Jammu University Non CBCS
Mathematics Previous Year Question Paper
6th Semester
MA-601
Jammu University NON-CBCS Mathematics 6th Semester Previous Year Question Paper
(Course Code: UGM-601 / MA-601 — Vector Spaces and Metric Spaces)
Mathematics is one of the most conceptual and scoring subjects in the B.A./B.Sc. 6th Semester (Non-CBCS) under Jammu University. The paper titled “Vector Spaces and Metric Spaces” focuses on core topics of linear algebra and real analysis — essential for higher studies in mathematics and data sciences.
Below, you’ll find the syllabus, exam structure, and important previous year questions from Jammu University’s official paper (Course MA-601).
🧾 Course Details
| Component | Details |
|---|---|
| Course Title | Vector Spaces and Metric Spaces |
| Course Code | UGM-601 / MA-601 |
| Semester | 6th Semester |
| Scheme | Non-CBCS |
| University | University of Jammu |
| Duration | 3 Hours |
| Maximum Marks | 100 (Theory – 80 + Sessional – 20) |
📚 Detailed Syllabus Overview
Unit I — Vector Spaces
- Definition and examples of vector spaces
- Subspaces and quotient spaces
- Linear dependence and independence of vectors
- Linear span and related exercises
Unit II — Basis & Dimension
- Basis and dimension of a vector space
- Isomorphic vector spaces
- Finite and infinite dimensional vector spaces
- Dual spaces and their dimensions
Unit III — Linear Transformations
- Linear transformations and their matrix representation
- Algebra of linear transformations
- Kernel, range and inverse transformations
- Examples and exercises on finite-dimensional vector spaces
Unit IV — Sets & Sequences
- Denumerable and non-denumerable sets
- Open and closed sets in ℝ
- Limit points, Heine–Borel Theorem, Bolzano–Weierstrass Theorem
Unit V — Metric Spaces
- Definition and examples of metric spaces
- Open and closed sets in a metric space
- Interior, closure, boundary of sets
- Convergence of sequences and continuous mappings
- Equivalent conditions and examples
🧮 Exam Pattern
| Section | Description | Marks |
|---|---|---|
| Total Questions | 10 (Attempt 5 — one from each unit) | 80 |
| Each Question | Long answer / derivation / proof | 16 |
| Time Allowed | 3 Hours | — |
| Internal Assessment | Class test (10 marks) + Assignments (10 marks) | 20 |
✍️ Important Questions (Based on PYQ – MA-601)
Unit I – Vector Spaces
- Define a vector space. Show that the set ( V = {(x, y) : x, y ∈ ℝ} ) forms a vector space under usual addition and scalar multiplication.
- Define subspace. Prove that the union of two subspaces of a vector space is a subspace if and only if one is contained in the other.
- Prove that the linear span of a set S of V(F) is the smallest subspace containing S.
- Test whether S = {(1, 0, 0), (–1, 0, 1), (0, 2, 0)} is linearly independent in ℝ³.
Unit II – Basis & Dimension
- Prove that a subset S of a finite-dimensional vector space V(F) is a basis of V iff every element of V can be uniquely expressed as a linear combination of S.
- Find a basis and dimension of the space of all 2×2 matrices over ℝ.
- If S is a subspace of V, prove that dim(S) ≤ dim(V).
- Prove that in a finite-dimensional vector space, a basis always exists.
Unit III – Linear Transformations
- Let T : ℝ³ → ℝ³ be defined by T(a,b,c) = (a–b, 3b–4c). Show that T is a linear transformation.
- For T(x,y,z) = (2y+z, x–4y, 3x), find its matrix representation relative to B = {(1,1,1), (1,1,0), (1,0,0)}.
- Prove that two vector spaces are isomorphic iff they have the same dimension.
- For T(x,y,z) = (3x, x–y, 2x+y+z), show that T is invertible and find T⁻¹.
Unit IV – Real Analysis / Sets
- Prove that if A and B are two denumerable sets, then A×B is also denumerable.
- Define the closure of a set A in ℝ. Prove that cl(A∩B) ⊆ cl(A) ∩ cl(B).
- State and prove Heine–Borel Theorem for closed and bounded intervals.
- Verify whether the intersection of finitely many open sets in ℝ is open or not.
Unit V – Metric Spaces
- Show that ( d(x,y) = \sqrt{(x₁–y₁)² + (x₂–y₂)²} ) defines a metric on ℝ².
- Define a convergent sequence in a metric space and prove that the limit is unique.
- Prove that a mapping f : X → Y between metric spaces is continuous iff the image of every open set of X is open in Y.
- Prove that the arbitrary union of open sets in a metric space is open.
📈 Preparation Tips
✅ Revise definitions and theorems thoroughly — many questions directly test conceptual clarity.
✅ Practice proofs and numerical examples from each unit.
✅ Memorize formulas for linear transformations, determinants, and distances.
✅ Use diagrams for open/closed sets and convergence proofs to simplify answers.
✅ Review previous year papers (like MA-601) to identify repetitive theorems and standard proofs.
📥 Download Section
📄 Download Jammu University Mathematics 6th Semester (Non-CBCS) PYQ – MA 601 (PDF)
You can view or download the full question paper along with other subject PDFs from our website.
🏁 Conclusion
The Mathematics 6th Semester Non-CBCS paper of Jammu University combines abstract algebra and real analysis through the study of Vector and Metric Spaces.
Mastering linear transformations, basis, dimension, and metric-space properties ensures high marks and conceptual strength for advanced mathematics, computer science, and research.
Keep revising, keep solving past papers — and stay connected with Jammu University Papers for more previous year question papers and study notes.