LU B.Sc Sem 3 Mathematics Linear and Abstract Algebra Important Questions 2026 | PYQ Pattern
LU B.Sc 3rd Semester Mathematics (Linear and Abstract Algebra) Important Questions 2026 – Lucknow University (LU) aur affiliated colleges ke B.Sc NEP Semester III (PCM group) students ke liye Mathematics – Linear and Abstract Algebra (Paper-I) ke sabse important questions, bilkul previous year question paper (PYQ) pattern me. Saare questions neeche page par hi hain – section-wise.
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Paper Details – B.Sc Sem 3 Mathematics (LU NEP 2026)
| University | University of Lucknow (LU) + affiliated colleges |
| Course | B.Sc. III Semester (NEP) – PCM |
| Paper | Mathematics – Linear and Abstract Algebra (Paper-I) |
| Time / Marks | 2 Hours / 75 Marks |
| Pattern | Kul 4 questions: Section A se 1 (15 marks) + Section B se 3 (20 marks each) |
Section A – Important Questions (15 Marks)
Note: Section A se koi ek question attempt karna hai.
Q1. (a) What is an inner automorphism? Show that Inn(G) is a normal subgroup of Aut(G) and that G/Z(G) ≅ Inn(G).
(b) For a commutative ring R with identity, show that an ideal M is maximal if and only if R/M is a field. [15 Marks]
Q2. (a) Give the statement and proof of Burnside’s lemma.
(b) Show that a Euclidean domain is a principal ideal domain. [15 Marks]
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Section B – Important Questions (20 Marks)
Note: Section B se koi teen questions attempt karne hain.
Q3. (a) Prove that the normalizer N(a) of an element a is a subgroup of G and that the number of conjugates of a equals the index of N(a) in G.
(b) Give the statement and proof of Cauchy’s theorem for finite groups. [20 Marks]
Q4. State the three Sylow theorems and use them to show that no group of order 12 is simple. State Eisenstein’s criterion. [20 Marks]
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Q5. (a) For finite dimensional subspaces W₁, W₂ of V, prove that dim(W₁) + dim(W₂) = dim(W₁ + W₂) + dim(W₁ ∩ W₂).
(b) Prove that α₁ = (1, 1, 0, 0), α₂ = (0, 0, 1, 1), α₃ = (1, 0, 0, 4), α₄ = (0, 0, 0, 2) form a basis of R⁴. [20 Marks]
Q6. For T(x₁, x₂, x₃) = (3x₁, x₁ – x₂, 2x₁ + x₂ + x₃) on R³, verify the rank-nullity theorem, decide whether T is invertible and, if so, find T⁻¹. [20 Marks]
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Q7. Obtain the eigenvalues and eigenvectors of A = [[3, 1, –1], [2, 2, –1], [2, 2, 0]]. Using the Cayley-Hamilton theorem, express A⁻¹ and A⁴ in terms of I, A and A². [20 Marks]
Q8. (a) Obtain the dual basis of {(1, –1, 3), (0, 1, –1), (0, 3, –2)} of R³.
(b) Find the symmetric bilinear form associated with q(x₁, x₂) = 4x₁² + 6x₁x₂ – 3x₂². [20 Marks]
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Q9. State and prove the Cauchy-Schwarz inequality and deduce the triangle inequality. Show that every orthonormal set is linearly independent. [20 Marks]
Q10. Apply the Gram-Schmidt process to the basis {1, x, x², x³} of polynomials of degree at most 3 with inner product (f | g) = ∫₀¹ f(t) g(t) dt. [20 Marks]
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Exam Tips – Mathematics (Linear and Abstract Algebra)
- Section A ke long answers ke short notes bana lo – 15 marks pakke karo.
- Section B ke derivations / numericals / diagrams ki practice karo.
- Upar ke questions ko 2 ghante me PYQ ki tarah solve karke dekho.
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FAQs
LU B.Sc 3rd Semester Mathematics (Linear and Abstract Algebra) exam me kitne questions attempt karne hote hain?
Ye 75 marks ka 2 ghante ka paper hai. Kul 4 questions attempt karne hote hain – Section A se 1 question (15 marks) aur Section B se 3 questions (20-20 marks).
Kya ye LU B.Sc Sem 3 Mathematics ka official previous year question paper (PYQ) hai?
Nahi. Ye LUUPDATE team ka banaya 2026 exam ke liye most important questions ka set hai, jo Lucknow University ke previous year papers ke pattern aur NEP syllabus par based hai.
B.Sc Sem 3 Mathematics Linear and Abstract Algebra ke answers / solutions kahan milenge?
LUUPDATE ke Most Important Questions course (BA, B.Sc, B.Com Sem I, III & V) me har question ka pura solution milta hai – sirf ₹99 me, LUUPDATE App par.
Ye paper kis group ke liye hai – PCM ya ZBC?
Mathematics (Linear and Abstract Algebra) LU B.Sc NEP Semester III ke PCM group ke students ke liye hai.
Disclaimer: Ye LUUPDATE team dwara taiyar kiye gaye expected important questions hain (previous year pattern par), Lucknow University ka official question paper nahi. Exam pattern aur syllabus ke liye university ka official notice zaroor dekhein.
