| # |
TEST TYPE |
MODULES |
SYLLABUS |
| 1 |
MWT-01 |
Real Analysis I |
Elementary set theory, finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property, supremum, infimum. Sequences and series, convergence, limsup, liminf. Bolzano Weierstrass theorem, Heine Borel theorem.Continuity, uniform continuity & metric spaces (continuity compactness, connectedness.) andTopology |
| 2 |
MWT-02 |
Modern Algebra (Group Theory) |
Permutations, combinations, Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem, Euler’s Ø- function, primitive roots. Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation groups, Cayley’s theorem, class equations, Sylow theorems. |
| 3 |
MWT-03 |
ODE & I.E |
Existence and uniqueness of solutions of initial value problems for first order ordinary differential equations, singular solutions of first order ODEs, system of first order ODEs. General theory of homogenous and non-homogeneous linear ODEs, variation of parameters, Sturm-Liouville boundary value problem, Green’s function. Linear integral equation of the first and second kind of Fredholm and Volterra type, Solutions with separable kernels. Characteristic numbers and eigenfunctions, resolvent kernel. |
| 4 |
MWT-04 |
Modern Algebra (Ring Theory) |
Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal domain, Euclidean domain. Polynomial rings and irreducibility criteria. Fields, finite fields, field extensions, |
| 5 |
MWT-05 |
Real Analysis II |
Differentiability, mean value theorem. Sequences and series of functions, uniform convergence. Riemann sums and Riemann integral, Improper Integrals. Monotonic functions, types of discontinuity, functions of bounded variation, Lebesgue measure, Lebesgue integral. Functions of several variables, directional derivative, partial derivative, derivative as a linear transformation, inverse and implicit function theorems.metric spaces (continuity compactness, connectedness.) and Topology |
| 6 |
MWT-06 |
Linear Algebra (section - I) |
Vector spaces, subspaces, linear dependence, basis, dimension, algebra of linear transformations, matrix representation of linear transformation, Algebra of matrices, rank and determinant of matrices, system of linear equations. |
| 7 |
MWT-07 |
Partial Differential Equation & COV |
Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for first order PDEs. Classification of second order PDEs, General solution of higher order PDEs with constant coefficients, Method of separation of variables for Laplace, Heat and Wave equations. Variation of a functional, Euler-Lagrange equation, Necessary and sufficient conditions for extrema. Variational methods for boundary value problems in ordinary and partial differential equations. |
| 8 |
25 May 2019 |
Linear Algebra (section - II) |
Eigenvalues and eigenvectors, Cayley-Hamilton theorem. Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms. Inner product spaces, orthonormal basis. Quadratic forms, reduction and classification of quadratic forms. |
| 9 |
MWT-09 |
Complex Analysis |
Algebra of complex numbers, the complex plane, polynomials, power series, transcendental functions such as exponential, trigonometric and hyperbolic functions. Analytic functions, Cauchy-Riemann equations. Contour integral, Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, Maximum modulus principle, Schwarz lemma, Open mapping theorem. Taylor series, Laurent series, calculus of residues. Conformal mappings, Mobius transformations. |
| 10 |
FLT-01 |
Full Length Test |
As per Exam Pattern. |
| 11 |
FLT-02 |
Full Length Test |
As per Exam Pattern. |
| 12 |
FLT-03 |
Full Length Test |
As per Exam Pattern. |
| 13 |
FLT-03 |
Full Length Test |
As per Exam Pattern. |