CSIR NET Mathematical Sciences Syllabus 2026: PDF Download, Unit-Wise Topics

Join CSIR NET Online Coaching - Enroll Now!

CSIR NET Mathematical Science Syllabus PDF

CSIR NET Mathematical Science Syllabus PDF

CSIR NET Mathematical Science Exam Pattern & Marking Scheme 2026

The CSIR NET Mathematics examination is conducted as a single-paper Computer-Based Test (CBT) of 3 hours (180 minutes) carrying a maximum of 200 marks. The paper presents 120 objective questions, out of which candidates must attempt 60 questions.

• In Part B, each question has four options with only one correct option. Marking an incorrect response leads to a deduction of 0.75 marks (25%).

• In Part C, each question may have one or more correct options. To earn full credit (+4.75 marks), candidates must identify all correct choices without selecting any incorrect choice. There is no partial credit, and no negative marks are deducted for incorrect responses in Part C.

• Candidates can attempt a maximum of 60 questions across the whole paper. If extra questions are attempted in any part, only the first set of attempted responses up to the allowed limit is evaluated.

• To review marking rules across Chemical, Life, and Physical Science papers, explore the comprehensive CSIR NET Exam Pattern.

CSIR NET Mathematics Syllabus: Unit-Wise Breakdown

The CSIR NET Mathematical Sciences curriculum is formally organized into 4 comprehensive units. Pure and Applied Mathematics candidates generally focus on Units 1, 2, and 3, whereas Statistics postgraduates focus on Unit 1 and Unit 4.

Unit 1: Analysis and Linear Algebra

• Analysis: Elementary set theory, finite, countable, and uncountable sets. Real number system as a complete ordered field, Archimedean property, supremum, and infimum. Sequences and series, convergence, limsup, liminf, Bolzano-Weierstrass theorem, and Heine-Borel theorem. Continuity, uniform continuity, differentiability, mean value theorem, and sequences/series of functions with pointwise and uniform convergence. Riemann sums and Riemann-Stieltjes integrals, improper integrals, and functions of several variables (directional derivatives, partial derivatives, Jacobian, and inverse function theorem). Metric spaces: compactness, connectedness, completeness, and Arzela-Ascoli theorem.

• Linear Algebra: Vector spaces, subspaces, linear dependence and independence, basis, dimension, and coordinates. Algebra of linear transformations, rank-nullity theorem, matrix representations of linear maps, and change of basis. Linear equations, determinants, eigenvalues, and eigenvectors. Cayley-Hamilton theorem, minimal polynomials, diagonalization, and canonical forms (triangular, Jordan, and rational canonical forms). Inner product spaces, orthonormal bases, Gram-Schmidt orthogonalization, and quadratic forms (classification, signature, and reduction).

Unit 2: Complex Analysis, Algebra, and Topology

• Complex Analysis: Algebra of complex numbers, complex plane, polynomials, and power series. Analytic functions, Cauchy-Riemann equations, and harmonic functions. Contour integrals, Cauchy theorem, Cauchy integral formula, Liouville theorem, and Morera theorem. Taylor and Laurent series expansions, singularities, and calculus of residues. Maximum modulus principle, Schwarz lemma, Rouche theorem, and conformal mapping (Möbius transformations).

• Algebra: Permutations, combinations, pigeonhole principle, inclusion-exclusion principle, and derangements. Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese remainder theorem, and Euler's phi function. Group theory: groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation groups, Cayley theorem, class equation, and Sylow theorems. Ring theory: rings, ideals, prime and maximal ideals, quotient rings, integral domains, Euclidean domains, PIDs, and UFDs. Polynomial rings, irreducibility criteria (Eisenstein criterion), field extensions, finite fields, and Galois theory basics.

• Topology: Basis, subbasis, dense sets, subspace topology, product topology, and quotient topology. Separation axioms (T0,T1,T2 Hausdorff spaces), connectedness, path-connectedness, and compactness.

Unit 3: Ordinary Differential Equations, Partial Differential Equations, and Applied Mathematics

• Ordinary Differential Equations (ODEs): First-order ODEs, existence and uniqueness theorems (Picard theorem), and singular solutions. Linear differential equations of higher order with constant coefficients, method of variation of parameters, Cauchy-Euler equations, and system of first-order linear equations. Sturm-Liouville boundary value problems and Green functions.

• Partial Differential Equations (PDEs): First-order linear and quasi-linear PDEs, method of characteristics, and Cauchy problems. Classification of second-order linear PDEs (elliptic, parabolic, hyperbolic), canonical forms, and method of separation of variables for Laplace, heat, and wave equations in Cartesian and polar coordinates.

• Numerical Analysis: Numerical solutions of algebraic and transcendental equations (Bisection, Regula-Falsi, Newton-Raphson, and Secant methods). Systems of linear equations (Gaussian elimination, LU decomposition, Gauss-Seidel, and Jacobi iteration). Interpolation (Lagrange, Newton divided differences, and Hermite). Numerical integration (Trapezoidal rule, Simpson's 1/3 and 3/8 rules) and numerical solutions of ODEs (Euler, Runge-Kutta 4th order, and Predictor-Corrector methods).

• Calculus of Variations: Functionals, Euler-Lagrange equations, geodesics, variational problems with moving boundaries, isoperimetric problems, and Rayleigh-Ritz method.

• Linear Integral Equations: Fredholm and Volterra integral equations of the first and second kinds, equations with separable kernels, Neumann series, resolvent kernels, and characteristic numbers/eigenfunctions.

• Classical Mechanics: Generalized coordinates, D'Alembert principle, Lagrange equations of motion, Hamilton canonical equations, Poisson brackets, canonical transformations, Hamilton-Jacobi theory, and small oscillations (normal modes).

Unit 4: Descriptive Statistics, Probability, and Applied Statistics

• Descriptive Statistics & Probability: Sample space, discrete and continuous random variables, probability mass/density functions, cumulative distribution functions, and joint/marginal/conditional distributions. Expectation, moments, Chebyshev inequality, law of large numbers, and central limit theorem. Standard distributions: Binomial, Poisson, Normal, Gamma, Beta, Cauchy, and Student's t.

• Markov Chains: Markov chains with discrete and continuous time parameters, transition probability matrices, classification of states, periodicity, stationary distributions, and Poisson/birth-death processes.

• Statistical Inference: Estimation theory: sufficiency, completeness, consistency, unbiasedness (UMVUE), Rao-Blackwell and Lehmann-Scheffe theorems, method of moments, and maximum likelihood estimation (MLE). Hypothesis testing: Neyman-Pearson lemma, likelihood ratio tests, confidence intervals, and non-parametric tests (Sign test, Wilcoxon signed-rank test, Mann-Whitney U test).

• Linear Models & Design of Experiments: Gauss-Markov theorem, linear regression (simple and multiple), analysis of variance (ANOVA for one-way and two-way layouts), completely randomized designs (CRD), randomized block designs (RBD), and Latin square designs (LSD).

• Multivariate Analysis & Sampling: Multivariate normal distribution, Wishart distribution, Hotelling's T2, and principal component analysis (PCA). Sampling techniques: simple random sampling (SRSWR and SRSWOR), stratified sampling, systematic sampling, and cluster sampling.

• Operations Research & Reliability: Linear programming problems (simplex method, duality, and sensitivity analysis), queuing models (M/M/1, M/M/c), hazard rates, failure time distributions, and series/parallel system reliability.

Part A General Aptitude Syllabus for Mathematics Aspirants

Part A is common to all five CSIR NET papers and tests core analytical aptitude:

• Numerical Ability: Number series, ratios, percentages, averages, work and time, speed and distance, and basic combinatorics.

• Logical Reasoning: Deductions, coding-decoding, Venn diagrams, syllogisms, and direction sense.

• Graphical Analysis: Coordinate graphs, line charts, pie charts, bar diagrams, and tabular data interpretation.

High-Weightage Mathematics Topics for Part C Scoring

Part C carries 95 marks through 20 attempted questions (each question carries 4.75 marks with zero negative marking). Candidates should prioritize these high-yield topics to secure a JRF qualifying rank:

1. Linear Algebra (Unit 1): Matrix diagonalization, minimal polynomials, Jordan canonical forms, and quadratic form signatures. This area offers direct scoring consistency.

2. Real Analysis (Unit 1): Uniform convergence of function sequences, metric space compactness, and multivariable differentiability.

3. Complex Analysis (Unit 2): Residue calculus, contour integrals, singularity classification, and Liouville/Rouche theorems.

4. Abstract Algebra (Unit 2): Sylow theorems, class equations, maximal/prime ideals, and irreducible polynomial factorizations.

5. Applied Mathematics (Unit 3): Boundary value problems (Sturm-Liouville), method of characteristics for PDEs, separable kernel integral equations, and extremal calculus of variations.

Join PW CSIR NET Online Coaching - Enroll Now!