Practices 1–4
Multivariable Taylor theory
Mixed coefficients, Hessian data, degenerate critical points and approximation control
Original GATE Mathematics practice
GATE Mathematics 2027 — Revised-Syllabus Mastery Workbook
Build working command of the 15 revised-focus areas with a 60-question original paper, complete derivations, verification stress tests and targeted remediation.
- 125 pages
- 60 original questions · 20 MCQ · 20 MSQ · 20 NAT · 15 modules
- English
- ₹1,499 held catalogue price
Official sources and editorial boundary
Official GATE 2027 Mathematics syllabusFree official source. Linked for verification; not bundled or sold by InshiHub.Official GATE 2026 Mathematics syllabusFree official source. Linked for verification; not bundled or sold by InshiHub.Official GATE 2027 question-paper patternUse the official pattern page for current response types, marking and later changes.Free 20-question revised-syllabus diagnosticStart with the shorter free diagnostic if you need to identify which revised-focus areas deserve priority.
Independent and unofficial InshiHub publication. Not affiliated with, endorsed by, sponsored by or approved by IIT Madras, any IIT or GATE. Every practice problem and solution is independently authored. No official question, answer choice, figure, logo or paper is included.
Format facts
- Official paper language and delivery
- English · Computer Based Test
- Official response formats
- MCQ, MSQ and NAT
- Official mark values
- One-mark and two-mark questions
- Negative marking
- Applies to incorrect MCQ responses; not to MSQ or NAT
- MSQ partial marking
- No partial marking under the official pattern
- Workbook boundary
- Original revised-focus practice; not a full-syllabus mock or official paper
Coverage map
Practices 5–8
Jacobians and change of variables
Forward and inverse determinants, transformed regions, integrals and singular-map conditions
Practices 9–12
Baire category
Closed covers, nowhere-dense sets, completeness and a missing-hypothesis counterexample
Practices 13–16
Uniform continuity
Sequential tests, compact domains, Lipschitz bounds and unbounded-domain failure
Practices 17–20
Lp spaces
Endpoint membership, norms, finite-measure inclusions and almost-everywhere equality
Practices 21–24
Laplace transforms
Definition, shifts, convolution, initial data and an ODE application
Practices 25–28
Finite Abelian groups
Primary decomposition, isomorphism classes, element orders and invariant factors
Practices 29–32
Bounded and compact operators
Identity counterexample, finite-rank approximation, diagonal errors and norm closure
Practices 33–36
Separability and dual spaces
Dense sets, nonseparable examples, functional norms and dual-basis reasoning
Practices 37–40
Maximum principle
Boundary extrema, the strong principle, uniqueness and explicit harmonic data
Practices 41–44
Path connectedness
Standard paths, continuous images, products, retracts and component counting
Practices 45–48
Sequential compactness
Subsequences, metric-space equivalence, continuous images and endpoint failures
Practices 49–52
Limit-point compactness
Infinite-subset criteria, accumulation points, metric equivalence and subspace traps
Practices 53–56
Tychonoff theorem
Arbitrary compact products, box topology, cylinder neighbourhoods and a metric trap
Practices 57–60
Tietze extension theorem
Normality, closed subsets, bounded extensions and failure when a condition is removed
Five complete HTML samples
Module 1 · Practice 1 · NAT · 1 mark
Sample 1 — One-mark NAT: Mixed coefficients in a multivariable Taylor polynomial
Find a mixed second-degree coefficient in the Taylor polynomial of an exponential function of two variables.
Core idea
Expand the one-variable exponential in the linear form 2x−y and track the cross term without halving it twice.
- 1Set h=2x−y and use e^h=1+h+h²/2 through total degree two.
- 2Expand (2x−y)²/2 as 2x²−2xy+y²/2.
- 3Read the coefficient of xy as −2.
- Final response
- −2.
- Independent check
- Multiply the linear terms 2x and −y in e^(2x)e^(−y); their product independently gives −2xy.
Common pitfall: In the Hessian form, the two symmetric off-diagonal terms already create the factor of two. Do not halve the cross term twice.
Module 2 · Practice 7 · NAT · 2 marks
Sample 2 — Two-mark NAT: Jacobian and the area of a rotated square
Transform a diamond-shaped region defined by x+y and x−y inequalities and compute its area.
Core idea
The substitution u=x+y, v=x−y turns the region into a square; the inverse Jacobian converts its area element back.
- 1Transform the inequalities into −2≤u≤2 and −2≤v≤2.
- 2Compute the uv-square area as 16.
- 3Multiply by |∂(x,y)/∂(u,v)|=1/2 to obtain area 8.
- Final response
- 8.
- Independent check
- The original rhombus has two diagonals of length 4, so its area is (1/2)·4·4=8.
Common pitfall: The transformed square lives in the uv-plane. Forgetting the inverse Jacobian gives an answer twice as large.
Module 6 · Practice 21 · MSQ · 1 mark
Sample 3 — One-mark MSQ: Definition and convergence of a Laplace transform
Audit four statements about the defining integral, value and convergence region of the Laplace transform of t.
Core idea
Evaluate the improper integral by parts and keep the condition s>0 attached to the formula 1/s².
- 1Write the defining integral ∫₀∞t e^(−st)dt.
- 2For s>0, integration by parts gives 1/s².
- 3At s=0 the integral diverges, and for s<0 the integrand grows exponentially.
- Final response
- A, C and D.
- Independent check
- Differentiate L{1}=1/s and use L{tf}=−dL{f}/ds to recover L{t}=1/s².
Common pitfall: A transform formula is incomplete without its region of convergence.
Module 11 · Practice 41 · MCQ · 1 mark
Sample 4 — One-mark MCQ: Path components separated by a coordinate sign
Identify the only non-path-connected set among four familiar subsets of the plane.
Core idea
The two branches of xy=1 cannot be joined because the continuous first coordinate cannot change sign without passing through zero.
- 1Separate the hyperbola into its x>0 and x<0 branches.
- 2Use the intermediate value theorem to rule out a path between them.
- 3Construct paths or use convexity for the punctured plane, circle and upper half-plane.
- Final response
- A — the hyperbola xy=1.
- Independent check
- Project a hypothetical path onto its first coordinate; it would be a continuous real function changing sign without taking the value zero.
Common pitfall: A set described by one equation need not have one path component.
Module 15 · Practice 57 · MSQ · 1 mark
Sample 5 — One-mark MSQ: Hypotheses and range control in Tietze extension
Select the correct theorem conditions and range conclusion for bounded real-valued Tietze extension.
Core idea
The ambient space is normal, the original domain is closed and a bounded interval range can be preserved; uniqueness is not asserted.
- 1State normality of X and closedness of A.
- 2Apply the range-preserving form to f:A→[−1,1].
- 3Reject uniqueness by exhibiting two distinct extensions of zero from a singleton.
- Final response
- A and C.
- Independent check
- On A={0} in R, both the zero function and x/(1+|x|) are [−1,1]-valued extensions, so uniqueness fails independently.
Common pitfall: Metric spaces are normal, but the subset must still be closed before Tietze applies.
Six-page exact-master preview
Cover plus one audited page for each HTML sample. The complete paid master is not stored in the public application.
Product boundary
Included
- A 60-question original practice paper with a balanced MCQ, MSQ and NAT mix.
- Four progressive questions for each of 15 revised-focus modules.
- Complete independently authored derivations and exact final responses.
- A useful verification or stress test for every question, using substitution, a second representation, boundary cases, option audits or counterexamples as appropriate.
- Problem-specific misconception diagnosis and a short remediation route.
- A revision map for converting setup, theorem-condition and execution errors into study actions.
Not included
- Any official GATE question, option, figure, scan, logo, watermark or paper layout.
- General Aptitude and complete coverage of every unrevised Mathematics syllabus area.
- A prediction of the GATE 2027 paper or a claim that a concept is examinable for the first time.
- A score, rank, admission, scholarship or result guarantee.
- Institutional affiliation, endorsement, sponsorship or approval.
Study loop
- 1Attempt one four-question module without the solution section.
- 2Mark the first step where your reasoning diverged, not only the final response.
- 3Read the decisive theorem or definition and reproduce the derivation from a blank page.
- 4Run the independent check and record the named misconception if it applies.
- 5Retry the full module after 48 hours, then return to complete-syllabus preparation.
Questions about this guide
What is inside the GATE Mathematics 2027 revised-syllabus workbook?
The held 125-page English workbook contains a 60-question original practice paper and 60 detailed solutions. Fifteen revised-focus modules receive four questions each, progressing from foundation to theorem-condition or counterexample audit.
Does this workbook reproduce any official GATE question?
No. Every prompt, answer choice, derivation, check, diagnosis and remediation note is independently authored by InshiHub. The official 2027 and 2026 syllabi are linked only as sources for the wording comparison.
Which GATE response formats are used?
The workbook uses an even mix of 20 MCQ, 20 MSQ and 20 NAT items with one-mark and two-mark labels. These labels follow the official format categories but the questions themselves are original.
Does it cover the complete GATE Mathematics syllabus?
No. It concentrates on 15 concepts newly named, broadened or made more explicit in the 2027 syllabus wording. It does not replace full-syllabus study or General Aptitude preparation.
Does a listed concept become examinable for the first time in 2027?
Not necessarily. The workbook makes no first-time-examinability claim. Some concepts were already implicit, named in a narrower form or examinable under earlier syllabus language.
How were the PDF and formulas checked?
The master passed qpdf syntax checks, PDF/UA-2 and both WTPDF profiles. All fonts are embedded with Unicode mappings, and all 1,282 mathematical fragments have attached MathML.
What is the catalogue price?
The held one-time catalogue price is ₹1,499, inclusive of applicable tax. It is not an active offer while seller, tax, payment, delivery, refund and reconciliation gates remain open.
Can I buy or reserve the workbook now?
No purchase, reservation, waitlist or release-alert collection is enabled. Checkout will appear only after the exact product and all commercial release gates have been independently verified.
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GATE Mathematics 2026 — Detailed SolutionsAudit a complete previous paper with independent derivations and key checks.GATE Mathematics 2025 — Detailed SolutionsUse another full Mathematics paper for transfer practice and timing.IIT JAM Mathematics 2025 — Detailed SolutionsBroaden proof and calculation practice across a related graduate mathematics exam.
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