ii

Reg. No,

r[t,] *f

Find rhe singrlar solution of Z = px t qy t

B.Tech. DEGREE EXAMINATION, NO!'EMBER 2015

Thid

(oR) Solve (mz - ny) p + (w

b.i.

ii.

sotve (o3

-

lz) q = ly

sl:z ur'

MA1OO3

jr(r)=r+

_

TRANSFORMS AND BOTJNDARY VALI]E PROBLEMS 2013 20l1and 2014 -2015)

(For the candidates admitted during the academic year

-zo2o')z =sin(x +2y)+3x2 y

Find the Fourier series of

29. a.

- nx Notei

(i)

x2

n (-r,x) of pedodicity 2zL Hence deduce

(iD

Part - A should be answered in OMR sheet within first 45 ftilutes and OMR sheet should over to hall invigilator at the end of45t minute. Part - B ard Part - C should be answercd in answer booklet

Time: Three Hours

6

-

(oR)

b.

The values of x and the conesponding values of that

30.

a.

/(x)

= 0.75 + 0.37 cos8 +

l.004sin, *1rn,

T

x

0

f(x)

i

1.98

1.30

(x) g

l.

-2+. T

T

T

,

,

I .05

over a period T are given below. Show

2T

.

-{.88

1_30

5T

T

6

4.25

1.98

b. A rod of length / has its ends A

and B kept at ooc and l00oc respectively until steady-state conditions prevail. Ifthe temperatue at B is reduced suddenly to 0'C and kept so, while that of A is maintained. Find the temperature u(x, t).

Find the Fourier tuansform

sln.t -x cos .]r

Ir.

7

[r-.2

t^, rl<1, r lor of f(x.) given oy 71x1=.]t--'-' ' " " 10. for xl>l hence evaluate

32. a.i.

ii.

The complete integral of pq =

I

15

(A) az=a2x+y+ac (C) az= )c+y+c 7(x2 + y2) is (A) xp= yq

(B) z=ar+aY+c @) z=x+Y+c

by

eliminating arbitmry function fiom

4.

Z=

f.,(y+x)+ fr(y-2x)+ fr(y+3x) @) z

=

7(y +x)+

fzj

+ 2x) + f.(y

-2x)

+

-3x)

fr(y +3x)

if

The partial differential equation is parabolic (B) 82 -4Ac o (D) 82 4AC +o 82 -4Ac =o

(c)

= I f $l aO (A) Ever

a*

(o*)-, +

"L,

a'

x

)\x'

+

(C)

b'-dx. ) *

''

6.

';{},

I

tx>t -a t.r"" .rur*t. d f [ 8" Use convolulion theorem lo Iind the in\ erse Z-bansform o I

b. Solve

(c)

2

Find rhe Fourier transform of

(-zl

3. sot'r" (o' -zoo'2 -6Dn)z =o ( ) z = ft(y - x)+ f,(y -2r)+ f,(y +3x) (B) z = f,(y - x) + fr(y

.x

* Use transform metiod to evaluate [,,

Find

(B) xy=pq (D) x+P=y+q

(C) py=sx

cos dx.

i)lx'

ii.

X#,'1h!;10,H:'*'

z=

(oR)

b.i.

b€ handed

Max. Marks: 100

2. The partial differential equation formed

A sting is stretched and fastened to two points / apa(. Motion is started by displacing the sting into the form y=*(A-r2) from which it is released at time t = 0. Find the displacementy(x,t).

ll.a.--

Semester

/(.x) {l-^0

'{ if

z

lf l, Y, ;t rt"t i"

Neither even nor odd

The constant

aa

ofthe Fourier

(A) k

(c)

|

0

(B) odd (D) Periodic series f (t)=k1n0
7.

(22 -r)l4z +L)

The RMS value of

(A)

(x)

in a < x < b is

(B)

0

VQ)a,

:!. by" usine" residue method. t))' l(z-a)(z

Z 'l -

(oR) the equation Jr',+ 2-7 y,4+12y, =2" given that yn = 1

is

lb

= 0. Using Z-transform.

lJtt,1'a.

\ t-o 2t

NT3MAlllil3

(D)

b

lr(,)a, b+a 21NI'MA1OO3

8.

Halfrange cosine series for

(A) a" g+ 3 ) a- cos /rr

(c)

.

(x)

in (0, n) is

17.

G) €,

Lbisr.lrw

rt

n-l

sLdncosw

a

9.

The one dimensional wave equation is

(A't

(c)

a"

.62u

At

Ax'

18. "

dt (D) a2,

au a2u at ax' -=a_

z+1

z n-l

(B) a2v

d2u

Z-transform of

(A) :

Ax'

1

nl

@\

(c) :

dzu

A\'

(B) , z-l @), z-l

(c\ z,

0) d*11,,

h.-l

Fi"d,[(-r)'] (A) z+1

At'

19. The inveme Z-ransform of l(z) 10.

l.

----"'" !d = o't1 - Ax2

How many initial and bomdary conditions are required to solv

(A) (C) I

@) @)

Four Three

20.

=-+I

(c)

10x

""

can be found out by

Synthetic division method Diagonalisation method

@) (D)

Long division method Euler method

@)

,2

-(I cosnz)It= zt (A)

I

(Dl

(Cr -' u lOx )l) 2.

(C)

Two Five

The steady state tempelature ofarod oflength / whose ends are kept at30'C and 40"C is (A) r 10r *30 @) , =2ot *30 =

I

1

(A)

",'

(D)

z2

-t

z2

+l

-

+l

z2

z2 +1

I

z z

(D)

,2

-+t

PART -B (5 x 4 = 20 Marks) Answer AI.[Y FM Questions

One dimensional wave equation is used to find (A) Temperature (3) Displacement (D) Mass (C) Time

21. Find the partial differential equation of all planes cutting equal intercepts form t}le x and y axes.

13. The Fourier inveme transform of f(x) is

(A) r

--

(C) - -;:r

I F(s)e '*ds

14. The Fourier sine transform of e

rAr t'6 " \'l ,

o2

(c)ta t_'l 15.

r

(D)

'.

l2n ;

(B)r'^

+ ! llf

--,-: I f GY* d, l2n ;

I

;

f GY',*e

23.

I

lG ["*a* ^

(B)

*r2 (D)

25.

t-

l fi s'+a'

,2 +r2

(c)

F(s+a) FG/a)

(B) (D)

x ' L0

of "flx)

=

1

f{l- z-

:

,

for

1xl < a

fotlxl>

a

uy *,e long division method.

JF+fi=t

Find Z-transform of flcosne.

E1s-a) F(s a)

PART-C(5x12:60Marks) A.nswer

16.

z

:

Find the inveme z-transfo.. or

26. Solve 2'7.


0 and x / is initially at rest in strctched string with fixed end points x equitibrium position. If it is set to vibmte by giving each point a velocity 3x(/-r), write down the boundary conditions of the problem. Find the Forrrier transform

E.! \, o'+r'

xirt}

A tightly

r

is

Fk'-l(r))is (A)

/(:r):

22. Find hatf-range Fouder sine series for

ALL Questioos

-

r(,r(,) c(4) (A) r(r)c(r)

(c) FG)-G(s)

28- a.i. Form the partial differential equation by eliminating the arbitrary function from the relation

(B) F(s)+G(r)

i(t'

(A r(s)rc(s)

2TNF3MAlOM

+

y'

+

"',

k

+

^y

+

*)

= 0. 2INI]MAIl)03

X#,'1h!;10,H:'

[ 0 if tx>t d f. Reg. No,. B.Tech. DEGREE EXAMINATION, NO!'EMBER 2015. Thid Semester. MA1OO3 _ TRANSFORMS AND BOTJNDARY VALI]E PROBLEMS.

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