Gauhati University 2007 Previous : thetica: I - Question Paper

R[a,b] / a Î


b
a
f (x)da (x)
( ) ( )
/ / lim x
n
f x f
n®¥
=
([ ] )
3
0
2
x d x - x
2007
MATHEMATICS
FIRST PAPER
(Real and Complex Analysis)
Full Marks: 80
Time: three hours
The figures in the margin indicate full marks for the ques.
PART-A (Objective-type Questions)
(Marks: 32)
every ques. (1-16) carries 4 codes (a), (b), (c) and (d), out of which 1 is for
accurate ans. select the accurate code: two x 16=32
1. The RS-integral
is equal to
a) 2
b) 3
c) 8
d) 5
2. If {x/(1+nx2)} converges uniformly to f on [0,1], then
is actual if
(a) x
0
(b) x>0
(c) x<0
(d) x=0
3. If f is of bounded variation on [a, b] and a  c  b, then
(a) V(f, a, b) – V(f, a, c) < V(f, c, b)
(b) V(f, a, b) – V(f, a, c) = V(f, c, b)
(c) V(f, a, b)- V(f, a, c)> V(f, c, b)
(d) V(f, a, b) + V(f, c, b)= V(f, a, c)
4. If f and
Then
is equal to
f x x dx
b
a
( ) ( ) /
a
¢
b
a
(b) a (x) f (x)dx

¢
b
a
(c) f (x)a (x)da
¢
b
a
(d) a (x) f (x)da
5. Let f (x, y) = xy , then f is
x dydz y dzdx z dxdy over the sphere
S
6. ( ) three 3 three

+ +
x y z a is two 2 two 2 + + =
3
3
4
(a) p a (b) 4p a2
3
4
3
(c) p a
5
5
12
(d) p a
7. A Í R
(a)m + (A + x) ¹ m (A)
(b)m (A + x) > m (A)
zz + bz + b z + c = 0
(a)
(a) Continuous at (0,0) and differentiable at (0,0)
(b) Both continuous and differentiable at (0,0)
(c) Continuous at (0,0) but not differentiable at (0,0)
(d) None of the above
8. Measure of A = {2, 4, 6, 8,…} is
(a) 
(b) 2
(c) 0
(d) 1
9. If c is real and b is complex, the formula
represents
(a) straight line
(b) circle
(c) ellipse
(d) hyperbola
is measurable and x € R, then
(c)m (A + x) = m (A)
(d)m (A + x) < m (A)
{ : 1}
1
2
( ) = =
+
= and A z z
z
z
f z
(1 )
3
1
(b) + i
( 1)
3
1
(c) i -
3
1
(d)

+
- +
i
x y ix dz
1
0
( two )
a i
3
1
( )
is
sin z cos z
1
-
4
( ) -p c
2
( )p
b
4
( )p
d
( 3)
1
2 -
+
z z
z
9
1
(a)
9
2
(b)
9
3
(c) 9
4
(d)
10. If f (z) is an analytic function with constant modulus, then f (z) is

Earning:   Approval pending.