Page 148 - Engineering Mathematics Workbook_Final

Complex Variables

(a) 2 (b) 5 the greatest integer less than or equal
to  is
(c) 2 + 5i (d) 5 + 2i

→
81. Let  be the circle given by z = 4e i 85. Let : f C C be non-zero and
, where  varies from 0 to 2 . Then analytic at all points in Z. If
( ) z =

( )cot 
f z
F
( ) z for
e
  z − z 2z dz = z C / Z , then the residue of F at

2

n Z is
)
(
−

i 
2
2
(a) 2 i e − ) 1 (b) (1 e
( )

f
(a) f n (b) ( ) n
2

−
(c) ( i e − ) 1 (d) 2 i  (1 e 2 ) f ( ) n
' f
(c)  (d) ( ) z z n
=
2

3 z
f
82. Let ( ) z = z e for z C and let
i
=
 be the circle z e , where  86. Which of the following is the
varies from 0 to 4 . Then imaginary part of the possible value
i
1   f 1 ( ) z of ln ( )
2 i  f ( ) z dz
(a)  (b) 
→
83. Let : f C C (the set of all complex 2
numbers) be defined by  
)
2
3
3
f ( , x y = x + 3xy + ( i y + 3x 2 ) y . (c) 4 (d) 8
' f
Let ( ) z denote the derivative of f
f z =
+
87. If ( ) 11 iv is analytic then, the
with respect to z. Then which one of
the following statements is TRUE? harmonic conjugate of
u = x − y + xy is
2
2
(a) ( ' 1f + ) i exist and 2 2
f ( ' 1 i+ ) = 3 5 (a) x − y − xy
(b) x + 2 y − 2 xy
(b) f is analytic at the origin
(c) f is not differentiable at I (c) 2xy + 1 ( y − 2 x 2 )
2
(d) f is differentiable at 1
xy + ( 2 y − 2 x 2 )
(d)


i z
84. Let  = C  2z − 2 e 5z + 2 dz , 2
+
C :cost i sint , 0 t  2 . Then

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