Section 1
Algebra
1.1–1.10
Groups, rings, linear algebra, field structure, and spectral arguments.
Doctoral archive · 2018English · IndiaComplete 103-page edition
NBHM Doctoral 2018 worked solutions
Prepare for NBHM Doctoral and PhD-level mathematics with independent derivations for all 50 numbered questions and 51 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 · 51 addressable promptsDotted official source locatorsIndependent cross-review
Direct format answer
How was the NBHM Doctoral 2018 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
20 January 2018
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 51 addressable solution prompts. This is because Question 5.1 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
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 1.7, 2.8, 2.10, 3.8, 4.9, 5.10 from the current 103-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 2018 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 2.8 · Fourier sine coefficients and missing endpoint periodicity
- Source locator
- Section 2: Analysis, official question 2.8, paper PDF page 5
- Review note
- Independently derived and cross-reviewed; explicit counterexample checked; literal-hypothesis conclusion differs from official key · 2026-07-28
Problem summary
Decide which decay and absolute-summability claims are forced for the sine coefficients of an arbitrary continuously differentiable function on the stated closed interval.
Core idea
Integration by parts leaves an endpoint term of order one over n. Because the hypotheses do not identify the two endpoint values, that term can prevent the stronger conclusions.
Derivation
- 1Integrate the sine coefficient by parts. It becomes an endpoint difference divided by n plus an oscillatory integral of the derivative divided by n.
- 2Continuity of the derivative bounds both contributions by a constant divided by n, so the coefficient itself tends to zero.
- 3Test the stronger claims with the admissible function f(t) = t. Its sine coefficient is minus 2 pi times (-1)^n divided by n.
- 4Then n times the coefficient alternates between two nonzero values, so it does not tend to zero.
- 5The absolute value of n cubed times the coefficient cubed is the nonzero constant 8 pi cubed, so the proposed absolute series cannot converge.
Final answer
Under the literal printed hypotheses, statement (a) is true and statements (b) and (c) are false; only (a) qualifies.
Free sample 2
Official question 3.8 · Compact image versus relatively compact image
- Source locator
- Section 3: Topology, official question 3.8, paper PDF page 7
- Review note
- Independently derived and cross-reviewed; closure counterexample and finite-dimensional case checked; literal-hypothesis conclusion differs from official key · 2026-07-28
Problem summary
Determine which of three linear maps sends the stated closed unit ball to a set that is itself compact.
Core idea
Arzelà–Ascoli can establish relative compactness without making an image closed. An explicit smooth sequence whose uniform limit is not continuously differentiable separates the two notions.
Derivation
- 1For the inclusion from continuously differentiable functions into continuous functions, build smooth approximations to the absolute-value cusp at one half.
- 2Each approximation lies in the stated unit ball and converges uniformly to a function that is not continuously differentiable.
- 3The image is therefore not closed in the target space and cannot itself be compact, even though it is relatively compact.
- 4For the right shift on square-summable sequences, shifted unit vectors remain pairwise a fixed positive distance apart, so no convergent subsequence exists.
- 5The coordinate truncation image is exactly a closed bounded ball in a ten-dimensional space and is compact.
Final answer
Under the literal phrase that the image T(B) is compact, statements (a) and (b) are false and statement (c) is true; only (c) qualifies.
Free sample 3
Official question 5.10 · Dual-norm optimisation with all equality cases
- Source locator
- Section 5: Miscellaneous, official question 5.10, paper PDF page 9
- Review note
- Independently derived and cross-reviewed; Hölder equality, multiplier check, uniqueness, and degenerate case verified; official-key match · 2026-07-28
Problem summary
Maximise a real linear functional on the unit sphere for the cubic norm, including the equality vector and the all-zero coefficient case.
Core idea
Hölder's inequality with conjugate exponents three halves and three gives the sharp bound, and its equality condition determines the maximising coordinates.
Derivation
- 1Let A be the sum of the three coefficient magnitudes raised to the three-halves power.
- 2Apply the sign estimate followed by Hölder's inequality to bound the functional by A raised to the two-thirds power.
- 3When A is positive, choose each coordinate with the coefficient's sign and square-root magnitude, normalised by A raised to one third.
- 4The cubic powers of those coordinates sum to one, and direct substitution attains the upper bound.
- 5If every coefficient is zero, the objective is identically zero and every feasible point is a maximiser.
Final answer
The maximum is (|a|^(3/2) + |b|^(3/2) + |c|^(3/2))^(2/3). For a nonzero coefficient vector the normalised signed square-root vector is the unique maximiser; if all coefficients vanish, every feasible point maximises.
Full-edition scope
- Original, step-by-step solutions for all 50 numbered questions
- Separate treatment of 51 addressable solution prompts, including both parts of Question 5.1
- 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 literal-hypothesis analysis for Questions 2.8, 2.10, and 3.8 where independent conclusions differ from the official key
- A source note distinguishing the paper's historical title from its Doctoral archive classification
- A dated correction process and public correction log after release
What is not included
- The official question paper, answer choices, logos, and answer-key text
- 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 the three answer-key differences
- 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.
2
Independent first derivation
Each result is proved or disproved from the printed hypotheses before the final answer is compared with the official key.
3
Cross-author mathematical audit
A reviewer who did not write the solution rechecks the theorem use, algebra, counterexamples, edge cases, and final conclusion.
4
Transparent source differences
Where literal hypotheses and the official key lead to different conclusions, the guide states the difference, gives a reproducible counterexample, and does not invent an official erratum.
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 20 January 2018 under Doctoral. This page uses “NBHM Doctoral 2018” as the searchable edition name while preserving the paper's historical title.
Independent answer-key comparison
Questions 2.8, 2.10, and 3.8 were independently rederived from the literal printed hypotheses before the official key was consulted. Reproducible counterexamples show that one or more labels in each key entry are not forced as written. The guide records the mathematical conclusion and the key comparison separately. No official erratum was located, and InshiHub does not present these findings as an official correction.
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 2018 question paperResearch Scholarships Screening Test PDFOfficial 2018 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
- c243b031fe9ad55c02f4a3a4effc808022bc6b2ac1efac33a9a949eb146fe42f
- Official answer-key SHA-256
- ff3e778550a5a13a09fd1a29078ec4047b98b868241ee42d3802ffe558417e9d
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 2018 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 51 addressable solution prompts completed
- Author review and independent cross-review recorded across all five sections
- Questions 2.8, 2.10, and 3.8 rederived from the literal printed hypotheses
- Historical title, Doctoral archive classification, and scoring rules verified
- Rights review completed for paid distribution
- Merchant and 2018 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
Frequently asked questions
NBHM Doctoral 2018 guide details
Does this guide reproduce the official NBHM 2018 paper?
No. It contains independently written solutions and short, non-substituting topic summaries. Use the official IMSc archive links on this page for the question paper and answer key.
What is covered in the NBHM Doctoral 2018 solution guide?
The current 103-page edition covers all 50 numbered questions and 51 addressable solution prompts, with dotted source labels, complete derivations, independent checks, and problem-specific error notes.
Why is the maximum score 40 when the paper has 50 questions?
The paper has five sections of ten one-point questions. Its printed rule counts the best four section scores, so the maximum counted score is 40.
Why are there 51 solution prompts for 50 numbered questions?
Question 5.1 asks for two separately labelled results. The guide explains both while preserving the paper's one-point, no-partial-credit treatment of that numbered question.
Why do three InshiHub conclusions differ from the official key?
For Questions 2.8, 2.10, and 3.8, direct derivation from the literal printed hypotheses produces explicit counterexamples to one or more key labels. The guide shows each construction and records the difference transparently. No separate official erratum was located, so InshiHub does not describe the differences as official corrections.
Is the complete 2018 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 2018 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 2018 answers checked?
Each solution was independently derived before comparison with the official key, then cross-reviewed by someone who did not author that range. Review covers theorem use, calculations, counterexamples, boundary cases, source locators, and rendered PDF layout.
Is InshiHub affiliated with NBHM or IMSc?
No. InshiHub is an independent publisher of worked explanations. IMSc hosts the official source files; this guide links to them and does not reproduce the official question text, answer choices, figures, logos, or key wording.