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BANGABASI EVENING COLLEGE
MATHEMATICS -HONOURS
(Module-III)
 denotes the set of real numbers and  denotes the set of all positive integers
Group-A
(Real analysis-I)
Answer any two questions
1. (a) State LUB Axiom of real numbers. Is the Axiom valid for the set of rational
numbers?- Justify your answer.
2+1
(b) Prove that “Every infinite bounded set of real numbers has a limit point”.
3
(c) Prove that product of a bounded sequence and a null sequence is a null sequence.
4
2. (a) Define ‘ Denumerable set’. Prove that the intersection of an arbitrary collection of
closed sets is closed.
1+3
n
13 ,n

}. Give an example of a
(b) Find the derived set of the following set: {
set which has an infinite number of limit points.
2+1
1
2
(c) Examine whether  
1
1
    is a Cauchy sequence. The sequence
4
2n  n
{n 2 } is not a Cauchy sequence. Why?
2+1
3. (a) Let f : [a, b]   be continuous in [a,b]. Prove that f is bounded on [a,b].
3
11 1
1
 
does not converges. 3
(b) Prove that the sequence S n  define by S
n
23 n
(c) If a function f (x) is continuous in [a, b] , then it attains its supremum and
infimum at least once in [a, b] . Let f :  be such that f (x)  k (constant) for all
x   . Show that f is continuous on  .
2+2
1/x
e
4. (a) Discuss the continuity and discontinuity of the following function: f(x)
,
1e1/x
when x  0 and f (0)  1 .
3
(b) Give an example to show that a function continuous in an open interval may fail to
be uniformly continuous in the interval.
3
f
:



(c) A function
satisfies the condition f(x+y)=f(x)f(y) for all x, y   . If f is
continuous at x=0, prove that f is continuous on  . 4
Group-B
Full Marks-05
Answer one question
cos
n

(
n

1
)(
I

I
)

2
sin(
n

1
)
d

5. If In 
n
n

2
cos
 , show that
2
3d.
Hence evaluate 4cos
5

/2
2
n

1
6. If Jnsin xdx
, n is positive integer, prove that
0
2n
Jn 
Jn1. Use this to evaluate
2n1
/2
sin xdx.
7
5
0

x sin x
dx .
2
0 1  cos x
7. Evaluate 
5
(Module-IV)
Group-A
(Linear Algebra-I)
Full Marks-20
Answer two questions
2 2 1


8. (a) Find the eigen values and all eigen vectors of A 1 3 1.
1 2 2


2 2
2
x
y
10
z
4
yz

10
zx
(b) Reduce the quadratic form 5
to the normal form and
show that it is positive definite.
5+5
9. (a) Prove the following:
(i) 0 is the eigen value of a singular matrix.
(ii) The eigen values of a diagonal matrix are its diagonal elements.
1
(iii) If  is the eigen value of a non-singular matrix A then  is the eigen value of
A 1 .
(iv) If A and P be both n  n matrices and P be non-singular, then A and P 1 AP have the
same eigen value.
3
3

{(
x
,
y
,
z
)


:
x

y

z

0
}
(b) Show that W
is a subspace of  . Find a basis and
the dimension of the subspace W of  3 .
5+5
2 0 1


10. (a) Show that the matrix  3 3 0  is nonsingular and express it as a product of
 6 2 3


elementary matrices.
(b). Expanding by Laplace’s method show that
0 a b

a 0 d

b
d 0

c 
e f
c
e
af
2.


be

cd
f
0
5+5
A
det
B0
11. (a) Let A and B are real orthogonal matrices of the same order and det
then show that A+B is singular matrix.
(b) Determine the conditions for which the system
xyz 1
x2yz b
5x7yazb2
admits of (i) only one solution (ii) no solution, (iii) many solutions.
3+7
12. (a) If  ,  be any two vectors in a Euclidean space V, then prove that
.
(b) Use Gram-Schmidt process to obtain an orthogonal basis from the basis set
{(
1
,0
,1
),
(
1
,1
,1
),
(
1
,3
,4
)}
of the Euclidean space  3 with standard inner product.
(c) State Cayley-Hamilton theorem. Verify Cayley-Hamilton theorem for the matrix
1 2 1 


A1 1 1 .Express A 1 as a polynomial in A and hence compute A 1 .
2+4+4
2 3 1


Group-B
(Vector Calculus-I)
Full Marks-05
.Answer any one question
2
2 2
(,y
,)
z
2
x

3
y

zat the point P(2,1,3)
13. Find the Directional derivative of fx
in the direction of the vector a  iˆ 2kˆ .
5

A
)
g
r
a
d
A

divA .
14. Prove that div (
5
 2 2
2ˆ
ˆ

xi
yˆ
j
z
kis irrotational. Hence
15. Prove that the vector field defined by F
 
find a scalar function (x, y, z) such that F   .
5
Test Examination-2016
BANGABASI EVENING COLLEGE
MATHEMATICS HONOURS
First Paper
Full Marks- 50
The figures in the margin indicate full marks
(Module-I)
Group-A
(Classical Algebra)
Full Marks-20
Answer any four questions
!
2
!
3
!
4
!


100
!is divided by 15?
1. (i) What is the remainder when 1
3) , x 3(mod
5) and x 1(mod
7) .
(ii) Solve the system of linear congruence’s: x 2(mod
2 .(i) Prove that there exists infinite number of prime integers.
(ii) Find the remainder when 1920 is divided by 181.

ib
ab
 a
 2
i
log
2 2.
3. (i) Prove that sin


ib
b
 a
 a
(ii) Find the principal value of 1  i  .
i
3+2

1

1

1
4. If cosh
, where x, y, a are real and a  1 , then
(
x

iy
)

cosh
(
x

iy
)

cosh
a
prove that point (x,y) lies on the ellipse.
5
5. If x, y, x are positive real numbers and xyz 1, prove that
8
8
xyz

(
1

x
)(
1

y
)(
1

z
)
.
27
5
12
12
6. (i) Show that a  b is divisible by 91 if a and b are both prime to 91.
(ii) Calculate  (2048) and  (5040) , where the function  , is called Euler’s Phi function.
3+(1+1)
7. Solve the equation x 3  3x 2  3  0 by Cardan’s method.
8. If  ,  ,  be the roots of the equation x 3  qx  r  0 , find the equation whose roots are
(   ) 2 , (   ) 2 , (   ) 2 .
Group-B
Full Marks-05
Answer any one question



(
a
,
b
)
:
a
,
b

Q
,a

b
is
an
ineger
9. Let  be a relation defined by 
where Q is the set

of all rational numbers. Is on Q is an equivalence relation? Justify.
10. Let S is the set of all 2x2 non-singular matrices whose elements are real numbers.
S is a proper subset of M 2 ( R) . Prove that S forms a non commutative group under
matrix multiplication “.”.
11. (a) Let A, B, C be non-empty sets. Prove that ( A  B)  (C  D)  ( A  C )  ( B  D) .
(b) Let f : N   N  be the mapping given by f(x)=2x, where N+ is a set of all
positive integers. Show that f is injective but not surjective.
(Module-II)
Group-A
(Analytical Geometry of Two Dimensions)
Full Marks-20
Answer any four questions
12. Show that the locus of the poles of the tangent to the director circle of the ellipse
x2 y2
2
4 2
4
2 2
 2 1 w.r.t this ellipse is x
a

y
b

(
a

b
)1
.
5
2
a b
2
13. If three normals from a point to the parabola y  4axcut the axis at the points
whose distances from the vertex are in A.P, then show that the point lies on the curve
2
27
ay

2
(x
2
a
)3.
5
2
14. Tangents are drawn to the parabola y  4ax at the points whose abscissae are in the
ratio p:1. Show that the locus of their point of intersection is a parabola.
5
2
2

2
hxy

by

2
gx

2
fy

c

0
15. If the equation ax
represents two straight lines
4
4
2
2
g
c
bf

ag
equidistant from the origin, show that f 
.
5


/r
a
cos

b
sin
may touch the
16. Show that the condition that the straight line 1
2
2
1.
circle r  cos is bk 2ak
5
2
2
x

4
xy

y

4
x

2
y

a

0
17. Reduce the equation 4
to its canonical form,
determine the nature of the conic for different values of a.
5
Group-B
Full Marks-05
Answer any one question
18. If the external bisectors of the angles of a triangle intersect the opposite sides at the
points P, Q, R, then prove that P, Q, R are collinear..
19. (i) In any triangle ABC, with usual notations, prove that
a
cos
B

b
cos
A
(i) c
,
(ii)
a
b
c


.
sin
A sin
B sin
C
20. If the diagonals of a quadrilateral bisect on another, then prove that the figure is a
parallelogram.