Download B. Sc(H)/Part-III Paper - Bangabasi Evening College

Bangabasi Evening College
MATHEMATICS-HONOURS
B. Sc(H)/Part-III
Paper- V
Module IX
Analysis
Answer any two Questions
1. (a) Prove or disprove: The range of any convergent sequence in  is a compact set.
e dt
(b) If e denoted by the equation   1 , prove that 2  e  3 .
1 t
(c) If S is a closed and bounded set of real numbers, then prove that every cover of S has
a finite subcover.
(d) Show that log( 1  x)  log( 1  x)  log( 1  x 2 )  log( 1  x 4 )   converges for x  1 .
2. (a) If a function f : [a, b]   has a bounded derivative on [a,b], then prove that f is a
bounded variation on [a,b].
(b) Construct a real-valued function on a compact interval which is continuous but is not
of bounded variation on that interval.
(c) If f is monotonic on [a,b], prove that f is bounded variation on [a,b] and
Vab ( f )  f (b)  f (a) .
3. (a) Let f : [a, b]   be a bounded function such that f is continuous on [a,b] except
at finitely many points in the open interval (a,b). Prove that f is Riemann integrable on
[a,b].
(b) Let f : [a, b]   be a continuous function. Show that the function F : [a, b]  
x
defined by F ( x)   f (t )dt , x  [a, b] is differential in its domain.
a
(c) Show that for k 2  1 ,
 1/ 2
dx

1
 

2
2
2
0
6
(1  x )(1  k x ) 6 1  k 2 / 4)
x2
e
(d) Prove that lim
x 0
1 t
0
x2
dt
 e.
Module X
Group-A
Answer one Question from each Section
Section-I
1. (a) Prove that a linear transformation T of a vector space V of dimension n into a
vector space W of dimension m over a same field F can be represented by m  n matrix
A over F with respect to a basis in V and a basis W.
(b) The matrix representation of a linear transformation T : 3  3 is
1 0 0 


 0 3  1
 0 2  1


3
relative to the standard ordered basis of  . Find the explicit representation of T and
matrix representation of T relative to the ordered basis {( 0,1,2), (1,0,1), (2,1,1)} .
2. (a) Let V and W be vector spaces of finite dimension over a field F and let T : V  W
be a linear mapping. Prove that T is nonsingular if and only if the matrix of T relative to
some chosen bases is nonsingular.
(b) A linear transformation T : 3  3 is defined by
T ( x, y , z )  ( x  y  z , 2 x  y  2 z , x  2 y  z ) .
Find KerT and dimensions of KerT and ImT
Section-II
3. (a) Let H be a subgroup of a group G such that [G:H]=2. Prove that H is a normal
subgroup of G. Is the converse is true? Justify your answer.
(b) Prove that an infinite cyclic group is isomorphic to the group (Z, +), ( Z is the set of
all integers).
4. (a) Let f : (G,)  (G / ,) be a isomorphism of groups. Prove that G / is cyclic if and
if only if G is cyclic.
(b) Let H and K be two normal subgroups of a group G such that H  K  {e} , where e is
the identity of G. Prove that hk  kh , h H and k  K .
Group-B
Differential Equations
Answer any one Question
d2y
 y  x near the ordinary point x=0.
dx 2
(b) (i) State a set of sufficient conditions for existence of Laplace transform.
(t  1) 2 , t  1
(ii) If F (t )  
, determine the Laplace transform of F(t).
0  t 1
0,
(c) Let F(t) be periodic function with period T (>0). Show that
1. (a) Obtain the series solution of
T
e  st F (t )dt
0
1  e  sT
L[ F (t )]  
d2y
dy
 3 y  0 in series near the ordinary point x=0.
dx
dx 2
(b) State Convolution theorem on Laplace transform and use it to find


1
L1 
.
2
 ( s  1)( s  1) 
2. (a) Solve the equation
 3x
(c) Solve, by using Laplace transform
y (0)  0, y ' (0)  3 .
d2y
dx
2

dy
 2 y  18e  x sin 3x , when
dx
Group-C
Tensor Calculus
Full Marks-10
Answer any one Question
1. (a) Define contravariant and covariant vector. Show that for the orthogonal
transformation x i  a ij x j (where a ij a kj   ki , Kronecker delta) the laws of transformation
of the covariant and the contravariant vectors are identical.
(b) Show that the partial derivative of a covariant vector is not a tensor.
(c) If a ik are components of a symmetric contravariant tensor prove that
g jk
1
a jk [ij , k ]  a ij
2
x i
2. (a) The line element of three dimensional Riemannian space
ds 2  (dx1 ) 2  ( x1 ) 2 (dx 2 ) 2  (dx 3 ) 2 .
1

Calculate 

2 2 
i 
i 

(b) Show that 
  j log g , where g  det( g ij )  0 and 
 is a
i j  x
i j 
Christoffel symbol.


(c) Sow that the covariant derivative of a tensor of type (1,0) is a tensor of type (1,1).
is