Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 17 Scheme 17MAT11 Engineering Mathematics I Question Paper
USN
17MAT11
(5'
?
Module-3
= (C-401+4
2
+401+ (8t
2
-3t
3
)k . Find the
of n
/,... A
components of velocity and acceleration in the direction of 1-3 j+ 2k at t = 0. (06 Marks)
-*
Find the constant a and b such that F = (axy + z
3
)i+ (3)(
2
? z)1+ (bxz' ? y)1
\
( is irrotational
and find scalar potential function d) such that t'
,
Vd). (07 Marks)
Important Note
-.,
._. ,-,i
r A particle moves along the cuvre
5 a.
rmitkoni
First Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics ? I
Time: 3 hrs.
Max. Marks: 100
Note: Answer any FIVE full questions, choosing
ONE full question from each module.
ai
0
?
ti Module-I
P
73
1 a.
Find the n
th
derivative of sin 2x sin 3x.
(06 Marks)
g
1 ? cos 0
-c
b.
Find the angle between the two curves r =
a
and r ---
b
.
1+ cosi)
(07 Marks)
cl
di
... ;-
c.
Find the radius of curvature for the curve x
3
+ y
-
' =3xy at (3/2, 3/2). (07 Marks)
,:. ,,
d.). ,
n
,
m =
.
OR
. ,
2 a.
If y = cos(m log x) then prove that x'y
n
_, + (2n + I )xy?
?
, + (m
2
+ n
2
)y =0. (06 Marks)
.? r-A
b.
With usual notation prove that tan st)= r
dO
. (07 Marks)
r
, cc
= +
m -,1- dl
-
.
tb
c.
Find the pedal equation of the curve rm = a' cos m0 .
(07 Marks)
,
?
0.)
=
O
?
F?
0.7
r
i' Module-2 . ,
= 0 ....
?
3 a.
Find the Taylor's series of log(cosx) in powers of (x ? n/3) upto fourth degrees terms.
2
L,
g
(06 Marks)
c.)
cr
u
7d '6
i 3 1 1
- ou au
b. If u = tan-
X ? y
' then prove that x + y
= sin 2u by using Euler's theorem.(07 Marks)
. gz x + y
i
ax ay
.
-
4
C.
a.
b.
yz
if u= ---, v ? xz , w ?
x
y
Evaluate lim
tan x \
a(uvw) xy
then find .1 =
z a(xyz)
OR
0
Using Maclaurin's series, prove that Ail + sin 2x =1+ x
x x3
+
?
x4?
? ? ? .(07 Marks)
1
2 3
1
4
a 'au 1 au
C.
If u 42x -3y, 3y 4 4z --
1 u 1
2x) then prove that + + = O.
2 'x 3 ay 4 az
(07 Marks)
(07 Marks)
(06 Marks)
c. Prove that curl(d) A) d)curl A+ gradd)x A (07 Marks)
of2
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'fff-N. ' MI
USN
17MAT11
(5'
?
Module-3
= (C-401+4
2
+401+ (8t
2
-3t
3
)k . Find the
of n
/,... A
components of velocity and acceleration in the direction of 1-3 j+ 2k at t = 0. (06 Marks)
-*
Find the constant a and b such that F = (axy + z
3
)i+ (3)(
2
? z)1+ (bxz' ? y)1
\
( is irrotational
and find scalar potential function d) such that t'
,
Vd). (07 Marks)
Important Note
-.,
._. ,-,i
r A particle moves along the cuvre
5 a.
rmitkoni
First Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics ? I
Time: 3 hrs.
Max. Marks: 100
Note: Answer any FIVE full questions, choosing
ONE full question from each module.
ai
0
?
ti Module-I
P
73
1 a.
Find the n
th
derivative of sin 2x sin 3x.
(06 Marks)
g
1 ? cos 0
-c
b.
Find the angle between the two curves r =
a
and r ---
b
.
1+ cosi)
(07 Marks)
cl
di
... ;-
c.
Find the radius of curvature for the curve x
3
+ y
-
' =3xy at (3/2, 3/2). (07 Marks)
,:. ,,
d.). ,
n
,
m =
.
OR
. ,
2 a.
If y = cos(m log x) then prove that x'y
n
_, + (2n + I )xy?
?
, + (m
2
+ n
2
)y =0. (06 Marks)
.? r-A
b.
With usual notation prove that tan st)= r
dO
. (07 Marks)
r
, cc
= +
m -,1- dl
-
.
tb
c.
Find the pedal equation of the curve rm = a' cos m0 .
(07 Marks)
,
?
0.)
=
O
?
F?
0.7
r
i' Module-2 . ,
= 0 ....
?
3 a.
Find the Taylor's series of log(cosx) in powers of (x ? n/3) upto fourth degrees terms.
2
L,
g
(06 Marks)
c.)
cr
u
7d '6
i 3 1 1
- ou au
b. If u = tan-
X ? y
' then prove that x + y
= sin 2u by using Euler's theorem.(07 Marks)
. gz x + y
i
ax ay
.
-
4
C.
a.
b.
yz
if u= ---, v ? xz , w ?
x
y
Evaluate lim
tan x \
a(uvw) xy
then find .1 =
z a(xyz)
OR
0
Using Maclaurin's series, prove that Ail + sin 2x =1+ x
x x3
+
?
x4?
? ? ? .(07 Marks)
1
2 3
1
4
a 'au 1 au
C.
If u 42x -3y, 3y 4 4z --
1 u 1
2x) then prove that + + = O.
2 'x 3 ay 4 az
(07 Marks)
(07 Marks)
(06 Marks)
c. Prove that curl(d) A) d)curl A+ gradd)x A (07 Marks)
of2
OR
A A
6 a.
xi+yj
Show that vector field F is both solenoidal and irrotational.
x
-
+ y
-
b. If F = (x + y +1) + j-(x + y)i then prove that = curlF = 0 .
c. Show that div(curl A) = 0 .
Module-4
7 a. Obtain reduction formula for sin" x dx(n > 0).
b. Solve the differential equation ?
dy
+ y cot x = cos x
dx
c. Find the orthogonal trajectory of the curve r = a(1+ sin 0).
OR
8
a. Evaluate I sin
7
0 cos
h
0 de
17MAT11,,
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
0
b. Solve the differential equation : (2xy + y - tan y)dx + (x
2
- x tan` y + sec
2
y)dy = 0 .
(07 Marks)
c. If the temperature of air is 30?C and the substance cools from 100?C to 70?C in 15 mins.
Find when the temperature will be 40?C_ (07 Marks)
Module-5
1 1 I 6
-
1 -1 2 s
9 a. Find the rank of the matrix by reducing to Echelon form. (06 Marks)
3 1 1 8
2
-2 3 7
6 -2 2
b.
Find the largest eigen value and egien vector of the matrix
-2 3 -1
by taking initial
2 -1 3
C.
vector as [ 1 1 1]
i
by using Rayleigh's power method. Carry out five iteration. (07 Marks)
Reduce 8x
2
+ 7y
2
+ 3z
2
- I 2xy + 4xz - 8yz into canonical form, using orthogona
transformation_ (07 Marks)
OR
10 a. Solve the system of equations
10x+ y+ z = 12
x+ 10y+ z= 12
x+ y+ 10z= 12
by using Gauss-Seidel method. Carry out three iterations. (06 Marks)
b. Diagonalise the matrix A =
5 4
(07 Marks)
c. Show that the transformation
x
i
+ 2x
2
+ 5X3
Y2 = 2X1 4x2 + 1 1 X3
y3 = ?X2 + 2X3
is regular. Write down inverse transformation.
CH!KODI ?
LUBIRARY
0
0,
EirvA (07 Marks)
* * * 2 of ") * *
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This post was last modified on 01 January 2020