Download Visvesvaraya Technological University (VTU) BE ( Bachelor of Engineering) Civil Engineering 15 Scheme 2020 January Previous Question Paper 3rd Sem 15MATDIP31 Additional Mathematics I
15MATDIP31
43
,
Third Semester B.E. Degree Examination, Dec.2019/Jan.2020
Additional Mathematics ? I
Time: 3 hrs.
Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE fill question from each !nodule.
Module-1
1 a. Find modulus and amplitude of I? cos 9 + i sin 9 .
(05 Marks)
3+4i .
b. Express in a +ib form.
(OS Marks)
3-4i
c. Find the value of 'X,' so that the points A(-1, 4, ?3), B(3., 2, ?5), C(-3, 8, ?5) and D(-3,
I),
may lie on one plane.
(06 Marks)
OR
2 a. Find the angle between the vectors a = 5 j+ k and b = 2 3 j+ 6k .
axb,bxc,cxa l=I abc
C.
Find the real part of
1
l+cose+isin 0
Module-2
3 a. Obtain the n
th
derivatiVe of sin(ax +b).
b. Find the pedal equation of = a" cosne
zx xy a(u, v, w)
c. If u =
Yz
, v = , w = , show that
x y z a(x, y,z)
OR
?
4 a. If u
og
(
X.
4
? y
?
show that x ?
all
+ y
ail
? =3 .?
x + y ax a
y
au:. au au
b. If u = y, y z ? x), show that + =0.
ax ay az
c. If y = a cos(logx) + bsin(log x), show that x
2
y?, + (2n +1)xy,
i
+ (n
2
+1)y,, =0
Module-3
5 a. Evaluate 1 xsin' xdx
J
0
b.
Evaluate f x
2
(1 x
2
)
3
'
2
dx
b a
c. Evaluate f f(x2 +
y
2 + )dzdydx
-h ?a
b.
Prove that
(05 Marks)
(05 Marks)
(06 Marks)
(05 Marks)
(05 Marks)
(06 Marks)
(OS Marks)
(05 Marks)
(06 Marks)
(05 Marks)
(05 Marks)
(06 Marks)
1 of 2
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15MATDIP31
43
,
Third Semester B.E. Degree Examination, Dec.2019/Jan.2020
Additional Mathematics ? I
Time: 3 hrs.
Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE fill question from each !nodule.
Module-1
1 a. Find modulus and amplitude of I? cos 9 + i sin 9 .
(05 Marks)
3+4i .
b. Express in a +ib form.
(OS Marks)
3-4i
c. Find the value of 'X,' so that the points A(-1, 4, ?3), B(3., 2, ?5), C(-3, 8, ?5) and D(-3,
I),
may lie on one plane.
(06 Marks)
OR
2 a. Find the angle between the vectors a = 5 j+ k and b = 2 3 j+ 6k .
axb,bxc,cxa l=I abc
C.
Find the real part of
1
l+cose+isin 0
Module-2
3 a. Obtain the n
th
derivatiVe of sin(ax +b).
b. Find the pedal equation of = a" cosne
zx xy a(u, v, w)
c. If u =
Yz
, v = , w = , show that
x y z a(x, y,z)
OR
?
4 a. If u
og
(
X.
4
? y
?
show that x ?
all
+ y
ail
? =3 .?
x + y ax a
y
au:. au au
b. If u = y, y z ? x), show that + =0.
ax ay az
c. If y = a cos(logx) + bsin(log x), show that x
2
y?, + (2n +1)xy,
i
+ (n
2
+1)y,, =0
Module-3
5 a. Evaluate 1 xsin' xdx
J
0
b.
Evaluate f x
2
(1 x
2
)
3
'
2
dx
b a
c. Evaluate f f(x2 +
y
2 + )dzdydx
-h ?a
b.
Prove that
(05 Marks)
(05 Marks)
(06 Marks)
(05 Marks)
(05 Marks)
(06 Marks)
(OS Marks)
(05 Marks)
(06 Marks)
(05 Marks)
(05 Marks)
(06 Marks)
1 of 2
15MATDIP31
OR
I Nix
6
a. Evaluate xydydx
0 x
I l l
b. Evaluate f(x + y + z)dxdydz
c. Evaluate X
4
+ x
2
)4
dx .
(05 Marks)
(05 Marks)
(06 Marks)
Module-4
a.
7 If r = (t
-
+ 1) i+ (4t ?3) j+ (2t
-
,
6t)k find the angle between the tangents at t = 1 and t = 2.
(05 Marks)
A
b. If r = e i+ 2cos3t j+ 2sin3tk , find the velocity and acceleration at any time t, and also
their magnitudes at t = 0. (05 Marks)
A
c. Show that F = (y+ z) i+ (z + x) j+ ( x + y) k is irrotational. Also find a scalar function 'y
such that F = V(1). (06 Marks)
OR
8 a.
Find the unit normal vector to the surface x
2
y+ 2xz = 4 at (2, ?2, 3). (05 Marks)
b. If F = xz
3
i? 2x
2
yz j+ 2yz
4
k find V F and V x F at (1, ?1, 1). (05 Marks)
d
a
-4 -4 d
c. If dt
dt dt
= wx a and
(-
-L
b
= wx b , then show that ?(ax b) = wx (ax b) (06 Marks)
9 a.
b.
c.
Module-5
Solve sec
2
x tan ydx + sec' y tan xdy = 0 .
Solve (y
3
?3x
2
y)dx +(3xy
2
?x
3
)dy = 0 .
Solve
dy
?+
y
xy
-
.
dx x
OR
10 a.
Solve
dy
? + y cot x = cos x
dx
b. Solve x
2
ydx ?(x
3
+ )dy = 0
c. Solve y(x + y)dx +(x + 2y ?1)dy =0
(05 Marks)
(05 Marks)
(06 Marks'
(05 Marks)
(05 Marks)
(06 Marks)
2 of 2
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This post was last modified on 02 March 2020