Download VTU BE 2020 Jan ME Question Paper 15 Scheme 3rd Sem 15MATDIP31 Additional Mathematics I

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,
Third Semester B.E. Degree Examination, Dec.2019/Jan.2020
Additional Mathematics ? I
Time: 3 hrs.
Max. Marks: 80

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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 .

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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),

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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

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C.
Find the real part of
1

l+cose+isin 0

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Module-2
3 a. Obtain the n
th
derivatiVe of sin(ax +b).
b. Find the pedal equation of = a" cosne

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zx xy a(u, v, w)
c. If u =
Yz
, v = , w = , show that
x y z a(x, y,z)

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OR
?
4 a. If u
og
(

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X.
4
? y
?
show that x ?

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all
+ y
ail

? =3 .?

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x + y ax a
y

au:. au au
b. If u = y, y z ? x), show that + =0.

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ax ay az
c. If y = a cos(logx) + bsin(log x), show that x
2
y?, + (2n +1)xy,
i

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+ (n
2
+1)y,, =0
Module-3
5 a. Evaluate 1 xsin' xdx

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J
0
b.
Evaluate f x
2

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(1 x
2
)
3
'

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2
dx
b a
c. Evaluate f f(x2 +
y

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2 + )dzdydx
-h ?a
b.
Prove that
(05 Marks)

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(05 Marks)
(06 Marks)
(05 Marks)
(05 Marks)
(06 Marks)

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(OS Marks)
(05 Marks)
(06 Marks)
(05 Marks)
(05 Marks)

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(06 Marks)
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15MATDIP31

43
,
Third Semester B.E. Degree Examination, Dec.2019/Jan.2020

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Additional Mathematics ? I
Time: 3 hrs.
Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE fill question from each !nodule.
Module-1

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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)

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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)

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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

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1

l+cose+isin 0
Module-2
3 a. Obtain the n

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th
derivatiVe of sin(ax +b).
b. Find the pedal equation of = a" cosne
zx xy a(u, v, w)
c. If u =

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Yz
, v = , w = , show that
x y z a(x, y,z)
OR
?

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4 a. If u
og
(
X.
4

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? y
?
show that x ?
all
+ y

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ail

? =3 .?
x + y ax a
y

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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

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2
y?, + (2n +1)xy,
i
+ (n
2

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+1)y,, =0
Module-3
5 a. Evaluate 1 xsin' xdx
J
0

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b.
Evaluate f x
2
(1 x
2

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)
3
'
2
dx

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b a
c. Evaluate f f(x2 +
y
2 + )dzdydx
-h ?a

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b.
Prove that
(05 Marks)
(05 Marks)
(06 Marks)

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(05 Marks)
(05 Marks)
(06 Marks)
(OS Marks)
(05 Marks)

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(06 Marks)
(05 Marks)
(05 Marks)
(06 Marks)
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15MATDIP31
OR
I Nix
6
a. Evaluate xydydx

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0 x
I l l
b. Evaluate f(x + y + z)dxdydz
c. Evaluate X
4

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+ x
2
)4
dx .
(05 Marks)

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(05 Marks)
(06 Marks)
Module-4
a.

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7 If r = (t
-
+ 1) i+ (4t ?3) j+ (2t
-
,

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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

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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

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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

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3
i? 2x
2
yz j+ 2yz
4

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k find V F and V x F at (1, ?1, 1). (05 Marks)
d
a
-4 -4 d
c. If dt

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dt dt
= wx a and
(-
-L
b

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= wx b , then show that ?(ax b) = wx (ax b) (06 Marks)
9 a.
b.
c.
Module-5

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Solve sec
2
x tan ydx + sec' y tan xdy = 0 .
Solve (y
3

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?3x
2
y)dx +(3xy
2
?x

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3
)dy = 0 .
Solve
dy

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?+
y
xy
-
.

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dx x
OR
10 a.
Solve
dy

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? + y cot x = cos x
dx
b. Solve x
2
ydx ?(x

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3
+ )dy = 0
c. Solve y(x + y)dx +(x + 2y ?1)dy =0
(05 Marks)
(05 Marks)

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(06 Marks'
(05 Marks)
(05 Marks)
(06 Marks)
2 of 2

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