Download Visvesvaraya Technological University (VTU) BE/B.Tech First And Second Semester (1st sem and 2nd sem) 2019-2020 Jan ( Bachelor of Engineering) 15 Scheme 15MAT11 Engineering Mathematics Previous Question Paper
15MAT11
Engineering Mathematics -
Time: 3
1
a.
b.
c.
2 a.
b.
c.
3
a.
b.
c.
4
a.
b.
c.
5 a.
b.
c.
Note:
hrs.
Answer any
x
If y =
2
cos
3
If y = a cos(log
Prove that the
Find the radius
Find the pedal
If y = ems
1
"
-!
' prove
Expand log cos
Evaluate lirn
x--)0
-
y
-
If = sin u show
FIVE full questions,
X , find y
n
.
x)+ bsin(log
curves r = a(1
of curvature
equation of r =
that -
x in powers
a a ' +b +c
choosing ONE full question
Module-1
Max. Marks: 80
from each module.
(05 Marks)
+ (11
2
+ 1)y = 0 . (06 Marks)
gonally. (05 Marks)
(05 Marks)
(06 Marks)
=0. (05 Marks)
(05 Marks)
(06 Marks)
(05 Marks)
of x. (05 Marks)
1
2
1 (au
Llu
j +
2
) . = ( ? 1
car r cv
(06 Marks)
a(u,v,w)
x) , prove that + (2n + 1)xy?
4
_,
+ sin 8) and r - a( I - sin 0) cut ortho
OR
of the curve rn = an cosnO.
2(1 + cos()) .
x
2
)y?_, - (2x +1)xy
n
_, -(n
2.
+ m )y
n
Module-2
of x -
7c
3
au au
?+ y ?
ax Oy
log(1
y = r sin
w =
using Taylor's series.
= 3 tan u .
OR
+ ex ) in ascending powers
_
ail all
0 prove that ( ?) + ?
ax ay
+ +
3
that x x
x + y
Using Maclaurin's series, expand
If u - f(x, y) and x = r cos 0 ,
,
x2 + y2 + z2
If u v=x+y+z,
A particle moves along the
components of velocity and
Find the directional derivatives
direction of 2i - j - 2k.
Prove that div(curlF) = 0 .
Module-3
xy yz zx , evaluate
a(x,
Marks) (05
y, z)
z = 2t - 5, determine the
+ j + 2k. (05 Marks)
(1, -2, -1) along the
(06 Marks)
(05 Marks)
curve x = 1 - t
3
, y = 1 + t
2
and
acceleration at t = 1 in the direction 2i
of (I) = x
2
yz +4xz' at the point
U
tr.
1:1
E >
o
1.2
71
.
c..
v
.- c
<
t"..t
V
0
z
C
.
E5
E
ti
_mo o
4-
U
c
?r
c
o
? -?
2 8-
iC
6
77,
v
z
f
0
a
.; 0
?-n* *1-11\C
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H
/K
(56)A
it
.1.-
I
1
USN
FirstRanker.com - FirstRanker's Choice
First Se . Degree Examination, Dec.20191Jan.2020
15MAT11
Engineering Mathematics -
Time: 3
1
a.
b.
c.
2 a.
b.
c.
3
a.
b.
c.
4
a.
b.
c.
5 a.
b.
c.
Note:
hrs.
Answer any
x
If y =
2
cos
3
If y = a cos(log
Prove that the
Find the radius
Find the pedal
If y = ems
1
"
-!
' prove
Expand log cos
Evaluate lirn
x--)0
-
y
-
If = sin u show
FIVE full questions,
X , find y
n
.
x)+ bsin(log
curves r = a(1
of curvature
equation of r =
that -
x in powers
a a ' +b +c
choosing ONE full question
Module-1
Max. Marks: 80
from each module.
(05 Marks)
+ (11
2
+ 1)y = 0 . (06 Marks)
gonally. (05 Marks)
(05 Marks)
(06 Marks)
=0. (05 Marks)
(05 Marks)
(06 Marks)
(05 Marks)
of x. (05 Marks)
1
2
1 (au
Llu
j +
2
) . = ( ? 1
car r cv
(06 Marks)
a(u,v,w)
x) , prove that + (2n + 1)xy?
4
_,
+ sin 8) and r - a( I - sin 0) cut ortho
OR
of the curve rn = an cosnO.
2(1 + cos()) .
x
2
)y?_, - (2x +1)xy
n
_, -(n
2.
+ m )y
n
Module-2
of x -
7c
3
au au
?+ y ?
ax Oy
log(1
y = r sin
w =
using Taylor's series.
= 3 tan u .
OR
+ ex ) in ascending powers
_
ail all
0 prove that ( ?) + ?
ax ay
+ +
3
that x x
x + y
Using Maclaurin's series, expand
If u - f(x, y) and x = r cos 0 ,
,
x2 + y2 + z2
If u v=x+y+z,
A particle moves along the
components of velocity and
Find the directional derivatives
direction of 2i - j - 2k.
Prove that div(curlF) = 0 .
Module-3
xy yz zx , evaluate
a(x,
Marks) (05
y, z)
z = 2t - 5, determine the
+ j + 2k. (05 Marks)
(1, -2, -1) along the
(06 Marks)
(05 Marks)
curve x = 1 - t
3
, y = 1 + t
2
and
acceleration at t = 1 in the direction 2i
of (I) = x
2
yz +4xz' at the point
U
tr.
1:1
E >
o
1.2
71
.
c..
v
.- c
<
t"..t
V
0
z
C
.
E5
E
ti
_mo o
4-
U
c
?r
c
o
? -?
2 8-
iC
6
77,
v
z
f
0
a
.; 0
?-n* *1-11\C
C
H
/K
(56)A
it
.1.-
I
1
USN
dy
?+ y ? = y
-,
x (06 Marks)
dx x
1 9
b. Apply Gauss-elimination method,
x-2y+3z=8, 2x+y?z=3.
c. Reduce the matrix A
=
[-1 3
?2 4
to solve the system of equations x+y+z=9,
(06 Marks)
(05 Marks' to diagonal form.
15MAT11
OR
A
6
a. If
-
F = (3x
2
3yz) i+ (3y
2
?3zx)j+ A (3z
2
?3xy)k , find (i) div F (ii) curl F. (05 Marks)
b. If F = (x + y + az)i +(bx + 2y ?z)j+ (x +cy +2z)k is irrotational, find a, b, c. (06 Marks)
c. Prove that curl(4A) = (curl A) + V x A (05 Marks)
Module-4
7
a. Find the reduction formula for 'sin' xdx (05 Marks)
b. Solve
C. Evaluate f
x
dx . (05 Marks)
0 V1? x -
OR
8 a. Find the orthogonal trajectory of the curve X
- --,- + =1, where X. is the parameter.
a
-
b
-
+X.
(05 Marks
( x
b. Solve 0 + e")dx + e' Y - dy = 0 . (06 Marks)
yI
c.
A body in air at 25?C cools from 100?C to 75? in one minute. Find the temperature of the
body at the end of three minutes.
Module-5
2 ?1 ?3 ?1
-
1 2 3 ?1
9 a. Find the Rank of the matrix A =
1 0 1 1
0 1 1 ?1
(05 Marks)
(05 Marks)
OR
2 0 1
10 a. Find the largest Eigen value and the corresponding Eigen vector of A = 0 2 0 and
1 0 2
X = (1 0 0)' as initial vectors.
(05 Marks)
b. Solve the system of equations 5x + 2y + z =12, x +4y+ 2z =15 , x + 2y +5z = 20 . Carry
out the 4
ffi
iterations, using Gauss-Seidal method. (06 Marks)
c. Reduce the quadratic form of x
2
+5y
2
+ z
2
+ 2xy + 6xz + 2yz into canonical form.
(05 Marks)
S
V.
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This post was last modified on 02 March 2020