QUESTION BANK
PADMASRI DR.B.V.R.I.C.E
LIST OF IMPORTANT QUESTIONS FOR MATHEMATICS PAPER-II
UNIT-1
ABSTRACT ALGEBRA
GROUPS
ESSAY QUESTIONS:
1. The order of every element of a finite group is finite and is less than or equal to order of the
2.
3.
4.
5.
6.
group.
A finite semi group satisfying the cancellation laws is a group.
In a group G ,if a Є G, then O(a)=O(a-1).
In a group G, for a ,b є G, O(a)=5,b≠e and aba-1 =b2 ,find O(b).
If a is an element of a group G such that O(a)=n, then am=e iff n/m.
Prove that the set all matrices Aα= cosα -sinα
Sinα
cosα
,αЄR form a group w.r.t. matrix
multiplication.
7.Prove that the set of nth roots of unity under multiplication form a finite group.
SHORT ANSWER QUESTIONS
1. If G is a group such that (ab)m=ambm for the three consecutive integers m,for all
a,b єG, show that G is abelian.
2.If a is an element of a group G such that O(a)=n, then am=e iff n/m.
3.Show that in a group G for a, b ЄG
,(ab)2=a2b2 iff G is abelian.
4.In a group G for a,b,x,yЄG the equation ax=b ,ya=b have unique solutions.
5.Show that the set Q+ of all +ve rational numbers form s an abelian group under the composition
defined b y ‘o’ such that aob= ab/3 for a,b ЄQ+.
6.Prove that the set G ={0,1,2,3,4} is an abelian group of order 5 w.r.t addition modulo 5.
7. Prove that the set G ={1,2,3,4,5,6} is an abelian group of order6 w.r.t multiplication modulo 7.
VERY SHORT ANSWER QUESTIONS
1. In a group G for every a€G,a2=e, prove that G is abelian.
2. Find the order of each element of the group G=Z6={0,1,2,3,4,5}, the composition being
addition modulo 6.
3. Find the order of each element of the multiplicative group G ={1, -1, i,-i }
4. If every element of a group (G,.) is its own inverse .Show that (G,.) is an abelian group .
SUBGROUPS
1. If H is a subgroup of G iff a,b є H =>ab єH.
2. If H,K are two subgroups of a group G, then HK is a subgroup of G iff
HK=KH.
3. The union of two subgroups of a group G is subgroup iff one is contained in the other.
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4 .If H is subgroup of G iff for a, b є H ab-1 є H.
5. Ihe intersection of two subgroups is a subgroup.
COSETS AND LAGRANGES THEOREM
ESSAY QUESTIONS
1. State and prove Lagranges theorem and show that its converse is not true by an example.
2. Any two left cosets of a subgroup are either disjoint or identical.
3.If H is a subgroup of a group G then there is 1-1 correspondence between the set of distinct
left cosets and the set of all distinct right cosets H in G.
4. Prove thet the ({0,3,6,9,12},+15) is a subgroup of (Z15 ,+15). Find the left coset s of above
subgroup in Z15.
NORMAL SUBGROUPS
ESSAY QUESTIONS
1. A subgroup H of a group G is a normal subgroup of G iff each left coset of
H in G is a right coset of H in G.
2. If H is a normal subgroup of G iff the product of two right cosets of H in G is again a right
coset of H in G.
3. If G is a group then the center Z of G is a normal subgroup of G.
4. If N, M are normal subgroups of G such that M∩ N={e}, then every element
of M commutes with every element of N.
5.A subgroup H of a group G is normal iff xHx-1=H for all x€G.
6.If G is a group and H is a subgroup of index 2 in G. then H is a normal subgroup of G.
SHORT ANSWER QUESTIONS
1. Every subgroup of an abelian group is normal.
2. Every group of prime order is simple.
3. definition of normal subgroup.
HOMOMORPHISMS ,ISOMORPHISMS OF GROUPS
1.
2.
State and prove Fundamental theorem of homomorphism of groups.
If f is a homomorphism of a group G into a group G1 then the kernel of f is normal subgroup
of G.
3. The necessary and sufficient condition for a homomorphism f of a group G onto a group
G1 with kernel K. to be an isomorphism of G into a G1 is that K ={e}.
4. Let G be multiplicative group and f:GG such that for a€G , f(a)=a-1 .prove that f is1 1,onto.also prove that fis homomorphism iff G is commutative.
5. The set of all automorphisms of a group G forms a group w.r.t. composition of mappings.
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VERY SHORT ANSWERS
1. In a group G for every a€G,a2=e,prove that G is abelian.
PERMUTATION GROUPS
ESSAY QUESTIONS
1. State and prove Cayleys theorem.
2. Let Sn be permutation group on n symbols . then the set of n! permutations in n!/2 are
even permutations and n!/2 are odd permutations.
SHORT ANSWER QUESTIONS
1. Find the regular permutation group isomorphic to the multiplicative group
{1,-1,i, -i}.
𝟏 𝟐 𝟑 𝟒 𝟓
𝟏 𝟐 𝟑 𝟒 𝟓
(
)
(
).
𝟏 𝟐 𝟓 𝟒 𝟑
𝟒 𝟑 𝟏 𝟐 𝟓
3.Express the following permutations as a product of disjoint cycles (2 5 4)(1 4 3)(2 1).
𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕
4.Show that (
) is odd.
𝟑 𝟐 𝟒 𝟓 𝟔 𝟕 𝟏
5.Find the order of a cycle (1 4 5 7).
6. Examine whether the following permutation is even or odd (1 2 3 4 5)(1 2 3)(4 5).
CYCLIC GROUPS
ESSAY QUESTIONS
2.Find the product of
1. Every subgroup of acyclic group is cyclic.
2. A group of prime order is cyclic.
3. The order of a cyclic group is equal to order of its generator.
SHORT ANSWER QUESTIONS
1. Show that the group (G={1,2,3,4,5,6},X7} is cyclic. also write down also of its generators.
2.A cyclic group of order n has Ø(n) generators.
3. Find the number of generators of cyclic groups of orders 5,6,8,12,15,60.
4. Every cyclic group is an abelian group.
5. If a is a generator of a cyclic group G , then a-1 is also a generator of G.
UNIT-II
RINGS INTEGRAL DOMAINS AND FIELDS
ESSAY QUESTIONS
1.
2.
3.
4.
Every finite integral domain is a field.
A field has no zero divisors.
A ring R has no zero divisors iff cancellation laws hold in R.
Prove that the set Z[i]={a+ib/a,b€Z,i2=-1} of Gaussian integers is an integral domain w.r.t.
addition and multiplication of numbers.
5. Prove that Q[√2]={a+b√2/a,b€Q} is a field w.r.t. addition and multiplication of numbers.
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6. The characteristic of an integral domain is either prime or zero.
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
6.
VERY SHORT ANSWER QUESTIONS
If R is a Boolean Ring then i)a+a=0 for all a€R ii) a+b=0 => 𝑎 = 𝑏 iii) Every Boolean ring is
abelian.
Prove that characteristic of a a Boolean ring is 2.
If the characteristic of a ring is 2 and the elements a,b of the ring commute. provethat
(a+b)2=a2+b2=(a-b)2.
Prove that ±1, ±𝑖 are the only four units in the domain of Gaussian integers.
Find the all the associates of (2-i) in the ring of Gaussian integers.
SUBRINGS ,IDEALS ,QUOTIENT RINGS& EUCLIDEAN RINGS
ESSAY QUESTIONS
The ideals of a field F are only {0} and F itself.
The intersection of two ideals of a ring R is an ideal of R.
If U1, U2 are two ideals of a ring R then U1+ U2={x+y/x€ U1, y€ U2 } is also ideal of R.
The ring of integers Z is principal ideal ring.
Prove that the ring of Gaussian integers is an Euclidean ring.
Every Euclidean ring is a Principal ideal ring.
SHORT ANSWER QUESTIONS
1.
2.
The ring of integers is an Euclidean ring.
𝑎 0
Show that the subset U={(
) /a, b € Z} is a left ideal but not right ideal of the ring of 2x2
𝑏 0
matrices over integers.
HOMOMORPHISM OF RINGS, MAXIMAL AND PRIME IDEALS
ESSAY QUESTIONS
1.Show that a ring with unity can be embedded in a ring of endomorphisms of some abelian
group.
2. If f is homomorphism of a ring R into the ring R1 then show that f is an into isomorphism if
kerf={0}.
3. An ideal in Z is a maximal ideal iff it is generated by a prime integer.
4.An ideal U of a commutative ring R with unity is maximal iff the quotient ring R/U is a
field.
5.An ideal U#R of a commutative ring R , is a prime ideal iff R/U is an integral domain.
VERY SHORT ANSWER QUESTIONS
1. Define Kernel of a homomorphism, maximal ideal, prime ideal.
2.Prove that f:J[i] J[i] defined by f(m+in) =m-in is an automorphism of the ring of Gaussian
integers
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UNIT-III (REAL ANALYSIS)
SEQUENCES
ESSAY QUESTIONS
1. State and prove Monotone convergence theorem.
2. Prove that the sequence {sn} is defined by sn=(1+1/n)n is convergent.
3 .Prove that the sequence is defined by Sn=1+1/1!+1/2!+…………….+1/n! is convergent.
4 .The sequence {Sn } is convergent iff it is a Cauchys sequence.
5 .Prove that the sequence Sn = 1+1/2+1/3+1/4+…………..+1/n is not convergent.
SHORT ANSWER QUESTIONS
1.
2.
3.
4.
5.
Every convergent sequence is bounded.
State and prove SANDWICH THEOREM.
Prove that lim [1/(n+1)2+ 1/(n+2)2+………….+1/(n+n)2] =0.
Prove that the sequence {Sn} wher e Sn=1/(n+1)+1/(n+2)+………..+1/(n+n) is convergent.
Show thatlim[(𝑛 + 1)(𝑛 + 2) … … . . (𝑛 + 𝑛)/𝑛
VERY SHORT ANSWER QUESTIONS
1. Prove that lim (sinn𝛱/3)/√n=0.
2. Prove that lim 1/n(1+1/3+………..+1/(2n-1))=0.
3.Prove that lim n!/nn=0.
INFINITE SERIES
ESSAY QUESTIONS
1. State and prove Leibnitz test.
2. Test for the convergence of ∑
1.3.5……2𝑛−1
2.4.6…..2𝑛
𝑥 𝑛−1
3
3
3. Test for the convergence of ∑∞
𝑛=1 √(𝑛 + 1) − 𝑛
1
1
1
1
1
𝑛
4. Examine the convergence of ∑∞
𝑛=1((−1) ) 𝑛 [1 + 2 + 3 + 4 + ⋯ . . 𝑛]
5. State and prove Cauchy nth root test.
6 .State and prove Ratio test.
SHORT ANSWER QUESTIONS
1. Prove that ∑(−1)n-1xn/n is convergent for -1<x≤1.
2𝑛 −2
2. Test for the convergence ∑ 2𝑛 +1 𝑥 𝑛 , 𝑥 > 0
VERY SHORT ANSWER QUESTIONS
1
1.Test for the convergence of ∑∞
𝑛=1 log(𝑛).
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2.Prove that 1-1/2+1/3-1/4+………. converges.
𝑛+1
3. Test for the convergence of ∑(𝑛+2)𝑛 𝑥 𝑛
4. Show that lim
2𝑛
𝑛!
.
=0.
LIMITS AND CONTINUITY
ESSAY QUESTIONS
1. Examine the continuity of the function f defined by f(x)=|x|+|x-1| at x=0,1.
sin(𝑎+1)𝑥+𝑠𝑖𝑛𝑥
𝑥
2.Let f:RR be such that f(x)=
(𝑥+𝑏𝑥 2 )−𝑥 1/2
𝑏𝑥 3/2
for x<0, f(x)=c for x=0 ,and f(x)=
for
x>0.Determine the values of a,b,c for which the function is continuous at x=0.
𝑥𝑛
3 .Examine the continuity of f defined by f(x)=lim1+𝑥𝑛 𝑒 𝑥 ∀ 𝑥 > 0.
4. .If f :[a,b]R is continuous on [a,b], then f is bounded on [a,b].
5. If f:[a,b]R is continuous on [a,b],then f Type equation here.is bounded on [a,b] and attains
its bounds or infimum and supremum .
6. If f is continuous on [a,b] and f(a) ,f(b) have opposite signs then there exists c€(𝑎, 𝑏) such
that f(c)=0.
7. If f is continuous on [a,b] and K is any real number between f(a)& f(b) then there exists
c€(𝑎, 𝑏) such that f(c)=K.
8.If a function f is continuous on [a,b] ,then it is uniformly continuous on [a,b].
SHORT ANSWER QUESTIONS
1. From the definition prove that f(x)=x2 is uniformly continuous on [-a,a].
2. Show that f:RR defined by f(x)=1 if x€Q and f(x)=-1 if x€R-Q is discontinuous for all
x€R.
𝑒 1/𝑥 −𝑒 −1/𝑥
3. Let f:RR be such that f(x)=𝑒 1/𝑥 +𝑒 −1/𝑥 if x≠o and f(0)=1. Discuss the continuity at x=0.
Unit-IV
DIFFERENTIATION
ESSAY QUESTIONS
1.Discuss the continuity and derivability of the function f(x)=|x-1|+|x-2| in the interval
[0,3].
2.State and prove Rolles theorem.
3.State and prove Lagranges Mean value theorem.
4.State and prove Cauchy’s Mean value theorem.
1
5.Show that the function f(x)= x2cos 𝑥 for x≠ 0
0
Derivative is not continuous at 0.
is derivable everywhere but the
for x=0
1
6. Show that the function f(x)= xsin( 𝑥) for x≠ 0
0
is continuous but not
for x=0
Derivable at x=0.
7.Prove that tanx> x>sinx ∀x€(o,𝜋/2)
2𝑥
8.Show that log(1+x) -2+𝑥 is increasing when x>0.
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QUESTION BANK
SHORT ANSWER QUESTIONS
1. Verify Rolles theorem in the interval [a,b] for the function f(x)=(x-a)m(x-b)n; m,n being
+ve integers.
𝑥 2 +𝑎𝑏
2. Discuss the applicability of Rolles theorem for f(x)=log𝑥(𝑎+𝑏) in [a,b],a>0.
3. Prove that
𝜋
⎷3
𝜋
1
+ 15 <Sin-1(0.6)< 6 + 8
6
4. Prove that x -
𝑥2
2
𝑥2
<log(1+x)<x - 2(1+𝑥) for x>0.
5. Find c of Cauchy’s mean value theorem for f(x)=⎷x and g(x)=1/⎷x in [a,b] where
0<a<b
6. Verify Cauchy’s mean value theorem for f(x)=x2 and g(x)=x3 in [1,2].
7. Write Taylor’s theorem with Lagranges form of remainder for f(x)=log(1+x) when
a=2,n=4.
8. Show that x-
𝑥3
3!
≤ 𝑠𝑖𝑛𝑥 ≤ 𝑥 −
𝑥3 𝑥5
3!
+ 5! .
9. Using Lagranges the orem ,show that x >log(1+x) >x / (x+1). If f(x)=log(1+x).for all x>o.
Riemann Integration
Essay questions
1. Show that the function f:[a,b]R is R-Integrable iff integrable.
𝑎
2. Prove that f(x)=x2 is integrable on [0,a] and ∫0 𝑥 2 𝑑𝑥=a3/3.
3. Prove that
⎷3
8
𝜋/3 𝑠𝑖𝑛𝑥
≤ ∫𝜋/4
𝑥
𝑑𝑥 ≤
⎷2
6
.
4. State and prove Fundamental theorem of integral calculus.
5. If f:[a,b]R is continuous on [a,b], then f is integrable on [a,b].
SHORT ANSWER QUESTIONS
𝑛
1. Show that lim∑𝑛𝑟=1 𝑛2 +𝑟 2=π/4.
𝑏
𝑏
2. If f€R[a.b] then show that |∫𝑎 𝑓(𝑥)𝑑𝑥 | ≤ ∫𝑎 |𝑓(𝑥)|𝑑𝑥.
3. Prove that 1/  0  1 sinx /(1+x2)dx  2/.
4. Evaluate 0  /4 (sec4x-tan4x)dx.
5. If f(x)=x2 if 0<x<1 and f(x)=1 if x>1 is integrable on [0,2].
𝑏
6. Prove that ∫𝑎 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑏 − 𝑒 𝑎 .
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