CSIR NET Mathematical Sciences Most Repeated Questions with Topic-wise Analysis

The CSIR NET June 2026 Mathematical Sciences examination is scheduled for 17 July 2026 from 03:00 PM to 06:00 PM. With the exam just around the corner, this is the ideal time to prioritise high-weightage questions and frequently repeated concepts instead of trying to revise the entire syllabus. 

An analysis of six previous year question (PYQ) papers (June 2023 to June 2025) reveals that Real Analysis, Linear Algebra, Complex Analysis, and Abstract Algebra carry the highest weightage year after year. 

This comprehensive analysis identifies the most repeated questions, topic-wise weightage, recurring concepts, and previous year trends to help you streamline your final revision, focus on high-scoring areas, and improve your performance in the CSIR NET Mathematical Sciences 2026 examination. 

Paper Pattern Analysis from CSIR NET Mathematical Sciences Previous Year Questions: 

A review of the previous six CSIR NET Mathematical Sciences papers shows that the examination follows a consistent three-part structure.

The biggest observation from previous year papers is the dominance of Part C. Nearly half of the Mathematics questions belong to this section, where candidates must identify multiple correct statements. These questions test conceptual clarity rather than direct formula application.

Another important trend is that Part A remains largely independent of the Mathematics syllabus, making it a scoring section through regular aptitude practice.

CSIR NET Mathematical Sciences High Weightage Questions

Not every subject contributes equally to the examination. Some topics consistently account for a major share of questions across every paper. The table below summarizes the observed weightage.

Topic-wise Analysis of CSIR NET Mathematical Sciences Questions

The analysis of previous year papers reveals that certain concepts are repeatedly tested within each subject.

Real Analysis

Real Analysis remains the most important subject in the examination. Frequently tested questions are:

Q: If lim sup and lim inf of a bounded sequence both exist and are equal, what follows?

A: The sequence converges, and its limit equals that common value.

Q: Does a monotone increasing surjective function ℝ→ℝ have to be continuous? 

A: Yes — a jump discontinuity would create a "gap" in the range, contradicting surjectivity.

Q: Is every monotonic function Riemann integrable on a closed interval?

 A: Yes — monotonic functions have at most countably many discontinuities, which is a measure-zero set, satisfying Lebesgue's criterion for Riemann integrability.

Q: A uniformly continuous function composed with a uniformly continuous function — is the composition uniformly continuous? 

A: Yes, the composition of uniformly continuous functions is uniformly continuous (a repeatedly tested closure property).

Linear Algebra

The majority of questions revolve around standard matrix properties and vector space concepts. Important questions include:

Q: When is a symmetric bilinear form / quadratic form positive definite? 

A: All eigenvalues of the associated symmetric matrix are strictly positive (equivalently, all leading principal minors are positive — Sylvester's criterion).

Q: If 0 is an eigenvalue of a matrix A, what can you say about A⁺ (or any operator built from A)? 

A: A is not invertible; if A is symmetric, 0 being an eigenvalue directly gives a non-trivial kernel vector, useful for constructing counterexamples in "does there exist a non-zero vector such that…" questions.

Q: How do you find the number of k-dimensional subspaces of an n-dimensional vector space over 𝔽_q containing a fixed subspace? 

A: Use the Gaussian binomial coefficient formula applied to the appropriate quotient space dimension.

Complex Analysis

Complex Analysis consistently features conceptual and application-based questions. Frequently asked questions are: 

Q: What is the order of the pole (or type of singularity) determined by the Laurent series?

A: Count negative-power terms — finitely many ⟹ pole of that order; infinitely many ⟹ essential singularity; none ⟹ removable.

Q: State the Argument Principle. 

A: For f meromorphic inside and on a simple closed contour C with no zeros/poles on C: (1/2πi)∮ f′/f dz = Z − P, where Z, P are the number of zeros and poles inside C (counted with multiplicity).

Q: Schwarz–Pick Lemma — what does it bound? 

A: For holomorphic f: 𝔻→𝔻, it bounds the hyperbolic-distance-preserving behavior: |f′(z)|/(1−|f(z)|²) ≤ 1/(1−|z|²), used to get sharp bounds on f′(0) or on |f(z₀)| for specific z₀.

Abstract Algebra

Questions are theorem-based and test concepts related to groups, rings, fields, homomorphisms, and finite fields rather than lengthy calculations.

Q: When is ℤ[X]/(f(X)) an integral domain? 

A: If (f(X)) is a prime ideal, i.e., f(X) is irreducible over the relevant ring/field context (careful: over ℤ[X], (p, f(X)) type ideals need checking modulo p first).

Q: What is the class equation, and what constraints does it impose? 

A: |G| = |Z(G)| + Σ [G:C(xᵢ)] over conjugacy class representatives xᵢ outside the center; each summand must divide |G| — used to rule out invalid "class equations" quickly (each term must be a valid group-order divisor and ≥1, with the center term ≥1).

Q: Multiplicative group of a finite field — cyclic or not? 

A: Always cyclic. (This is one of the single most repeated true facts across all 6 papers.)

ODE / PDE

Differential Equations regularly contribute high-weightage conceptual questions in both ODE and PDE. Most questions revolve around standard theorems, solution techniques, and classification methods rather than lengthy derivations. 

Q: For y″+p(x)y′+q(x)y=0 with p≡0, how does the Wronskian behave? 

A: It is constant (Abel's formula collapses since the integral of p is zero).

Q: How do you classify a 2nd order linear PDE Au_xx+Bu_xy+Cu_yy=0? 

A: Compute the discriminant B²−4AC: positive → hyperbolic, zero → parabolic, negative → elliptic.

Q: Lagrange's method for Pp+Qq=R — what are the "subsidiary/auxiliary equations"? 

A: dx/P = dy/Q = dz/R; solve for two independent first integrals, then combine.

Topology & Metric Spaces

Topology questions focus on understanding definitions and applying standard results rather than proving advanced theorems.

Q: Is a compact subset of a general topological space always closed? 

A: Only if the space is Hausdorff. Not true in general (finite-complement topology on an infinite set is the standard counterexample).

Q: Are open connected subsets of ℝⁿ path-connected? 

A: Yes, always — this is a genuinely true (not a trap) recurring statement.

Numerical Analysis

Although Numerical Analysis carries comparatively lower weightage, its questions are highly predictable and revolve around a few recurring numerical methods. 

Q: What is the order of convergence of Newton–Raphson at a root of multiplicity m > 1?

A: Linear (order 1), not quadratic — quadratic convergence requires a simple root (m=1)

Most Repeated CSIR NET Mathematical Sciences PYQ Concepts

While exact questions rarely repeat, many core concepts appear consistently across multiple examination sessions. The table below summarizes the most frequently repeated concepts from the six-paper analysis.

These concepts have appeared repeatedly across multiple exam sessions and should be prioritised during revision. While exact questions may change, the underlying concepts continue to be tested in different forms.