable: limit, continuity, differentiation, Rolle’s theorem, mean value theorem, Taylor’s theorem, maxima and minima. Functions of two variables: limit, continuity, partial derivatives, differentiability, maxima and minima, Lagrange multipliers, homogeneous functions.

Integration as the inverse of differentiation. Definite integrals and their properties. Fundamental theorem of integral calculus. Double and triple integrals. Change of order of integration. Surface areas and volumes using integrals.

Rank, inverse, and determinant of a matrix. Systems of linear equations — existence and nature of solutions. Eigenvalues and eigenvectors. Cayley-Hamilton theorem. Symmetric, skew-symmetric, and orthogonal matrices.

Ordinary differential equations of first and second order. Homogeneous and non-homogeneous equations. Bernoulli’s equation. Exact equations and integrating factors. Method of variation of parameters.

Axiomatic definition of probability and its properties. Conditional probability and multiplication rule. Theorem of total probability. Bayes’ theorem. Independence of events.

Probability mass functions (pmf) and probability density functions (pdf). Cumulative distribution functions (CDF) and their properties. Mathematical expectation, variance, and moments. Moment-generating functions (MGF). Characteristic functions.

Discrete distributions: Binomial, Negative Binomial, Geometric, Poisson, Hypergeometric. Continuous distributions: Uniform, Exponential, Gamma, Beta, Normal. Poisson approximation to Binomial. Normal approximation to Binomial. Lognormal distribution.

Joint pmf and pdf. Marginal and conditional distributions. Independence of random variables. Covariance and correlation. Functions of random variables and their distributions. Distribution of order statistics. Sampling distributions: chi-squared, t, and F distributions and their properties.

Unbiasedness, consistency, sufficiency, and completeness. Uniformly minimum variance unbiased (UMVU) estimators. Rao-Blackwell theorem. Lehmann-Scheffé theorem. Cramér-Rao inequality. Methods of estimation: method of moments, maximum likelihood estimation, and least squares. Confidence intervals for normal population parameters.

Statistical hypotheses: simple and composite. Type I and Type II errors. Level of significance and power of a test. Neyman-Pearson lemma. Uniformly most powerful (UMP) tests. Likelihood ratio tests. Tests for one-sample and two-sample problems based on normal, chi-squared, t, and F distributions. Chi-squared goodness-of-fit test. Tests for independence in contingency tables.

Taxonomy of plants and animals. Heredity and genetic variation. Conservation principles. Principles of ecology. Evolutionary theory. Key laboratory techniques: light and electron microscopy, spectroscopy, centrifugation (differential and density gradient), gel electrophoresis (SDS-PAGE, agarose), PCR techniques (RT-PCR, real-time PCR), Southern, Northern, and Western blotting.

Structure and function of carbohydrates, proteins, lipids, and nucleic acids. Enzyme kinetics: Michaelis-Menten equation, competitive and non-competitive inhibition. Glycolysis, TCA cycle, and oxidative phosphorylation. Photosynthesis: light and dark reactions. Plant physiology: transpiration, mineral nutrition, phytohormones. Animal physiology: digestion, circulation, immunity, hormonal regulation, nervous system basics.

DNA structure (Watson-Crick model). DNA replication in prokaryotes and eukaryotes. Transcription and post-transcriptional modifications (capping, polyadenylation, splicing). Translation and the genetic code. Gene regulation: lac operon, trp operon. Mutation types and DNA repair mechanisms. Restriction endonucleases. Cloning vectors: plasmids, bacteriophages, cosmids, BACs, YACs.

Structure and function of prokaryotic and eukaryotic cells. Cell organelles: nucleus, mitochondria, chloroplast, endoplasmic reticulum, Golgi apparatus, lysosomes. Cell membrane structure and transport mechanisms (active, passive, facilitated). Cell cycle regulation: cyclin-CDK complexes, checkpoints. Mitosis and meiosis. Cell signalling: receptor types, second messengers, signal transduction cascades. Apoptosis pathways.

Mendelian inheritance: monohybrid and dihybrid crosses. Linkage, crossing over, and recombination. Chromosome mapping using recombination frequencies. Multiple alleles and codominance. Extrachromosomal inheritance. Sex-linked traits. Principles of population genetics. Hardy-Weinberg equilibrium and its applications in evolutionary biology.

Innate and adaptive immunity. Antigens and antibody structure. Immunoglobulin classes and their functions. Antigen-antibody interactions. Major Histocompatibility Complex (MHC class I and II). Complement system. B and T lymphocyte activation. Vaccines and types. Immunological techniques: ELISA, immunofluorescence, immunoblotting, flow cytometry.

Tools of rDNA technology: restriction enzymes, DNA ligases, and polymerases. Cloning strategies and gene libraries: genomic and cDNA libraries. Expression systems in bacteria, yeast, and mammalian cells. PCR and its variants (RT-PCR, real-time PCR, digital PCR). DNA sequencing: Sanger and next-generation sequencing. Transgenic plants and animals. Applications in medicine and agriculture.

Concepts of stress and strain in crustal rocks. Brittle and ductile deformation. Fractures, joints, and faults: types and nomenclature. Folds: geometry, classification, and mechanisms. Geological maps: symbols, strike, dip, reading geological sections. Stereographic projection and its applications in structural analysis. Unconformities: types and significance. Time-stratigraphic concepts.

Crystal symmetry and crystal systems. Physical properties of minerals: hardness (Mohs scale), cleavage, fracture, lustre, colour, streak, specific gravity. Classification of minerals: silicates (nesosilicates, sorosilicates, cyclosilicates, inosilicates, phyllosilicates, tectosilicates), oxides, sulphides, carbonates, sulphates, halides. Optical properties in thin section: refractive index, birefringence, extinction angle, pleochroism. Identification of common rock-forming minerals in hand specimen and thin section.

Igneous rocks: classification by texture and composition. Bowen’s reaction series. CIPW norm calculations. Magmatic differentiation and assimilation. Metamorphic rocks: types (contact, regional, dynamic). Metamorphic zones and facies. P-T conditions of metamorphism. Sedimentary rocks: classification, textures, sedimentary structures, and environments of deposition. Diagenesis and lithification processes.

Consumer theory: preferences, utility functions, budget constraints, demand functions. Slutsky equation. Revealed preference theory. Producer theory: production functions (Cobb-Douglas, CES), cost curves, profit maximisation. Market structures: perfect competition, monopoly, monopolistic competition, oligopoly. Game theory: Nash equilibrium, dominant strategies, prisoner’s dilemma. General equilibrium: Walras law. Welfare economics: Pareto efficiency. Market failure: externalities, public goods, information asymmetry, adverse selection, moral hazard.

National income accounting: GDP, GNP, NNP, and measurement methods. IS-LM model. Aggregate demand and supply framework. Classical versus Keynesian models. Money creation and banking. Monetary policy and fiscal policy tools. Inflation: types, causes, and measures. Unemployment: types and theories. Phillips curve: short-run and long-run. Economic growth models: Harrod-Domar and Solow. Open economy macroeconomics: balance of payments, exchange rate systems.