Download VTU BE 2020 Jan Question Paper 15 Scheme 15MAT11 Engineering Mathematics First And Second Semester

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a.
b.
c.
2 a.
b.

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c.
3
a.
b.
c.

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4
a.
b.
c.
5 a.

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b.
c.
Note:
hrs.
Answer any

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x
If y =
2
cos
3

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If y = a cos(log
Prove that the
Find the radius
Find the pedal

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If y = ems
1
"
-!
' prove

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Expand log cos
Evaluate lirn
x--)0
-
y

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-
If = sin u show
FIVE full questions,
X , find y
n

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.
x)+ bsin(log
curves r = a(1
of curvature
equation of r =

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that -
x in powers
a a ' +b +c
choosing ONE full question
Module-1

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Max. Marks: 80
from each module.
(05 Marks)
+ (11
2

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+ 1)y = 0 . (06 Marks)
gonally. (05 Marks)
(05 Marks)
(06 Marks)
=0. (05 Marks)

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(05 Marks)
(06 Marks)
(05 Marks)
of x. (05 Marks)
1

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2
1 (au
Llu
j +
2

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) . = ( ? 1
car r cv
(06 Marks)
a(u,v,w)
x) , prove that + (2n + 1)xy?

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4
_,
+ sin 8) and r - a( I - sin 0) cut ortho
OR
of the curve rn = an cosnO.

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2(1 + cos()) .
x
2
)y?_, - (2x +1)xy
n

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_, -(n
2.
+ m )y
n

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Module-2
of x -
7c
3
au au

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?+ y ?
ax Oy
log(1
y = r sin
w =

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using Taylor's series.
= 3 tan u .
OR
+ ex ) in ascending powers
_

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ail all
0 prove that ( ?) + ?
ax ay
+ +
3

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that x x
x + y
Using Maclaurin's series, expand
If u - f(x, y) and x = r cos 0 ,
,

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x2 + y2 + z2
If u v=x+y+z,
A particle moves along the
components of velocity and

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Find the directional derivatives
direction of 2i - j - 2k.
Prove that div(curlF) = 0 .
Module-3
xy yz zx , evaluate

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a(x,
Marks) (05
y, z)
z = 2t - 5, determine the
+ j + 2k. (05 Marks)

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(1, -2, -1) along the
(06 Marks)
(05 Marks)
curve x = 1 - t
3

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, y = 1 + t
2
and
acceleration at t = 1 in the direction 2i
of (I) = x

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2
yz +4xz' at the point
U
tr.
1:1

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E >
o
1.2
71
.

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c..
v
.- c
<
t"..t

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V
0
z
C
.

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E5
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ti
_mo o
4-

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U
c
?r
c
o

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? -?
2 8-
iC
6
77,

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v
z
f
0
a

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.; 0
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H
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(56)A
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USN
FirstRanker.com - FirstRanker's Choice
First Se . Degree Examination, Dec.20191Jan.2020
15MAT11
Engineering Mathematics -

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Time: 3
1
a.
b.
c.

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2 a.
b.
c.
3
a.

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b.
c.
4
a.
b.

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c.
5 a.
b.
c.
Note:

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hrs.
Answer any
x
If y =
2

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cos
3

If y = a cos(log
Prove that the

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Find the radius
Find the pedal
If y = ems
1
"

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-!
' prove
Expand log cos
Evaluate lirn
x--)0

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-
y
-
If = sin u show
FIVE full questions,

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X , find y
n
.
x)+ bsin(log
curves r = a(1

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of curvature
equation of r =
that -
x in powers
a a ' +b +c

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choosing ONE full question
Module-1
Max. Marks: 80
from each module.
(05 Marks)

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+ (11
2
+ 1)y = 0 . (06 Marks)
gonally. (05 Marks)
(05 Marks)

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(06 Marks)
=0. (05 Marks)
(05 Marks)
(06 Marks)
(05 Marks)

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of x. (05 Marks)
1
2
1 (au
Llu

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j +
2
) . = ( ? 1
car r cv
(06 Marks)

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a(u,v,w)
x) , prove that + (2n + 1)xy?
4
_,
+ sin 8) and r - a( I - sin 0) cut ortho

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OR
of the curve rn = an cosnO.
2(1 + cos()) .
x
2

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)y?_, - (2x +1)xy
n
_, -(n
2.
+ m )y

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n

Module-2
of x -
7c

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3
au au
?+ y ?
ax Oy
log(1

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y = r sin
w =
using Taylor's series.
= 3 tan u .
OR

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+ ex ) in ascending powers
_
ail all
0 prove that ( ?) + ?
ax ay

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+ +
3
that x x
x + y
Using Maclaurin's series, expand

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If u - f(x, y) and x = r cos 0 ,
,

x2 + y2 + z2
If u v=x+y+z,

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A particle moves along the
components of velocity and
Find the directional derivatives
direction of 2i - j - 2k.
Prove that div(curlF) = 0 .

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Module-3
xy yz zx , evaluate
a(x,
Marks) (05
y, z)

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z = 2t - 5, determine the
+ j + 2k. (05 Marks)
(1, -2, -1) along the
(06 Marks)
(05 Marks)

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curve x = 1 - t
3
, y = 1 + t
2
and

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acceleration at t = 1 in the direction 2i
of (I) = x
2
yz +4xz' at the point
U

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tr.
1:1
E >
o
1.2

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71
.
c..
v
.- c

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<
t"..t
V
0
z

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C
.
E5
E
ti

--- Content provided by‌ FirstRanker.com ---

_mo o
4-
U
c
?r

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c
o
? -?
2 8-
iC

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6
77,
v
z
f

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0
a
.; 0
?-n* *1-11\C
C

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H
/K
(56)A
it
.1.-

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I
1
USN
dy
?+ y ? = y

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-,
x (06 Marks)
dx x
1 9
b. Apply Gauss-elimination method,

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x-2y+3z=8, 2x+y?z=3.
c. Reduce the matrix A
=
[-1 3
?2 4

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to solve the system of equations x+y+z=9,
(06 Marks)
(05 Marks' to diagonal form.
15MAT11
OR

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A
6
a. If
-
F = (3x

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2
3yz) i+ (3y
2

?3zx)j+ A (3z

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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)

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Module-4
7
a. Find the reduction formula for 'sin' xdx (05 Marks)
b. Solve
C. Evaluate f

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x
dx . (05 Marks)
0 V1? x -
OR
8 a. Find the orthogonal trajectory of the curve X

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- --,- + =1, where X. is the parameter.
a
-
b
-

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+X.
(05 Marks
( x
b. Solve 0 + e")dx + e' Y - dy = 0 . (06 Marks)
yI

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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

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-

1 2 3 ?1
9 a. Find the Rank of the matrix A =
1 0 1 1

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0 1 1 ?1
(05 Marks)
(05 Marks)
OR
2 0 1

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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

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out the 4
ffi
iterations, using Gauss-Seidal method. (06 Marks)
c. Reduce the quadratic form of x
2

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+5y
2
+ z
2
+ 2xy + 6xz + 2yz into canonical form.

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(05 Marks)
S
V.
2of 2
(1 t LIBRARY

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