CNET (CSIR-UGC NET)
- 1
- Year
- 2
- Papers
- 240
- Questions
For JRF and Assistant Professor eligibility. The Mathematical Sciences paper covers analysis, algebra, complex analysis, topology, differential equations, numerical methods, mechanics and — in Unit 4 — statistics and operations research.
Exam pattern
- Part A: general aptitude
- Part B and Part C: Mathematical Sciences (Units 1–4, Unit 4 is Statistics)
- Part C questions may have more than one correct option
The most recent paper solved here, CNET July 2025 Mathematical Sciences, ran to 120 questions: 20 in Section A (part a), 40 in Section B (part b) and 60 in Section C (part c).
Conducted by National Testing Agency (NTA). Patterns change from year to year, so confirm the current one at csirnet.nta.ac.in.
Topic analysis
How many questions each topic was worth, year by year, across the 2 solved CNET papers (240 questions). Sittings of the same year are counted together.
| Subject · topic | 2025 | Total |
|---|---|---|
| General Aptitude | 40 | 40 |
| Quantitative Aptitude | 24 | 24 |
| Logical Reasoning | 10 | 10 |
| Data Interpretation | 6 | 6 |
| Analysis | 33 | 33 |
| Sequences and Series | 10 | 10 |
| Elementary Set Theory, Real Number System, Supremum, Infimum | 7 | 7 |
| Continuity, Uniform Continuity, Differentiability, Mean Value Theorem | 4 | 4 |
| Functions of Several Variables, Inverse & Implicit Function Theorems | 4 | 4 |
| Metric Spaces, Compactness, Connectedness, Normed Linear Spaces | 3 | 3 |
| Riemann Sums and Riemann Integral, Improper Integrals | 3 | 3 |
| Monotonic Functions, Bounded Variation, Lebesgue Measure & Integral | 2 | 2 |
| Linear Algebra | 28 | 28 |
| Eigenvalues and Eigenvectors, Cayley-Hamilton Theorem | 9 | 9 |
| Inner Product Spaces, Orthonormal Basis | 4 | 4 |
| Vector Spaces, Subspaces, Linear Dependence, Basis, Dimension | 4 | 4 |
| Algebra of Linear Transformations | 3 | 3 |
| Algebra of Matrices, Rank, Determinant, Linear Equations | 3 | 3 |
| Quadratic Forms, Reduction and Classification | 3 | 3 |
| Matrix Representation, Change of Basis, Canonical and Jordan Forms | 2 | 2 |
| Algebra | 18 | 18 |
| Groups, Subgroups, Sylow Theorems, Cayley's Theorem | 8 | 8 |
| Rings, Ideals, UFD, PID, Euclidean Domain | 4 | 4 |
| Polynomial Rings and Irreducibility Criteria | 3 | 3 |
| Fields, Finite Fields, Field Extensions, Galois Theory | 1 | 1 |
| Fundamental Theorem of Arithmetic, Congruences, Chinese Remainder Theorem | 1 | 1 |
| Permutations, Combinations, Pigeon-Hole Principle, Inclusion-Exclusion | 1 | 1 |
| Statistical Inference | 18 | 18 |
| Methods of Estimation, Properties of Estimators | 7 | 7 |
| Elementary Bayesian Inference | 4 | 4 |
| Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio Tests | 4 | 4 |
| Confidence Intervals | 3 | 3 |
| Complex Analysis | 16 | 16 |
| Liouville's Theorem, Maximum Modulus Principle, Open Mapping Theorem | 5 | 5 |
| Taylor Series, Laurent Series, Calculus of Residues | 5 | 5 |
| Algebra of Complex Numbers, Power Series, Transcendental Functions | 2 | 2 |
| Contour Integral, Cauchy's Theorem, Cauchy's Integral Formula | 2 | 2 |
| Analytic Functions, Cauchy-Riemann Equations | 1 | 1 |
| Conformal Mappings, Mobius Transformations | 1 | 1 |
| Ordinary Differential Equations | 10 | 10 |
| General Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's Function | 7 | 7 |
| Existence & Uniqueness, Singular Solutions, Systems of First Order ODEs | 3 | 3 |
| Partial Differential Equations | 9 | 9 |
| Separation of Variables (Laplace, Heat and Wave Equations) | 5 | 5 |
| Lagrange and Charpit Methods, Cauchy Problem for First Order PDEs | 3 | 3 |
| Classification of Second Order PDEs | 1 | 1 |
| Probability | 9 | 9 |
| Random Variables and Distribution Functions, Expectation and Moments | 4 | 4 |
| Sample Space, Discrete Probability, Independent Events, Bayes Theorem | 2 | 2 |
| Characteristic Functions, Probability Inequalities | 1 | 1 |
| Modes of Convergence, Laws of Large Numbers, Central Limit Theorem | 1 | 1 |
| Modes of Convergence, Laws of Large Numbers, Central Limit Theorems | 1 | 1 |
| Numerical Analysis | 6 | 6 |
| Numerical Solutions of Algebraic Equations, Newton-Raphson, Rate of Convergence | 2 | 2 |
| Finite Differences | 1 | 1 |
| Finite Differences, Lagrange/Hermite/Spline Interpolation | 1 | 1 |
| Numerical Differentiation and Integration | 1 | 1 |
| Numerical Solutions of ODEs (Picard, Euler, Runge-Kutta) | 1 | 1 |
| Calculus of Variations | 6 | 6 |
| Variation of a Functional, Euler-Lagrange Equation, Conditions for Extrema | 5 | 5 |
| Variational Methods for Boundary Value Problems | 1 | 1 |
| Linear Integral Equations | 6 | 6 |
| Fredholm and Volterra Type, Separable Kernels, Resolvent Kernel | 6 | 6 |
| Linear Models & Regression | 6 | 6 |
| Gauss-Markov Models, Estimability, Best Linear Unbiased Estimators | 2 | 2 |
| Simple and Multiple Linear Regression, Logistic Regression | 2 | 2 |
| Analysis of Variance and Covariance, Fixed/Random/Mixed Effects | 1 | 1 |
| Gauss-Markov Models, Estimability, BLUE, Tests for Linear Hypotheses | 1 | 1 |
| Topology | 5 | 5 |
| Separation Axioms, Connectedness and Compactness | 3 | 3 |
| Basis, Dense Sets, Subspace and Product Topology | 2 | 2 |
| Multivariate Analysis | 5 | 5 |
| Partial and Multiple Correlation Coefficients | 2 | 2 |
| Data Reduction: Principal Component Analysis, Discriminant, Cluster, Canonical Correlation | 1 | 1 |
| Multivariate Normal Distribution, Wishart Distribution, Distribution of Quadratic Forms | 1 | 1 |
| Multivariate Normal Distribution, Wishart Distribution, Quadratic Forms | 1 | 1 |
| Stochastic Processes | 4 | 4 |
| Markov Chains: Classification of States, Stationary Distribution | 4 | 4 |
| Distribution Theory | 4 | 4 |
| Standard Discrete and Continuous Univariate Distributions | 2 | 2 |
| Distribution of Order Statistics and Range | 1 | 1 |
| Sampling Distributions, Standard Errors, Asymptotic Distributions | 1 | 1 |
| Classical Mechanics | 3 | 3 |
| Hamilton's Canonical Equations, Hamilton's Principle | 2 | 2 |
| Generalized Coordinates, Lagrange's Equations | 1 | 1 |
| Operations Research | 3 | 3 |
| Linear Programming Problem, Simplex Method, Duality | 2 | 2 |
| Steady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1) | 1 | 1 |
| Descriptive Statistics | 2 | 2 |
| Descriptive Statistics | 1 | 1 |
| Descriptive Statistics, Exploratory Data Analysis | 1 | 1 |
| Sampling Theory | 2 | 2 |
| Simple Random, Stratified and Systematic Sampling | 2 | 2 |
| Design of Experiments | 2 | 2 |
| 2^K Factorial Experiments: Confounding and Construction | 1 | 1 |
| Completely Randomized, Randomized Block and Latin-Square Designs; Connectedness, Orthogonality, BIBD | 1 | 1 |
| Nonparametric Inference | 1 | 1 |
| Simple Nonparametric Tests for One and Two Sample Problems | 1 | 1 |
| Reliability | 1 | 1 |
| Hazard Function and Failure Rates, Censoring and Life Testing, Series and Parallel Systems | 1 | 1 |
| Queueing Theory | 1 | 1 |
| Steady-State Solutions of Markovian Queuing Models: M/M/1, M/M/C, M/G/1 | 1 | 1 |
| Non-parametric Inference | 1 | 1 |
| Rank and Sign Tests, Run Test | 1 | 1 |
| Survival Analysis | 1 | 1 |
| Series and Parallel Systems | 1 | 1 |
| All topics | 240 | 240 |
Syllabus
From the National Testing Agency (NTA) Mathematical Sciences syllabus. Taken from the MSET Mathematical Sciences syllabus, which states that it follows the CSIR-UGC NET syllabus. Confirm against the current CSIR-UGC NET notification. Always check it against the current notification.
Unit 1
Analysis
- Elementary Set Theory, Real Number System, Supremum, InfimumElementary set theory, finite, countable and uncountable sets · Real number system as a complete ordered field · Archimedean property, supremum, infimum
- Sequences and SeriesSequences and series, convergence, limsup, liminf · Bolzano Weierstrass theorem, Heine Borel theorem · Sequences and series of functions, uniform convergence
- Continuity, Uniform Continuity, Differentiability, Mean Value TheoremContinuity, uniform continuity · Differentiability, mean value theorem
- Riemann Sums and Riemann Integral, Improper IntegralsRiemann sums and Riemann integral · Improper Integrals
- Monotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMonotonic functions, types of discontinuity · Functions of bounded variation · Lebesgue measure, Lebesgue integral
- Functions of Several Variables, Inverse & Implicit Function TheoremsFunctions of several variables, directional derivative, partial derivative · Derivative as a linear transformation · Inverse and implicit function theorems
- Metric Spaces, Compactness, Connectedness, Normed Linear SpacesMetric spaces, compactness, connectedness · Normed linear Spaces · Spaces of continuous functions as examples
Linear Algebra
- Vector Spaces, Subspaces, Linear Dependence, Basis, DimensionVector spaces, subspaces, linear dependence · Basis, dimension
- Algebra of Linear TransformationsAlgebra of linear transformations
- Algebra of Matrices, Rank, Determinant, Linear EquationsAlgebra of matrices · Rank and determinant of matrices · Linear equations
- Eigenvalues and Eigenvectors, Cayley-Hamilton TheoremEigenvalues and eigenvectors · Cayley-Hamilton theorem
- Matrix Representation, Change of Basis, Canonical and Jordan FormsMatrix representation of linear transformations · Change of basis, canonical forms, diagonal forms, triangular forms · Jordan forms
- Inner Product Spaces, Orthonormal BasisInner product spaces, orthonormal basis
- Quadratic Forms, Reduction and ClassificationQuadratic forms, reduction and classification of quadratic forms
Unit 2
Complex Analysis
- Algebra of Complex Numbers, Power Series, Transcendental FunctionsAlgebra of complex numbers, the complex plane · Polynomials, power series · Transcendental functions such as exponential, trigonometric and hyperbolic functions
- Analytic Functions, Cauchy-Riemann EquationsAnalytic functions, Cauchy-Riemann equations
- Contour Integral, Cauchy's Theorem, Cauchy's Integral FormulaContour integral, Cauchy's theorem · Cauchy's integral formula
- Liouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremLiouville's theorem · Maximum modulus principle, Schwarz lemma · Open mapping theorem
- Taylor Series, Laurent Series, Calculus of ResiduesTaylor series, Laurent series · Calculus of residues
- Conformal Mappings, Mobius TransformationsConformal mappings, Mobius transformations
Algebra
- Permutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionPermutations, combinations · Pigeon-hole principle, inclusion-exclusion principle, derangements
- Fundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremFundamental theorem of arithmetic, divisibility in Z · Congruences, Chinese Remainder Theorem · Euler's phi-function, primitive roots
- Groups, Subgroups, Sylow Theorems, Cayley's TheoremGroups, subgroups, normal subgroups, quotient groups, homomorphisms · Cyclic groups, permutation groups, Cayley's theorem · Class equations, Sylow theorems
- Rings, Ideals, UFD, PID, Euclidean DomainRings, ideals, prime and maximal ideals, quotient rings · Unique factorization domain, principal ideal domain, Euclidean domain
- Polynomial Rings and Irreducibility CriteriaPolynomial rings and irreducibility criteria
- Fields, Finite Fields, Field Extensions, Galois TheoryFields, finite fields, field extensions · Galois Theory
- Lattice Theory (Distributive/Modular/Boolean Lattices) asked, not named in the syllabusLattices and Boolean algebras
Topology
- Basis, Dense Sets, Subspace and Product TopologyBasis, dense sets · Subspace and product topology
- Separation Axioms, Connectedness and CompactnessSeparation axioms · Connectedness and compactness
Unit 3
Ordinary Differential Equations
- Existence & Uniqueness, Singular Solutions, Systems of First Order ODEsExistence 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 Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionGeneral theory of homogenous and non-homogeneous linear ODEs · Variation of parameters · Sturm-Liouville boundary value problem, Green's function
Partial Differential Equations
- Lagrange and Charpit Methods, Cauchy Problem for First Order PDEsLagrange and Charpit methods for solving first order PDEs · Cauchy problem for first order PDEs
- Classification of Second Order PDEsClassification of second order PDEs · General solution of higher order PDEs with constant coefficients
- Separation of Variables (Laplace, Heat and Wave Equations)Method of separation of variables for Laplace, Heat and Wave equations
Numerical Analysis
- Numerical Solutions of Algebraic Equations, Newton-Raphson, Rate of ConvergenceNumerical solutions of algebraic equations · Method of iteration and Newton-Raphson method · Rate of convergence
- Solution of Linear Systems: Gauss Elimination and Gauss-SeidelSolution of systems of linear algebraic equations using Gauss elimination and Gauss-Seidel methods
- Finite Differences, Lagrange/Hermite/Spline InterpolationFinite differences · Lagrange, Hermite and spline interpolation
- Numerical Differentiation and IntegrationNumerical differentiation and integration
- Numerical Solutions of ODEs (Picard, Euler, Runge-Kutta)Numerical solutions of ODEs using Picard, Euler, modified Euler and Runge-Kutta methods
Calculus of Variations
- Variation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaVariation of a functional, Euler-Lagrange equation · Necessary and sufficient conditions for extrema
- Variational Methods for Boundary Value ProblemsVariational methods for boundary value problems in ordinary and partial differential equations
Linear Integral Equations
- Fredholm and Volterra Type, Separable Kernels, Resolvent KernelLinear integral equation of the first and second kind of Fredholm and Volterra type · Solutions with separable kernels · Characteristic numbers and eigenfunctions, resolvent kernel
Classical Mechanics
- Generalized Coordinates, Lagrange's EquationsGeneralized coordinates, Lagrange's equations
- Hamilton's Canonical Equations, Hamilton's PrincipleHamilton's canonical equations · Hamilton's principle and principle of least action
- Motion of Rigid Bodies, Euler's Dynamical Equations, Small OscillationsTwo-dimensional motion of rigid bodies · Euler's dynamical equations for the motion of a rigid body about an axis · Theory of small oscillations
Integral Transforms
- Laplace Transform asked, not named in the syllabusLaplace transform
- Fourier Transform asked, not named in the syllabusFourier transform
Fluid Dynamics
- Potential Flow, Streamlines, Doublets asked, not named in the syllabusPotential flow, streamlines, sources, sinks and doublets
Differential Geometry
- Curves and Surfaces, Curvature asked, not named in the syllabusCurves and surfaces, curvature and torsion
Graph Theory
- Graphs, Trees, Connectivity asked, not named in the syllabusGraphs, trees, connectivity and colouring
Unit 4
Descriptive Statistics
- Descriptive Statistics, Exploratory Data AnalysisDescriptive statistics · Exploratory data analysis
Probability
- Sample Space, Discrete Probability, Independent Events, Bayes TheoremSample space, discrete probability · Independent events · Bayes theorem
- Random Variables and Distribution Functions, Expectation and MomentsRandom variables and distribution functions (univariate and multivariate) · Expectation and moments · Independent random variables, marginal and conditional distributions
- Characteristic Functions, Probability InequalitiesCharacteristic functions · Probability inequalities (Tchebyshef, Markov, Jensen)
- Modes of Convergence, Laws of Large Numbers, Central Limit TheoremModes of convergence · Weak and strong laws of large numbers · Central Limit theorems (i.i.d. case)
Stochastic Processes
- Markov Chains: Classification of States, Stationary DistributionMarkov chains with finite and countable state space · Classification of states · Limiting behaviour of n-step transition probabilities · Stationary distribution
- Poisson and Birth-and-Death ProcessesPoisson process · Birth-and-death processes
- Martingales asked, not named in the syllabusMartingales, stopping times and martingale convergence
Distribution Theory
- Standard Discrete and Continuous Univariate DistributionsStandard discrete univariate distributions · Standard continuous univariate distributions
- Sampling Distributions, Standard Errors, Asymptotic DistributionsSampling distributions · Standard errors · Asymptotic distributions
- Distribution of Order Statistics and RangeDistribution of order statistics · Distribution of range
Statistical Inference
- Methods of Estimation, Properties of EstimatorsMethods of estimation · Properties of estimators
- Confidence IntervalsConfidence intervals
- Tests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsTests of hypotheses: most powerful and uniformly most powerful tests · Likelihood ratio tests
- Analysis of Discrete Data and Chi-Square Test of Goodness of FitAnalysis of discrete data · Chi-square test of goodness of fit
- Large Sample TestsLarge sample tests
- Nonparametric Tests, Rank Correlation, Test for IndependenceSimple nonparametric tests for one and two sample problems · Rank correlation and test for independence
- Elementary Bayesian InferenceElementary Bayesian inference
Linear Models & Regression
- Gauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsGauss-Markov models · Estimability of parameters · Best linear unbiased estimators · Confidence intervals and tests for linear hypotheses
- Analysis of Variance and Covariance, Fixed/Random/Mixed EffectsAnalysis of variance and covariance · Fixed, random and mixed effects models
- Simple and Multiple Linear Regression, Logistic RegressionSimple and multiple linear regression · Elementary regression diagnostics · Logistic regression
Multivariate Analysis
- Multivariate Normal Distribution, Wishart Distribution, Quadratic FormsMultivariate normal distribution · Wishart distribution and their properties · Distribution of quadratic forms
- Partial and Multiple Correlation CoefficientsInference for parameters · Partial and multiple correlation coefficients and related tests
- Data Reduction: PCA, Discriminant, Cluster AnalysisPrinciple component analysis · Discriminant analysis · Cluster analysis · Canonical correlation
Sampling Theory
- Simple Random, Stratified and Systematic SamplingSimple random sampling · Stratified sampling · Systematic sampling
- Probability Proportional to Size SamplingProbability proportional to size sampling
- Ratio and Regression MethodsRatio and regression methods
Design of Experiments
- Completely Randomized, Randomized Block and Latin-Square DesignsCompletely randomized designs · Randomized block designs · Latin-square designs
- Connectedness and Orthogonality of Block Designs, BIBDConnectedness and orthogonality of block designs · BIBD
- 2^K Factorial Experiments: Confounding and Construction2K factorial experiments: confounding · 2K factorial experiments: construction
- Response Surface Designs asked, not named in the syllabusResponse surface designs and rotatability
Survival Analysis
- Hazard Function and Failure Rates, Censoring and Life TestingHazard function and failure rates · Censoring and life testing
- Series and Parallel SystemsSeries and parallel systems
Operations Research
- Linear Programming Problem, Simplex Method, DualityLinear programming problem · Simplex methods · Duality
- Elementary Queuing and Inventory ModelsElementary queuing models · Elementary inventory models
- Steady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)M/M/1 · M/M/1 with limited waiting space · M/M/C · M/M/C with limited waiting space · M/G/1
- Transportation and Assignment Problems asked, not named in the syllabusTransportation problem: feasibility, basic feasible solutions and optimality · Assignment problem
- Sequencing and Project Scheduling asked, not named in the syllabusSequencing jobs on machines (Johnson's rule) · Project scheduling: CPM and PERT
- Replacement Models asked, not named in the syllabusIndividual and group replacement policies
- Dynamic and Integer Programming asked, not named in the syllabusDynamic programming and Bellman's principle of optimality · Integer programming and branch and bound
- Game Theory asked, not named in the syllabusTwo-person zero-sum games, saddle points and mixed strategies
Demography / Vital Statistics
- Life Tables, Mortality Rate asked, not named in the syllabusLife tables, mortality and fertility rates
Index Numbers & Time Series
- Time Series Models asked, not named in the syllabusStationarity, autocorrelation and partial autocorrelation · Autoregressive, moving average and ARIMA models · Forecasting from a fitted model
Statistical Quality Control
- Control Charts asked, not named in the syllabusControl charts for variables and attributes · Average run length and chart performance
- Acceptance Sampling asked, not named in the syllabusSampling inspection plans, average outgoing quality and rectifying inspection