CUET PG Syllabus 2027: Subject-Wise Topics, Topic-Wise Weightage and PDF Download

Group Theory: Groups, subgroups, cyclic groups, permutation groups, Lagrange’s theorem, normal subgroups, quotient groups, group homomorphisms, isomorphism theorems (first, second, third), Sylow’s theorems and their applications. Ring Theory: Rings, subrings, ideals, quotient rings, ring homomorphisms, integral domains, fields, polynomial rings, irreducibility criteria (Eisenstein’s criterion). Linear Algebra: Vector spaces, subspaces, linear independence, basis and dimension, linear transformations, rank-nullity theorem, matrix representation, eigenvalues and eigenvectors, diagonalisation, minimal polynomial, inner product spaces, Gram-Schmidt orthogonalisation, self-adjoint and unitary operators.

Real number system and completeness axiom; sequences and their convergence (limit, Cauchy sequence); series of real numbers and convergence tests (comparison, ratio, root, integral tests); continuity and uniform continuity of functions; differentiability; Rolle’s theorem, Lagrange’s mean value theorem, Cauchy’s mean value theorem; Taylor’s theorem with remainder; Riemann integration (definition, properties, fundamental theorem of calculus); improper integrals; sequences and series of functions; pointwise and uniform convergence; Weierstrass M-test; power series and radius of convergence; term-by-term differentiation and integration.

Complex numbers and arithmetic; modulus, argument, and polar form; complex functions and limits; analytic functions; Cauchy-Riemann equations and harmonic functions; elementary complex functions (exponential, logarithmic, trigonometric); complex integration and contour integrals; Cauchy’s theorem and Cauchy’s integral formula; Taylor series; Laurent series; classification of singularities (removable, pole, essential); residues and residue theorem; evaluation of real integrals using contour integration; conformal mappings and bilinear transformations.

Ordinary Differential Equations (ODEs): First-order ODEs — separable, homogeneous, exact, linear, Bernoulli; second-order linear ODEs with constant and variable coefficients; variation of parameters; power series solutions; Frobenius method; Bessel’s and Legendre’s equations; systems of first-order linear ODEs; Laplace transforms and their use in solving ODEs. Partial Differential Equations (PDEs): Formation and classification; first-order PDEs — Lagrange’s method and Charpit’s method; second-order linear PDEs — parabolic, elliptic, and hyperbolic types; wave equation (D’Alembert’s solution and separation of variables); heat equation; Laplace equation; Fourier series solutions of boundary value problems.

Limits, continuity, and differentiability; applications of single-variable derivatives (maxima, minima, curve sketching, L’Hôpital’s rule); partial derivatives and total derivatives; chain rule for multivariable functions; gradient, divergence, and curl; directional derivatives; maxima and minima of functions of two variables; Lagrange multipliers; double and triple integrals and their evaluation; change of order of integration; Jacobian and change of variables in multiple integrals; line integrals and surface integrals; Green’s theorem; Stokes’ theorem; Gauss’s divergence theorem.

Probability axioms; conditional probability and independence; Bayes’ theorem; random variables (discrete and continuous); probability mass function and probability density function; standard distributions — Binomial, Poisson, Normal, Exponential, Uniform, Gamma, Beta; expectation, variance, and higher moments; moment generating functions and characteristic functions; joint distributions; marginal and conditional distributions; correlation and regression analysis; sampling distributions — t, chi-square, and F distributions; point estimation (MLE, method of moments, unbiasedness, consistency, efficiency); confidence intervals; hypothesis testing (one-sample and two-sample tests for mean and proportion); chi-square goodness-of-fit test and test of independence; analysis of variance (ANOVA).

Types of errors — absolute, relative, round-off, truncation; solution of nonlinear equations — bisection method, Newton-Raphson method, secant method, fixed-point iteration and convergence; interpolation — Newton’s forward and backward difference formulae, Lagrange’s interpolation, Newton’s divided difference formula, cubic spline interpolation; numerical differentiation; numerical integration — trapezoidal rule, Simpson’s 1/3 and 3/8 rules, Gaussian quadrature; numerical solution of ODEs — Euler’s method, modified Euler’s method, Runge-Kutta method (4th order); solution of systems of linear equations — Gauss elimination, Gauss-Jordan elimination, LU decomposition, Gauss-Seidel iterative method.

Statics: Forces and equilibrium — resultant, components, Lami’s theorem; moments and couples; centre of gravity of uniform laminae and solid bodies; laws of friction and applications. Dynamics: Newton’s laws of motion; rectilinear motion and projectile motion; circular motion; simple harmonic motion; work-energy theorem; conservation of linear momentum and energy; angular momentum and torque; moment of inertia and theorems of parallel and perpendicular axes; rotation of a rigid body about a fixed axis.

Vector algebra and vector calculus; gradient, divergence, curl; divergence theorem and Stokes’ theorem; matrices and determinants; linear differential equations; Fourier series; Laplace and Fourier transforms; complex analysis fundamentals; special functions — Legendre polynomials and Bessel functions.

Newton’s laws and conservation laws; Lagrangian mechanics and Euler-Lagrange equations; Hamiltonian mechanics; central force problem and Kepler’s laws; small oscillations; rigid body dynamics; Euler’s equations; moment of inertia tensor; free, damped, and forced oscillations; special theory of relativity basics.

Electrostatics — Coulomb’s law, Gauss’s law, electric potential, Laplace and Poisson equations; magnetostatics — Biot-Savart law, Ampere’s law, magnetic vector potential; electromagnetic induction and Faraday’s law; Maxwell’s equations in integral and differential form; electromagnetic waves, Poynting vector, radiation pressure; reflection, refraction, and polarisation of electromagnetic waves; waveguides and resonant cavities.

Wave-particle duality and de Broglie hypothesis; uncertainty principle; Schrödinger equation (time-dependent and time-independent); particle in a box; harmonic oscillator; step potential and tunnelling; hydrogen atom and energy eigenvalues; orbital and spin angular momentum; addition of angular momenta; time-independent perturbation theory; variational method; WKB approximation; identical particles and Pauli exclusion principle.

Zeroth to third laws of thermodynamics; entropy and its properties; thermodynamic potentials (internal energy, enthalpy, Helmholtz free energy, Gibbs free energy); Maxwell’s thermodynamic relations; phase transitions and Clausius-Clapeyron equation; Maxwell-Boltzmann statistics; Bose-Einstein distribution and applications (blackbody radiation, Bose-Einstein condensation); Fermi-Dirac distribution and applications (free electron gas); partition function and its connection to thermodynamic quantities.

Semiconductor devices — p-n junction, transistors (BJT, FET), logic gates; operational amplifiers — inverting, non-inverting, integrator, differentiator; atomic structure — Bohr model, quantum numbers, spectroscopy, Zeeman and Stark effects; nuclear structure — binding energy, liquid drop model, magic numbers; radioactivity — alpha, beta, and gamma decay laws; nuclear reactions — fission and fusion basics, Q-value.

Laws of thermodynamics; Gibbs free energy and equilibrium; Hess’s law; phase rule; chemical kinetics — rate laws, Arrhenius equation, reaction mechanisms, catalysis; quantum mechanical model of the atom — Schrödinger equation, atomic orbitals, molecular orbital theory, hybridisation; electrochemistry — Nernst equation, EMF calculations, electrolysis; colligative properties; UV-Vis, IR, 1H NMR, and mass spectrometry basics.

IUPAC nomenclature; stereochemistry — chirality, enantiomers, diastereomers, R/S and E/Z configuration, optical activity; inductive and resonance effects on reactivity; reaction mechanisms — SN1, SN2, E1, E2, electrophilic and nucleophilic addition, electrophilic aromatic substitution; functional group chemistry — alkanes, alkenes, alkynes, alcohols, ethers, aldehydes, ketones, carboxylic acids, amines, arenes; important named reactions — Aldol, Cannizzaro, Diels-Alder, Grignard, Wittig, Beckmann rearrangement, Baeyer-Villiger; spectral interpretation — key IR frequencies, 1H NMR chemical shifts and coupling constants, mass fragmentation patterns.

Periodic trends — ionisation energy, electron affinity, electronegativity, atomic and ionic radii; acid-base theories — Lewis, Brønsted-Lowry, HSAB principle; coordination compounds — Werner’s theory, crystal field theory, spectrochemical series, d-d transitions, colour and magnetic properties, isomerism; organometallic compounds — 18-electron rule, oxidative addition, reductive elimination, catalytic cycles; bioinorganic chemistry — haemoglobin, chlorophyll, nitrogenase, vitamin B12; qualitative and quantitative analytical techniques.