CSIR NET Mathematics December 2025 Syllabus with Complete Unit-Wise Topics
CSIR NET syllabus 2025 covers a wide range of topics essential for advanced mathematical research and teaching. One of the smartest strategies to crack the CSIR NET Mathematics Exam is to focus on topics that carry the most weight. Understanding topic-wise weightage helps you prioritize your study plan, maximize your score, and manage your time effectively. To get a good score in the exam, candidates need to study and learn all the topics in the course thoroughly.
Overview of the CSIR NET 2025 Exam Structure
The CSIR NET Maths Exam for Junior Research Fellowship and Lecturership Eligibility is a single paper test with multiple choice questions (MCQs). The exam can be divided into three parts.
1. Part A: This section will have 20 questions
covering General Science, Quantitative Reasoning & Analysis, and Research
Aptitude. Candidates must answer any 15 questions. Each question will be worth
two marks. This segment is of 30 marks out of 200.
2. Part B: This part will have 40 Multiple Choice
Questions (MCQs) generally covering the topics given in the syllabus.
Candidates must answer any 25 questions. Each question shall be of three marks.
The total marks allocated to this section shall be 75 out of 200.
3. Part C: Part C is the most challenging section and includes high-level, analytical, and research-oriented questions. The questions shall be of an analytical nature where a candidate is expected to apply scientific knowledge to solve the given scientific problem. A candidate shall be required to answer any 20 questions. Each question shall be 4.75 marks. The total marks allocated to this section shall be 95 out of 200.
CSIR NET
Mathematics December 2025 Syllabus
The Syllabus for CSIR NET Mathematics December 2025 remains aligned with previous years, but the trend shows a growing emphasis on depth of understanding and analytical thinking. Below is the refined, unit-wise breakdown.
1. Analysis
a) Real Analysis:
Real Analysis is one of the most important sections of the syllabus and also
one of the most scoring if prepared correctly. You will cover:
● Sequences, subsequences, and series
● Convergence tests and limit properties
● Continuity and uniform continuity
● Differentiability, Mean Value Theorem
● Riemann integration and improper integrals
● Sequences of functions and uniform convergence
Real Analysis questions often appear in both Parts B and C, especially those on convergence and integration.
b) Complex Analysis:
This unit blends beauty and logic. Topics include:
● Analytic and harmonic functions
● Cauchy-Riemann equations
● Contour integration
● Cauchy's theorem and Cauchy's integral formula
● Taylor and Laurent series
● Residue theorem and its applications
Expect tough Part C questions involving residues, contour evaluation, or tricky series expansions.
2. Linear Algebra
Linear Algebra carries weight, reliability, and high scoring potential. The important areas are:
● Vector spaces, subspaces
● Basis, dimension, linear independence
● Linear transformations
● Matrices and determinants
● Eigenvalues and eigenvectors
● Cayley-Hamilton theorem
● Diagonalization and minimal polynomials
● Inner product spaces and orthogonality
3. Algebra (Abstract Algebra)
Abstract Algebra requires theoretical understanding but pays off well if you focus on core theorems. Topics include:
● Groups and subgroups
● Cyclic groups, permutation groups
● Cosets and Lagrange's theorem
● Normal subgroups and quotient groups
● Homomorphisms and isomorphism theorems
● Rings, ideals, ring homomorphisms
● Fields and field extensions
Group theory and ring theory problems often show up in conceptual Part C questions.
4. Ordinary Differential Equations (ODE)
This unit is fundamental for applied mathematics. Key topics:
● First-order differential equations
● Linear differential equations
● Power series solutions
● Systems of differential equations
● Sturm-Liouville boundary value problems
● Existence and uniqueness theorem
5. Partial Differential Equations (PDE)
A Partial
Differential Equation, commonly denoted as PDE, is a differential equation
containing partial derivatives of the dependent variable (one or more) with
more than one independent variable.
● First-order PDE
● Classification: hyperbolic, parabolic, elliptic
● Method of characteristics
● Heat, wave, and Laplace equations
● Fourier series and boundary value techniques
This unit rewards strong conceptual clarity and practice with classical PDE methods.
6. Numerical Analysis
Numerical Analysis offers some scoring opportunities if you understand the procedures clearly. Topics include:
● Error analysis
● Iterative solutions of equations
● Interpolation
● Numerical differentiation and integration
● Numerical solutions of ODEs and PDEs
● Polynomial approximations
7. Integral Equations & Calculus of Variations
a) Integral Equations
● Volterra and Fredholm equations
● Kernels, resolvent kernels
● Eigenfunction expansions
b) Calculus of Variations
● Functionals
● Euler-Lagrange equations
● Isoperimetric problems
Though the weightage fluctuates, these topics often appear in Part B.
8. Topology
Topology questions are tricky but usually direct. Topics include:
● Topological spaces
● Bases, subbases
● Continuity
● Compactness and connectedness
● Metric spaces
Understanding
definitions deeply will serve you well.
9. Probability & Statistics
● Probability spaces
● Random variables
● Expectation, variance
● Distributions
● Joint distributions
● Law of large numbers
● Central limit theorem
This unit typically contains a mix of direct and analytical questions.
Knowing the syllabus is not enough, you need a clear preparation strategy. Start by identifying your strongest and weakest areas. Build a schedule that allows slow, deep learning in the beginning and high intensity problem practice as the exam date approaches. Solve past papers, take mock tests, revise short notes weekly, and keep your fundamentals clean.
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