### IIT-JAM Maths Syllabus

**Sequences and Series of Real Numbers**Sequence of real numbers, convergence of sequences, bounded and monotone sequences, convergence criteria for sequences of real numbers, Cauchy Sequences, subsequences, Bolzano-Weierstrass theorem. Series of real numbers, absolute convergence, tests of convergence for series of positive terms – comparison test, ratio test, root test, Leibniz test for convergence of alternating series.*:***Functions of One Real Variable**: Limit, continuity, intermediate value property, differentiation, Rolle’s Theorem, mean value theorem, L’Hospital rule, Taylor’s theorem, maxima and minima.**Functions of Two or Three Real Variables**: Limit, continuity, partial derivatives, differentiability, maxima and minima.**Integral Calculus**: Integration as the inverse process of differentiation, definite integrals and their properties, fundamental theorem of 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**: Ordinary differential equations of the first order of the form y’=f(x, y), Bernoulli’s equation, exact differential equations, integrating factor, orthogonal trajectories, homogeneous differential equations, variable separable equations, linear differential equations of second order with constant coefficients, method of variation of parameters, Cauchy-Euler equation.**Vector Calculus**: Scalar and vector fields, gradient, divergence, curl, line integrals, surface integrals, Green, Stokes and Gauss theorems.**Group Theory**: Groups, subgroups, Abelian groups, non-Abelian groups, cyclic groups, permutation groups, normal subgroups, Lagrange’s Theorem for finite groups, group homomorphism’s and basic concepts of quotient groups.**Linear Algebra**: Finite dimensional vector spaces, linear independence of vectors, basis, dimension, linear transformations, matrix representation, range space, null space, rank-nullity theorem. Rank and inverse of a matrix, determinant, solutions of systems of linear equations, consistency conditions, eigen values and eigen vectors for matrices, Cayley-Hamilton theorem.**Real Analysis**: Interior points, limit points, open sets, closed sets, bounded sets, connected sets, compact sets, completeness of R. Power series (of real variable), Taylor,s series, radius and interval of convergence, term-wise differentiation and integration of power series.

### TIFR (For Int PhD Mumbai/Bengaluru)

**Algebra:**Definitions and examples of group (finite and infinite, commutative and non-commutative), cyclic groups, subgroups, homo-morphisms, quotients. Definitions and example of rings and fields. Basic facts about finite dimensional vector spaces, matrices, determinants, ranks of linear transformations, characteristic and minimal polynomials, symmetric matrices. Integers and their basic properties. Polynomials with real or complex coefficients in one variable.**Analysis:**Basic facts about real and complex numbers, convergence of sequences and series of real and complex numbers, continuity, differentiation and Riemann integration of real valued functions defined on an interval (finite or infinite), elementary functions (polynomial functions, rational functions, exponential and log, trigonometric functions).

### IIT-JAM Physics Syllabus

**Mathematical Methods**:**Mechanics and General Properties of Matter**:**Oscillations, Waves and Optics**:**Electricity and Magnetism**:**Kinetic theory, Thermodynamics**:**Modern Physics**:**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.