Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 2015 June-July 14MAT11 Engineering Mathematics I Question Paper
14MAT11
First Semester B.E. Degree Examination, June/July 2015
Engineering Mathematics - I
Time: 3 hrs.
Max. Marks:100
a
0
0
6 If y ym + y
4
. 2x prove that
C
1 a.
O
....,
{.7
14
(x2 ?
1
)Y n+2 + (2n +1)xy?,
1
+ (n
2
- m
2
)y. = 0 \ (07 Marks)
A el = am sin m0 + bm cos m0
(31b\<13
(06 Marks)
( 1
71
E
b. Find the pedal equation for the curve
o
-I
C.
Derive an expression to find radius of curvature in cartesian forntd (07 Marks)
#
?s
ci
,
0 ,} OR
ba
c =
CL ? -. 2 a. Find the n
th
derivative of sin
2
x cos
3
x (07 Marks)
3
16
b. Show that the curves r = a(1+cos 0) and r = b (1-cos0 niehect at right angles.
(06 Marks)
..17
tri
.= e)
4,1)
II c. Find the radius of curvature when x = a log (sect + t ct. (07 Marks)
,=
l
., ..
1
-
E
t
z
MODULEnl)
.
'
O
al .1-
2 0
g
i 2.
3 a. Using Maclaurin's series expand tan x upto ilNitirm c
(07 Marks)
? i
?
g
au au x
3
; y
b. Show that x ? + y? = 2u log u wh 6 log u = (06 Marks)
Q
ti.)
?
"".
c =
5X ay
= .9.
3x +4y
o
C.
Find the extreme values
i14.
5
11
(x - 02
(07 Marks)
? ?..)
i
1
8
0 14
/16- 11
OR
O-,0
--
r,
,
b
..,
aluate lim
ex sin x X"L x
2
0 0
?'
.-).0
x
2
+ x log(1- x)
41
If u = x log xy
\
Okre x
3
+ y
3
+ 3xy =1 Find du
dx
xz xy, _, a(u,v,w)
v = ? , w = ?, lino .
Y
z a(x, y, z)
MODULE- III
rr)
USN
a
Note: Answer FIVE full questions, selecting
at least TWO questions from each part.
MODULE- I
0
If u =
(07 Marks)
(06 Marks)
(07 Marks)
U
Find div F and Curl F where
-
F. grad (x
3
+ y
3
+ z
3
-3xyz)
Using differentiation under integral sign,
Evaluate
xa -1
dx (a, 0)
o
log
x
f
Hence find
1 3
X ? 1
dx
log x
0
c. Trace the curve y
2
(a - x
3
, a > Ouse general rules.
OR
1 of 2
(07 Marks)
(06 Marks)
(07 Marks)
C
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PF
14MAT11
First Semester B.E. Degree Examination, June/July 2015
Engineering Mathematics - I
Time: 3 hrs.
Max. Marks:100
a
0
0
6 If y ym + y
4
. 2x prove that
C
1 a.
O
....,
{.7
14
(x2 ?
1
)Y n+2 + (2n +1)xy?,
1
+ (n
2
- m
2
)y. = 0 \ (07 Marks)
A el = am sin m0 + bm cos m0
(31b\<13
(06 Marks)
( 1
71
E
b. Find the pedal equation for the curve
o
-I
C.
Derive an expression to find radius of curvature in cartesian forntd (07 Marks)
#
?s
ci
,
0 ,} OR
ba
c =
CL ? -. 2 a. Find the n
th
derivative of sin
2
x cos
3
x (07 Marks)
3
16
b. Show that the curves r = a(1+cos 0) and r = b (1-cos0 niehect at right angles.
(06 Marks)
..17
tri
.= e)
4,1)
II c. Find the radius of curvature when x = a log (sect + t ct. (07 Marks)
,=
l
., ..
1
-
E
t
z
MODULEnl)
.
'
O
al .1-
2 0
g
i 2.
3 a. Using Maclaurin's series expand tan x upto ilNitirm c
(07 Marks)
? i
?
g
au au x
3
; y
b. Show that x ? + y? = 2u log u wh 6 log u = (06 Marks)
Q
ti.)
?
"".
c =
5X ay
= .9.
3x +4y
o
C.
Find the extreme values
i14.
5
11
(x - 02
(07 Marks)
? ?..)
i
1
8
0 14
/16- 11
OR
O-,0
--
r,
,
b
..,
aluate lim
ex sin x X"L x
2
0 0
?'
.-).0
x
2
+ x log(1- x)
41
If u = x log xy
\
Okre x
3
+ y
3
+ 3xy =1 Find du
dx
xz xy, _, a(u,v,w)
v = ? , w = ?, lino .
Y
z a(x, y, z)
MODULE- III
rr)
USN
a
Note: Answer FIVE full questions, selecting
at least TWO questions from each part.
MODULE- I
0
If u =
(07 Marks)
(06 Marks)
(07 Marks)
U
Find div F and Curl F where
-
F. grad (x
3
+ y
3
+ z
3
-3xyz)
Using differentiation under integral sign,
Evaluate
xa -1
dx (a, 0)
o
log
x
f
Hence find
1 3
X ? 1
dx
log x
0
c. Trace the curve y
2
(a - x
3
, a > Ouse general rules.
OR
1 of 2
(07 Marks)
(06 Marks)
(07 Marks)
C
c. Find the orthogonal trajectories of-5.he
g
family of confocal conics
where k is parameter.
1
efc
\
MODULE? V ,
Solve by Gauss elimin p method
5x
1
+ X
2
+ X
3
+ X
4
4
.
> x
1
+ 7X
2
+ X
3
+ X
4
=12, x, +X
2
+6X
3
+ X
4
= ?5 ,
x
1
+ x
2
+X
3 +4b9-6
Diagonali*Ore matrix A =
?19 7
? 42 16
.
Find: the eigen value and the corresponding eigen vector of the matrix
X
2
y
2
/- 2
a
-
b +
=1
via
b.
c.
ipPF
14MAT11
then prove that V rn = n r'
2
r (07 Marks) 6 a. If r = xi + yj + z k and r= r
b. Find the constants a, b, c such that F = x + y + az)i + (bx + 2y ? z)j + + cy + 2z)k is
irrotational. Also find (I) such that F = V4 (06 Marks)
C.
Using differentiation under integral sign,
sin x
x
dx Evaluate
J.
e''
0
MODULE? IV
7 a.
Obtain reduction formula for cos' x dx
b. Solve : (1 + 2xy cos x
2
? 2xy)dx + (sin x
2
? x
2
)dy = 0
(07
M
a*
(07 Marks)
(06 Marks)
A body originally at 80?C cools down to 60?C in 20 minutes, the temperature of the air being
40?C. What will be temperature of the body after 40 minutes from the original ? (07 Marks)
OR
* ?
tS15:
4
;V#
14
?
1
(07 Marks)
(07 Marks)
(06 Marks)
6 ?2 2
?2 3 ?1
2 ?1 3
by power method taking the initial eigen vector (1, 1, 1)
1
(07 Marks)
OR
10 a. Solve by L U decomposition method
x+5y+z=14, 2x+y+3z=14, 3x+y+4z=17 (07 Marks)
b. Show that the transformation y
i
= 2x
1
? 2x
2
? x
3
, y
2
= ?4x, + 5x, + 3x
3
, (06 Marks)
y
3
= x
l
? x
2
? x
3
is regular and find the inverse transformation.
c. Reduce the quadratic form 2x
1
2
+ 2x
2
2
+ 2x
3
2
+ 2x
1
x
3
into canonical form by orthogonal
transformation. (07 Marks)
* * * * *
2 of 2
c.
2a
8
a. Evaluate J x
2
2a) 11
-
(
2
dx
b. Solve : xy(1+ x y
2
4'
d
=1
dx
(07 Marks)
(06 Marks)
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This post was last modified on 01 January 2020