Across Mathematics Q11–Q65
Linear algebra
Vector spaces, matrices, eigenvalues, canonical forms and numerical linear algebra
Complete GATE Mathematics subject guide
GATE Mathematics 2025 — Detailed Solutions
Work through the complete 85-mark Mathematics subject section with 55 independent derivations, key checks and transparent treatment of two literal-source caveats.
- 59 pages
- 55 questions · 85 marks · 24 MCQ · 17 MSQ · 14 NAT
- English
- ₹1,499 held catalogue price
Official sources and editorial boundary
Official GATE 2025 Mathematics master paperFree official source. Linked for verification; not bundled or sold by InshiHub.Official GATE 2025 Mathematics final answer keyFree official source. Linked for verification; not bundled or sold by InshiHub.Official GATE 2025 paper and key indexUse the official archive for the free source paper, final key and later corrections.Free GATE 2027 revised-syllabus diagnosticCheck readiness for concepts newly named, broadened or made more explicit after these papers.
Independent and unofficial InshiHub study guide. Not affiliated with, endorsed by, sponsored by or approved by IIT Roorkee, any IIT or GATE. The official master paper and final answer key remain free at the linked official source; they are not included in this product.
Format facts
- Guide scope
- Complete Mathematics subject section Q11–Q65; General Aptitude Q1–Q10 excluded
- Questions and marks
- 55 Mathematics questions carrying 85 marks
- Response mix
- 24 MCQ · 17 MSQ · 14 NAT
- Source and key
- Official master paper and official final answer key, linked free and not bundled
- Editorial method
- Original summaries, derivations, checks and pitfalls; no official problem or option prose reproduced
- Source caveat policy
- Q62 and Q63 distinguish the literal printed conditions from the additional intended condition needed for the official numerical key
Coverage map
Across Mathematics Q11–Q65
Real and complex analysis
Sequences, series, continuity, integration, power series and analytic-function arguments
Across Mathematics Q11–Q65
Calculus and differential equations
Multivariable derivatives, transforms, ODEs, PDEs and numerical methods
Across Mathematics Q11–Q65
Abstract algebra
Groups, rings, fields, finite structures and polynomial reasoning
Across Mathematics Q11–Q65
Topology and functional analysis
Compactness, connectedness, normed spaces and operator-theoretic criteria
Across Mathematics Q11–Q65
Probability and optimisation
Probability models, statistics, linear programming and operations-research methods
Five complete HTML samples
Mathematics Q11 · MCQ · 1 mark
Sample 1 — One-mark MCQ: Rank-one matrices and diagonalizability
Classify the parameter set for which a nonzero rank-one matrix is diagonalizable, then determine its compactness and connectedness.
Core idea
For A=uwᵀ, the only possibly nonzero eigenvalue is wᵀu; the exceptional zero case is a nonzero nilpotent matrix.
- 1Use A²=(wᵀu)A to identify the spectral split.
- 2Remove the great circle wᵀu=0 from the unit sphere.
- 3Recognise the remaining two open hemispheres as disconnected and noncompact.
- Final response
- B — neither compact nor connected.
- Independent check
- When wᵀu=0, A is nonzero with A²=0, which cannot be diagonalizable; otherwise its zero and nonzero eigenspaces span the full space.
Common pitfall: A bounded subset of the sphere need not be compact when it is not closed.
Mathematics Q22 · MCQ · 1 mark
Sample 2 — One-mark MCQ: Proper quadratic maps
Test properness for one singular and one positive-definite quadratic form on the plane.
Core idea
A positive-definite quadratic form is coercive, whereas a singular form has an unbounded nullspace over the compact target set {0}.
- 1Use the null line of the singular form to disprove properness.
- 2Bound the positive-definite form below by its smallest eigenvalue times the squared norm.
- 3Combine boundedness with closedness of a compact-set inverse image.
- Final response
- B — only the positive-definite form is proper.
- Independent check
- The symmetric matrix has positive eigenvalues 3/2 and 5/2, yielding a direct coercive lower bound.
Common pitfall: Positive semidefinite is insufficient when a nontrivial nullspace survives.
Mathematics Q35 · NAT · 1 mark
Sample 3 — One-mark NAT: Euler’s method for a nonlinear initial-value problem
Run three half-step Euler updates and evaluate the requested combination of mesh values.
Core idea
Evaluate each slope at the current pair before advancing both the independent variable and the approximation.
- 1Start from the supplied initial pair and use step size 1/2.
- 2Compute the three trigonometric slope values at successive mesh points.
- 3Substitute the approximations at x=1 and x=1.5 into the requested combination.
- Final response
- 1.
- Independent check
- The sine values at the three mesh arguments are 0, 1 and −1, so the arithmetic can be checked exactly.
Common pitfall: Do not evaluate the slope at the newly advanced x-coordinate before updating y.
Mathematics Q47 · MCQ · 2 marks
Sample 4 — Two-mark MCQ: Algebraic closure over a finite field
Determine whether the elements algebraic over a finite subfield form an algebraically closed and infinite field inside an algebraically closed ambient field.
Core idea
Place finitely many coefficients in one finite extension and use transitivity of algebraicity; then use the nested finite fields of sizes qⁿ.
- 1Collect a polynomial’s finitely many coefficients inside a finite extension of the base field.
- 2Use transitivity to keep every root algebraic over the base field.
- 3Exhibit finite subfields of arbitrarily large cardinality inside the algebraic closure.
- Final response
- C — both statements are true.
- Independent check
- Roots of x^(qⁿ)−x produce finite subfields of size qⁿ, proving infinitude without introducing transcendental elements.
Common pitfall: An infinite algebraic extension need not contain any transcendental element.
Mathematics Q62 · NAT · 2 marks
Sample 5 — Two-mark NAT with source caveat: Asymptotic Rayleigh quotients
Diagonalise a shifted symmetric matrix and audit when the quotient of successive quadratic forms approaches the official numerical key.
Core idea
The limit is the largest eigenvalue represented in the chosen vector’s spectral support, not automatically the matrix’s largest eigenvalue.
- 1Compute the shifted matrix eigenvalues as 1, 3 and 6.
- 2Expand the vector in an orthonormal eigenbasis.
- 3Show that powers of 6 dominate only when the eigenvalue-6 coefficient is nonzero.
- Final response
- Official key: 6, under the necessary nonzero top-eigenspace condition.
- Independent check
- Vectors in the eigenvalue-1 or eigenvalue-3 eigenspaces give constant quotients 1 or 3, proving the literal printed condition is insufficient.
Common pitfall: Do not assume every nonzero vector has a component in the top eigenspace.
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
- All 55 Mathematics-subject questions, official Q11–Q65, carrying 85 marks.
- Official MCQ, MSQ and NAT response types, marks and final-key results for every locator.
- Independently written mathematical setups, full derivations and exact final responses.
- A separate check or proof idea and a problem-specific exam pitfall for every question.
- Explicit treatment of the literal-source caveats affecting Q62 and Q63.
Not included
- General Aptitude Q1–Q10 and its 15 marks.
- Official question statements, option prose, figures, scans, logos and paper layout.
- A claim of official affiliation, endorsement, sponsorship or approval.
- A score, rank, admission, scholarship or examination-result guarantee.
Study loop
- 1Open the free official paper and attempt one question under a fixed time limit.
- 2Compare the first step where your reasoning diverged, not only the final response.
- 3Reproduce the independent derivation from a blank page without copying the guide.
- 4Run the separate check and record the named problem-specific pitfall.
- 5Retry the question after 48 hours, then assemble mixed timed sets from the full paper.
Questions about this guide
What does the GATE Mathematics 2025 guide cover?
It covers the complete Mathematics subject section, official Q11–Q65: 55 questions carrying 85 marks. General Aptitude Q1–Q10 is outside the product boundary.
Does the guide reproduce the official GATE 2025 paper?
No. It uses question-number locators, independent mathematical summaries, original derivations, checks and final-key mappings. The free official paper and key are linked but not bundled.
How are the official answers verified?
Every question’s type, marks and key are mapped to the official final answer key. The mathematics is derived independently and checked before comparison with that key.
Why does the guide discuss caveats for Q62 and Q63?
The literal printed conditions do not uniquely justify the official numerical results. The guide shows the literal reading, the counterexample or vacuity, and the additional intended condition needed for the official key.
Is purchase available now?
Yes. The complete guide is available as a one-time, tax-inclusive purchase for ₹1,499 with Dodo-hosted checkout and automatic PDF delivery after confirmed payment.
Continue your mathematics preparation
GATE Mathematics 2026 — Detailed SolutionsTransfer the same methods to another complete 85-mark Mathematics section.GATE Mathematics 2027 — Revised-Syllabus Mastery WorkbookPractise 15 revised-focus areas through 60 original MCQ, MSQ and NAT problems.IIT JAM Mathematics 2025 — Detailed SolutionsBroaden proof and calculation practice with a complete related Mathematics paper.
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