Download Visvesvaraya Technological University (VTU) BE ( Bachelor of Engineering) Civil Engineering 2018 Scheme 2020 January Previous Question Paper 3rd Sem 18MATDIP31 Additional Mathematics I
I
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1. 8MATDIP31
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Third Se .E. Degree Examination, Dec.2019/Jan.2020
Additional Mathematics ? I
Time: 3 hrs.
Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
4.)
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1
2
3
4
5
a.
b.
c.
a.
b.
c.
a.
b.
c.
a.
b
c.
a.
b.
c.
Express the
Prove that
If a = (3,-1,4),
Find the angle
Prove that
Find the fourth
Obtain the
If u =
If u = x(1-
Obtain the
If u = f(x
Find the
r = e
--`
i +2cos5tj
Find div F
Show that
following complex
cos0+ isin 00
and
a
i
expansion
vectors,
that
=
a(u,v)
number
cos80 + i sin
-4
c = (4.2,-1)
a =
a, b, c
in
and represent
Module-2
the form of
80 .
, find ax
OR
Si - j+ k and
them on
x + iy
bx c
b = 2i
the
.
log
e
.
0+41+30
: (06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
the curve,
(06 Marks)
(07 Marks)
and find is conservative
1+5i
- 3j + 6k .
argand diagram.
sec x .
moves along
force field
sin 0+icos0)
-4
b =
between
ax b , bx
roots
Maclaurin's
X
3
+ y
3-
(1,2,3)
the
c, cx
of -1+
, prove
, find
z -x)
and
F , where
+ z' )i
+Ssin2tk
--*
of log
e
(1
x +
ax
+ x).
y
aU
? = 2 tan u
ay
OR
of the function
au au
+
ay az
+ =
Module-3
It u=x2-2y;v=x+yfind
x + y
y), v = xy
Maclauvin's
y, y - z,
velocity
and curl
F = (2xy
series
+ (x
2
3
(x,y)
expansion
a(u,v)
, prove
at
F
a(x,y)
au
that
ax
acceleration of a particle
any time t.
= V(x
3
+ y
3
+ z
3
-3xyz).
+ 2yz)j + (y
2
+ 2xz)k
the scalar potential.
(07 Marks)
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H KUU%
I
USN
1. 8MATDIP31
Sou
s
\,
,
* LIBRARY
,
s
e,4 ?
Third Se .E. Degree Examination, Dec.2019/Jan.2020
Additional Mathematics ? I
Time: 3 hrs.
Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
4.)
CL)
ti
C 00
E (?,
-
c
7
- :
L Q)
a9
17- ? r
0
cn
0
cn
2 0-
Tc
0 -0
t i
-6
-
a
7- a
t
'
?-
.4 1
CI,
c.4
cz
0 ?-
>,
tt.0
t.1)
? ?
Cr
E
0
c.J
5
O
z
C
s
t
0
E
1
2
3
4
5
a.
b.
c.
a.
b.
c.
a.
b.
c.
a.
b
c.
a.
b.
c.
Express the
Prove that
If a = (3,-1,4),
Find the angle
Prove that
Find the fourth
Obtain the
If u =
If u = x(1-
Obtain the
If u = f(x
Find the
r = e
--`
i +2cos5tj
Find div F
Show that
following complex
cos0+ isin 00
and
a
i
expansion
vectors,
that
=
a(u,v)
number
cos80 + i sin
-4
c = (4.2,-1)
a =
a, b, c
in
and represent
Module-2
the form of
80 .
, find ax
OR
Si - j+ k and
them on
x + iy
bx c
b = 2i
the
.
log
e
.
0+41+30
: (06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
the curve,
(06 Marks)
(07 Marks)
and find is conservative
1+5i
- 3j + 6k .
argand diagram.
sec x .
moves along
force field
sin 0+icos0)
-4
b =
between
ax b , bx
roots
Maclaurin's
X
3
+ y
3-
(1,2,3)
the
c, cx
of -1+
, prove
, find
z -x)
and
F , where
+ z' )i
+Ssin2tk
--*
of log
e
(1
x +
ax
+ x).
y
aU
? = 2 tan u
ay
OR
of the function
au au
+
ay az
+ =
Module-3
It u=x2-2y;v=x+yfind
x + y
y), v = xy
Maclauvin's
y, y - z,
velocity
and curl
F = (2xy
series
+ (x
2
3
(x,y)
expansion
a(u,v)
, prove
at
F
a(x,y)
au
that
ax
acceleration of a particle
any time t.
= V(x
3
+ y
3
+ z
3
-3xyz).
+ 2yz)j + (y
2
+ 2xz)k
the scalar potential.
(07 Marks)
OR
Solve : (5x
4
+3x
2
y
2
?2xyldx +(2x
3
y-3x'y
2
?5y1dy =0.
10 a.
b.
c.
dy
Solve :
+ x sin 2y = x
3
cos' y
dx
Solve : [I+ (x + y)tan + I - 0 .
dx
18MATDIP31
OR
6 a. Show that the vector field, F = (3x + 3y + 4z)i + x ? 2y + 3zij + (3x + 2y ? )1c is solenoidal.
(06 Marks)
b.
xz
Find the directional derivative of = , at (
x
-
+y
-
A A A
?1, 1 ) in the direction of A = i ? 2j+ k .
c.
Find the constant 'a' such that the vector
irrotational.
Module-4
2
7 a. Find the reduction formula for 'sin' xdx
1 3
Evaluate f f x
3
y'dxdy.
00
3 2 1
Evaluate f f f (x + y + z)dzdxdy
0 0 0
OR
6
Evaluate : isin
6
(3x)dx
b. Evaluate : f xy dydx
,
1 1--x1-x-
y
c. Evaluate : f f ixyzdzdydx
0 0 0
Module-5
(07 Marks)
field F = 2xy
2
z
2
i + x yz
-
j+ax'y'zi is
(07 Marks)
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b.
c.
8 a.
9
a.
d
y
Solve : + y cot x = sin x
dx
b. Solve : (2x
s
?xy
2
? 2y +3)dx ?(x
2
y + 2x)cly = 0 .
c. Solve : 3x(x + )dy + (x
3
?3xy ? 2y
3
)dx =0.
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
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This post was last modified on 02 March 2020