Section 1
Algebra
1.1–1.10
Groups, rings, linear algebra, field structure, and spectral arguments.
Doctoral archive · 2019English · IndiaComplete 109-page edition
NBHM Doctoral 2019 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 locatorsFive-section mathematical review
Direct format answer
How was the NBHM Doctoral 2019 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
19 January 2019
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 because Question 2.9 contains two separately explained parts inside one numbered, one-point question. The official source does not establish a count of physical answer fields, 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 eigenvalue problems.
Section 5
Miscellaneous
5.1–5.10
Probability, combinatorics, geometry, number patterns, and mixed reasoning.
Six-page visual preview
Inspect one guide page from each selected problem
The six preview slots cover Questions 1.3, 2.9, 3.10, 4.3, 4.9, and 5.10 from the current 109-page manuscript. Every preview is an InshiHub explanation page, never a scan of the official paper or answer key.
Crawlable HTML samples
Three free NBHM Doctoral 2019 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 1.10 · Spectral structure and a real matrix cube root
- Source locator
- Section 1: Algebra, official question 1.10, paper PDF page 3
- Review note
- Independently derived; spectral-theorem and determinant checks completed; official-key match · 2026-07-28
Problem summary
Test diagonalisation, orthogonal-eigenbasis, and real odd-root claims for a specified invertible symmetric two-by-two matrix.
Core idea
The real spectral theorem resolves the eigenvector claims and reduces the cube-root construction to taking real cube roots of two nonzero real eigenvalues.
Derivation
- 1Write the matrix as A with rows (19, 2019) and (2019, 1). It is real and symmetric.
- 2By the real spectral theorem, an orthogonal matrix Q diagonalises A with real eigenvalues. This proves real diagonalisation and supplies an orthonormal eigenbasis.
- 3The determinant is 19 minus 2019 squared, which is nonzero, so neither eigenvalue is zero.
- 4Take the real cube root of each eigenvalue, including the negative one, and conjugate that diagonal matrix by Q.
- 5The resulting real matrix is invertible and its cube is A, so all three tested claims hold.
Final answer
Statements (a), (b), and (c) are all true.
Free sample 2
Official question 2.9 · Laurent expansion on an annulus and a contour integral
- Source locator
- Section 2: Analysis, official question 2.9 parts (a) and (b), paper PDF page 5
- Review note
- Independently derived; convergence regions and residue calculation checked; official-key match · 2026-07-28
Problem summary
Expand a rational function in the annulus between two poles, then evaluate its integral around a positively oriented intermediate circle.
Core idea
Partial fractions separate the poles; the inner pole contributes negative powers, the outer pole contributes nonnegative powers, and the coefficient of the inverse first power fixes the contour integral.
Derivation
- 1Decompose negative one over (z minus 1)(z minus 2) as one over z minus 1, minus one over z minus 2.
- 2For modulus greater than 1, expand the first term as the sum of z to the power negative n minus 1 for n starting at zero.
- 3For modulus less than 2, expand the second contribution as the sum of z to the power n divided by 2 to the power n plus 1.
- 4Both expansions are valid together precisely on the annulus 1 less than the modulus of z less than 2.
- 5A counter-clockwise circle of radius 3/2 encloses only the pole at 1, whose residue is 1, so the integral is 2 pi i.
Final answer
Part (a): the Laurent series is the sum over n at least zero of z^(-n-1), plus the sum over n at least zero of z^n/2^(n+1), valid for 1 < |z| < 2. Part (b): the integral is 2 pi i.
Free sample 3
Official question 4.9 · Mixed-boundary eigenvalues and eigenfunctions
- Source locator
- Section 4: Calculus and Differential Equations, official question 4.9, paper PDF page 8
- Review note
- Independently derived; all parameter signs and both boundary conditions checked; official-key match · 2026-07-28
Problem summary
Classify every nonzero solution of a second-order eigenvalue equation with a derivative condition at one endpoint and a value condition at the other.
Core idea
Separate positive, zero, and negative spectral parameters; the mixed boundary conditions retain exactly the half-integer cosine frequencies.
Derivation
- 1For a positive parameter written as mu squared, the general solution is A cos(mu x) plus B sin(mu x).
- 2The derivative condition at zero forces B to vanish, while the value condition at one requires cos(mu) to vanish for a nonzero solution.
- 3Thus mu equals (2n + 1) pi/2 for n at least zero, giving eigenvalue (2n + 1)^2 pi^2/4.
- 4At parameter zero, the affine solution is forced to vanish. For a negative parameter, the hyperbolic solution is also forced to vanish because cosh(mu) is positive.
- 5Every nonzero scalar multiple of the retained cosine is an eigenfunction, and the sign analysis proves that no other real parameters work.
Final answer
For n = 0, 1, 2, ... the eigenvalue is (2n + 1)^2 pi^2/4 and the eigenfunction is C cos((2n + 1) pi x/2), where C is any nonzero real constant.
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 2.9
- 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
- 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, figures, logos, and answer-key text
- A fabricated count of physical answer fields, which the source does not establish
- 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 mathematics
Each result is derived from first principles before the final answer is compared with the official key.
3
Format-aware review
Review preserves the one-point, all-or-nothing rule, including the two requested parts inside Question 2.9.
4
Release accountability
The edition is versioned, material corrections are dated, and paid access remains separate from editorial publication.
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 19 January 2019 under Doctoral. This page uses “NBHM Doctoral 2019” as the searchable edition name while preserving the paper's historical title. The two labels describe the same official source and are not presented as an erratum.
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, figures, logos, or answer-key text.
Official NBHM past-paper archiveDoctoral classification and source indexOfficial 2019–20 PhD scholarship noticeNBHM scheme source hosted by IMScOfficial 2019 question paperResearch Scholarships Screening Test PDFOfficial 2019 answer keyAnswer-annotated PDF hosted by IMSc
- Exam date
- Source accessed
- 2026-07-28
- Source status
- Publicly accessible source; no reuse licence located
- Official paper SHA-256
- a6cc980e2142d672b31edf6dc170576d2970f6a788ed90c8f2f07cf106d7aef3
- Official answer-key SHA-256
- 1d5a5da1daf1e11b7f08ac99edc9a43d5f28517ff10da54ff1131585b7c22723
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 2019 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
- Independent mathematical review recorded across all five sections
- Historical title, Doctoral archive classification, and scoring rules verified
- Rights review completed for paid distribution
- Merchant and 2019 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 2019 guide details
Does this guide reproduce the official NBHM 2019 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 2019 solution guide?
The current 109-page edition covers all 50 numbered questions and 51 addressable solution prompts, with exact decimal source labels, complete derivations, verification, 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 does the page say Doctoral when the paper title says Research Scholarships Screening Test?
The cover uses the historical title Research Scholarships Screening Test, while the official NBHM past-paper archive classifies the January 2019 file under Doctoral. InshiHub preserves both facts.
How were questions with more than one requested answer scored?
Each numbered question was worth one point, and the printed instructions provided no partial credit when multiple answers were requested. The guide therefore treats both parts of Question 2.9 separately while preserving its single-question scoring context.
Is the complete 2019 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 2019 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 2019 answers checked?
Each solution was independently derived before comparison with the official key. Review records theorem use, calculations, counterexamples, boundary cases, and source locators separately.
Is InshiHub affiliated with NBHM or IMSc?
No. InshiHub is an independent publisher of worked explanations. NBHM owns its examination materials, and IMSc hosts the official source files.