Section 1
Algebra
1.1–1.10
Groups, rings, linear algebra, field structure, and spectral arguments.
Doctoral archive · 2013English · IndiaComplete 106-page edition
NBHM Doctoral 2013 worked solutions
Prepare for NBHM Doctoral and PhD-level mathematics with independent derivations for all 50 numbered questions and 50 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 · 50 addressable promptsDotted official source locatorsIndependent cross-review
Direct format answer
How was the NBHM Doctoral 2013 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 2013
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 50 addressable solution prompts. Each of the 50 numbered questions maps to one complete solution chapter. 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 1.4, 2.9, 3.7, 4.1, 4.4, 5.7 from the current 106-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 2013 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 · Integral domains and the n=1 matrix boundary
- Source locator
- Section 1: Algebra, official question 1.4, paper PDF page 3
- Review note
- Independently derived and cross-reviewed; all three ring tests and the n=1 boundary checked · 2026-07-28
Problem summary
Classify a polynomial ring, a full matrix ring, and an analytic-function ring by the absence of zero divisors, while preserving the dimension-one edge case.
Core idea
Leading terms settle the polynomial ring, matrix units expose zero divisors for n at least 2, and the identity theorem forces one analytic factor to vanish on the connected disc.
Derivation
- 1Two nonzero real polynomials have a nonzero product because their leading coefficients multiply and their degrees add.
- 2For n at least 2, the nonzero matrix units E11 and E22 satisfy E11E22=0, so the full matrix ring is not an integral domain.
- 3At n=1 the matrix ring is the real field itself, so the matrix statement changes truth value at this literal boundary.
- 4If analytic functions f and g satisfy fg=0 and f is nonzero somewhere, continuity makes f nonzero on a neighbourhood.
- 5The second factor vanishes on that neighbourhood, and the identity theorem makes it vanish on the whole connected disc.
Final answer
For the intended n≥2 case, the integral domains are the polynomial ring and the analytic-function ring. If n=1 is admitted, the matrix ring is also an integral domain.
Free sample 2
Official question 4.1 · Mixed-boundary eigenvalues and a local key typo
- Source locator
- Section 4: Applied Mathematics, official question 4.1, paper PDF page 9
- Review note
- Independently derived and cross-reviewed; all spectral cases, energy identity, and archived-key typography checked · 2026-07-28
Problem summary
Solve a second-order eigenvalue problem with a Dirichlet condition at one endpoint and a Neumann condition at the other, then verify the variable that the archived key omits.
Core idea
The two boundary conditions select half-integer sine modes. An energy identity independently excludes zero and negative eigenvalues.
Derivation
- 1For a positive eigenvalue λ=μ², the condition at x=0 removes the cosine term.
- 2The derivative condition at x=1 requires cos μ=0, so μ=(2n+1)π/2 for n=0,1,2,….
- 3For λ=0 the two boundary conditions force the affine solution to vanish identically.
- 4For λ<0 the hyperbolic solution also collapses to zero because cosh μ is positive.
- 5Multiplying the equation by the eigenfunction and integrating gives λ∫u²=∫(u′)², confirming that every nonzero mode has λ>0.
Final answer
λn=((2n+1)π/2)² and un(x)=C sin((2n+1)πx/2), for n=0,1,2,… and C≠0.
Free sample 3
Official question 5.7 · Shortest constrained broken path by reflection
- Source locator
- Section 5: Miscellaneous, official question 5.7, paper PDF page 10
- Review note
- Independently derived and cross-reviewed; reflection, equality, and segment feasibility checked · 2026-07-28
Problem summary
Minimise a three-segment path that must meet two parallel horizontal segments in order, while verifying that the unfolded straight line remains feasible.
Core idea
Reflect the endpoints across the two constraint lines. The broken path then has the same length as a path from one reflected endpoint to the other, so the triangle inequality supplies a sharp straight-line lower bound.
Derivation
- 1Reflect the starting point across the first horizontal line, preserving the first segment length for every permitted contact point.
- 2Reflect the endpoint across the second horizontal line, preserving the last segment length.
- 3The three-segment length is at least the distance between the reflected endpoints, which is √37.
- 4The straight line between those reflected endpoints meets the first line at x=1/6 and the second at x=2/3.
- 5Both abscissae lie in the permitted unit segments, so equality is feasible and the lower bound is attained.
Final answer
The minimum is √37, attained at P=(1/6,0) and Q=(2/3,3).
Full-edition scope
- Original, step-by-step solutions for all 50 numbered questions
- One complete solution chapter for each of the 50 addressable prompts
- 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 n=1 boundary analysis for the matrix-ring and compactness questions
- A branch-complete treatment of complex exponentiation in Question 2.9
- A corrected eigenfunction derivation and transparent archived-key typography note for Question 4.1
- A strict-maximum and unbounded-below classification for Question 4.4
- The transform domain in Question 4.5 and the corrected Simpson-formula punctuation in Question 4.9
- The regular-endpoint or vanishing-bracket hypothesis required for the weighted-orthogonality conclusion in Question 4.7
- 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 fabricated count of physical answer fields, because the separate answer booklet is not in the archive
- A claim that an official erratum exists for any dimension, notation, domain, or key-typography 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, branch, endpoint, transform domain, extremum class, or key typography 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 19 January 2013 under Doctoral. This page uses “NBHM Doctoral 2013” as the searchable edition name while preserving the paper's historical title.
Independent answer-key comparison
Questions 1.4, 3.5, and 3.7 have dimension-one boundaries: the archived selections use the intended n≥2 matrix setting, while the guide also states what changes at n=1. Question 2.9 uses the multivalued logarithm convention for complex exponentiation. In Question 4.1, the archived key gives the correct eigenvalues but omits x from the sine argument; the guide derives the required x-dependent eigenfunction and records this as a local key typo. Question 4.4's stationary path is the unique strict maximiser and the functional is unbounded below. Question 4.5 requires s>0 in the real Laplace convention, or the corresponding right half-plane for complex s. Question 4.7's weighted-orthogonality conclusion uses the standard regular-endpoint, or vanishing Lagrange-bracket, interpretation; the weaker literal open-interval assumptions do not by themselves control the boundary term or weighted-integral convergence. Question 4.9 uses the standard Simpson formula without the key's harmless extra closing parenthesis. 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 2013 question paperResearch Scholarships Screening Test PDFOfficial 2013 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
- d518dcec18fbac406d4a12469f880573f33c182f42ca87f9a3fdfa1a6ebc5519
- Official answer-key SHA-256
- fe968b29877ea87f70577abc040cafd31d5bf53827007a189394f0980c149fe4
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 2013 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 50 addressable prompts completed
- Author review and independent cross-review recorded across all five sections
- Questions 1.4, 2.9, 3.5, 3.7, 4.1, 4.4, 4.5, 4.7, and 4.9 independently rederived with every material boundary explicit
- Historical title, Doctoral archive classification, and scoring rules verified
- Rights review completed for paid distribution
- Merchant and 2013 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 · logical, convergence, iteration, and archived-key boundariesNBHM Doctoral 201450 questions · 53 prompts · notation, transform, and admissibility boundariesNBHM Doctoral 201550 questions · 54 prompts · scalar, singular-solution, and endpoint boundariesNBHM Doctoral 201650 questions · 53 prompts · archived-key and interpretation analysisNBHM Doctoral 201750 questions · 51 prompts · explicit convention and quantifier analysis
Frequently asked questions
NBHM Doctoral 2013 guide details
Does this guide reproduce the official NBHM 2013 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 2013 solution guide?
The current 106-page edition covers all 50 numbered questions and 50 addressable prompts, with dotted source labels, complete derivations, independent checks, and problem-specific error notes.
How was the 2013 paper scored?
The paper had five sections of ten one-point questions. Only the candidate's best four section scores counted, so the maximum counted score was 40. No partial credit was provided for multiple-selection questions.
Why does the Question 1.4 solution discuss n=1?
For n at least 2 the full real matrix ring has zero divisors, matching the archived selection. At n=1 it is just the real field and is an integral domain. The guide states both the intended reading and the literal edge case.
What is different about the Question 4.1 eigenfunction?
The archived key gives the correct eigenvalues but visibly omits x from the sine argument. The guide derives the boundary-condition-compatible function C sin((2n+1)πx/2) and identifies the omission as a local key typo, not a paper error.
Why are Questions 4.4, 4.5, and 4.7 qualified?
The stationary curve in Question 4.4 is the unique strict maximiser and the functional is unbounded below. The Laplace-transform identity in Question 4.5 requires s>0 in the real convention, or the corresponding right half-plane in the complex convention. The weighted-orthogonality conclusion in Question 4.7 uses the standard regular-endpoint, or vanishing Lagrange-bracket, interpretation.
Is the complete 2013 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 2013 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 2013 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.