Part A
20 single-correct multiple-choice items
+2 correct · −1 incorrect · 0 blank
The official instructions state that the TIFR CAM Integrated PhD route is evaluated on Part A only.
TIFR GS 2026 · Mathematics Stage IEnglish · IndiaComplete 81-page edition
TIFR GS 2026 Mathematics Stage I — Independent Detailed Solutions
Study all 40 items through independently written proofs, counterexamples, option-level conclusions, and second-route checks. Read five complete solutions in searchable HTML and inspect six pages from the exact full-edition PDF.
20 Part A · 20 Part BExact official source locatorsIndependent cross-review
The official paper and its embedded final indicators are free from TIFR.The configured price is solely for InshiHub's original explanations, verification, organisation, and controlled digital delivery.
Direct exam-pattern answer
What was the TIFR GS 2026 Mathematics Stage I pattern?
The central GS-2026 schedule placed the Mathematics examination on 14 December 2025 from 2:00 pm to 5:00 pm. The Stage I paper has 20 single-correct items in Part A and 20 true-or-false items in Part B. Both parts use negative marking, with a larger two-mark penalty for an incorrect Part B response.
Admission cycle
GS 2026
Central exam date
14 December 2025
Mathematics session
2:00 pm–5:00 pm
Working time
180 minutes
Paper structure
Part A: 20 · Part B: 20
Maximum score
80 marks
Part B
20 true-or-false items
+2 correct · −2 incorrect · 0 blank
The other mathematics routes covered by this Stage I paper answer both parts, subject to the current official programme instructions.
GS-2026 is the admission-cycle label
The central GS advertisement and schedule give 14 December 2025 as the examination date. A School of Mathematics page displays 4 December 2025. This page uses the central schedule and records the discrepancy rather than silently merging the two dates.
Six-page visual preview
Inspect the cover and five representative solutions
The preview shows the cover plus A6, A8, A11, B13, and B20 from the exact 81-page master. Every page is an InshiHub-authored page, never a scan of TIFR's paper, choices, figures, or layout.
Crawlable HTML samples
Five free TIFR GS 2026 Mathematics solutions
Each summary identifies the task without replacing the official question. Open TIFR's paper at the source locator, attempt the item, and then audit your method against the derivation, verification, and common-error note below.
Free sample 1
Official item A6 · Nilpotent matrices and induced linear maps
- Source locator
- Part A, official Question 6, paper PDF page 3
- Review note
- Independently derived and cross-reviewed; induced nilpotence, trace, determinant, locator, and final indicator checked · 2026-07-28
Problem summary
Determine the trace and determinant of left multiplication on a matrix space when the multiplying matrix satisfies a fixed nilpotence relation.
Core idea
Composition by left multiplication converts powers of the induced operator directly into powers of the original matrix, so nilpotence passes to the full 100-dimensional operator without writing its matrix.
Step-by-step reasoning
- 1Define the linear operator T_A on the 10 by 10 real matrix space by T_A(X)=AX.
- 2Repeated composition gives T_A raised to the k-th power applied to X equal to A raised to the k-th power times X.
- 3The given relation A raised to the tenth power equals zero therefore makes T_A nilpotent.
- 4Every eigenvalue of a nilpotent finite-dimensional operator is zero, so its trace is zero.
- 5A nilpotent operator on this nonzero space is not invertible, so its determinant is also zero.
Final result
Choice (a): for every admissible A, both det(T_A) and trace(T_A) are zero.
Free sample 2
Official item A8 · Ultrametric triangles and equal-radius balls
- Source locator
- Part A, official Question 8, paper PDF pages 3–4
- Review note
- Independently derived and cross-reviewed; both statements, inclusions, locator, and final indicator checked · 2026-07-28
Problem summary
Test two consequences of the strong triangle inequality: the shape of a distance triple and the intersection pattern of equal-radius open balls.
Core idea
Apply the maximum-form triangle inequality first to the largest of three distances, then twice around a shared point of two balls.
Step-by-step reasoning
- 1Order the three pairwise distances from largest to smallest.
- 2Applying the ultrametric inequality to the largest distance forces it to equal the second-largest distance.
- 3Thus an ultrametric triangle cannot have a uniquely longest side.
- 4If two radius-r open balls meet, a shared point shows that their centres are less than r apart.
- 5Any point in the first ball is then less than r from the second centre; symmetry proves the two balls are equal.
Final result
Choice (a): both statements are true.
Free sample 3
Official item A11 · Differential inequalities and a long-time limit
- Source locator
- Part A, official Question 11, paper PDF page 4
- Review note
- Independently derived and cross-reviewed; monotonicity, integral bound, locator, and final indicator checked · 2026-07-28
Problem summary
Determine whether a positive solution of a first-order initial-value problem has a finite long-time limit and locate that limit in the required range.
Core idea
Use the differential equation twice: positivity proves monotonicity, and the resulting lower bound on the solution produces an integrable upper bound for its derivative.
Step-by-step reasoning
- 1The derivative is positive for x at least 1, so the solution is increasing and remains at least its initial value 1.
- 2Substituting that lower bound into the denominator gives 0<g'(x)≤1/(x squared+1).
- 3Integrating from 1 to x bounds g(x) above by 1 plus the corresponding arctangent integral.
- 4Hence g(x) is at most 1+pi/4, which is strictly below 2.
- 5An increasing function with this finite upper bound has a limit in the interval from 1 to 1+pi/4.
Final result
Choice (b): the limit exists and lies in the required interval ending at 2; in fact it lies in [1, 1+pi/4].
Free sample 4
Official item B13 · A Sylow-subgroup covering counterexample
- Source locator
- Part B, official Question 13, paper PDF page 7
- Review note
- Independently derived and cross-reviewed; counterexample, exact union count, locator, and final indicator checked · 2026-07-28
Problem summary
Test a prime-power conclusion for a finite group covered by the collection of all its Sylow subgroups.
Core idea
Use the smallest nonabelian symmetric group: its transpositions and 3-cycles separate cleanly into its Sylow subgroups, yet its order contains two different primes.
Step-by-step reasoning
- 1The group S3 has three Sylow 2-subgroups, each generated by one of its three transpositions.
- 2Its unique Sylow 3-subgroup contains the identity and both 3-cycles.
- 3The union of these four subgroups therefore contains all six elements of S3.
- 4However, the order of S3 is 6=2 times 3, which is not a power of one prime.
- 5The covering condition therefore does not imply the proposed prime-power conclusion.
Final result
False: S3 is covered by its Sylow subgroups but has order 6.
Free sample 5
Official item B20 · Divisibility of a long power sum modulo seven
- Source locator
- Part B, official Question 20, paper PDF page 8
- Review note
- Independently derived and cross-reviewed; direct count, block check, locator, and final indicator checked · 2026-07-28
Problem summary
Prove or refute a divisibility claim modulo seven for a long sum of powers whose exponents are multiples of six.
Core idea
Separate multiples of seven, reduce every other term to one using the six-element multiplicative group modulo seven, and count the surviving terms exactly.
Step-by-step reasoning
- 1For k not divisible by 7, Fermat's theorem gives k raised to the sixth power congruent to 1 modulo 7.
- 2Raising this congruence to a positive integer power makes every such summand congruent to 1.
- 3For k divisible by 7, the summand is congruent to 0.
- 4There are floor(2025/7)=289 divisible terms and 2025-289=1736 remaining terms.
- 5Because 1736=7 times 248, the entire sum is congruent to zero modulo 7.
Final result
True: 7 divides the sum from k=1 to 2025 of k raised to 6n for every positive integer n.
Complete topic map
Which topics appear in the 2026 Mathematics Stage I paper?
The paper moves across undergraduate mathematics rather than isolating one subject. The full guide preserves the official A1–A20 and B1–B20 labels so each topic can be located immediately.
Analysis and calculus
A1–A5, A10–A11, A13, A18, A20; B1, B7, B11, B16
Series, products, power series, uniform convergence, uniform continuity, critical points, differential inequalities, fractional parts, Taylor data, monotone limits, differentiability, periodic functions, and differential operators.
Linear algebra and matrices
A6, A9, A12, A14–A15; B5–B6, B9, B14
Nilpotent operators, range and kernel, products of endomorphisms, matrix topology, linear operators, diagonalizability, matrix powers, stochastic matrices, and matrix functions.
Topology and metric structure
A7–A8; B2–B4, B10, B12, B19
Dense-set extensions, ultrametrics, topology of the rationals, function-space compactness, continuous bijections, finite metric embeddings, finite topologies, and parameter topology.
Algebra, geometry, and discrete mathematics
A16–A17, A19; B8, B13, B15, B17–B18, B20
Commutative algebra, plane geometry, group theory, polynomial composition, Sylow subgroups, finite abelian groups, symmetric groups, combinatorial expectation, and modular arithmetic.
Full-edition scope
- Original, step-by-step solutions for all 20 Part A and all 20 Part B items
- Complete proof or counterexample for every true-or-false conclusion
- Letter-only Part A option analysis without reproducing official option text
- Official part, question number, and source-page locator for every solution
- Independent numerical, structural, or edge-case verification
- Problem-specific strategy notes and common-error warnings
- Transparent comparison with the final indicators embedded in TIFR's official paper
- Coverage across algebra, analysis, topology, geometry, probability, combinatorics, and differential equations
- A dated correction process and public correction log after release
What is not included
- The official question paper, question wording, answer choices, figures, logos, or source layout
- A mirrored or substitute copy of any TIFR source file
- The separate CAM Mathematics PhD examination paper
- Admissions counselling, application evaluation, or score guarantees
- Affiliation with, endorsement by, sponsorship by, or approval from TIFR
Editorial quality system
Every result is derived, challenged, and source-mapped
1
Exact source mapping
Every solution names its official part, item number, and PDF page while leaving the official question and choices at TIFR.
2
Independent first derivation
The mathematical result is established from the stated structure before the embedded final indicator is consulted.
3
Cross-author mathematical audit
A different reviewer rechecks theorem use, calculations, counterexamples, option conclusions, source locators, and rendered pages.
4
Two-route verification
Where useful, a second calculation, structural argument, limiting case, or explicit counterexample tests the conclusion independently.
Official sources, provenance, and rights
TIFR keeps the official paper and final indicators free
The official PDF is titled “Mathematics: Stage I Questions” and contains the final indicators used for the post-derivation comparison. InshiHub links to TIFR's source files but does not republish question text, answer choices, figures, logos, or paper layout.
Official TIFR past-paper archiveSource index for Graduate School papersOfficial GS 2026 Mathematics paperQuestion paper with embedded final indicatorsOfficial mathematics instructionsProgramme boundary, format, and scoring rulesOfficial GS-2026 advertisementCentral examination date and session
- Central exam date
- Source accessed
- 2026-07-28
- Source status
- Publicly accessible official sources; no reuse licence located
- Official paper SHA-256
- 7448bf233aa6987c75e8ae2e81ece799f07588596341e41b63bdd0b0fd786014
- Frozen InshiHub edition
- 81 A4 pages · 640,961 bytes · SHA-256 df695cc0b7cbd25a703cf482fab7c7348ef8055a9ebbee12f63a8dd996dfe990
Independently produced by InshiHub. InshiHub is not affiliated with, endorsed by, sponsored by, or approved by Tata Institute of Fundamental Research. TIFR and related names identify the examination source only. This guide contains InshiHub's original explanations and does not reproduce the official question text, answer choices, figures, logos, or paper layout.
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Publishing this page does not accept payment or create an order. Paid access may appear only after every mathematical, seller, tax, payment, delivery, refund, and reconciliation gate passes for this exact edition.
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Build breadth across examinations
Compare a different advanced mathematics paper
TIFR Stage I has its own format and negative-marking logic. For a second preparation style, compare the independently worked NBHM Combined paper rather than treating unrelated examinations as one interchangeable syllabus.
Open the NBHM Combined 2026 guideFrequently asked questions
TIFR GS 2026 Mathematics Stage I guide details
What is included in this TIFR GS 2026 Mathematics guide?
The current 81-page edition contains original worked solutions to all 20 Part A and all 20 Part B items, with source locators, full reasoning, independent checks, final results, and problem-specific error notes.
Does the guide include the official TIFR question paper?
No. The guide contains InshiHub's independently written explanations and short, non-substituting topic summaries. The official paper remains available free from TIFR through the direct links on this page.
Is this an official TIFR publication?
No. InshiHub is an independent publisher and is not affiliated with, endorsed by, sponsored by, or approved by Tata Institute of Fundamental Research.
How many GS-2026 Mathematics Stage I items are covered?
All 40 items are covered: 20 multiple-choice items in Part A and 20 true-or-false items in Part B.
Which TIFR mathematics programmes use this Stage I paper?
The official instructions assign this Mathematics Stage I paper to the mathematics routes covered by the GS process. They also state that the TIFR CAM Integrated PhD route evaluates Part A only, while the other covered routes answer both parts. Candidates should verify the current programme instructions directly with TIFR.
Is this the separate Mathematics PhD examination at CAM?
No. This guide is for the GS 2026 Mathematics Stage I paper. It does not cover the separate CAM Mathematics PhD examination.
Why was the GS-2026 examination held in December 2025?
GS-2026 names the admission cycle. TIFR's central advertisement scheduled the examination for 14 December 2025, so the cycle label and calendar date refer to different things.
What were the Part A and Part B marking rules?
Part A had 20 single-correct multiple-choice items scored +2 for correct, -1 for incorrect, and 0 for blank. Part B had 20 true-or-false items scored +2 for correct, -2 for incorrect, and 0 for blank.
Are the solutions checked against TIFR's final answers?
Yes. Each result was independently derived first, then independently cross-reviewed and compared with the final indicators embedded in TIFR's official paper.
What language and format is the guide supplied in?
It is an English, one-time digital PDF. The frozen edition has 81 A4 pages.
Is ₹1,499 the final tax-inclusive price?
The configured one-time catalogue price is ₹1,499, inclusive of applicable tax. It is not a temporary discount or crossed-out-price promotion, and no payment control is present while validation remains incomplete.
How are corrections and updated versions handled?
Material corrections receive a dated public note. The delivery system is designed to keep an exact content version attached to each entitlement so that replacement files remain traceable.