ENGINEERING MATHEMATICS I QPaper Jan 2014

Question Paper 6 Pages

Anna University

Civil Engineering

Contributed by

Naresh Tella

Re
g. Ko. :
IL
_,____,_---1----1_:_.i__..1-.i........1-.....1--'---'
Qu
est
ion
Pap
er
Code:
37001
B.E.
/B
.
Tech
.
DEGREE
EXAMINATION, JANUARY 2014.
First
Semester
Civil En
gineering
MA615l-MATHEl\1A'I'ICS
- 1
(Commo
n
to
all
Branches
except
Marine
Engmeering)
(Regulation
20
13)
Tune
:
Three
hours
Maximum
: 100
marks
A
nswer
ALL
questions
.
PART
A -
(10
x 2 =
20
mark
s)
1.
If
the
eigen
values
of
the
ma
true A
of
order
3 x 3
arc
2, 3
and
1,
then
find
the
e1gen
,·alues
of
adjoint
of
A.
2.
If
;_
is
th
e
eigen
value
of
the
matrix
A,
then
prove
that
J.
3
is
the
cigen value
of
A
1
•
3.
Give
an
exam
ple
for conditionally
convergent
series.
4.
.
111111
Te
st
the
convergence
of
the
series
l - - - - + - + - - - - -
....
to
oo
.
2
2
s
2
4
2
5
2
1
2
s
2
5. \'{h
at
is
t
he
c
urv
atu
re
of
the
circle
(x
- 1
)2
+ (y +
2)
2
=
16
at
any
po
in
t on
it?
6.
Find
th
e envelope
of
the
family
of
curves
y = nu: +
.!..
,
whore
m
is
the
m
7.
parameter
.
If
:c
1
1 y' = I .
then
find
dy
.
dx
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Page 1
I I , "
''
,0)
, -1110,th,11fi11d •
;,(y, y)
n 1· I 1 1 I I I l I
f>
I
·,
111
,l
.,
x,
11
111
" ,J,,ublr-
''
· ' 1
0£
I
\P
'ln·n
,o
u1
1<
r,
>Y
t H'
1n
P·~ \" ) ,
..,,
11\ll
'
\:1,ll!OJ\
• u
10.
Evalual
c J
J,.
rL,r/0
11.
u
I)
PAltT
13
- (5 x
JG==
80
ma
rk
s)
(a)
(i)
. [2
Find
th
e e
1g
en va
lu
es
and the e
1g
en v
ecto
rs of t
he
malrix
1
. l
: ~1-
2 2
(
8)
(
1i
) Us
in
g
Cny
l
ey
-Ham1.lt
on
th
eor
em find A -
1
o
nd
,1
•.
tf
A ·H
_:
-;1 (
8)
Or
(b) R
ed
uce
the
qu
a
drat
ic
fo
rm
6x
2
+
3y
2
+
3z
2
-
4xy
-
2yz
+
tic-a
mt
o a
cano
ni
ca
l
fo
rm by a n o
rth
ogo
nal
r
ed
uc
ti
on.
Hen
ce
find
it
s r
ank and
n
at
ur
e. (
16
)
12. (a) (
i)
Examine
the
con
ve
rgence
and
the
divergence
of
th
e fo
ll
owm g
se
ri
es
2 6 2
jtj
3
2"-2(•-I)
( )
) + - X + - X + - X + .... +
---
X + .... X > 0 . (8)
G 9
17
2" + I
(n)
Di
scu
ss
the
co
nv
erge
n
ce
nnd
the
divergence
of
th
e follo
wmg
se
ri
es
I J I l
3
-
3
(1
+ 2) +
3
(1
+ 2 +
3)
-
3
(1 + 2 + 3 + ,j) + " " (8)
2 3 4 5
Or
(b)
(1)
'!'est
the
co
nv
erge
nce of
the
se
ne
s I, nc-•' .
(8)
n oO
(
11
)
Te
st
the
co
nv
ergence
of
the
se
ri
es
x x
2
x
1
x'
_ _ _ _ _
..,
_ _
___
+
(0
< X <
1)
.
1 + x I • x
2
I + x
3
1 +
x'
....
(
8)
2
37001
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Page 2
,.
13.
fol
(il
Fmd
the
radiu
s
of
curvature
of
the
cyclmd x =
a(O
+
sm
0),
Y =
a(l
-
cosO).
(8)
(n) F
md
the
equation
of
the
evolutes
of
the
parabola
y
2
=
4ax
. (8)
(
h)
(i)
Or
Find
the
equation
of
circle
of
curvature
at
(.9..
E.)
on
4'4
fx
+
,JY=
✓
a
.
(8)
(ii)
Find
the
enve
lope
of
tho
family
of
straight
lines
y =
mx
-
2am
-
am
3
,
where
111
is
the
parameter.
(8)
14. (a) (i)
Expand
c'
log(! +
y)
in
powers
of
x
and
y
upto
the
third
degree
terms
using
Taylor•~ theorem. (8)
(11)
Y
•
•x
xy
a
(u.
V,
W
If
u =
_::.
,
,,
=
.::....
, w = - , find
X
)'
Z
ax,y,z}'
(8)
Or
(b)
(
i)
J)1scuss
the
maxima
and
min
i
ma
of
f(x,
y)
= x
3
y
2
(!_
- x -
y}.
(8)
(
u)
(
)
cw
C/L
aw
If
w = f y -
z,::
-
x,
x - y ,
then
show
that
- + - + - =
0.
(8)
ax
O:>
az
1
2-r
15.
(a) (i)
By
changi
ng
the
or<l
er
of
mtegranon
evaluate
J J xy
dydx.
(8)
0
.,
:
(ii
)
By
changing
to
polar
coordmatcs,
evaluate
ff
e (,• ·
,')
dxdy
.
(8)
0 0
Or
(b) (i)
Evaluate
fJ
xy
dxdy
over
the
positive
quadrant
of
the
circle
(8)
(ii
)
Evaluate
HJ
(
dzdydx
r .
where
V
is
the
re
gion
bounded
by
v
x+y+z.+l
x = 0 , y =
0.
z = 0
and
.Y
+ y + z =
l.
(8)
3
37001
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Page 3
---
Hcg. No.
Qu
est
ion
Pap
er
Code:
6077-1
ll.E./13.Tcch.
DEGREE
EXAMINATION, NOVEMBER/DECEMBER 2016.
Seco
nd
Semester
Civil Engmccr111g
MA
2161/MA 22/080030004
-MATHEMATICS-II
(Common
to
nil
Branches)
(Regulations
2008)
Tim
e :
'l'hree
hours
Maximum
:
100
mark
s
Answer
ALL
questions
.
PART
A - (10 x 2 =
20
ma
r
ks)
J . T ,nd l
lu
"p
nrt
1
culnrintegr.1l
of
(D-
2)
2
-=
e
2
'
sin
2
x.
2,
Su
lv
e.
(D',
1
)y
= 0
3.
Find
the
unit
vector
normal
to
the
surface
x
2
+
xy
+ y
2
+
xyz
at
the
point
(1,
- 2,1).
,J,
5.
G.
7.
8.
9.
10
.
Pr
o
ve
thnt
ft
=
(2x
+ y
z)i
;1-
(4y
+
zx)]
-
(6z
- x
y)l~
is
solenoidal.
Find
the
constants
a,b
and
c
if
/(z)
= x +
ay
+
i(bx
+
cy)
is
analytic.
find
the
fixed
points
of
the
tr
ansformation
w = - z +
1
.
z+l
Evaluate
2
•
/.
dz
1
,.
2
z
-7z
+
l2
Find
the
re
s
idue
of
the
function
,
4
at
a
simple
pole.
z
(z
-3)
Us
ing
the
initial
value
theorem,
find
Lt
sL(f(t)) for
the
function
/(t)
= e-•
cost.
•
➔
"'
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Page 4
l'AltT
B - (r,
~
JG
::
f:!O
mnrk~J
I I. (n)
(1)
Solv, ( IJl • ,1
/y
x III x
(8)
/11)
(b) (IJ
(11)
Solv,
,J',
rly
Jx
' rlx
r/x
• •
Solve,
dt
- 2x -
ly
(BJ
Or
(8/
(
8)
12
. (a)
(1)
Verify Stoke's
th
eorem for fr
..
(:x
2
...
y
2
)
l -
2:x:;
j
taken
ar
oun
d
the
re
c
tan
gle bounded by tho
Im
es
x =
=:
a,
y =
0,:;
= b . (1
0/
(1i)
Fmd
the
angle between
the
su
rfaces x
2
-
:l
- z
2
= 9
and
z = x
2
..
y
2
- 3 ot
th
e
point
(2, - 1, 2
).
(6)
Or
(b)
(1)
Find
the
values of
the
constants
a,b,c so
that
F "'(axy + b z
3
)i
+
{3:x
2
-
cz
)J
•
{3.xz
2
- y
~
may
be
irr
otat
1onaL For
th
ese
value
s of
a,b,c,
find
the
sca
lar
potential
of
f:.
(8)
(ii)
Using
Gauss
divergence theorem
evaluate
JJ
f:.n.ds where
s
F =
(x
2
-
yz
)l
+
(l
-z x
)J
+
{z
2
-x
y
}k
and
S
1s
the
s
urfa
ce formed
by
:x
=
0,
:x
= 1, y = 0, y = 2, z = 0 a
nd
z = 3 . (8)
13
. (a) (i)
Pr
ove
that
u = x
3
-3
:xy
2
+
3x
2
-3y
2
-1
is
harmonic
and
also find
th
e
analytic
function
/(z)
= u
...
iv . (8)
(ii)
Under
the
transformation
w =
.!.,
find
the
image
of
lz
-2 •1 = 2 . (8)
z
Or
(b) (
i)
Find
the
bilinear
mapping
which
map
s
-1,0,
1
of
the
z-
plane
onto
-1,
- i, l
of
the
w -
plane.
(8)
(ii)
Show
that
an
analytic
function
with
constant
modulus
is
constant.
(8)
2
60771
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Page 5
14. (u) (i)_ • ~x
dO
Usmg
the
method
of
contour
inte<>ration
evaluate
J
----
. (8)
0
0
5T4mnB
(ii)
F
8z+3
'ind
the
Laurent's
series
expansion
of
..,......-.------
- -
in
the
(z+
5)(z -
2)(z
-3
)
region 1 <
lz+
s1
< 8 .
(8)
Or
(b)
(i)
Using
the
method
of
contour
integration,
evaluate
"'J x2
2 •
dx.
0
(x
+
1)
(x·
+
4)
(
8)
(ii)
E\·aluate
J • = +
1
dz.
where C
is
the circle !z + 1
..-
ij =
2,
using
z·
+
2z
+ 4
<
Cauchy's
integral
formula.
15. (a)
(i)
Find
the
Laplace
transform
of
t
2
e
2
'
sin t .
(ii)
Using convolution theorem find
L·
1
[ ( •
1
• j].
s·
+
2s
+o
Or
(8)
(8)
(b)
(i)
Find
the
Laplace
transform
of
the
half-wave
rectifier given
by
1
asinwt
f(t)
=
0
;r
,O<t<-
{
2
)
,r
~
and
t + ; =
{(t).
,-<t<-
w w
(ii)
Solve using Laplace transforms:
d
2
y
dy
dx
2
+ 4
dx
+
4y
=
te·
1
,y(O) = O,y'(O) =
1.
3
(8)
(8)
60
771
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Page 6