Gauhati University MATHEMATICS FIRST PAPER 2007 Question Paper

University :  Gauhati University

Degree : B.Sc Mathematics 

Question Paper Subject: MATHEMATICS FIRST PAPER Question paper

Question Paper Year : 2007

Question Paper Details : Previous Years Question Papers Types Version

(Real and Complex Analysis)

PART-A (Objective-type Questions)

(Marks: 32)

Each question (1-16) carries four codes (a), (b), (c) and (d), out of which one is for
correct answer. Choose the correct code: 2 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 true 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. ( ) 3 3 3

+ +
x y z a is 2 2 2 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 equation
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
( 2 )
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
(a) non-zero real
(b) complex
(c) zero
(d) positive real
11. If
(a) straight line
(b) circle
(c) ellipse
(d) hyperbola
12. The value of
along the straight line from z =0 and z =1+i is
13. A simple pole of
(a) 0
14. Residue of
at z = 3 is
9 2
( )
z
z
f z
-
=
z = 2

+ C z i
dz
f (z)
5
( )p
a
5
( ) -p b
5
( ) i c
p
5
( )
i
d
-p
( (1 ) (1 ))
1
( )
3 3
2 2
f z x i y i
x y
+ - -
+
=
ÎR(a )
fd k{ (b) (a)}
b
a

a £ a -a
15. If
and C is the circle
is
16. Let
If z
0 and f (0) =0, then at the origin f is
(a) continuous and f
(0) exists
(b) continuous and C-R equations are satisfied
(c) discontinuous and C-R equations are satisfied
(d) discontinuous and f
(0) does not exist
PART-B (Subjective-type Questions)
(Marks: 48)
Answer any three parts of each of the Question Nos. 17, 18, 19, 20
17. (a) If f
and k is a number such that
for all x [a, b], then prove that
+ x + x + x + × × × 2 3
4
1
3
1
2
1
1
2 2
1
a bn
n
n + 
³
dx
x y
x y
dy dy
x y
x y

dx


+
¹ -
+
- 1
0
3
1
0
1
0
1
0
( )3 ( )
dxdy
E


a - x - y2 2 2
u e (x cos y y sin y) x = -
(b) Prove that a monotonic increasing function which is bounded in closed interval
[a,b] is a function of bounded variation. State its total variation.
(c) State and prove Weierstrass M-test of uniform convergence of a series of
function.
(d) Show that the power series
Is uniformly convergent on [-1, k], 0(c) Prove that every continuous function is measurable. 4×3=12
18. (a) If f and g are bounded measurable functions defined on a measurable set E of
finite measure, then prove that

+ =
+

E E E
(af bg) a f b g
(b) If f is bounded and integrable on [-, ] and if an, bn are its Fourier coefficients,
then prove that
is convergent
(c) Show that
(d) State and prove a theorem connecting a line integral along a closed contour with a
double integral over the domain bounded by that contour.
(e) Compute
Where E is the region bounded by the circle x2+y2= ax. 4×3=12
19. (a) Find the analytical function f(x) = u + iv of which the real part is
(b) Prove that the Cauchy-Riemann equations can be written in polar form as
q ¶q
= - ¶
¶
¶
¶
= ¶
¶
¶ u
r
v
and
v
r r
u 1
( 1)( 3)
1
( )
+ +
=
z z
f z
1< z < 3
0 1 2 2
4
= p
+


¥
x
dx
z = 1 and z = 2
(c) Show that
16
21
)
6
(
sin
3
6 i
dz
z
z
C
p
p =
-


(d) Expand
In a Laurent series valid for
(e) Show that 4×3=12
20. (a) State and prove Liouville’s theorem.
(b) Prove that all the roots of
z7-5z3+12 = 0
lie between the circle
.
(c) State Rouche’s theorem and use it to show that every polynomial of degree n has
exactly n zeros.
(d) If f: G  C is analytical, then prove that f preserves angles at each point z0 of G,
where f( z0)
0.
(e) Find a bilinear transformation which maps the upper half of the z-plane into the
unit circle in the  = 0 and  is mapped into  = -1. 4×3=12

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