Section 1
Algebra
1.1–1.10
Groups, rings, linear algebra, field structure, and spectral arguments.
Doctoral archive · 2012English · IndiaComplete 107-page edition
NBHM Doctoral 2012 worked solutions
Prepare for NBHM Doctoral and PhD-level mathematics with independent derivations for all 50 numbered questions and 54 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 · 54 addressable promptsDotted official source locatorsIndependent cross-review
Direct format answer
How was the NBHM Doctoral 2012 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
28 January 2012
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 54 addressable solution prompts. This is because Questions 1.8, 2.4, and 2.5 contain 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 1.9, 2.8, 3.7, 4.8, 4.9, 5.6 from the current 107-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 2012 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 1.9 · Minimal polynomials and an impossible odd-dimensional premise
- Source locator
- Section 1: Algebra, official question 1.9, paper PDF page 4
- Review note
- Independently derived and cross-reviewed; diagonalisation, counterexample, determinant obstruction, and logical convention checked · 2026-07-28
Problem summary
Test three diagonalisation implications, including a real three-dimensional premise that cannot be satisfied and therefore needs an explicit logical convention.
Core idea
Square-free annihilating polynomials settle the realizable cases, while determinants prove that no real 3 by 3 matrix can square to minus the identity.
Derivation
- 1If A squared equals A, then the minimal polynomial divides t(t-1), which has distinct roots, so A is diagonalizable.
- 2A nonzero nilpotent shift gives a matrix whose square is diagonalizable although the matrix itself is not, disproving the second implication.
- 3If a real 3 by 3 matrix satisfied A squared equals minus the identity, taking determinants would give det(A) squared equals minus one.
- 4That determinant equation has no real solution, so the third premise describes no real 3 by 3 matrix.
- 5Under the ordinary exam reading that asks for a realizable mathematical assertion, only the first statement is selected; under strict material implication, the third statement is vacuously true.
Final answer
Under the intended non-vacuous exam reading, only statement (a) is selected. The premise in (c) is impossible over real 3 by 3 matrices; if it is formalised purely as a material implication, (c) is vacuously true.
Free sample 2
Official question 4.8 · Affine iteration under literal and common-limit readings
- Source locator
- Section 4: Applied Mathematics, official question 4.8, paper PDF page 9
- Review note
- Independently derived and cross-reviewed; literal counterexamples and the stronger common-limit implication checked · 2026-07-28
Problem summary
Analyse convergence of an affine recurrence for every initial vector and separate the literal wording from the stronger interpretation implicit in the archived key.
Core idea
Subtracting two trajectories isolates powers of the iteration matrix. The literal statement permits different limits, whereas a common limit for all starting values forces those powers to vanish.
Derivation
- 1The recurrence has the closed form x sub n equals B to the n x sub zero plus the finite geometric sum applied to b.
- 2The claim that every initial trajectory merely converges does not say that all trajectories have the same limit.
- 3With B equal to the identity and b equal to zero, every trajectory is constant and convergent, yet I minus B is singular and 1 is an eigenvalue.
- 4Therefore the invertibility and spectral conclusions are false under the literal wording; the instability claim is also false, for example with B equal to 2I and b equal to zero.
- 5Under the stronger common-limit reading, differences B to the n times x sub zero tend to zero for every vector, so B to the n tends to zero, the spectral radius is below one, and I minus B is invertible.
Final answer
Literal wording: none of the three claims follows; B=I and b=0 refutes the last two, and B=2I and b=0 refutes the first. Stronger common-limit reading: the archived selections (b) and (c) follow because B to the n tends to zero.
Free sample 3
Official question 4.9 · Smallest listed spectral disc by Cayley and Routh-Hurwitz analysis
- Source locator
- Section 4: Applied Mathematics, official question 4.9, paper PDF page 10
- Review note
- Independently derived and cross-reviewed; characteristic polynomial, transform, stability inequalities, and archived-key discrepancy checked · 2026-07-28
Problem summary
Locate the spectrum of a four-dimensional matrix sharply enough to distinguish a convenient Gershgorin disc from the smallest listed guaranteed disc.
Core idea
After shifting the spectral parameter by 1 and scaling by 4, a Cayley transform converts the unit-disc question into a left-half-plane stability test that can be certified by strict Routh-Hurwitz inequalities.
Derivation
- 1With mu equal to lambda minus 1, direct determinant expansion gives mu to the fourth plus 15 mu squared minus 14 mu plus 16.
- 2Set mu equal to 4z; after harmless scaling the polynomial becomes 32z to the fourth plus 30z squared minus 7z plus 2.
- 3Apply the Cayley substitution z=(1+w)/(1-w) and clear the denominator.
- 4The transformed polynomial is 71w to the fourth plus 106w cubed plus 144w squared plus 134w plus 57.
- 5Its coefficients and the two strict quartic Routh-Hurwitz determinants are positive, so every transformed root has negative real part and every original root satisfies absolute value of lambda minus 1 less than 4.
Final answer
Choice (c), the closed disc centred at 1 with radius 4, is the smallest listed disc guaranteed to contain the spectrum. The proof in fact gives strict radius 4. The archived key's radius-6 selection is a sufficient Gershgorin bound, but it is not the smallest listed answer.
Full-edition scope
- Original, step-by-step solutions for all 50 numbered questions
- One complete solution chapter for each of the 54 addressable prompts, including every subpart of Questions 1.8, 2.4, and 2.5
- 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
- Separate ordinary-language and material-implication readings for the impossible premise in Question 1.9
- The exact open convergence disc and complete boundary-divergence check for Question 2.8
- The r>0 domain of the radial differential equation and the origin regularity condition in Question 4.5
- Literal and stronger common-limit readings of the affine iteration in Question 4.8
- An independent Cayley-transform and Routh-Hurwitz proof for Question 4.9, with the archived key discrepancy stated precisely
- A dated correction process and public correction log after release
What is not included
- The official question paper, answer choices, source layout, logos, diagrams, 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 1.9's impossible premise or Question 4.8's unstated convergence interpretation
- A claim that an official erratum exists for Question 4.9 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 logic, convergence domain, origin regularity, quantifier scope, or an archived-key selection 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 28 January 2012 under Doctoral. This page uses “NBHM Doctoral 2012” as the searchable edition name while preserving the paper's historical title.
Independent answer-key comparison
Question 1.9 has an impossible premise over real 3 by 3 matrices: the ordinary non-vacuous exam reading selects only (a), while strict material implication makes the impossible-premise statement vacuously true. Question 2.8 converges exactly on |z-2|<2 and diverges at every boundary point because the terms do not tend to zero. Question 4.5's radial equation is derived for r>0; smoothness at the origin supplies the additional regularity condition, making every global smooth radial harmonic function constant. In Question 4.8, the literal wording is refuted by B=I and b=0, whereas the archived selections follow from the stronger common-limit reading that forces B^n to tend to zero. For Question 4.9, an independent characteristic-polynomial, Cayley-transform, and Routh-Hurwitz proof gives the smallest listed disc, centred at 1 with radius 4; the archived key's radius 6 is a sufficient Gershgorin bound but not the smallest listed answer. 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 2012 question paperResearch Scholarships Screening Test PDFOfficial 2012 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
- 3872d5f331ed12be385a77c8d63da29ed9e433d97285d3892faf1ec89af20684
- Official answer-key SHA-256
- 94f250210227c77782e829a1d7d6a20704ac05aed43b04611ac613ad535e90ff
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 2012 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 54 addressable prompts completed
- Author review and independent cross-review recorded across all five sections
- Questions 1.9, 2.8, 3.7, 4.5, 4.8, 4.9, and 5.6 independently rederived with every material boundary explicit
- Historical title, Doctoral archive classification, and scoring rules verified
- Rights review completed for paid distribution
- Merchant and 2012 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 201150 questions · 52 prompts · dimension, logic, extremum, and convention 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 2012 guide details
Does this guide reproduce the official NBHM 2012 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 2012 solution guide?
The current 107-page edition covers all 50 numbered questions and 54 addressable prompts, with dotted source labels, complete derivations, independent checks, and problem-specific error notes.
Why are there 54 solution prompts for 50 numbered questions?
Questions 1.8 and 2.4 each contain two separately addressable parts, and Question 2.5 contains three. Counting those subparts produces 54 complete solution prompts without changing the official count of 50 numbered questions.
Why does the Question 1.9 sample give two logical readings?
No real 3 by 3 matrix can square to minus the identity because its determinant would have to satisfy a real square equals minus one. The ordinary non-vacuous exam reading therefore selects only (a), while a purely formal material implication would also make the impossible-premise statement vacuously true.
What boundary checks are included for Questions 2.8 and 4.5?
For Question 2.8 the Taylor series converges exactly on the open disc |z-2|<2 and diverges at every boundary point because its terms do not tend to zero. For Question 4.5 the radial differential equation is valid for r>0; a smooth global solution must also satisfy the origin regularity condition and is constant.
Why do the Question 4.8 and 4.9 conclusions differ from the archived key?
Question 4.8's literal wording allows B=I and b=0, while the key's selections follow under the stronger unstated requirement that all starting values share one limit. For Question 4.9 an independent Cayley and Routh-Hurwitz proof places every eigenvalue strictly inside radius 4; the key's radius 6 is only a looser Gershgorin bound. These are independent findings, not claimed official errata.
Is the complete 2012 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 2012 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 2012 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.