07a4bs02 mathematics -iii
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R07 SET-1Code.No: 07A4BS02
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
II B.TECH II SEM – REGULAR/SUPPLEMENTARY EXAMINATIONS MAY - 2010
MATHEMATICS –III
(METALLURGY & MATERIAL TECHNOLOGY)
Time: 3hours Max.Marks:80
Answer any FIVE questions All questions carry equal marks
- - -
1.a) Show that( ) ( )
( , )( )
m nm n
m n β
Γ Γ=
Γ +
b) Show that1
0
( , )(1 )
m
m n
xm n dx
x β
∞ −
+=
+∫ [8+8]
2.a) Find whether 2( ) 2 x iy
f z x y
−
= + is analytic or not.
b) Find the analytic function f(z) = u(r, θ ) + iv(r, θ ) such that u(r, θ ) = -r 3 sin 3 θ .
[8+8]
3.a) Find the real part of the principal value of ilog (1 + i)
b) Separate into real and imaginary parts of sech ( x + i y ) . [8+8]
4.a) Evaluate ∫ ng the path y = x and y = x1
2
0
( )
i
x iy d
+
+ z alo 2.
b) Evaluate, using Cauchy’s integral formula2
( 1)( 2)
x
C
e dz z z− −∫
, where c is the circle
z = 3. [8+8]
5.a) Expand f(z) = sinz in Taylor’s series about z =4
π
b) Determine the poles of the function f(z) =2
2( 1) ( 2)
z
z z− +. [8+8]
6.a) Find the residue of f(z) =3
4( 1) ( 2)( 3) z
z z z− − − at z = 1.
b) Evaluate2
3
2 5C
z
z z
−
+ +∫ where c is a circle given by
i) z = 1
ii) 1 z i+ − = 2
iii) 1 z i+ + = 2 [8+8]
7.a) State and prove Fundamental theorem of Algebra.
b) State and prove Liouville's theorem. [8+8]
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8.a) Find the invariant (fixed) points of the transformation1
1
zw
z
−=
+.
b) Determine the bilinear transformations whose fixed points are 1, -1. [8+8]
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R07 SET-2Code.No: 07A4BS02
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
II B.TECH II SEM – REGULAR/SUPPLEMENTARY EXAMINATIONS MAY - 2010
MATHEMATICS –III
(METALLURGY & MATERIAL TECHNOLOGY)
Time: 3hours Max.Marks:80
Answer any FIVE questions All questions carry equal marks
- - -
1.a) Evaluate
i)
1
4 2
0
(1 ) x x dx−∫
ii)4 2 2
0
( )a
x a x d − x∫
iii)
2
3 1/3
0
(8 ) x x dx−∫
b) Show that 2 ( 1/ 2) 1.3.5........(2 1).n
n n π Γ + = − , where n is a positive integer.[12+4]
2.a) Show that both the real and imaginary parts of an analytic function are harmonic.
b) Find the analytic function f(z) = u + iv if u + v =sin2
(cosh 2 cos 2 )
x
y x−. [6+10]
3.a) Find all zeros of
i) sinh z
ii) cosh z
b) Prove that sinh z , cosh z are periodic functions of imaginary
period 2 π i. [8+8]
4.a) Use Cauchy’s integral formula to evaluate2( 2)( 1)
z
C
edz
z z+ +∫ where c is the circle
z = 3.
b) Use Cauchy’s integral formula to evaluate2
3( 1)
z
C
edz
z −∫ where c is the circle z = 3/2.
[8+8]
5.a) Expand f(z) =2
1
6 z z− − about
i) z = -1.
ii) z = 1.
b) Find the poles of the functions.
i)2
1
( 2
z
z z
+
− )
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ii)2
2( 1) ( 2)
z
z z− + [8+8]
6.a) Find the residue of2
4 1
z
z + at these singular points which lie inside the circle z = 2.
b) Find the residue of2
21
z
z− at these singular points which lie inside the circle z = 1.5.
[8+8]
7. Show that all the roots of z5+3z2 = 1 lie inside the circle 3 4 z < and that two of its
roots lie inside the circle 3/ 4 z < . [16]
8.a) Determine and graph the image of z a a− = under the transformation w = z2.
b) Find the Bilinear map of the points z = -1, 0, 1 on to w= 0, i, 3i. Find the fixed points
of the transformation. [8+8]
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R07 SET-3Code.No: 07A4BS02
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
II B.TECH II SEM – REGULAR/SUPPLEMENTARY EXAMINATIONS MAY - 2010
MATHEMATICS –III
(METALLURGY & MATERIAL TECHNOLOGY)
Time: 3hours Max.Marks:80
Answer any FIVE questions All questions carry equal marks
- - -
1.a) Evaluate
i)6 2
0
x x e dx
∞−
∫
ii)4 3/ 2
0
xe x dx
∞−
∫
iii)24
0
3 x dx∞
−
∫ b)
Evaluate
i)
/ 2
6 7
0
sin cos d
π
θ θ θ ∫
ii)
/ 2
10
0
sin d π
θ θ ∫ [10+6]
2.a) Find where the function
i) w =1
z
ii) w =1
z
z − ceases (fails) to be analytic.
b) In a two dimensional fluid flow, the stream function ψ = tan-1(y/x), then, find
velocity potential function φ . [8+8]
3.a) Find the principle values of(1 3)
(1 3) i
i +
+
b) If u = log tan4 2
π θ ⎛ +⎜
⎝ ⎠
⎞⎟ then prove that tanh tan
2 2
u θ ⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠. [8+8]
4. Use Cauchy’s integral formula to evaluate3(1 )
z
C
edz
z z−∫ where c is
a)1
2 z =
b)1
1
2
z − =
c) 2 z = [16]
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5.a) Expand in Taylor’s series f(z) =4 9
z
z + about the point z = 0.
b) Find the poles of the functions 2 2
iz ze
z a+
c) Find the poles of the functions2 3 2
z
z z− +. [16]
6. Evaluate2 2
0( 9)( 4)
dx
x x
∞
+ +∫ 2 using Residue theorem. [16]
7. Prove, by using Rouche's theorem, that the equation ez = azn has n roots inside the unit
circle. [16]
8. Show that the transformation1
w z z= + , converts that the radial lines θ = constant in
the z-plane in to a family of confocal hyperbolar in the w-plane. [16]
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R07 SET-4Code.No: 07A4BS02
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
II B.TECH II SEM – REGULAR/SUPPLEMENTARY EXAMINATIONS MAY - 2010
MATHEMATICS –III
(METALLURGY & MATERIAL TECHNOLOGY)
Time: 3hours Max.Marks:80
Answer any FIVE questions All questions carry equal marks
- - -
1.a) Show that when n is a positive integer .( ) ( 1) ( )nn n J x J x− = −
b) Show that 1/ 22
( ) cos J x xπ
− = x . [8+8]
2.a) Determine whether the function f(z) = 2xy + i (x2 - y2), is analytic or not.
b) Find the analytic function whose real part u = sin2
(cosh 2 cos 2 )
x
y x−. [8+8]
3.a) Find the principal value of : (1-i)1+i.
b) Find the real and imaginary parts of sec z. [8+8]
4.a) Evaluate ∫ along y = x(1,1)
2 2
(0,0)
(3 4 ) x xy ix d + + z 2.
b) Evaluate3
3
sin3
2
C
z zdz
z
π
−
⎛ ⎞
−⎜ ⎟⎝ ⎠
∫ with c: z = 2 Cauchy’s integral formula. [8+8]
5. Determine and classify all singularities of the given functions.
a)3
1
z z−
b)4
41
z
z+
c)1 1
co t z z
−
d)2
41
x
e z−
e)1 c
os z
z
−
f)/ 2 z z
e [16]−
6. Evaluate
2
0
cos2
5 4cosd
π θ
θ θ +∫ using calculus of residues. [16]
7. Prove that all the roots of
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a) 16z5 – z + 8 = 0 lie between the circles1
2 z = and z = 1.
b) z6 – 9z2 + 11 = 0 lie between the circles z = 1 and z = 3. [16]
8. Find the bilinear transformation which maps the points z = 1, i, -1 on to the pointsw = i, 0, -i. [16]
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