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'
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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