mathematics iii1

8/10/2019 Mathematics Iii1

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R09 Set No.2Code No: A109210201

1

3

n

3 2

II B.Tech I Semester Examinations,November2010

MATHEMATICS-III

Common to ICE, E.COMP.E, ETM, EIE, ECE, EEE

Time: 3hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

? ? ? ??

1. (a) EvaluateR

(x+y)dx+x2y dy along y=3x between (0,0) and (3,a)c

(b) EvaluateR

ezdz where C: |z| = 1 [15 ]c

2. (a)FindtheResiduesof f(z) = z2z(z+2)

at z =-2.

(b) FindtheResiduesof f(z) = z3

(z1)4(z2)(z3)at |z|=1. [8+7]

3. (a) Define conformalTransformation. Show that a bilinear transformationis con-

formal.

(b) Show that circles areinvariantunder linear transformationw =az +c . [8+7]

4. (a) P.T. xJ1(x) = nJn(x)- x Jn+1(x).

(b) EvaluateR x dx Using functions. [15]

0 1+x6

5. (a) S.T. the real & imaginary parts of the function w = log z satisfy the C-Requations when z is not zero.

(b) S.T. f(z) =z+2z is not analytic anywhere in the complex plane. [15]

6. (a) Prove that, for all x, x7 = 16 P7(x) + 8 P5(x) +

14P3(x) +1P1(x)429 39 33 3

(b) Show thatR 1

1xkPn(x)dx =0 for k=0,1,2,n-1 [15]

7. (a)Represent thefunction f(z) = 1z(z+2) (z+1)

inLaurent series within

5 7

4 |z|

4

(b) Define for a complex function(z)

i. Isolated Singularity

ii. Removable Singularity

iii. Essential singularity [15]

8. (a) How many vertices do the following graphs have if they contain (explain your

answer)

i. 16 edges and all vertices of degree 2

ii. 21 edges, 3 vertices of degree 4 and others each of degree 3

(b) Determine whether the following graphs (Figures 1) are isomorphic ornot.

[8+7]

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Fig -1
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2

(z /6)3

0

x2+y2

x

Code No: A109210201 R09 SetNo. 4


MATHEMATICS-III





? ? ? ??

1

1. (a) Evaluate

R

1

1

x4

(1x2

)

2

T2(x)dx.

1

(b) Prove thatR

1

x2Pn 1(x)Pn+1(x)dx =0 [15]

2. (a) EvaluateR

sin2 z dz whereC:

|z| = 1

(b) EvaluateR

log z dz where C: |z 1|=1/ [15](z1)3 2

c

q23. (a) S.T. J3/2(x) = x {1/xsin xcos x}

(b) S.T. R1

x3(1x)5dx=2(8, 6) [15]

4.Expand 7z2 about thepointz =-1 in the region 1


4/7

3


7. (a) Find the image of|z| = 2 under the transformationw =3z.

(b) Show that the transformationw=

z

2

maps the circle |z 1| =

1 in to thecardiod r=2(1+cos) [8+7]

8. (a) Draw a corresponding binary tree for the following algebraic expression.

(b) Show that a tree is abipartitegraph. [8+7]

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4


(z2 +2z+5)

dx

n

z+i i+z

z(z2 z 2)


MATHEMATICS-III





? ? ? ??

1. (a) Find the residue of 1(zSinz)

at z = 0.

(b) EvaluateR

(z3)dzwhere c is |z+1i|=2. [7+8]

2. (a) P.T. Jn

(x)=0 has no repeated roots except at x=0

(b) P.T. d {xJ1(x)} =xJ0(x) [15]

z+i

3. (a) EvaluateR

1i

(2x+iy+1)dz along thestraight line joining (1,-i) &(2,i)

(b) If C is the boundary of the square with vertices at the points z =0, z =1, z

=1+i and z =i show thatR

(3z+1)dz =0 [15]c

4. (a) P.T.R 1

1(P1)2dx=n(n + 1)

(b) P.T. P 1P 1 =(2n 1)Pn 1 [15]n n2

5. (a) Find the map of the circle|z| = c under the transformationw =z2+4ii z

b)Show that both the transformationsw = zi andw= transform |w|1intoupper

half planeI(z)>0. [7+8]

6. (a) S.T. tan1 z = 1 log i+z2 iz

(b) Find the roots of sinz =cosh4 [15]

7. (a) Write the Kruskals algorithm for minimal spanning tree.

(b) Consider the followinggraph.Determinewhich of the following sequenc sare

paths; simple paths; cycle and simple cycle.

i. v1 e1 v2 e6v4 e3v3 e2 v2

ii. v5 e5 v1 e8v4 e3v3 e2 v2 e6v4 e4v5

iii. v1 e8v4 e3v3 e7 v1 e8v4

iv. v5 e5 v1 e8v4 e3v3 e2 v2 e6v4 e4v5 [7+8]

8.Expandf(z)= z+3

in powers of z.

(a) With in the unit circle about the origin

(b) With in the annular region between the concentric circles about the origin

having radii 1 and 2respectively.(c) The exterior to the circle of radius 2. [15]

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5


0

z2+z

2

= 1


MATHEMATICS-III





? ? ? ??

1. (a) EvaluateR a

x4

a2x2dx

(b) P.T.R/2d

R/2

sind= [15]0 sin 0

2. For the function f(z) =2z3 +1

find

(a) A tailors expansion valid in the neighborhood of thepoint i.

(b) A Laurents series valid within the annulus of which centre is origin. [15]

3. (a) Find the image of the triangle with vertices at i,1+i,1-i in the z-plane,under5i

thetransformation e 3 .(z2+4i)

(b) Find the image of the infinite strip, 0


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6


(z 1)3

(z+1)(z i)2

(b) Evaluate by Residue theoremR

c

4z24z+1(z2)(z2+4)

dz whereC: |z|=1. [8+7]

6. (a) EvaluateR

c

z3 +z2+2z1dz whereC:

|z| = 3

(b) EvaluateR

c

z4 dz where Cis the ellipse 9x2+4y2=36 [15]

7. (a) Prove that Tan1z=h

log(i+z)

i

.(iz)

(b) Find the analytic function whose real part is r4 cos 4. [15]

0 ; 1

mathematics iii1

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