Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 15 Scheme 15MAT11 Engineering Mathematics I Question Paper
1
LIBRARY
USN
Illig111111
SOCiftt
y
15MAT11
First Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics - I
Time: 3 hrs. Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
6
1
7x + 6
?
ti
a. Find the n
th
derivative of (05 Marks)
2x
2
+ 7x + 6
p.,
.7.' h Find the angle between the radius vector and the tangent for the curve
r'" = a' (cosm0 +sinm0). (05 Marks)
-o
1.)
c. Show that the radius of curvature at any point 0 on the cycloid x = a (0 + sing),
tns
c.,
y = a(l - cos 0) is 4 acos(0/2)
1-_, -
(06 Marks)
,,,
wi
,c,
OR
... -
.5. .
2 a. If x = sint and y = cosmt, prove that (1- x
2
)y,,,2-
5 d
b. Find the pedal equation of the curve r = a
-
sec 20.
76
tr,
II
:19 . rd0
.:-... - +
c. Prove with usual notation tan 4) - .
= ,
..a
.
dr
t i,
fu ,
Module-2
Expand e
l
' using Maclaurin's series upto third degree term.
Ai au
C.
If u = e
(
"
?b
Y
)
flax - by), prove that b ?
C
+ a ? = 2abu
ax ay
OR
4 a. Expand sin x in ascending power ofit/2 upto the term containing x
4
.
(
x
3 j_ 3
b. If u tan' , show that x u
x
+ y u
y
= sin2u.
x y
yz zx xy a(u, v, w)
c. If u = v = ? , w = . Find
a(x,y,z)
Module-3
5 a. Find the angle between the surfaces x
2
+ y
-
+ z
-
= 9 and x
-
+ y
-
z = 3 at the point
(2, -1, 2).
(05 Marks)
b. Show that F = (y + z)i +(x + z)j+(x + y)k is irrotational. Also find a scalar function 4) such
that P = V4).
(05 Marks)
c. Prove that V - (4)X (I)( V ? A) + V4) ? A .
(06 Marks)
OR
6 a. Prove that Curl (4)
-
A) = 4)(Curl A) + grad4) x X.
(05 Marks)
b.
A particle moves along the curve C ; x = t
3
- 4t, y = t
2
+ 4t, z = 8t
2
- 3t
3
where `e denotes
the time. Find the component of acceleration at t = 2 along the tangent.
(05 Marks)
r .
V
a_
.z.-.
I
1
liM
b. Evaluate
x ?>. 0
X `
[
sin
2
X
va
cr
(05 Marks)
(05 Marks)
(06 Marks)
(05 Marks)
(05 Marks)
(06 Marks)
(05 Marks)
(05 Marks)
(06 Marks)
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1
LIBRARY
USN
Illig111111
SOCiftt
y
15MAT11
First Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics - I
Time: 3 hrs. Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
6
1
7x + 6
?
ti
a. Find the n
th
derivative of (05 Marks)
2x
2
+ 7x + 6
p.,
.7.' h Find the angle between the radius vector and the tangent for the curve
r'" = a' (cosm0 +sinm0). (05 Marks)
-o
1.)
c. Show that the radius of curvature at any point 0 on the cycloid x = a (0 + sing),
tns
c.,
y = a(l - cos 0) is 4 acos(0/2)
1-_, -
(06 Marks)
,,,
wi
,c,
OR
... -
.5. .
2 a. If x = sint and y = cosmt, prove that (1- x
2
)y,,,2-
5 d
b. Find the pedal equation of the curve r = a
-
sec 20.
76
tr,
II
:19 . rd0
.:-... - +
c. Prove with usual notation tan 4) - .
= ,
..a
.
dr
t i,
fu ,
Module-2
Expand e
l
' using Maclaurin's series upto third degree term.
Ai au
C.
If u = e
(
"
?b
Y
)
flax - by), prove that b ?
C
+ a ? = 2abu
ax ay
OR
4 a. Expand sin x in ascending power ofit/2 upto the term containing x
4
.
(
x
3 j_ 3
b. If u tan' , show that x u
x
+ y u
y
= sin2u.
x y
yz zx xy a(u, v, w)
c. If u = v = ? , w = . Find
a(x,y,z)
Module-3
5 a. Find the angle between the surfaces x
2
+ y
-
+ z
-
= 9 and x
-
+ y
-
z = 3 at the point
(2, -1, 2).
(05 Marks)
b. Show that F = (y + z)i +(x + z)j+(x + y)k is irrotational. Also find a scalar function 4) such
that P = V4).
(05 Marks)
c. Prove that V - (4)X (I)( V ? A) + V4) ? A .
(06 Marks)
OR
6 a. Prove that Curl (4)
-
A) = 4)(Curl A) + grad4) x X.
(05 Marks)
b.
A particle moves along the curve C ; x = t
3
- 4t, y = t
2
+ 4t, z = 8t
2
- 3t
3
where `e denotes
the time. Find the component of acceleration at t = 2 along the tangent.
(05 Marks)
r .
V
a_
.z.-.
I
1
liM
b. Evaluate
x ?>. 0
X `
[
sin
2
X
va
cr
(05 Marks)
(05 Marks)
(06 Marks)
(05 Marks)
(05 Marks)
(06 Marks)
(05 Marks)
(05 Marks)
(06 Marks)
b. Find the largest eigen value and corresponding eigenvector of the matrix
6 ?2 2
A = ? 2 3 ?1 by power method taking X
(
''
)
= [1, 1
2 ? 1 3
c. Reduce the matrix A =
?1
? 2
3
4
to the diagonal form.
15MAT11
c.
Show that F = (2xy
2
+ yz)i + (2x
2
y + xz + 2yz
2
)j + (2y
2
z + xy)k is a conservative force
field. Find its scalar potential. (06 Marks)
Module-4
7
a. Obtain the reduction formula for j sing' x dx (05 Marks)
b.
Solve (y
2
ex''' + 4x
3
)dx +(2xye'''' ?3y
2
)dy = 0. (05 Marks)
c. Find the orthogonal trajectories of r = a (l+sin0). (06 Marks)
9 a.
OR
2
Evaluate x-si2x ? x
2
dx (05 Marks)
J
Solve (y
3
? 3x
2
y)dx ? (x
3
?3xy
2
)dy = 0. (05 Marks)
A bottle of mineral water at a room temperature of 72?F is kept in a refrigerator where the
temperature is 44?F. After half an hour, water cooled to 61?F
i) What is the temperature of the mineral water in another half an hour?
ii) How long will it take to cool to 50?F? (06 Marks)
Find the rank of the matrix
?1
--3
?1
1
Module-5
1 2 3 ?1
A = (05 Marks)
1 0 1 1
0 1 1 ?1
8 a.
b.
c.
10 a. Use Gauss elimination method to solve
2x+y+4z=12
4x+11y?z=33
8x ?3y + 2z = 20
b. Find the inverse transformation of the following lin
y, = x, +2x, +5x
3
y
2
= 2x, +4x, +l lx
3
y
3
=?x, +2x
;
c. Reduce the quadratic form 2x; + 2x + 2x + 2x
i
x.
(05 Marks)
(06 Marks)
ear transformation.
to the Cannonical form.
Socipf",
CHIKODI *
1
1"..
... ir T ril Irb a v...?
(05 Marks)
(05 Marks)
(06 Marks)
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This post was last modified on 01 January 2020