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Candidates are required to give their answers in their own words as far

as practicable.

All question carry equal marks.

Attempt any SIX questions.

Q. [1] [a] State Cauchy Reimann differential equation for the function

f(z) to be analytic in the complex plane. Using it shows thatfunction f(z) = z2 is analytical in the entire z plane and find

its derivative.

[b] Define complex integration. Evaluate c

dzzf )( , where f(z)

= cosz and c is the line segment form i to +iQ. [2] [a] State and prove Taylors theorem.

[b] Define a pole of order m. Calculate the residues of the

function f(z) =)3()2()1(

3

++ zzz

zat its poles.

Q. [3] [a] Determine the Z transform of the signals and Roc x(n) =

(coswon) u(n)

[b] Prove the linearity of z-transforms.

Q. [4] [a] Find the inverse z-transform of the function.

11

1

23131)(

+++=

zzzzx

[b] Determine the one-sided z-transform of the signals;

(i) x(n) = an u(n) (ii) x(n) = x(n-2) when x(n) = an

Q. [5] [a] Find a Fourier series to represent f(x) = -x; for 0

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as practicable.



Q. [1] [a] Prove that Cauchys Riemann equation are necessary

condition for a function f(z) to be analytic.

[b] Evaluate dzz

zz

c

++3

24

)1(

63where C is a circle |z| = 2 .

Q. [2] [a] State and prove Eauretns theorem.

[b] Define singularity of a function and discuss its different

types. Find the poles and residues at each pole of the

function. f(z) = )3)(2()1( 2

3

+ zzz

z

Q. [3] [a] Find the z-transform of the following:

[i] x(k) = ak-1 [ii] x(t) = e-at sinwt

[b] Prove that:

[i] z[x(t-nT)] = z-nx(z)

[ii] z[x(t+nT)] = zn

=

1

0

)()(n

k

kzkTxzx

Q. [4] [a] Find the inverse z-transform of the function:

[i] x(z) =)()1( 2

2

nTezz

z

by partial fraction method.

[ii] x(z) =)1()2(

22

2

+

zz

zzby inversion integral method.

[b] Solve the difference equation

x(k+2)+3x(k+1)+2x(k)=0

x(0) = 0 , x(1) = 1

Q. [5] [a] Find the half range Cosine series for f(x) defined by

f(x) = 0 , a

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as practicable.



Q. [1] [a] State Cauchys Riemann equations. Prove that if a function

f(z) of a complex variable is analytic, the Cauchy-Riemannequations must be satisfied by and the real and imaginary

parts of f(z).

[b] Evaluate the integral

C z

z3

2

6

sin

Where C is a circle |z| =1

Q. [2] [a] State and prove Taylors Theorem.

[b] What do you mean by a pole and a residue? Find the residue

at each pole of the function f(z) =)3()1( 3

3

+ zz

z

Q. [3] [a] Find the z-transform of:

[i] x(n) = an u(n) [ii] x(n) = cosu(n)

[b] [i] If z[x1(n)] = x1(z) and z[x2(n)] = x2(z), prove that

x1(z)*x2(z)= x1(z)x2(z)

[ii]Determine the z-transform and sketch the ROC for x(n)

= jn

n

3

1

Q. [4] [a] Solve the different equation: y(n+2)+y(n) = 1 if y(0) = 0[b] Starting form a trigonometric series, obtained a Fourier

series of f(x) in the interval (0,2).Q. [5] [a] Write down the half range sine series in (0,2) for f(x)=0.

[b] Obtained the Fourier cosine transform of e-x

, (x>0) and

hence evaluate( )

+0221 x

dx

Q. [6] Solve the boundary value problem that arises in the heat

conduction in rod.

1

100)0,(,0),1(),0(,

2

22 x

xututx

ua

t

u===

=

Q. [7] An infinitely long metal plate in the form of the area enclosed

between the lines y=0 and y =1 for positive values of x. Thetemperature is along the edge y =0 and y =1 and at infinity.

In the steady state, the equation 02

2

2

2

=

+

y

T

x

T. If the edge

x =0 is kept at constant temperature to show that its solution

is T =1

)12sin(4

1

1

)12(y

neT

n

n

o

+

=

+

Q. [8] Use simplex method to maximize the following function z=f(z)

=150x1+300x2 subject to the constraints x1 0, and 2x1+x2

16, x1+x2 8, x2 3.5

PURWANCHAL UNIVERSITYIV SEMESTER FINAL EXAMINATION- 2004

LEVEL : B. E. (Electronics & communication)SUBJECT: BEG204HS, Applied Mathematics

Full Marks: 80TIME: 03:00 hrs Pass marks: 32

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as practicable.


Attempt any SIX questions. Q. No. [1] is compulsory.

Q. [1] [a] Define the analytic function. Find the real and imaginary

part of the following. (i) Tanhz (ii) logsinz. [8][b] Show that v(x,y) are harmonic functions if

f(z). [7]

Q. [2] [a] Evaluate

C

dzzzz

z

)2)(1(

34where c is circle |z|=3/2 [7]

[b] State Taylors theorem and find the Laurents series of the

function )1( zz

ez

about the center a where a=1. [6]

Q. [3] [a] (i) If z(x(k)) = X(z) then show that Z(kx(k)) = -z ( ))(zxdz

d

[7]

(ii) Find the z-transform of akcosk for k 0.

[b] Find the inverse z-transform of

)1()2(

222

3

+

zz

z[6]

Q. [4] Solve the difference equation: x(k+2) 1.3x(k+1) + 0.4x(k)

= (k), x(0)= x(1) =0 and x(k) = 0 for k /2 and hence show that

=

0

22/

1

cos)2/cos(

w

wxwtcosx for |x| < /2

0= for |x|> /2 [7]

[b] Find the Fourier sine transform of f(x) = 0>

xx

e ax

and

hence show that

=0

1 2/sin/ aexdxaxTan [6]

Q. [6] A tightly stretched string with fixed ends pointer at x =0

and is initially the position is given by

Also it is initially at rest and suddenly reached. Find thedeflection u(x,t). [13]

Q. [7] If 02

2

2

2

=

+

y

u

x

u, which satisfy the condition u(x,0) = sinx

for all x , u(0, ) = 0 for all x shown that its solution isu(x,y) = e-ysinx. [13

Q. [8] Solve linear programming problems:

Minimize Z = 3x1+2x2Subject to 2x1+4x2 10

4x1+2x2 10x2 4

x1, x2 0 by using simplex method(constructing duality). [13]


LEVEL : B. E. (Electronics & communication)

SUBJECT: BEG204HS, Applied Mathematics

Full Marks: 80TIME: 03:00 hrs Pass minarks:

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as practicable.



Q. [1] [a] State the Cauchy Riemann equations. Using the Cauchy

Fiemann equations, show f(z) = z2

is analytica in z-plane.[b] Define complex integration. Evaluate f(z) dz where f(z) =

sinz and c is the line segment from 0 to i.

Q. [2] [a] State and prove Taylors theorem.

[b] Determine the poles and residue at each pole of the function.

[i] Cotz [ii]3

2

)( az

ze

Q. [3] [a] Obtain the z transform of[i] x(t) = e-at coswt [ii] x(k) = k2 ak-1, k 1

[b] Prove that

[i] z[x(t-nT)] = z-n x(z)

[ii] z[x(t+nT)] = zn[x(z)- k=0 x(kT)z-k]

Q. [4] [a] Determine the inverse z transform of

[i]

23

1232

2

++

++

zz

zz[ii]

2

)2)(1(

4

zz

z

[b] Solve the difference equation by z-transform method:

X(n+2)+3x(n+1)+2x(n) = 0, x(0) = 0 , x(1) = 1

Q. [5] [a] Find the Fourier sine integral of f(x) = e-kx , x>0 , k>0 and

hence show that

=+0

22 2

sin kxedw

wk

wxw , x>0, k>0.

[b] Find the Fourier transform of2

xe

Q. [6] Solve2

22

2

2

x

y

t

z

=

for a string of length l fixed at both ends

with y(x,0) = k(lx-x2), 0 x 1

Q. [7] If 02

2

2

2

=

+

y

z

x

z

z= sinx, at y =0. for all values of x and z =0

at y= for all values of x, show that z= e-ysinxQ. [8] Maximize

Z = 4x1+3x2

Subject to 2x1 +3x2 6-3x1 +2x2 32x1 +x2 4

x1, x2 0, by using simplex method.


LEVEL : B. E. (Electronics & communication)

SUBJECT: BEG204HS, Applied Mathematics

Full Marks: 80TIME: 03:00 hrs Pass minarks:

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