Section 1
Algebra
1.1–1.10
Groups, rings, linear algebra, field structure, and spectral arguments.
Doctoral archive · 2011English · IndiaComplete 104-page edition
NBHM Doctoral 2011 worked solutions
Prepare for NBHM Doctoral and PhD-level mathematics with independent derivations for all 50 numbered questions and 52 addressable solution prompts. Read three complete solutions in searchable HTML, inspect six guide pages, and understand the original best-four-of-five scoring rule before practising.
50 questions · 52 addressable promptsDotted official source locatorsIndependent cross-review
Direct format answer
How was the NBHM Doctoral 2011 paper scored?
The 150-minute paper contained five subject sections of ten one-point questions. All 50 questions could be attempted, but only the best four section totals counted, giving a maximum counted score of 40. When one numbered question requested multiple answers, every requested answer was required for that point; no partial credit was provided.
Test date
22 January 2011
Working time
150 minutes
Paper structure
5 sections × 10 questions
Question value
1 point per numbered question
Counted result
Best 4 section scores
Maximum counted score
40 points
Question count is not a physical-answer-field count
The guide has 52 addressable solution prompts. This is because Question 5.3 contains separately explained parts inside the numbered, one-point question entries. The archived paper refers to a separate answer booklet but does not establish its physical field count, so this page does not invent one.
Five-section subject map
Each section contributes ten numbered questions. The guide keeps the printed dotted labels so a learner can move directly between an official question and the matching explanation.
Section 2
Analysis
2.1–2.10
Real and complex analysis, measure, sequences, series, and function spaces.
Section 3
Topology
3.1–3.10
Continuity, compactness, connectedness, quotient ideas, and metric spaces.
Section 4
Applied Mathematics
4.1–4.10
Multivariable calculus, integration, ordinary differential equations, and boundary-value problems.
Section 5
Miscellaneous
5.1–5.10
Probability, combinatorics, geometry, optimisation, and mixed reasoning.
Six-page visual preview
Inspect representative pages before deciding
The preview covers Questions 1.2, 2.6, 3.6, 4.1, 4.2, 5.3 from the current 104-page manuscript. Every image is an InshiHub explanation page, never a scan of the official paper or answer key.
Crawlable HTML samples
Three free NBHM Doctoral 2011 solutions
These summaries identify the mathematical task without substituting for the official question. Open the source paper at the precise locator, attempt the problem, and then audit your work against the derivation, verification, and common-error note.
Free sample 1
Official question 2.6 · Uniform continuity and an impossible subsequence premise
- Source locator
- Section 2: Analysis, official question 2.6, paper PDF page 5
- Review note
- Independently derived and cross-reviewed; modulus, subsequence contradiction, endpoint extension, and logical convention checked · 2026-07-28
Problem summary
Test three proposed sufficient conditions for uniform continuity, including two incompatible limits along a sequence and one of its subsequences.
Core idea
A direct modulus proves the first condition, subsequence uniqueness exposes the impossible second premise, and a continuous compact-interval extension settles the oscillatory third function.
Derivation
- 1The estimate |f(x)-f(y)| at most the square root of |x-y| gives a global modulus of continuity, so choosing delta equal to epsilon squared proves uniform continuity.
- 2If f(1/n) converges to one half, then its square-indexed subsequence f(1/m squared) must converge to the same limit.
- 3The proposed limit one quarter along that subsequence is therefore incompatible with the first limit, so no function satisfies both.
- 4For the oscillatory example, |f(x)| at most the square root of x forces f(x) to zero at the left endpoint despite increasingly rapid oscillation.
- 5Extending the function continuously to the compact interval [0,1] and applying Heine-Cantor proves uniform continuity of its restriction.
Final answer
Under the intended non-vacuous exam reading, statements (a) and (c) imply uniform continuity. Statement (b) is internally inconsistent; under purely formal material implication it would also be vacuously sufficient.
Free sample 2
Official question 4.2 · Fixed-volume cylinder and a reversed extremum
- Source locator
- Section 4: Applied Mathematics, official question 4.2, paper PDF page 9
- Review note
- Independently derived and cross-reviewed; endpoint growth, stationary equation, global classification, and archived-key relationship checked · 2026-07-28
Problem summary
Determine what actually happens to the total surface area of a closed cylinder when the official paper asks for a maximum at fixed positive volume.
Core idea
Eliminate the height using the volume constraint, inspect both endpoints before differentiating, and distinguish the unbounded-above printed problem from the unique minimum encoded by the archived key.
Derivation
- 1With fixed volume V and radius r greater than zero, the height is V divided by pi r squared.
- 2The total surface area becomes 2 pi r squared plus 2V divided by r.
- 3This area tends to infinity as r tends to zero from above and also as r tends to infinity, so it is unbounded above and has no maximum.
- 4For the evidently intended minimum, differentiating gives 4 pi r minus 2V divided by r squared.
- 5The stationary equation gives V equal to 2 pi r cubed and therefore h equal to 2r; the second derivative is positive everywhere.
Final answer
For the problem as printed, no maximum exists. For the intended minimum problem, h=2r, which is the archived key's result.
Free sample 3
Official question 5.3 · Countability across finite sets, sequences, and algebraic roots
- Source locator
- Section 5: Miscellaneous, official question 5.3, paper PDF page 10
- Review note
- Independently derived and cross-reviewed; encodings, diagonal argument, finite-root union, and zero-polynomial convention checked · 2026-07-28
Problem summary
Classify three families using finite encodings, Cantor's diagonal argument, and a necessary convention about the zero polynomial.
Core idea
Finite tuples create countable unions in the positive cases, binary-valued sequences already force uncountability in the second family, and each nonzero polynomial contributes only finitely many roots.
Derivation
- 1A finite subset of the natural numbers has a unique increasing listing, so the k-element subsets inject into a countable finite Cartesian power.
- 2Taking the union over all finite sizes keeps the family countable.
- 3Binary-valued sequences form a subfamily of positive-integer sequences, and Cantor diagonalisation proves that subfamily uncountable.
- 4Integer-coefficient polynomials are encoded by finite integer tuples, so there are countably many of them.
- 5Each nonzero polynomial has finitely many complex roots, making their union countable; admitting the zero polynomial as having every complex root would reverse this conclusion.
Final answer
Part (a) is countable, part (b) is uncountable, and part (c) is countable under the standard nonzero-polynomial convention.
Full-edition scope
- Original, step-by-step solutions for all 50 numbered questions
- Complete coverage of all 52 addressable prompts, including the three explicit classification parts of Question 5.3
- Coverage of all five subject sections with the best-four-of-five scoring rule preserved
- Verification, counterexamples, boundary cases, and problem-specific common-error notes
- Official section, dotted question label, and paper-page locators for every solution
- Dimension-one qualifications for the triangular-subgroup and spectral-radius compactness questions
- Separate ordinary-language and material-implication readings for the impossible premise in Question 2.6
- Standard degree-of-precision and literal individual-polynomial readings for Simpson quadrature in Question 4.1
- A proof that the printed maximum in Question 4.2 does not exist, together with the intended unique-minimum result
- The nonzero-polynomial convention required by the countability conclusion in Question 5.3(c)
- A dated correction process and public correction log after release
What is not included
- The official question paper, answer choices, source layout, logos, and answer-key text
- Scan crops, mirrored official files, or a substitute copy of the official paper
- A claim that the archived key resolves Question 2.6's impossible premise or the dimension-one boundaries
- A claim that the printed maximum in Question 4.2 exists
- A claim that an official erratum exists for Question 4.2 or any other mathematical boundary
- Coaching, admissions advice, or any score guarantee
- Affiliation with NBHM, the Department of Atomic Energy, or IMSc
Editorial quality system
Every answer is derived, challenged, and source-mapped
1
Exact source mapping
Every solution identifies its official section, dotted question label, and paper page without copying the question or answer choices.
2
Independent first derivation
Each result is proved or disproved from the printed mathematical hypotheses before the final answer is compared with the archived key.
3
Cross-author mathematical audit
A different reviewer rechecks theorem use, calculations, counterexamples, boundary cases, source locators, and the rendered explanation.
4
Transparent source boundaries
Where dimension, vacuous truth, quantifier scope, extremum type, or an implicit algebraic convention changes the conclusion, the guide states the precise boundary and does not invent an official correction.
Review boundary
Public mathematics and paid access are separate states
The guide page and free samples can be public while commercial validation remains incomplete. Seller accountability and end-to-end delivery approval are required before any payment control may be introduced.
Official sources, provenance, and rights
Historical paper title and Doctoral archive classification
The paper itself is titled “Research Scholarships Screening Test.” The official NBHM past-paper archive classifies the file dated 22 January 2011 under Doctoral. This page uses “NBHM Doctoral 2011” as the searchable edition name while preserving the paper's historical title.
Independent answer-key comparison
Question 1.2's upper-triangular subgroup is normal when n=1 but not when n is at least 2, and Question 1.4's quotient-field conclusion needs the usual proper-ideal convention. Question 2.6 contains two incompatible limits along a sequence and its subsequence: the ordinary non-vacuous exam reading excludes that condition, while formal material implication makes it vacuously sufficient. In Question 3.6, eigenvalue bounds do not control nonnormal matrix norms when n is at least 2, although the n=1 family is compact. Question 4.1 has Simpson degree of precision 3 under the standard definition, while arbitrarily high individual odd polynomials are exact under a literal reading. Question 4.2 is unbounded above as printed; h=2r is the unique minimum condition recorded by the archived key. Question 5.3(c) is countable under the standard nonzero-polynomial convention, while including the zero polynomial would make every complex number a root. These are independent mathematical qualifications and are not presented as official errata.
National Board for Higher Mathematics publishes the source documents, which are hosted by The Institute of Mathematical Sciences. InshiHub links to that archive but does not republish its paper, answer choices, logos, or answer-key text.
Official NBHM past-paper archiveDoctoral classification and source indexOfficial 2011 question paperResearch Scholarships Screening Test PDFOfficial 2011 answer keyAnswer-key PDF hosted by IMSc
- Exam date
- Source accessed
- 2026-07-28
- Source status
- Publicly accessible source; no reuse licence located
- Official paper SHA-256
- e76fd395c4b3bdc2beb3b9bd2f6fb7c6603b0ed80bd620754947929a21905375
- Official answer-key SHA-256
- 495896d7b5e2d61aa314429333e2059658525da3b97feb585f7b9127bb78027b
InshiHub is independent and is not affiliated with, endorsed by, or sponsored by the National Board for Higher Mathematics, the Department of Atomic Energy, or The Institute of Mathematical Sciences. Official names identify the examination source only.
Transparent release status
Free samples are public. Paid access remains disabled.
The 2011 edition remains in validation. Publishing this page does not accept payment or create an order. Paid access may appear only after every recorded gate below passes for this exact edition.
- All 50 numbered questions and 52 addressable prompts completed
- Author review and independent cross-review recorded across all five sections
- Questions 1.2, 1.4, 2.6, 3.6, 4.1, 4.2, and 5.3 independently rederived with every material boundary explicit
- Historical title, Doctoral archive classification, and scoring rules verified
- Rights review completed for paid distribution
- Merchant and 2011 edition approval recorded
- Exact transaction-seller entity and India responsibility allocation confirmed
- Current buyer terms and privacy notice linked
- INR final-price display, payment, delivery, refund, and order reconciliation tested
Related years
Continue the NBHM Doctoral paper sequence
NBHM Doctoral 201250 questions · 54 prompts · convergence, regularity, iteration, and archived-key boundariesNBHM Doctoral 201350 questions · 50 prompts · dimension, branch, and key-typography boundariesNBHM Doctoral 201450 questions · 53 prompts · notation, transform, and admissibility boundariesNBHM Doctoral 201550 questions · 54 prompts · scalar, singular-solution, and endpoint boundaries
Frequently asked questions
NBHM Doctoral 2011 guide details
Does this guide reproduce the official NBHM 2011 paper?
No. It contains independently written solutions and short, non-substituting topic summaries. Use the direct IMSc archive links on this page for the official question paper and archived key.
What is covered in the NBHM Doctoral 2011 solution guide?
The current 104-page edition covers all 50 numbered questions and 52 addressable prompts, with dotted source labels, complete derivations, independent checks, and problem-specific error notes.
Why are there 52 solution prompts for 50 numbered questions?
Question 5.3 contains three separately addressable classification parts. Counting those subparts produces 52 complete prompts without changing the official count of 50 numbered questions.
Which dimension boundaries are disclosed in the guide?
Question 1.2's upper-triangular subgroup is normal when n=1 but not when n is at least 2. Question 3.6's spectral-radius-bounded family is the compact interval [-2,2] when n=1 but is unbounded for every fixed n at least 2.
Why does the Question 2.6 sample give two logical readings?
The proposed values along 1/n and 1/n squared are incompatible because the latter is a subsequence of the former. The ordinary non-vacuous exam reading excludes that condition; a purely formal material implication makes it vacuously sufficient.
What qualifications are needed for Questions 4.1, 4.2, and 5.3?
Question 4.1 has standard degree of precision 3, although individual exact odd polynomials exist in arbitrarily high degree. Question 4.2 has no printed maximum; h=2r is the unique minimum condition. Question 5.3(c) is countable only under the usual nonzero-polynomial convention.
Is the complete 2011 guide available for paid access now?
Not yet. The editorial page and free samples are public, but paid access remains disabled while seller, tax, payment, delivery, refund, and reconciliation validation is incomplete.
What is the configured price for the complete 2011 guide?
The configured one-time catalogue price is ₹999, 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 were the 2011 answers checked, and is InshiHub affiliated with the source institutions?
Each solution was independently derived, checked against edge cases and source locators, and cross-reviewed before final comparison with the archived key. InshiHub is an independent publisher and is not affiliated with NBHM, the Department of Atomic Energy, or IMSc.