Section 1
Algebra
1.1–1.10
Groups, rings, linear algebra, field structure, and spectral arguments.
Doctoral archive · 2015English · IndiaComplete 105-page edition
NBHM Doctoral 2015 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 2015 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
24 January 2015
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 1.3, 1.7, 2.9, and 3.5 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
Calculus and Differential Equations
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 1.4, 2.7, 3.10, 4.8, 4.10, 5.9 from the current 105-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 2015 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 1.4 · Sylow counting and elements of prime order
- Source locator
- Section 1: Algebra, official question 1.4, paper PDF page 3
- Review note
- Independently derived and cross-reviewed; Sylow conditions, uniqueness, cyclicity, and the converse subgroup argument checked · 2026-07-28
Problem summary
Count elements of order seven in a finite group by first determining the number of Sylow seven-subgroups.
Core idea
The Sylow divisibility and congruence conditions force a unique subgroup of order seven, whose nonidentity elements are exactly the elements being counted.
Derivation
- 1Let n_7 denote the number of Sylow seven-subgroups. Sylow's theorems give n_7 dividing 4 and n_7 congruent to 1 modulo 7.
- 2Among the positive divisors 1, 2, and 4 of 4, only 1 is congruent to 1 modulo 7. The subgroup P of order seven is therefore unique.
- 3A group of prime order is cyclic. Hence P has six nonidentity elements, and every one of them has order seven.
- 4Conversely, any element of order seven generates a subgroup of order seven.
- 5That generated subgroup must equal the unique subgroup P, so no additional elements of order seven can occur outside P.
Final answer
There are exactly 6 elements of order seven.
Free sample 2
Official question 3.10 · Compact matrix families and the scalar boundary
- Source locator
- Section 3: Topology, official question 3.10, paper PDF page 7
- Review note
- Independently derived and cross-reviewed; finite-dimensional compactness, nilpotent counterexample, Hermitian bound, unitary bound, and n=1 qualification checked · 2026-07-28
Problem summary
Classify three finite-dimensional matrix families by closedness and boundedness while separating the nonnormal case from Hermitian and unitary cases.
Core idea
A spectral-radius bound does not control the norm of a general nonnormal matrix when n is at least two, but it does control a Hermitian matrix; unitary matrices have fixed Frobenius norm.
Derivation
- 1The real vector space underlying M_n(C) is finite-dimensional, so a subset is compact exactly when it is closed and bounded.
- 2For n at least two, place a parameter t in the upper-right entry of a two-by-two nilpotent block. Its spectral radius stays zero while its operator norm equals t, so the general spectral-radius family is unbounded.
- 3For a Hermitian matrix, the spectral theorem gives operator norm equal to spectral radius. The radius bound therefore makes the family bounded, and the defining conditions are closed.
- 4The unitary family is closed under the equations A*A=AA*=I and bounded because every member has Frobenius norm sqrt(n). It is therefore compact.
- 5When n=1, the first family is just the closed unit disk in C and is compact too. The archived selection (b),(c) thus matches the intended nontrivial case n at least two, with the scalar case recorded separately.
Final answer
For n at least two, the compact families are (b) and (c). If n=1 is admitted, (a) is compact as well.
Free sample 3
Official question 4.8 · Singular solutions and a reciprocal-square substitution
- Source locator
- Section 4: Calculus and Differential Equations, official question 4.8, paper PDF page 8
- Review note
- Independently derived and cross-reviewed; singular solution, linear transformation, integrating factor, real-domain restrictions, and archived-key coverage boundary checked · 2026-07-28
Problem summary
Solve a nonlinear first-order equation without losing the identically zero solution excluded by the suggested reciprocal-square substitution.
Core idea
Record the zero solution first, then linearise each nonzero branch with z=1/y^2 and impose the positivity and interval restrictions required for a real solution.
Derivation
- 1The identically zero function satisfies the original equation and must be separated before dividing by a power of y.
- 2On a connected interval where y is nonzero, set z=y^(-2). The equation becomes z'-(2/x)z=-2x^3 on intervals that do not contain x=0.
- 3Multiplication by the integrating factor x^(-2) gives (x^(-2)z)'=-2x, hence z=Cx^2-x^4.
- 4A real nonzero branch requires Cx^2-x^4>0. Thus C is positive, 0<|x|<sqrt(C), and y has a constant sign on each connected branch.
- 5The resulting branch is y=sigma/(|x|sqrt(C-x^2)), with sigma equal to 1 or -1. It diverges at x=0, so it cannot be joined to a finite nonzero solution there; uniqueness rules out other branches.
Final answer
The complete real solution set contains y identically equal to zero. Every nonzero branch satisfies 1/y^2=Cx^2-x^4>0, equivalently y=sigma/(|x|sqrt(C-x^2)) for C>0 on a connected interval inside (0,sqrt(C)) or (-sqrt(C),0).
Full-edition scope
- Original, step-by-step solutions for all 50 numbered questions
- Separate treatment of 54 addressable prompts, including both parts of Questions 1.3, 1.7, 2.9, and 3.5
- 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
- Explicit treatment of the n=1 boundary in Question 3.10
- Separate treatment of the zero solution omitted from the archived Question 4.8 key coverage
- Fixed-endpoint and free-endpoint interpretations distinguished for Question 4.10
- 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 fabricated count of physical answer fields, because the separate answer booklet is not in the archive
- A claim that an official erratum exists for Questions 3.10, 4.8, or 4.10
- 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 matrix size, a singular substitution, or endpoint convention changes the conclusion, the guide states both readings 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 24 January 2015 under Doctoral. This page uses “NBHM Doctoral 2015” as the searchable edition name while preserving the paper's historical title.
Independent answer-key comparison
Question 3.10 was rederived in finite-dimensional matrix space: for n at least two, a nilpotent nonnormal family has spectral radius zero and unbounded norm, while the Hermitian and unitary families are compact. If n=1 is admitted, the first family is the compact closed unit disk, so the archived (b),(c) selection is reported for the intended nontrivial matrix-size case. For Question 4.8, the archived relation 1/y^2=Cx^2-x^4 correctly describes nonzero real branches where the right side is positive, but the reciprocal-square substitution excludes the additional solution y identically equal to zero. Question 4.10 gives the archived trigonometric family under the standard fixed-endpoint extremal convention; free endpoints impose natural boundary conditions and leave only the zero member. These are independent analytical 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 2015 question paperResearch Scholarships Screening Test PDFOfficial 2015 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
- 2fec7a4c2b2ff7aba4dd269edfe8f74a8a1bc681c3c4421db32ade249b24483b
- Official answer-key SHA-256
- 21b1ce8e9525f4bd86f593c03b543975ae0b5904147c14048c2c15a1e34cdcfe
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 2015 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 3.10, 4.8, and 4.10 independently rederived with every boundary explicit
- Historical title, Doctoral archive classification, and scoring rules verified
- Rights review completed for paid distribution
- Merchant and 2015 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 201450 questions · 53 prompts · notation, domain, regularity, and admissibility analysisNBHM Doctoral 201650 questions · 53 prompts · archived-key and interpretation analysisNBHM Doctoral 201750 questions · 51 prompts · explicit convention and quantifier analysisNBHM Doctoral 201850 questions · 51 prompts · three literal-hypothesis comparisons
Frequently asked questions
NBHM Doctoral 2015 guide details
Does this guide reproduce the official NBHM 2015 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 2015 solution guide?
The current 105-page edition covers all 50 numbered questions and 54 addressable solution 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 1.3, 1.7, 2.9, and 3.5 each contain two separately requested parts. The guide explains every part while preserving the one-point, no-partial-credit treatment of each numbered question.
Why does the Question 3.10 answer separate n=1 from n at least two?
For n at least two, nilpotent nonnormal matrices can have spectral radius zero and unbounded norm. When n=1, the same condition describes the compact closed unit disk. The guide records this scalar boundary while treating the archived (b),(c) selection as the intended nontrivial-size result.
Why does the Question 4.8 solution include y identically equal to zero?
The reciprocal-square substitution is defined only where y is nonzero. The archived relation correctly describes nonzero branches on their permitted intervals, while direct substitution shows that y identically equal to zero is an additional solution not listed in the archived key.
Why does the Question 4.10 answer discuss endpoint conventions?
The standard fixed-endpoint interpretation gives the archived Euler-Lagrange family. If both endpoints are instead free, natural boundary conditions reduce that family to the zero solution. The guide states both readings without claiming an official erratum.
Is the complete 2015 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 2015 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 2015 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.