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Syllabus

SEMESTER-V
MATHEMATICS
(OMR)
PAPER-I: Group and Ring Theory & Linear Algebra
UNIT-I
:
SYLLABUS

PART A

Group and Ring Theory
Introduction to Indian ancient Mathematics and
Mathematicians should be included under Continuous
Internal Evaluation (CIE).
Automorphism, inner automorphism, Automorphism
groups, Automorphism groups of finite and infinite
cyclic groups, Characteristic subgroups, Commutator
subgroup and its properties; Applications of factor
groups to automorphism groups.
UNIT-II : Conjugacy classes, The class equation, p-groups, The
Sylow theorems and consequences, Applications of
Sylow theorems; Finite simple groups, Non-
simplicity tests; Generalized Cayley’s theorem, Index
theorem, Embedding theorem and applications.
UNIT-III: Polynomial rings over commutative rings, Division
algorithm and consequences, Principal ideal domains,
Factorization of polynomials, Reducibility.tests,
Irreducibility tests, Eisenstein criterion, Unique
factorization in Z [x].

UNIT-IV
: Divisibility in integral domains, Irreducibles, Primes.
Unique factorization domains, Euclidean domains.
UNIT-V
PART-B
Linear Algebra
: Vector spaces, Subspaces, Linear independence and
dependence of vectors, Basis and Dimension,
Quotient space.
UNIT-VI : Linear transformations, The Algebra of linear
transformations, rank nullity theorem, their
representation as matrices.
UNIT-VII : Linear functionals, Dual space, Characteristic values,
Cayley Hamilton Theorem.
UNIT-VIII: Inner product spaces and norms, Cauchy-Schwarz
inequality, Orthogonal vectors, Orthonormal sets and
bases, Bessel’s inequality for finite dimensional
spaces, Gram-Schmidt orthogonalization process,
Bilinear and Quadratic forms.

पिछले साल का exam paper

B.Sc. (Semester-V) Examination, 2023-24
MATHEMATICS
Paper: I
(Group and Ring Theory & Linear Algebra)
1. If I is a ideal in ring R then :
(a) R/L is a ring
(c) RI is ring
(b) R+I is a ring
(d) None of these
2. If O(G)=27, then the number of conjugate classes of G is:
(b) 11
(c) 12
(d) 13
Ans. (a)
(a) 10
Ans. (b)
3. The elements α = (a, b,) and ẞ=(a, b) of the vector space V₂(R) are
linearly independent if and only if:
(a) a, a, b,b₂ 0
(b) a,a-bb-0
(c) ab₂-ab₁ = 0
Explanation:
(@ab₂-ab₁ +0
Ans. (d)
Let
a, b = R then,
aa + b
= 0 gvien
aa + ba₂ =0
ab, + bb₂ =0
(1)
(2)
On elliminating b
ab₁
aa+a₂
= 0
b₂
a[a,b₂-ab₁]=0
For L.I, so
ab₂-ab₂ #0
4. Which of the following function T: p²
(a) T(x,, x)=(x,-x2,0)
(c) T(x, x)=(1+x2, x2)
Explanation:
Let
2 is a linear tansformation?
(b) 7(x, x)=(x,.-x)
(d) 7(x, x)=(x. 1+x₂)
Ans. (a)
a = (x1, x2); ẞ=(y₁₂)
7(a) =T(x,,x)=(x₁-x2, 0)
T(B) = Ty₁y)=(y₁,-y2, 0)
Let a, b are any two scalars then
aa+bẞ a(x1, x2) + by₁, y2) = (ax + by₁, ax + by₂)
T(aa+bẞ)=(ax, + by₁ – ax + by,, 0)
=(a(x-x)+b(y-y2),
0)=a7(a)+hT(B)
5. Every cyclic group is :
Hence it is a linear transformation.
(a) Group
Abelian group
(b) Sub-group
(d) None of these
Ans. (c)
6. Let G be a cyclic group and a = G, then which of the following is true:
(a) 0(a)<0(G)
(c) 0(a)≥0(G)
60(a)=0(G)
(d) 0(a)≤0(G)
7. The number of generator of a infinite cyclic group is:
LAT2
(c) ∞
(d) 3
Ans. (b)
Ans. (a)

8. Let G be a cyclic group of ordern. The order of the group Aut (G) is:
La) (n), Euler-phi-function (b) n
(c) n+1
1
(d) n-1
9. Which of the following is the class equation of a group G:
Ans. (a)
O(G)
O(G) (b)O(G)= O(Z(G)) + O(N(a))
O(Z/G)
(a) O(G)= O(N(G))+-
O(G)
O(G)
) O(G)= O(Z(G))+
agZ(6) O(N(a))
(d)O(G)= Σ
deZ((i) O(N(a))
Ans. (c)
10. The group of Automorphisms of a cyclic group is:
(a) abelian
(b) identity
(c) non-abelian
(d) None of these
Ans. (a)
11. If O(G)=p” where p is prime number then the centre:
(a) Z= {e}
b) Z {e}
(c) Z= {o]
(d) None of these
Ans. (b)
12. The relation of conjugacy is an……………….relation on the group G:
(a) trivial (b) simple ) equivalence (d) normal
Ans. (c)
(a) 2
(b) 3
√(c) 5
(d) 15
Ans. (c)
13. A group of order 30 has Sylow 5-Subgroup of order
14. A group G is a p-group iff:
(a) O(G)=np (b) O(G)=1
(c) O(G)= p² (d)O(G)=p” Ans. (d)
15. Which of the following is correct:
Va (p-1)!=-1 (modp)
(c) (p-1)=1 (modp)
(b) (p-1)!=1 (modp)
(d) (p-1)=-1 (modp)
16. Which of the following order is of order of a cyclic group G :
(c) O(G)=35 (d) O(G)=10
Ans. (a)
Ans. (b)
(a) O(G) 14 (b)(G)=4
17. Let f(x)=2+6x+4x², g(x)=2x+4x² be two polynomials over the ring
(I8, +8,-xg). then deg [f(x) + g(x)] is:
(a) 0
Explanation:
(b) 1
(c) 2
(d) 3
Ans. (a)
f(x)+g(x)=2+(6+2)x+(4+4)=2+8x+8x² since mod is 8
=2+0x+0x² = 2 = Constant so deg [f(x) + g(x)] is zero
18. If F is a field and f(x).g(x) = f(x), then deg. (f(x).g(x)) is equal to :
(a) deg. f(x)
(c) deg. f(x)+ deg.g(x)
19. In a commutative ring R, ab is equal to :
(b) b
(b) deg. g(x)
(d) deg. f(x)-deg.g(x)
(d) o
(c) f(a)=0 (d)f (a) 0
(c) a
Ans. (c)
Ans. (a)
Ans. (a)
a) ba
20. If(x) e F(x) and a = F, for a field F, then (x-a) divides f(x) iff:
(a) f(a)=0
(b) f(a)0
21. If R is a Euclidean Ring and a, b are two non-zero elements of R then:
(a) d(ab) = d(a)
22. f(x)=-2 € Z[x] is:
(c) d(ab)<d(a)
(a) reducible in Z
(c) irreducible in Q
(b) (ab)>d(a)
None of these
Ans. (d)
irreducible in Z
Ans. (b)
(d) None of these

(a) Euclidean domain
VP.I.D. E.D.
23. The units of Z[i] are:
(a) ±1
Ve) (a) & (b)
24. Which of the following is not correct:
25. Which of the following is not an example of vector space :
(b) ±1
(d) None of these
Ans. (c)
U.F.D. (b) P.I.DU.F.D.
(d) E.D. P.I.D.
Ans. (c) –
(a) Rover R (b) Rover Q
(c) Cover R (dyQ over R
Explanation:
(dy2 over R
Ans. (d)
Let
a e R and ve Q
then av will be a real number and not belongs to Q. (set of rational
number)
Let
√2 (1)
#Q It is irrational.
Hence Q over R is not a vector space.
26. A division ring and a field must contain of least:
(a) One element
رکھا
two element
(c) Infinite number of element (d) Finite Number of Element Ans. (b)
27. The ring of integers modulo P is an integral domain iff P:
a) Prime
(b) Rational
(c) Natural
(d) None of these
Ans. (a)
28. The unity element of an ordered integral domain is :
1
(b) -1
(c) 0
(d) None of these
Ans. (a)
29. Every finite integral domain is a :
Field
(b) Ring with zero divisor
(d) None of these
Ans. (a)
a) a
(c) a¹
(b) a²
(d) None of these
Ans. (a)
(c) Group
30. A ring R is to be boolean ring if a²=
31. Which of the algebraic structure in not a ring?
(a) (R₁+,.)
(c) (C,+..)
Explanation:
f) {(a+b√2+c√3): a,b,ce I,+;}
(d) (Q.+..)
R={(a+b√2+c√3):a,b,c € 1,+..}
Since (R₁) It is an abelian group w.r.t. addition
Here e=0 when a=b=c=0
(R2).(R.) is not a semi group because
(a+b√2+c√3).(a+b₂ √2+c₂ √√3)
Hence it is not a Ring.
Ans. (b)
=(aa₂+2bb₂+3c₁₂+ √2 (ba₂+ab₂))
+√3(a₁₂+α₂)+ √6 (bc+qb₂) € R
32. The composite of two isomorphism, is also:
Va) An isomorphism
(c) An automorphism
(b) Homomorphsim
(d) An automorphism