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DEPARTMENT OF MATHEMATICS
B.A. / B.Sc. II Year (Practical) Examination
Subject : MATHEMATICS
Paper : II
QUESTION BANK
W.E.F. Annual 2010
Time : 3 hours}
{Marks : 50
UNIT – I GROUPS
1.
Let S be the set of all real numbers except –1. Define * on S by
a * b = a + b + ab.
Show that (S, *) is an abelian group and find the solution of the equation
2*x*3 = 7 is S.
2.
Let R* be the set of all real numbers except 0. Define * on R by a*b = |a|b.
Is (R, *) a group ? Justify your answer.
3.
Prove that a non empty set G together with an associative binary
operation * on G such that the equations a*x = b and y* a = b have a solution
in G for all a, b  G is a group.
4.
Determine which of the following subsets of the complex numbers are sub
groups under addition of group C of complex numbers under addition.
a. R
b. Q +
c. i R set of pure imaginary numbers including 0.
5.
Let G be a group and a be a fixed element of G. Show that Ha = { x  B : xa=ax}
is a subgroup of G.
6.
Let H be a subgroup of a group G. For a, b  G. Let a ~ b if and only if ab-1= H.
Show that ~ is an equivalence relation on G.
7.
Which of the following functions from R into R are permutations of R
a. f(x) = x + 1
b. f(x) = ex
123456 
123456 
If ,   S6 and   

=

524316 
314562


a. 2 
b. 2009
c. 2009
12345 
If S = { , 2, 3, 4, 5, 6} with = 
 then by using multiplication table,
24513 
prove that S forms an Abelian group.
8.
9.
10.
Compute the indicated product of cycles that are permutations of {1,2,3,4,5,6,7,8}
a. (1, 4, 5) (7, 8) (2, 5, 7)
b. (1, 3, 2, 7) (4, 8, 6)
11.(a) Find the subgroup generated by 30 and also order of cyclic additive group of Z 42.
(b) In Z30 find the order of the subgroup generated by 18 and 24.
..2
..2..
12.
Prove that nth roots of unity form a cyclic group of order n.
13.
Find all the generators of the cyclic group Z28.
14.(a) Find all the cyclic subgroups of Z2009.
(b) Find the number of subgroups of Z2009.
15.
Determine which of the following maps are homomorphisms. If the map is
homomorphism describe its kernel.
a.  : Z  R under addition given by (n) = n.
b.  : Z  R under addition given by = the greatest integer  x.
c.  : Z6  Z2 given (x) = the remainder of x when divided by 2.
16.
Define a mapping between groups G = p{1, -1, I, -I} and Ĝ = {1, -1} such that
it is a homomorphism and find its kernel.
17.
Let R under addition and R* under multiplication are groups.  : R  R* defined
by  (x) = ex. Show that  is Isomorphism.
18.
Let (S,*) be the group of all real numbers except –1 under the operation * defined
by a * b = a + b + ab. Show that (S, *) is isomorphic to the group R* of non zero
real numbers under multiplication.
19.
If N = {,} then prove that N is not a normal subgroup of S 3 in which
1 2 3 
1 2 3 
= 
,

=

1 2 3  .
1 2 3 


20.
Prove that 3Z is normal subgroup of Z. Find factor group Z/3Z and its order.
Show that it is cyclic.
UNIT – II GROUPS
21.
Let T = {a, b, c, d} Addition and Multiplication are defined by
+
a
b
c
d
a
a
b
c
d
b
b
a
d
c
c
c
d
a
b
d
d
c
b
a
x
a
b
c
d
a
a
a
a
a
b
a
b
c
d
c
a
a
a
a
d
a
b
c
d
Show that (T, +, x) is non commutative ring without unity.
22.
(R, +, .) is a ring. Define ⊕,O on R by V (a, r), (b, s)  R x Z
(a, r)  (b, s) = (a + b, r +s)
(a, r) O (b, s) = (a.b +rb+sa,rs)
Show that (R, , O) is a ring.
23.
Find all the units of
a. Z14
b. Z x Z
..3
..3..
24.(a) Describe all the ring homomorphisms of Z x Z into Z.
(b) Describe all the ring homomorphisms of Z into Z.
25.(a) Solve the equation x2 – 5x + 6 = 0 in Z12
(b) Solve the equation x3 – 2x2 – 3x = 0 in Z12
26.(a)
(b)
(c)
(d)
Z4 x 4Z
Z6 x Z15
Z3 x 3Z
Z3 x Z3
27.
Define the quaternions of Hamilton and explain how it is a skew filed.
28.
Show that the set of all nilpotent elements in a commutative ring R forms an ideal
of R and R / N has no nonzero nilpotent elements.
29.
R = {0, 2, 4, 6} (R, +8, X8) is a ring. Let M = {0, 4} show that M is a maximal ideal
of R but not prime ideal.
30.
Define a Boolean ring. Show that Z2 and Z2 x Z2 are Boolean rings.
31.(a) Prove the left distributive law in M2 (F).
0 0 
(b) Show that 
 is not only a left divisor of '0' but also a right divisor of '0'
0 1
in M2 (F).
32.
Let R be a commutative ring with unity of characteristic 4. Compute.
(a) (a+b)4
(b) (a+b)3
Z12
and find all ideals of Z12.
N
33.
If N is an ideal of Z12, calculate
34.
Find all prime ideals and maximal ideals of Z6.
35.
 : Z  Z x Z be defined by n = (n, n), then show that
(a) 2Z is an ideal of Z,
(b) 2Z is not an ideal of Z x Z
36.
Define 2 : Q[x}  R as (a0 + a1x+ …….+anxn)2 = a0 + a1 2+ …… +an2n
Show that
(a) x2 + x – 6 is the kernel N of 2
Q[ x ]
(b) Show that
is isomorphic to Q.
N
37.
Let 2 : Z7 [x]  Z7, Calculate
(a) (x2 + 3) 2
(b) [(x4+2x)(x3 - 3x2 + 3) 3
38.
Find four elements in the kernel of each of the following homomorphisms
(a) 5 : Q[x]  R
(b) 4 : Q[x]  R
..4
..4..
39.
Find the sum and product of the given polynomials in the given polynomial ring
(a) f(x) = 4x – 5
g(x) = 2x2 – 4x + 2 in z8 [x]
(b) f(x) = 2x2 + 3x + 4
g(x) = 3x2 + 2x + 3 in z6 [x] .
40.(a) Find how many polynomials are there of degree  3 in Z2[x] including zeroes.
(b) Find all zeroes of x2 + 1 in Z2.
UNIT - III REAL NUMBERS
41.
Use the definition of limit to show that
 3n  1  3
(a) lim 

 2n  5  2
(b) lim
42.
( 1)n
0
n2
Use limit theorems to establish the convergence of
 2n 
(a)  2

 n  1
 n2  1 
(b)  2

 2n  3 
 n  1
(c) 

 n  1
43.
Use squeeze theorem to find limit of
1
(a)  n ! n2
 sin n 
(b) 

 n 
44.
x 
Let (xn) be a sequence of positive real numbers such that L = lim  n 1  exists. If
 xn 
L  1, then (xn) converge and lim(xn) = 0, apply this theorem to show that the
sequence.
 n 
(a)  n 
2 
 23n
(b)  2n
3

 converge to zero

45.(a) Apply monotone convergence theorem to show that
3
1
where (yn) is defined by y = 1, yn+1 =
(2yn + 3) for n  1.
n 
2
4
x
(b) Let x1 = 8, xn+1 = n  2 for n  N, show that (xn) is bounded and monotone.
2
Find its limit.
lim( y n ) 
..5
46.
..5..
1 1
1
Let (xn) =  2  2  .....  2 for each n  N. Prove that (xn) is increasing and
1 2
n
bounded.
47.
Show that the sequence
1

(a)  1  ( 1)n  
n

n 

(b)  sin
4 

are divergent by applying divergence criteria (i.e., (xn) is divergent if it has two
convergent subsequences whose limit are not equal.
48.
Show directly from definition that
 n  1
(a) 
 is a Cauchy sequence
 n 
(b) (-1)n is not a Cauchy sequence.
49.
Show that the sequence (xn) defined by x1 = 1, x2 =2, xn =
1
(xn-2 + xn-1) for n > 2
2
is a Cauchy sequence.
50.
Define a contractive sequence. If x1 = 2 and xn+1 = 2 +
1
for n > 1, prove that
xn
(xn) is a contractive sequence.
51.
Using comparison test show that the series

(a)
n
n 1
3
1
n

(b)
52.
cos n
are convergent
2
n 1 n

Test for convergence or divergence the following series by applying limit
comparison test

(a)
1
 n(n  1)(n  2)
n 1

(b)

n 1
1
n 1

53.
Apply Leibnitz test to establish convergence or divergence of

n 1
54.
1
n
.
The equation x3 – 7x + 2 = 0 has a root between 0 and 1. Use an appropriate
1
contractive sequence (xn) to approximate the root. Taking x1 = , calculate the
2
root upto 4 iterations.
..6
..6..
x2  x  6
55.(a) Let f be defined for all x  IR, x  2, by f(x) =
. Can f be defined at x = 2
x 2
in such a way that f is continuous at this point.
0
for x = 0
1
1
- x for 0 < x <
2
2
f(x) =
1
2
for x =
1
2
3
1
- x for
<x<1
2
2
1
56.
for
x=1
Prove that the Dirichlets function defined by
f(x) = 1 if x is rational
f(x) = 0 if x is irrational
is not continuous at any point of IR.
1
57.
Discuss the continuity of f(x) =
ex 1
1
x
, x  0, f(0) = 0, at x = 0.
e 1
58.(a) Show that the polynomial P(x) = x4 + 73 – 9 has at least two real roots.
1

(b) Let f be continuous on the interval [0, 1] such that f(x) = f  c   .
2

59.
If f is a continuous function of x satisfying the functional equation
f(x + y) = f(x) + f(y), show that f(x) = ax, where a is a contant.
60.
Give an example of a function which is continuous in an open interval but fails to
be uniformly continuous on that interval . Justify your answer.
UNIT –IV Differentiation and Integration
61.
Verify Rolle's theorem for f(x) = 2 + (x - 1)2/3 in [0, 2].
62.
Discuss the applicability of Rolle's theorem to the function f(x) = |x| in [-1, 1}.
63.
Verify Lagrange's Mean Valve theorem for f(x) = log x in [1, e].
Evaluate the following limits
1
64.
65.
 tan x  x 2
lim 
 ;(0, ) .
x 0
 x 
 x3 
lim  x  ;(0, )
x 0 e
 
..7
..7..
66.

 log( x  1) 
lim 
;(0, )

x 0
2
 sin x 
67.
 3
lim  1   ;(0, )
x 0
x

68.
lim x1n sin x;(0, )
69.
x  1nx
;(0,  )
x  x1nx
70.
lim
71.
x 0
lim
1nx
x 
x
;(0, ) .
Find the values of a and b in order that lim
x 0
x (1  a cos x )  b sin x
, may
x3
be equal to 1.
72.
1
1
1
x  x 2  1  x . Using this inequality, approximate
2
8
2
2 , and find the best accuracy.
Show that if x > 0, then 1 +
the values of
1.2 and
73.
Use Taylor's theorem with n = 2 to obtain more accurate approximations
for 1.2 and 2 .
74.
If f(x) = x2 for x  [0, 4], calculate the Riemann Sum with the partition
P = {0, 1, 2, 4} with the tags at the left end points of the sub interval.
75.
If I = [0, 4] in a closed interval calculate norms of the following partitions of I
(a) P1 = (0, 1, 2, 4)
(b) P2 = (0, 1, 1.5, 2, 3.4, 4)
(c) P3 = (0, 0.5, 2.5, 3.5, 4)
(d) P4 = (0, 2, 3, 4)
76.
Show that every constant function on [a, b] is Riemann integrable.
77.
Show that the function f(x) defined by
for 0  x  1
2
g(x) =
is Riemann integrable in [0,3].
for 1 < x  3
3
78.
Use Mean Value Theorem of differentiation to prove that
x 1
 log x < x – 1 for x > 1.
x
79.
Use the Substitution Theorem to evaluate the following integrals
4
(a)

1
sin t
t
4
dt
(b)

1
cos t
t
dt
9
80.
Apply the Second Substitution Theorem to evaluate the integral
dt
2
1
t
.
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