Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 2015 Jan 2014 Dec 10MAT21 Engineering Mathematics Question Paper

Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 2015 Jan 2014 Dec 10MAT21 Engineering Mathematics Question Paper

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USN
10MAT21
F
g
? 4-1
Second Semester B.E. Degree Examination, Dec.2014/Jan.2015
Engineering Mathematics -
Time: 3 hrs.
(.1
Cd
Note: 1. Answer any FIVE full questions, choosing at least two from each part.
CL.
2. Answer all objective type questions only on OMR sheet page 5 of the answer booklet.
3. Answer to objective type questions on sheets other than OMR will not be valued.
-0
PART ? A
1 a. ,Choose the correct answers for the following : (04 Marks)
1) A differential equation of the first order but of higher degree, solvable for x, has the
solution as:
A) F(y, p, c) = 0 B) F(x, p, c) C) F(x, y, c) = 0 D) CI, C2) = 0
ii) If xy 4
-
C = C
2
X is the general solution of a differential equation then its singular
solution is
A) y = x
iii) The general solution of Clairant's equation is,
A) y = Cf (x) + f(C) . B) y = Cx + f(C) C) x = Cf(y) + f(C) D) x = Cy +g(C)
iv) The general solution of p
2
?7p +12 = 0 is,
A) (y ?3x ? c)(y ?4x ?c) = 0
C) (3x c)(4x ? c)
b. Solve : xp
2
?(2x +3y)p +6y = 0.
C. Solve : y =3x +logp
d. Obtain the general solution and the singular solution of the equation
Clairant's equation.
Choose the correct answers for the following :
i) The particular integral of (D
2
+ a
2
)y = sin ax is,
A)
? xcosax
B) C)
xcosax x sin ax D) ? x sin ax
2a 2a 2a 2a
ii) The solution of the differential equation (D
4
? 5D
2
+4)y = 0 is,
A) y =C
i
e' +C
2
e
-
B) y= (C
I
+C,x+C
3
x
2
+C,x
3)e2x
C) y = C
I
cos x +C
2
sin x + C
3
cos 2x + C
4
sin 2x
D) None of these
iii) The particular integral of (D ? 1)
2
y --.3ex is,
A) --xex B) --
3
X
2
e
x
C)
?
3
2
x
2
ex D) ?x
-
ex
2 2 2 3
iv) The roots of auxiliary equation of D' (D
2
+ 2D)
2
y = 0 are:
Max. Marks:100
B) y=-x C) 4x
2
y+1=0 D) 4x
2
y-1=0
14
?)
B) (y ? c)(x ?c) = 0
D) (y +3x + c)(y ?4x ? c) = 0
(05 Marks)
(05 Marks)
xp
3
yp
2
+1 = 0 as
(06 Marks)
(04 Marks)
+C
3
e
2
x +C
4
e
-2
x
d.
A) 0,0,0,0,2,2 B) 0,0,0,0,-2,-2
b. Solve : (D
2
?2D +1)y = xex + x
c.
Solve : (D
2
? 4D + 4)y = e
2
x + cos 2x + 4 .
Solve : ?
dx
-7x + y = 0, ?
dy
-2x ?5y = O.
dt dt
C) 0,0,2,2,-2,-2 D) 2,2,2,2,0,0
(05 Marks)
(05 Marks)
(06 Marks)
1 of 4
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USN
10MAT21
F
g
? 4-1
Second Semester B.E. Degree Examination, Dec.2014/Jan.2015
Engineering Mathematics -
Time: 3 hrs.
(.1
Cd
Note: 1. Answer any FIVE full questions, choosing at least two from each part.
CL.
2. Answer all objective type questions only on OMR sheet page 5 of the answer booklet.
3. Answer to objective type questions on sheets other than OMR will not be valued.
-0
PART ? A
1 a. ,Choose the correct answers for the following : (04 Marks)
1) A differential equation of the first order but of higher degree, solvable for x, has the
solution as:
A) F(y, p, c) = 0 B) F(x, p, c) C) F(x, y, c) = 0 D) CI, C2) = 0
ii) If xy 4
-
C = C
2
X is the general solution of a differential equation then its singular
solution is
A) y = x
iii) The general solution of Clairant's equation is,
A) y = Cf (x) + f(C) . B) y = Cx + f(C) C) x = Cf(y) + f(C) D) x = Cy +g(C)
iv) The general solution of p
2
?7p +12 = 0 is,
A) (y ?3x ? c)(y ?4x ?c) = 0
C) (3x c)(4x ? c)
b. Solve : xp
2
?(2x +3y)p +6y = 0.
C. Solve : y =3x +logp
d. Obtain the general solution and the singular solution of the equation
Clairant's equation.
Choose the correct answers for the following :
i) The particular integral of (D
2
+ a
2
)y = sin ax is,
A)
? xcosax
B) C)
xcosax x sin ax D) ? x sin ax
2a 2a 2a 2a
ii) The solution of the differential equation (D
4
? 5D
2
+4)y = 0 is,
A) y =C
i
e' +C
2
e
-
B) y= (C
I
+C,x+C
3
x
2
+C,x
3)e2x
C) y = C
I
cos x +C
2
sin x + C
3
cos 2x + C
4
sin 2x
D) None of these
iii) The particular integral of (D ? 1)
2
y --.3ex is,
A) --xex B) --
3
X
2
e
x
C)
?
3
2
x
2
ex D) ?x
-
ex
2 2 2 3
iv) The roots of auxiliary equation of D' (D
2
+ 2D)
2
y = 0 are:
Max. Marks:100
B) y=-x C) 4x
2
y+1=0 D) 4x
2
y-1=0
14
?)
B) (y ? c)(x ?c) = 0
D) (y +3x + c)(y ?4x ? c) = 0
(05 Marks)
(05 Marks)
xp
3
yp
2
+1 = 0 as
(06 Marks)
(04 Marks)
+C
3
e
2
x +C
4
e
-2
x
d.
A) 0,0,0,0,2,2 B) 0,0,0,0,-2,-2
b. Solve : (D
2
?2D +1)y = xex + x
c.
Solve : (D
2
? 4D + 4)y = e
2
x + cos 2x + 4 .
Solve : ?
dx
-7x + y = 0, ?
dy
-2x ?5y = O.
dt dt
C) 0,0,2,2,-2,-2 D) 2,2,2,2,0,0
(05 Marks)
(05 Marks)
(06 Marks)
1 of 4
b.
c.
Solve (3x + 2)
2
y" +3(3x + 2)y
l
?36y = 8x
2
+ 4x + 1
Solve : (D
2
+a
2
)y = sec ax by method of variation of parameters.
d.
Solve : y" + xy' + y = 0 in series solution.
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1OMAT21
3 a. Choose the correct answers for the following : (04 Marks)
i) The complementary function of x
2
y" ? 3xy' + 4y = x is:
A) (C
I
+ C
2
log x)x
2

C) C
I
+C
2
x
B) (C
1
+ C
2
x)e
2
x
C
D) 1+ C4
x x
g
2x +1)
B) (D
2
? 4D + 4)y = 2e
2
z
D) None of.these
2x
2
y" ? xy' 4- (1? X
2
)y = 0, we assume the
r-1
a. Choose the correct answers for the following : (04 Marks)
i) 2z = ?+ y ? , 31
X
2
a
2
b
2
'
2
d' e arbitrary constants is a solution of:
-
40e '
A) 2z=p
2
x+qy .- B) 2z =px+q
2
y C) 2z = px + qy D) None of these
Trrauxiliary equations of Lagrange's linear equation Pp + Qq = R are:
ii) By the method of variation parameters, the value of 'W' is called,
A) Euler's function B) Wronskian of the function
C) Demorgan's function D) Cauchy's function
iii) The equation (2x +1)
2
y" ?6(2x +1)y' +16y = 8(2x +1)
2
by putting
with D = reduces to
dz
A) (D
2
+ 4D + 4)y = 3e
2
z
C) (D
2
?4D +4)y = 0
To find the series solution for the equation
solution as,
iv)
A) y= Ea
r
x S) y=Ea
r+i
xr
+1
r
_
r=1:1 r=0
=Ea
r
x
K
"
(05 Marks)
(05 Marks)
(06 Marks)
A
dx dy dz
B
dx dy dz
t-k) ? = ? ? .D) ? = ? = ? )
P Q R p q R
iii) General solution of the equation
a2z
?
ax2
= x + y is,
dx dy dz
x y z
D) ?
dx
+?
dy
+ ?
dz
= 0
x y z
A)
?
x3
+
x
+ f(y)+ g(y)
6 2
B)
?
x3
-
x 2y
+ f(y) + yg(y)
6 2
x3
-
x2y
+ xf(y)+ g(y) D)
x3
+?
x2
C) -+
y
+ xf(y) + yg(y)
6 2 6 2
iv) Suitable set of multipliers to solve x
2
(y ? z)p+ y
2
(z ? x)q = z
2
(X - y) is,
A) (0, 1, 1) B) (x, y, z) C) (0,1,1) D) 1,1,1)
x y x y z
b.
Form a partial differential equation by eliminating arbitrary function from the relation,
(1)(xy+z
2
,x+y+z)=0 (05 Marks)
c. Solve : (y
2
+z
2
)p+ x(yq ? z)= 0 (05 Marks)
d.
Solve by the method of separation of variables ?
82z
? 2?
az
+?
az
= 0 (06 Marks)
ax
2
ax ay
2 of 4
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USN
10MAT21
F
g
? 4-1
Second Semester B.E. Degree Examination, Dec.2014/Jan.2015
Engineering Mathematics -
Time: 3 hrs.
(.1
Cd
Note: 1. Answer any FIVE full questions, choosing at least two from each part.
CL.
2. Answer all objective type questions only on OMR sheet page 5 of the answer booklet.
3. Answer to objective type questions on sheets other than OMR will not be valued.
-0
PART ? A
1 a. ,Choose the correct answers for the following : (04 Marks)
1) A differential equation of the first order but of higher degree, solvable for x, has the
solution as:
A) F(y, p, c) = 0 B) F(x, p, c) C) F(x, y, c) = 0 D) CI, C2) = 0
ii) If xy 4
-
C = C
2
X is the general solution of a differential equation then its singular
solution is
A) y = x
iii) The general solution of Clairant's equation is,
A) y = Cf (x) + f(C) . B) y = Cx + f(C) C) x = Cf(y) + f(C) D) x = Cy +g(C)
iv) The general solution of p
2
?7p +12 = 0 is,
A) (y ?3x ? c)(y ?4x ?c) = 0
C) (3x c)(4x ? c)
b. Solve : xp
2
?(2x +3y)p +6y = 0.
C. Solve : y =3x +logp
d. Obtain the general solution and the singular solution of the equation
Clairant's equation.
Choose the correct answers for the following :
i) The particular integral of (D
2
+ a
2
)y = sin ax is,
A)
? xcosax
B) C)
xcosax x sin ax D) ? x sin ax
2a 2a 2a 2a
ii) The solution of the differential equation (D
4
? 5D
2
+4)y = 0 is,
A) y =C
i
e' +C
2
e
-
B) y= (C
I
+C,x+C
3
x
2
+C,x
3)e2x
C) y = C
I
cos x +C
2
sin x + C
3
cos 2x + C
4
sin 2x
D) None of these
iii) The particular integral of (D ? 1)
2
y --.3ex is,
A) --xex B) --
3
X
2
e
x
C)
?
3
2
x
2
ex D) ?x
-
ex
2 2 2 3
iv) The roots of auxiliary equation of D' (D
2
+ 2D)
2
y = 0 are:
Max. Marks:100
B) y=-x C) 4x
2
y+1=0 D) 4x
2
y-1=0
14
?)
B) (y ? c)(x ?c) = 0
D) (y +3x + c)(y ?4x ? c) = 0
(05 Marks)
(05 Marks)
xp
3
yp
2
+1 = 0 as
(06 Marks)
(04 Marks)
+C
3
e
2
x +C
4
e
-2
x
d.
A) 0,0,0,0,2,2 B) 0,0,0,0,-2,-2
b. Solve : (D
2
?2D +1)y = xex + x
c.
Solve : (D
2
? 4D + 4)y = e
2
x + cos 2x + 4 .
Solve : ?
dx
-7x + y = 0, ?
dy
-2x ?5y = O.
dt dt
C) 0,0,2,2,-2,-2 D) 2,2,2,2,0,0
(05 Marks)
(05 Marks)
(06 Marks)
1 of 4
b.
c.
Solve (3x + 2)
2
y" +3(3x + 2)y
l
?36y = 8x
2
+ 4x + 1
Solve : (D
2
+a
2
)y = sec ax by method of variation of parameters.
d.
Solve : y" + xy' + y = 0 in series solution.
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1OMAT21
3 a. Choose the correct answers for the following : (04 Marks)
i) The complementary function of x
2
y" ? 3xy' + 4y = x is:
A) (C
I
+ C
2
log x)x
2

C) C
I
+C
2
x
B) (C
1
+ C
2
x)e
2
x
C
D) 1+ C4
x x
g
2x +1)
B) (D
2
? 4D + 4)y = 2e
2
z
D) None of.these
2x
2
y" ? xy' 4- (1? X
2
)y = 0, we assume the
r-1
a. Choose the correct answers for the following : (04 Marks)
i) 2z = ?+ y ? , 31
X
2
a
2
b
2
'
2
d' e arbitrary constants is a solution of:
-
40e '
A) 2z=p
2
x+qy .- B) 2z =px+q
2
y C) 2z = px + qy D) None of these
Trrauxiliary equations of Lagrange's linear equation Pp + Qq = R are:
ii) By the method of variation parameters, the value of 'W' is called,
A) Euler's function B) Wronskian of the function
C) Demorgan's function D) Cauchy's function
iii) The equation (2x +1)
2
y" ?6(2x +1)y' +16y = 8(2x +1)
2
by putting
with D = reduces to
dz
A) (D
2
+ 4D + 4)y = 3e
2
z
C) (D
2
?4D +4)y = 0
To find the series solution for the equation
solution as,
iv)
A) y= Ea
r
x S) y=Ea
r+i
xr
+1
r
_
r=1:1 r=0
=Ea
r
x
K
"
(05 Marks)
(05 Marks)
(06 Marks)
A
dx dy dz
B
dx dy dz
t-k) ? = ? ? .D) ? = ? = ? )
P Q R p q R
iii) General solution of the equation
a2z
?
ax2
= x + y is,
dx dy dz
x y z
D) ?
dx
+?
dy
+ ?
dz
= 0
x y z
A)
?
x3
+
x
+ f(y)+ g(y)
6 2
B)
?
x3
-
x 2y
+ f(y) + yg(y)
6 2
x3
-
x2y
+ xf(y)+ g(y) D)
x3
+?
x2
C) -+
y
+ xf(y) + yg(y)
6 2 6 2
iv) Suitable set of multipliers to solve x
2
(y ? z)p+ y
2
(z ? x)q = z
2
(X - y) is,
A) (0, 1, 1) B) (x, y, z) C) (0,1,1) D) 1,1,1)
x y x y z
b.
Form a partial differential equation by eliminating arbitrary function from the relation,
(1)(xy+z
2
,x+y+z)=0 (05 Marks)
c. Solve : (y
2
+z
2
)p+ x(yq ? z)= 0 (05 Marks)
d.
Solve by the method of separation of variables ?
82z
? 2?
az
+?
az
= 0 (06 Marks)
ax
2
ax ay
2 of 4
B) r(--
1
) C) r
2
r)
2
is:
D) r(? ?
3
)
2
A) If ?
1
)
2
iv) The value of p(5, 3) + p(3, 5)
2
Show that 1
de
x sin Ode = TE
0
d.
sin 0
0

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10MAT21
PART? B
a. Choose the correct answers for the following :
2x
i) f f(X y)dydx =
as
A) 3 B) 4 C) 5
2 3 2
ii) if f Xy
2
zdzdydx =
oi l
A) 36 B) 16 C) 26
iii) The integral 21
2
dx is,
(04 Marks)
D) 6
D) 46
A)
2

?
35
B) -
3
-
5
4
C) 3
35
r
.?
4 Li)
35
b. Evaluate fi
.

z

xy(x + y)dydx taken over the area between
I x+z
y = 'x and y = x. (M5 harks)
C. Evaluate : 1 1 1(x + y + z)dydxdz
-1 0 x-z
. 0 (05 Marks)
i ?
(06 Marks)
(04 Marks)
volume integral
6 a. Choose the correct answers for the following :
i) Gauss Divergence theorem is a relation between:
A) a line integral and a surface integral B) a surface integral and a
C) a line integral and a volume integral D) two volume integrals
y
ill.
1 ii) f Mdx + Ndy is also equal to,
c
R.
er(aN am'
zIxd c) jj y
ax ay
0
0
A) 1
.
1
-(am _aN)
cixdy

ay ax
B) f f
a
M
+
dxdy
R aY ax
aN
am
if( + dxdy
R OX ay
iii) Using the following integral, work done by a force F can be calculated:
A) Surface integral B) Volume integral C) Both (A) and (B) D) Line integral
iv) If F = x
2
i + xyj then the value of SF. dr from (0, 0) to (1, 1) along the line y ? x is,
A) 2 / 3 B) 3 / 2 C) 1 / 3 D) 112
b. Find the area between the parabolae, y
2
= 4x and x
2
= Lly with the help of Green's theorem
in a plane. (05 Marks)
c. Evaluate f xydx + xy
2
dy by Stoke's theorem where C is the square in the x-y plane with
vertices (1, 0), (-1, 0), (0, 1) and (0, -1) (05 Marks)
d. Evaluate HF.fids given F = xi + yj + zk over the sphere x
2
+ y
2
+ z
Z
= a
2
by using Gauss
divergence theorem. (06 Marks)
3 of 4
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USN
10MAT21
F
g
? 4-1
Second Semester B.E. Degree Examination, Dec.2014/Jan.2015
Engineering Mathematics -
Time: 3 hrs.
(.1
Cd
Note: 1. Answer any FIVE full questions, choosing at least two from each part.
CL.
2. Answer all objective type questions only on OMR sheet page 5 of the answer booklet.
3. Answer to objective type questions on sheets other than OMR will not be valued.
-0
PART ? A
1 a. ,Choose the correct answers for the following : (04 Marks)
1) A differential equation of the first order but of higher degree, solvable for x, has the
solution as:
A) F(y, p, c) = 0 B) F(x, p, c) C) F(x, y, c) = 0 D) CI, C2) = 0
ii) If xy 4
-
C = C
2
X is the general solution of a differential equation then its singular
solution is
A) y = x
iii) The general solution of Clairant's equation is,
A) y = Cf (x) + f(C) . B) y = Cx + f(C) C) x = Cf(y) + f(C) D) x = Cy +g(C)
iv) The general solution of p
2
?7p +12 = 0 is,
A) (y ?3x ? c)(y ?4x ?c) = 0
C) (3x c)(4x ? c)
b. Solve : xp
2
?(2x +3y)p +6y = 0.
C. Solve : y =3x +logp
d. Obtain the general solution and the singular solution of the equation
Clairant's equation.
Choose the correct answers for the following :
i) The particular integral of (D
2
+ a
2
)y = sin ax is,
A)
? xcosax
B) C)
xcosax x sin ax D) ? x sin ax
2a 2a 2a 2a
ii) The solution of the differential equation (D
4
? 5D
2
+4)y = 0 is,
A) y =C
i
e' +C
2
e
-
B) y= (C
I
+C,x+C
3
x
2
+C,x
3)e2x
C) y = C
I
cos x +C
2
sin x + C
3
cos 2x + C
4
sin 2x
D) None of these
iii) The particular integral of (D ? 1)
2
y --.3ex is,
A) --xex B) --
3
X
2
e
x
C)
?
3
2
x
2
ex D) ?x
-
ex
2 2 2 3
iv) The roots of auxiliary equation of D' (D
2
+ 2D)
2
y = 0 are:
Max. Marks:100
B) y=-x C) 4x
2
y+1=0 D) 4x
2
y-1=0
14
?)
B) (y ? c)(x ?c) = 0
D) (y +3x + c)(y ?4x ? c) = 0
(05 Marks)
(05 Marks)
xp
3
yp
2
+1 = 0 as
(06 Marks)
(04 Marks)
+C
3
e
2
x +C
4
e
-2
x
d.
A) 0,0,0,0,2,2 B) 0,0,0,0,-2,-2
b. Solve : (D
2
?2D +1)y = xex + x
c.
Solve : (D
2
? 4D + 4)y = e
2
x + cos 2x + 4 .
Solve : ?
dx
-7x + y = 0, ?
dy
-2x ?5y = O.
dt dt
C) 0,0,2,2,-2,-2 D) 2,2,2,2,0,0
(05 Marks)
(05 Marks)
(06 Marks)
1 of 4
b.
c.
Solve (3x + 2)
2
y" +3(3x + 2)y
l
?36y = 8x
2
+ 4x + 1
Solve : (D
2
+a
2
)y = sec ax by method of variation of parameters.
d.
Solve : y" + xy' + y = 0 in series solution.
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1OMAT21
3 a. Choose the correct answers for the following : (04 Marks)
i) The complementary function of x
2
y" ? 3xy' + 4y = x is:
A) (C
I
+ C
2
log x)x
2

C) C
I
+C
2
x
B) (C
1
+ C
2
x)e
2
x
C
D) 1+ C4
x x
g
2x +1)
B) (D
2
? 4D + 4)y = 2e
2
z
D) None of.these
2x
2
y" ? xy' 4- (1? X
2
)y = 0, we assume the
r-1
a. Choose the correct answers for the following : (04 Marks)
i) 2z = ?+ y ? , 31
X
2
a
2
b
2
'
2
d' e arbitrary constants is a solution of:
-
40e '
A) 2z=p
2
x+qy .- B) 2z =px+q
2
y C) 2z = px + qy D) None of these
Trrauxiliary equations of Lagrange's linear equation Pp + Qq = R are:
ii) By the method of variation parameters, the value of 'W' is called,
A) Euler's function B) Wronskian of the function
C) Demorgan's function D) Cauchy's function
iii) The equation (2x +1)
2
y" ?6(2x +1)y' +16y = 8(2x +1)
2
by putting
with D = reduces to
dz
A) (D
2
+ 4D + 4)y = 3e
2
z
C) (D
2
?4D +4)y = 0
To find the series solution for the equation
solution as,
iv)
A) y= Ea
r
x S) y=Ea
r+i
xr
+1
r
_
r=1:1 r=0
=Ea
r
x
K
"
(05 Marks)
(05 Marks)
(06 Marks)
A
dx dy dz
B
dx dy dz
t-k) ? = ? ? .D) ? = ? = ? )
P Q R p q R
iii) General solution of the equation
a2z
?
ax2
= x + y is,
dx dy dz
x y z
D) ?
dx
+?
dy
+ ?
dz
= 0
x y z
A)
?
x3
+
x
+ f(y)+ g(y)
6 2
B)
?
x3
-
x 2y
+ f(y) + yg(y)
6 2
x3
-
x2y
+ xf(y)+ g(y) D)
x3
+?
x2
C) -+
y
+ xf(y) + yg(y)
6 2 6 2
iv) Suitable set of multipliers to solve x
2
(y ? z)p+ y
2
(z ? x)q = z
2
(X - y) is,
A) (0, 1, 1) B) (x, y, z) C) (0,1,1) D) 1,1,1)
x y x y z
b.
Form a partial differential equation by eliminating arbitrary function from the relation,
(1)(xy+z
2
,x+y+z)=0 (05 Marks)
c. Solve : (y
2
+z
2
)p+ x(yq ? z)= 0 (05 Marks)
d.
Solve by the method of separation of variables ?
82z
? 2?
az
+?
az
= 0 (06 Marks)
ax
2
ax ay
2 of 4
B) r(--
1
) C) r
2
r)
2
is:
D) r(? ?
3
)
2
A) If ?
1
)
2
iv) The value of p(5, 3) + p(3, 5)
2
Show that 1
de
x sin Ode = TE
0
d.
sin 0
0

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10MAT21
PART? B
a. Choose the correct answers for the following :
2x
i) f f(X y)dydx =
as
A) 3 B) 4 C) 5
2 3 2
ii) if f Xy
2
zdzdydx =
oi l
A) 36 B) 16 C) 26
iii) The integral 21
2
dx is,
(04 Marks)
D) 6
D) 46
A)
2

?
35
B) -
3
-
5
4
C) 3
35
r
.?
4 Li)
35
b. Evaluate fi
.

z

xy(x + y)dydx taken over the area between
I x+z
y = 'x and y = x. (M5 harks)
C. Evaluate : 1 1 1(x + y + z)dydxdz
-1 0 x-z
. 0 (05 Marks)
i ?
(06 Marks)
(04 Marks)
volume integral
6 a. Choose the correct answers for the following :
i) Gauss Divergence theorem is a relation between:
A) a line integral and a surface integral B) a surface integral and a
C) a line integral and a volume integral D) two volume integrals
y
ill.
1 ii) f Mdx + Ndy is also equal to,
c
R.
er(aN am'
zIxd c) jj y
ax ay
0
0
A) 1
.
1
-(am _aN)
cixdy

ay ax
B) f f
a
M
+
dxdy
R aY ax
aN
am
if( + dxdy
R OX ay
iii) Using the following integral, work done by a force F can be calculated:
A) Surface integral B) Volume integral C) Both (A) and (B) D) Line integral
iv) If F = x
2
i + xyj then the value of SF. dr from (0, 0) to (1, 1) along the line y ? x is,
A) 2 / 3 B) 3 / 2 C) 1 / 3 D) 112
b. Find the area between the parabolae, y
2
= 4x and x
2
= Lly with the help of Green's theorem
in a plane. (05 Marks)
c. Evaluate f xydx + xy
2
dy by Stoke's theorem where C is the square in the x-y plane with
vertices (1, 0), (-1, 0), (0, 1) and (0, -1) (05 Marks)
d. Evaluate HF.fids given F = xi + yj + zk over the sphere x
2
+ y
2
+ z
Z
= a
2
by using Gauss
divergence theorem. (06 Marks)
3 of 4
c.
Find +
-li
ft sin 3t
0
D) _
1
s
2 s
2
+ 4
7 )
,
Potd
,

e
-4s

s - 3
D) e'
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10MAT21
a. Choose the correct answers for the following : (04 Marks)
i) If L{f(t)} = F(s) then L{
f
M is,
A)
IF(ods
B) 1F(s)ds
C) IF(s)ds D) 1F(s)ds
0
ii)
If L
tcos at - cos bt}
__ leg
1 (s
2
+13
2

- ?) then L{
sin 2 t
1 .
t 2 s' +a
2
t
A) Ilog(
s2
+ 4)
B)
1 log
s2 +
1
C) !
s2
log(
s

4 s
2
2 s
2
4 + 4
iii) L{e
3I
H(t -4)1=
e
12-4s
e
12-4s
e
12+4s
A} B) C)
s + 3 s -3 s +3
iv) 4'6(t- a)}=
A) (-a)
t,
e' B) aneas C) ane
-
as
b. Find the Laplace transform of t
5
e
4t
cosh3t. (05 Marks)
(05 Marks)
(06 Marks)
(04 Marks)
E, 0 < t <
loomk. li Oak
d. Given f(t) =
2
where gt+a) - go
E
, show that Ltf(t)i= ?tanh ?).
, as
? E, ?
a
?t *
* - s 4
2
8 a. Choose the correct answer for th ?flowing :
?
et -
1
+1
2 6
t2 1+t Dh2e-.(14
0
)
- 6) 2 6
i)
tl
e
' (-
1
B)
2 6
ii} 1:4
s
+ ?
(s - 4
A)
1
'
t

iii) L
-1
{
2
s

s +5
A) sin it
e
4 t ?
e
-3t
m e
-4 t _
e
-3t
B) 1 ?e4t
C)
t t t
B)
7
1
cos j t C)
1
cos5t
5
D) cos ,5 t
l[s
2
+
S
a
2
y }
iv) L
-I
,
N
, =
A) t sin at B) t cos at / 2a C) t sin at / 2a D) tcosat
{ b. Find L
-I s 2
-
+4
s(s + 4)(s ? 4)
1(s ? 1)
1
0
2
+ 0}
c. Find L'
by using convolution theorem.
d.
Solve by using Laplace transform y"(t)+ y(t) = 0 ; y(0) = 2, y(Tr/ 2)=1
* * * * *
(05 Marks)
(05 Marks)
(06 Marks)
4 of 4
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