Section 1
Algebra
1.1–1.10
Groups, rings, linear algebra, field structure, and spectral arguments.
Doctoral archive · 2016English · IndiaComplete 101-page edition
NBHM Doctoral 2016 worked solutions
Prepare for NBHM Doctoral and PhD-level mathematics with independent derivations for all 50 numbered questions and 53 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 · 53 addressable promptsDotted official source locatorsIndependent cross-review
Direct format answer
How was the NBHM Doctoral 2016 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
23 January 2016
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 53 addressable solution prompts. This is because Questions 1.2, 1.6, and 4.9 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 1.2, 2.3, 3.8, 4.9, 5.4, 5.10 from the current 101-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 2016 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 2.3 · Approximate identities and the sign of a shrinking scale
- Source locator
- Section 2: Analysis, official question 2.3, paper PDF page 5
- Review note
- Independently derived and cross-reviewed; compact-support convergence, Jacobian orientation, one-sided limits, and constant-function counterexample checked · 2026-07-28
Problem summary
Evaluate a compactly supported normalized kernel against a continuous function while distinguishing positive and negative approaches of the scale parameter.
Core idea
Rescale to a fixed interval and retain the absolute Jacobian. The sign that remains outside the normalized integral determines whether the printed two-sided limit exists.
Derivation
- 1After the change of variables x equals epsilon times t, the whole-line Jacobian is the absolute value of epsilon. The normalized expression therefore equals the sign of epsilon times the integral of rho(t)f(epsilon t).
- 2The kernel is supported on a fixed compact interval. Continuity at zero makes f(epsilon t) converge uniformly there to f(0), and normalization makes the integral converge to f(0).
- 3The positive-scale limit is f(0), while the negative-scale limit is minus f(0). The printed two-sided limit therefore exists exactly when f(0) is zero.
- 4For the constant test function f equals one, the expression is exactly the sign of epsilon. This directly disproves an unconditional two-sided value.
- 5The archived value f(0) is recovered under the customary approximate-identity convention that epsilon approaches zero through positive values.
Final answer
As printed, the right-hand limit is f(0) and the left-hand limit is -f(0); the two-sided limit exists only when f(0)=0. Under epsilon approaching zero from above, the value is f(0).
Free sample 2
Official question 3.8 · Lower semicontinuity, indicator functions, and closed graphs
- Source locator
- Section 3: Topology, official question 3.8, paper PDF page 8
- Review note
- Independently derived and cross-reviewed; all indicator sublevels, closed-graph proof, counterexample, and neutral archived-key comparison checked · 2026-07-28
Problem summary
Test lower semicontinuity through sublevel sets and determine whether a closed graph alone controls lower limits.
Core idea
List every possible sublevel set for a zero-one indicator, then use a function that diverges near one point to separate closed-graph behaviour from lower semicontinuity.
Derivation
- 1For the indicator of a set E and a threshold between zero and one, the sublevel set is the complement of E.
- 2A closed E can have an open, nonclosed complement, so the first assertion fails. An open E has a closed complement, making every indicator sublevel set closed.
- 3Define f(0)=0 and f(x)=1/x away from zero. Values along nonzero points diverge near zero, so no missing finite graph-limit point is created and the graph is closed.
- 4Its sublevel set at -1 is the half-closed interval [-1,0), which is not closed. The function is therefore not lower semicontinuous.
- 5The independent mathematical selection is (b) only. This differs from the archived selection (b),(c), and the guide reports the difference without claiming an official erratum.
Final answer
Statement (a) is false, statement (b) is true, and statement (c) is false. The independently correct selection is (b) only.
Free sample 3
Official question 5.4 · Vandermonde convolution and coefficient nonuniqueness
- Source locator
- Section 5: Miscellaneous, official question 5.4, paper PDF page 10
- Review note
- Independently derived and cross-reviewed; combinatorial proof, generating-function proof, valid-index convention, and nonuniqueness family checked · 2026-07-28
Problem summary
Construct coefficients for a binomial identity, make the out-of-range convention explicit, and determine whether one scalar relation fixes every coefficient.
Core idea
Split an r-element selection across groups of sizes k and n-k to obtain a natural Vandermonde choice, then compare the number of equations with the number of unknown coefficients.
Derivation
- 1Counting an r-element selection by the number l chosen from the k-element group yields the Vandermonde sum.
- 2A natural coefficient is C(n-k,r-l), interpreted as zero whenever r-l is outside the interval from zero to n-k.
- 3The same identity follows by taking the coefficient of z^r in the product (1+z)^k(1+z)^(n-k).
- 4The displayed relation supplies one scalar equation for k+1 coefficients, so it cannot determine them uniquely without another condition.
- 5Changing alpha_0 by kt and alpha_1 by -t preserves the sum because C(k,0)=1 and C(k,1)=k, giving an explicit family of alternatives.
Final answer
The coefficients are underdetermined. One natural choice is alpha_l=C(n-k,r-l), with value zero outside the valid lower-index range.
Full-edition scope
- Original, step-by-step solutions for all 50 numbered questions
- Separate treatment of 53 addressable prompts, including both parts of Questions 1.2, 1.6, and 4.9
- 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
- Transparent analysis of the archived-key conflict in Question 3.8
- Explicit scale, sign, trajectory, and coefficient qualifications for Questions 2.3, 4.2, 4.9, and 5.4
- 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
- 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 Question 3.8 or any interpretation-sensitive item
- 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 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 a sign, convention, or missing hypothesis 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 23 January 2016 under Doctoral. This page uses “NBHM Doctoral 2016” as the searchable edition name while preserving the paper's historical title.
Independent answer-key comparison
Question 2.3 was rederived with both signs of the printed scale: the positive-side limit is f(0), the negative-side limit is -f(0), and the archived f(0) form assumes the customary positive approximate-identity parameter. In Question 3.8, a function with f(0)=0 and f(x)=1/x away from zero has a closed graph but is not lower semicontinuous, so the independent mathematical selection is (b) only rather than the archived (b),(c). Question 4.2 has exact arc length pi|a| unless a is assumed nonnegative; Question 4.9 separates nonzero phase curves from singleton equilibria; and Question 5.4 records one natural Vandermonde choice without claiming uniqueness. 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 2016 question paperResearch Scholarships Screening Test PDFOfficial 2016 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
- 9d623b15d2b7670eefa8b03ec06e5837fbf00f9bdf79bdb1f024e9b997aab9ad
- Official answer-key SHA-256
- d4c195ca2e8051be1a77d3c17ac8f74f29295c1dc37c9dbe84b3300285555175
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 2016 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 53 addressable prompts completed
- Author review and independent cross-review recorded across all five sections
- Questions 2.3, 3.8, 4.2, 4.9, and 5.4 independently rederived with interpretation boundaries explicit
- Historical title, Doctoral archive classification, and scoring rules verified
- Rights review completed for paid distribution
- Merchant and 2016 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 201550 questions · 54 prompts · scalar, singular-solution, and endpoint boundariesNBHM Doctoral 201750 questions · 51 prompts · explicit convention and quantifier analysisNBHM Doctoral 201850 questions · 51 prompts · three literal-hypothesis comparisonsNBHM Doctoral 201950 questions · 51 addressable solution prompts
Frequently asked questions
NBHM Doctoral 2016 guide details
Does this guide reproduce the official NBHM 2016 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 2016 solution guide?
The current 101-page edition covers all 50 numbered questions and 53 addressable solution prompts, with dotted source labels, complete derivations, independent checks, and problem-specific error notes.
Why are there 53 solution prompts for 50 numbered questions?
Questions 1.2, 1.6, and 4.9 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 2.3 answer discuss two one-sided limits?
The printed scale approaches zero without a positive-side restriction. The normalized expression tends to f(0) from the positive side and -f(0) from the negative side. The archived f(0) value assumes the customary positive approximate-identity scale.
Why does the Question 3.8 result differ from the archived key?
A real-valued function can have a closed graph without being lower semicontinuous. The guide gives a complete counterexample and concludes that only (b) is mathematically valid, while describing the archived (b),(c) selection neutrally and without claiming an official erratum.
Why is the Question 5.4 coefficient formula described as one natural choice?
Vandermonde convolution supplies the binomial coefficients, but the printed relation is one equation in k+1 unknowns. The guide proves both the natural formula and its nonuniqueness, including the required zero-extension convention.
Is the complete 2016 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 2016 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 2016 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.