Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 2014 June-July 10MAT21 Engineering Mathematics II Question Paper
USN
10MAT21
Second Semester B.E. Degree Examination, June / July 2014
Engineering Mathematics - II
Max. Marks:100
04
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PART - A
ca.:7
3
,-..,
5 1 a. Choose tieUorrect answer : (04 Marks)
.7i '
00 1I
i) The generat,Solution of the equation x
2
p
2
+ 3xyp + 2y
2
= 0 is
O00
'4,
-
1,
-
, (A) (y
2
x - C),(Xy - c) - 0 (B) (x-y-o) (x
2-
+7
-
-
-
- c) = 0
,
:r
a
(C) (xy - c)
(x
y-c) = 0 (D) (y-x-b) (x
2
+ y
2
+ c) = 0
a)
6
ii) The given differenti41 equation is solvable for y, if it is possible to express y in terms of
'''
(A) y and p (14)' x and p (C) x and y (D) y and x
....
.a.'
iii) The singular solution of Clairaut's equation is
(A) y = xg(x) + f [g(x)] ' ?:,:_''.: + = (B) y = cx + f(c)
(C) cy
-
I
-
f(c) - .,.., ' (D) y g (x) + f [g(x)]
It.:fir:--
,
,
iv) The singular solution of the 9,44401
y
= px - log p is
(A) y
2
= 4ax (B) x'= 1, log x
A
(C) y = 1- log (!) (D) x
2
- y log x
Ns,
x
a. Choose the correct answer (04 Marks)
i) The complementary function of [D
4
+ 4] x = 0 is
(A)
:
.X7,e
4
[ci cos t + c
2
sin t] + [ c
3
cos t + c
4
sin t]
[ci cos t + C2 sin t] + [C3 cos t + C
4
sin t]
x = [ci+ c2 t] e
4
1) Find the particular integral of (D
3
- 3D
2
+ 4) y = e
2x
is
(D) x = [c
i
+ c2 t]
.T...
Time: 3 hrs.
It
-a
e
0
Note4. Answer FIVE full questions choosing at least two from each part.
Answer all objective type questions only in OMR sheet page 5 of the Answer Booklet.
3. Aittwers to objective type questions on sheets other than OMR will not be valued.
I.
0
0
"C)
7
?
3
b. Solve p
2
- 2p sin h x -1 - O. (04 Marks)
8 $2 C. Solve y = 2px + tan-I (xp2). (06 Marks)
.c.v.
d. Obtain the general solution and singular solution of Clairaut's equation is (y - px) (p-1) = p.
(06 Marks)
G
6
.2
o
0.0 0
o
8
(A) x
ze zx
(B)
x2e3x
(C)
x 2ex
(D)
x
2
e
4x
--;ci
6 6 6 6
z
o
a
41)
2
iii) Roots of ?
d2y
+4?
dy
+5y = 0 are
dx
2
dx
(A) 2 ? i (B) 3 + i (C) 2 ? 2i
iv) Find the particular integral of (D
3
+ 4D) y - sin 2x is
(A)
x sin x
(B)
? xsinx (C)
? x sin 2x
8 8 8
b. Solve ?
d
x
y
3
+6?
d2y
+11?
dy
+6y = ex +1.
d dx
2
dx
c. Solve d
d
2
x
2
- 4y = cos h (2x - 1) + r.
Y
(D)
x sin 2x
8
(04 Marks)
(06 Marks)
(D) -2 +
1 of 4
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USN
10MAT21
Second Semester B.E. Degree Examination, June / July 2014
Engineering Mathematics - II
Max. Marks:100
04
,
-al
co
8
b
?
0
,
tio
w
^c'
cks ???
PART - A
ca.:7
3
,-..,
5 1 a. Choose tieUorrect answer : (04 Marks)
.7i '
00 1I
i) The generat,Solution of the equation x
2
p
2
+ 3xyp + 2y
2
= 0 is
O00
'4,
-
1,
-
, (A) (y
2
x - C),(Xy - c) - 0 (B) (x-y-o) (x
2-
+7
-
-
-
- c) = 0
,
:r
a
(C) (xy - c)
(x
y-c) = 0 (D) (y-x-b) (x
2
+ y
2
+ c) = 0
a)
6
ii) The given differenti41 equation is solvable for y, if it is possible to express y in terms of
'''
(A) y and p (14)' x and p (C) x and y (D) y and x
....
.a.'
iii) The singular solution of Clairaut's equation is
(A) y = xg(x) + f [g(x)] ' ?:,:_''.: + = (B) y = cx + f(c)
(C) cy
-
I
-
f(c) - .,.., ' (D) y g (x) + f [g(x)]
It.:fir:--
,
,
iv) The singular solution of the 9,44401
y
= px - log p is
(A) y
2
= 4ax (B) x'= 1, log x
A
(C) y = 1- log (!) (D) x
2
- y log x
Ns,
x
a. Choose the correct answer (04 Marks)
i) The complementary function of [D
4
+ 4] x = 0 is
(A)
:
.X7,e
4
[ci cos t + c
2
sin t] + [ c
3
cos t + c
4
sin t]
[ci cos t + C2 sin t] + [C3 cos t + C
4
sin t]
x = [ci+ c2 t] e
4
1) Find the particular integral of (D
3
- 3D
2
+ 4) y = e
2x
is
(D) x = [c
i
+ c2 t]
.T...
Time: 3 hrs.
It
-a
e
0
Note4. Answer FIVE full questions choosing at least two from each part.
Answer all objective type questions only in OMR sheet page 5 of the Answer Booklet.
3. Aittwers to objective type questions on sheets other than OMR will not be valued.
I.
0
0
"C)
7
?
3
b. Solve p
2
- 2p sin h x -1 - O. (04 Marks)
8 $2 C. Solve y = 2px + tan-I (xp2). (06 Marks)
.c.v.
d. Obtain the general solution and singular solution of Clairaut's equation is (y - px) (p-1) = p.
(06 Marks)
G
6
.2
o
0.0 0
o
8
(A) x
ze zx
(B)
x2e3x
(C)
x 2ex
(D)
x
2
e
4x
--;ci
6 6 6 6
z
o
a
41)
2
iii) Roots of ?
d2y
+4?
dy
+5y = 0 are
dx
2
dx
(A) 2 ? i (B) 3 + i (C) 2 ? 2i
iv) Find the particular integral of (D
3
+ 4D) y - sin 2x is
(A)
x sin x
(B)
? xsinx (C)
? x sin 2x
8 8 8
b. Solve ?
d
x
y
3
+6?
d2y
+11?
dy
+6y = ex +1.
d dx
2
dx
c. Solve d
d
2
x
2
- 4y = cos h (2x - 1) + r.
Y
(D)
x sin 2x
8
(04 Marks)
(06 Marks)
(D) -2 +
1 of 4
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15 LIBRARY
49
1OMAT2 1
4
e
!".?ift afro)
d. Solve
d
Y (06 Marks)
x dz
?+y=ze 7
dx dx
3 a. Choose the correct answer : (04 Marks)
i) The Wronskian of x and x e
x
is
(A) ex
(B)
e2x
(C) e
-2x
(D) ex.
ii) The complementary function of x
2
y" - xy' ? 3y = x
2
log x is
(A) ci cos (log x) + c2 sin (log x) (B) c
1
x
-1
+ c2x
3
.
(C) c ix + c2x
3
(D) cicos x + c2 sin x.
iii) To transform (1+x)
2
y" + (1+ x)y' + y = 2 sin log (1 + x) into a line& differential
equation with constant coefficient
(A) (1+ x) = e
t
(B) (1+ x) = e
-t
(C) (1 + x)
2
? e
t
r
(D)'.(1-x)
2
= e
t
.
iv) The equation ao(ax + b)
2
y" + ao(ax + b) y' + a2y = 4(x) is
:',,
(A) Simultaneous equation (B) Cauchy's linearvpation
(C) Legendre linear equation (D) Euler's equag174#:'
b. Using the variation of parameters method to solve the equatio 4,7 1- 2y' + y = ex log x.
(04 Marks)
<
j
.,
d
2
v
c. Solve x
2
- ?(2m-1)x?
d
+
y
(
n
2 +
n2)
2) y ,
n
2
x
in log x,
(06 Marks)
d
2
clx
d. Obtain the Frobenius method solve the equation
,.,
C
1
(06 Marks)
(04 Marks)
i) Partial differential equation by elitninating a and b from the relation
Z = (x-a)
2
+ (y ?b)
2
is
?
(A)
p
2
q2
4z
(B) .04==
ii) The Lagranges's linear partial differential Pp + Qq = R the subsidiary equation
is
(A)
dx dy dz
(B)
dx dy dz
(C)
dx-
?:
dz
(D)
dx
+
dy dz
R P Q P Q R Q R P P Q R
iii) By the method of separation of variable we seek a solution in the form is
(A) x = x + (B) z x
2
+ y
2
(C) x z + (D) x ? x(x) y(y)
iv) The solution of
z
= sin (xy) is
ax
2
(A) z = -x
2
sin (xy) + y f(x) + 0(x)
(C) z
xy
y f(x) + (1)(x)
x
b. Form the partial differential equation of all sphere of radius 3 units having their centre in the
xy - plane. (04 Marks)
c. Solve x (y
2
+ z) p-y (x
2
+ z) q = z (x
2
-y
2
). (06 Marks)
d. Use the method of separation of variables to solve
aZ 2 aZ
y
3
x ?=u?
(06 Marks)
ax ay
PART - B
5 a. Choose the correct answer :
x2
i) The value of j r
e
7
xdydx is
0 0
(A) 0 (B) 1
(04 Marks)
(C) 3
2 of 4
(D) 'A .
(C) r ? 4z (D) t = 4
-
7
-
(B)
? sin(xy)
x f(Y)
(
licY)
Y
2
(D) None of these.
d2y dy?y=0? x?
dx2 + dx
4 a. Choose the correct answer :
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USN
10MAT21
Second Semester B.E. Degree Examination, June / July 2014
Engineering Mathematics - II
Max. Marks:100
04
,
-al
co
8
b
?
0
,
tio
w
^c'
cks ???
PART - A
ca.:7
3
,-..,
5 1 a. Choose tieUorrect answer : (04 Marks)
.7i '
00 1I
i) The generat,Solution of the equation x
2
p
2
+ 3xyp + 2y
2
= 0 is
O00
'4,
-
1,
-
, (A) (y
2
x - C),(Xy - c) - 0 (B) (x-y-o) (x
2-
+7
-
-
-
- c) = 0
,
:r
a
(C) (xy - c)
(x
y-c) = 0 (D) (y-x-b) (x
2
+ y
2
+ c) = 0
a)
6
ii) The given differenti41 equation is solvable for y, if it is possible to express y in terms of
'''
(A) y and p (14)' x and p (C) x and y (D) y and x
....
.a.'
iii) The singular solution of Clairaut's equation is
(A) y = xg(x) + f [g(x)] ' ?:,:_''.: + = (B) y = cx + f(c)
(C) cy
-
I
-
f(c) - .,.., ' (D) y g (x) + f [g(x)]
It.:fir:--
,
,
iv) The singular solution of the 9,44401
y
= px - log p is
(A) y
2
= 4ax (B) x'= 1, log x
A
(C) y = 1- log (!) (D) x
2
- y log x
Ns,
x
a. Choose the correct answer (04 Marks)
i) The complementary function of [D
4
+ 4] x = 0 is
(A)
:
.X7,e
4
[ci cos t + c
2
sin t] + [ c
3
cos t + c
4
sin t]
[ci cos t + C2 sin t] + [C3 cos t + C
4
sin t]
x = [ci+ c2 t] e
4
1) Find the particular integral of (D
3
- 3D
2
+ 4) y = e
2x
is
(D) x = [c
i
+ c2 t]
.T...
Time: 3 hrs.
It
-a
e
0
Note4. Answer FIVE full questions choosing at least two from each part.
Answer all objective type questions only in OMR sheet page 5 of the Answer Booklet.
3. Aittwers to objective type questions on sheets other than OMR will not be valued.
I.
0
0
"C)
7
?
3
b. Solve p
2
- 2p sin h x -1 - O. (04 Marks)
8 $2 C. Solve y = 2px + tan-I (xp2). (06 Marks)
.c.v.
d. Obtain the general solution and singular solution of Clairaut's equation is (y - px) (p-1) = p.
(06 Marks)
G
6
.2
o
0.0 0
o
8
(A) x
ze zx
(B)
x2e3x
(C)
x 2ex
(D)
x
2
e
4x
--;ci
6 6 6 6
z
o
a
41)
2
iii) Roots of ?
d2y
+4?
dy
+5y = 0 are
dx
2
dx
(A) 2 ? i (B) 3 + i (C) 2 ? 2i
iv) Find the particular integral of (D
3
+ 4D) y - sin 2x is
(A)
x sin x
(B)
? xsinx (C)
? x sin 2x
8 8 8
b. Solve ?
d
x
y
3
+6?
d2y
+11?
dy
+6y = ex +1.
d dx
2
dx
c. Solve d
d
2
x
2
- 4y = cos h (2x - 1) + r.
Y
(D)
x sin 2x
8
(04 Marks)
(06 Marks)
(D) -2 +
1 of 4
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OM OP
15 LIBRARY
49
1OMAT2 1
4
e
!".?ift afro)
d. Solve
d
Y (06 Marks)
x dz
?+y=ze 7
dx dx
3 a. Choose the correct answer : (04 Marks)
i) The Wronskian of x and x e
x
is
(A) ex
(B)
e2x
(C) e
-2x
(D) ex.
ii) The complementary function of x
2
y" - xy' ? 3y = x
2
log x is
(A) ci cos (log x) + c2 sin (log x) (B) c
1
x
-1
+ c2x
3
.
(C) c ix + c2x
3
(D) cicos x + c2 sin x.
iii) To transform (1+x)
2
y" + (1+ x)y' + y = 2 sin log (1 + x) into a line& differential
equation with constant coefficient
(A) (1+ x) = e
t
(B) (1+ x) = e
-t
(C) (1 + x)
2
? e
t
r
(D)'.(1-x)
2
= e
t
.
iv) The equation ao(ax + b)
2
y" + ao(ax + b) y' + a2y = 4(x) is
:',,
(A) Simultaneous equation (B) Cauchy's linearvpation
(C) Legendre linear equation (D) Euler's equag174#:'
b. Using the variation of parameters method to solve the equatio 4,7 1- 2y' + y = ex log x.
(04 Marks)
<
j
.,
d
2
v
c. Solve x
2
- ?(2m-1)x?
d
+
y
(
n
2 +
n2)
2) y ,
n
2
x
in log x,
(06 Marks)
d
2
clx
d. Obtain the Frobenius method solve the equation
,.,
C
1
(06 Marks)
(04 Marks)
i) Partial differential equation by elitninating a and b from the relation
Z = (x-a)
2
+ (y ?b)
2
is
?
(A)
p
2
q2
4z
(B) .04==
ii) The Lagranges's linear partial differential Pp + Qq = R the subsidiary equation
is
(A)
dx dy dz
(B)
dx dy dz
(C)
dx-
?:
dz
(D)
dx
+
dy dz
R P Q P Q R Q R P P Q R
iii) By the method of separation of variable we seek a solution in the form is
(A) x = x + (B) z x
2
+ y
2
(C) x z + (D) x ? x(x) y(y)
iv) The solution of
z
= sin (xy) is
ax
2
(A) z = -x
2
sin (xy) + y f(x) + 0(x)
(C) z
xy
y f(x) + (1)(x)
x
b. Form the partial differential equation of all sphere of radius 3 units having their centre in the
xy - plane. (04 Marks)
c. Solve x (y
2
+ z) p-y (x
2
+ z) q = z (x
2
-y
2
). (06 Marks)
d. Use the method of separation of variables to solve
aZ 2 aZ
y
3
x ?=u?
(06 Marks)
ax ay
PART - B
5 a. Choose the correct answer :
x2
i) The value of j r
e
7
xdydx is
0 0
(A) 0 (B) 1
(04 Marks)
(C) 3
2 of 4
(D) 'A .
(C) r ? 4z (D) t = 4
-
7
-
(B)
? sin(xy)
x f(Y)
(
licY)
Y
2
(D) None of these.
d2y dy?y=0? x?
dx2 + dx
4 a. Choose the correct answer :
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ii) The value off ( )is
-?"4.9., or Eracr
(A) 2 ?ri (B) n (C) (D) 427c .
iii) The integral
a
x
dxdy after changing the order of integration is
j
a
y
x2 +y2
(A)
j
a
fa
(B)
f
a x
-,1,dxdy
o x +y 0 0 X +y
(C) f
.
1
0
+y
4v) The value of 13(3, ) is
) 15
(B)
16
16 15
(D) f f
0 0
x +y
(D)
b. Change-
4a ax
order of integration in dydx and hence evaluate the same. (04 Marks)
X
4a
c. Evaluate r f
j
y
2
+ z
2
) dx dy dz.
j
e
b
-.
d. Prove that
r
i x
x
I
.
,
1
n
d
1/1 - x k 1 1+x? x4
x
- 4,5
6 a. Choose the correct answer
(06 Marks)
(06 Marks)
(04 Marks)
0
(
i) Let S be the closed boundary surface of a region of volume V then for a vector field f
defined in V and in S $ E
n
d
s
is
C
-qh. A\
(A) curl f dv (B) div f dv
(C) I grad f dv
(D) None of these
ii) If SE& where f = 3xy1
7
- j and C is the part of the parabola y # 2x
2
from the region
(0, 0) to the point 2) is
(A) 7/
6
(B) -% (C) 3x -E 3y (D) -35
iii) In the Green'slheorem in the plane fMdx
+ Ndy =
1i
(A). fir +?
l a
)dxdy (B) ff(
am
?Y
aN
N
)cixdy
ay
.]dxdy
:: (C) iliV
N
? ? '
It
)dxdy (D) 111
?N
+
am
) dxdy
1
iv) A necessary and sufficient condition that the line integral p..
d
-
r
. for any closed curve C
is
(A) divF = 0 (B) divF # 0 (C) curIF = 0 (D) grad F = 0
b. Using the divergence theorem, evaluate if .nds where f = 4xzi - y
2
3 + yzk and S is the
surface of the cube bounded by x - - - - - - - - - - - - - - - - - - - - - - - 0, x - 1, y - 0, y - 1, z 0, z - 1. (04 Marks)
c. Use the Green's theorem, evaluate
if (2x
2
? y
2
)dx +(x
2
+ y
2
)dy
where C is the triangle formed
by the lines x = 0, y = 0 and x + y = 1. (06 Marks)
d. Verify the Stoke's theorem for f = -y
3
1+ xl where S is the circle disc x
2
+ y
2
< 1, z = 0.
(06 Marks)
3 of 4
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This post was last modified on 01 January 2020