Download JNTU Kakinada B.Tech 1-1 2014 Feb R10 MATHEMATICS I Question Paper

Download JNTUK (Jawaharlal Nehru Technological University Kakinada) B.Tech Supplementary 2014 Feb-March R10 I Semester (1st Year 1st Sem) MATHEMATICS I Question Paper.

Code No: R10102/R10 Set No. 1
I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014
MATHEMATICS-I
( Common to Civil Engineering, Electrical & Electronics Engineering,
Mechanical Engineering, Electronics & Communication Engineering,
Computer Science & Engineering, Chemical Engineering, Electronics &
Instrumentation Engineering, Bio-Medical Engineering, Information
Technology, Electronics & Computer Engineering, Aeronautical
Engineering, Bio-Technology, Automobile Engineering, Mining and
Petroliem Technology)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
?????
1. (a) Solve (x
2
+y
2
a
2
)xdx + (x
2
y
2
b
2
)ydy = 0. [7+8]
(b) If air is maintained at 20
0
Cand the temperature of the body cools from 80
0
C
to 60
0
C in 10 minutes, nd the temperature of the body after 30 minutes.
2. (a) Solve (D
2
+a
2
)y =Secax
(b) Solve (D
2
+ 4)y =e
x
+Sin 2x [8+7]
3. (a) If V = log (x
2
+y
2
) +x 2y nd
@V
@x
:,
@V
@y
;
@
2
V
@x
2
:
@
2
V
@y
2
:
(b) If U = xe
xy
where x
2
+y
2
+ 2xy = 1; nd
@
2
U
@x
2
: [8+7]
4. (a) Trace the curve r = 2 + 3 sin.
(b) Trace the curve y
2
(2ax) =x
3
. [8+7]
5. (a) Find the surface of the solid generated by revolution of the lemniscate r
2
=
a
2
cos
2
 about the initial line.
(b) Show that the whole length of the curve x
2
(a
2
x
2
) = 8a
2
y
2
isa
p
2. [8+7]
6. (a) Show that
R
4a
0
R
y
y
2
4a
x
2
y
2
x
2
+y
2
dx dy = 8a
2


2

5
3


.
(b) Evaluate
RR
R
ydxdy where R is the domain bounded by y-axis, the curve
y=x
2
and the line x +y = 2 in the rst quadrants . [8+7]
7. (a) If V= e
xyz
(i+j+k), nd curl V.
(b) Find the constants a and b so that the surface ax
2
-byz = (a+2)x will be
orthogonal to the surface 4x
2
y +z
3
=4 at the point (1,-1,2) [8+7]
8. (a) Show that the area of the ellipse x
2
/a
2
+ y
2
/b
2
= 1 is ab
(b) If f = (2x
2
{ 3z)i { 2xyj { 4xzk, evaluate
(i)
R
v
r:fdV and
(ii)
R
v
rfdV where V is the closed region bounded by x = 0, y = 0, z = 0,
2x + 2y +z = 4. [8+7]
?????
1 of 1

FirstRanker.com - FirstRanker's Choice
Code No: R10102/R10 Set No. 1
I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014
MATHEMATICS-I
( Common to Civil Engineering, Electrical & Electronics Engineering,
Mechanical Engineering, Electronics & Communication Engineering,
Computer Science & Engineering, Chemical Engineering, Electronics &
Instrumentation Engineering, Bio-Medical Engineering, Information
Technology, Electronics & Computer Engineering, Aeronautical
Engineering, Bio-Technology, Automobile Engineering, Mining and
Petroliem Technology)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
?????
1. (a) Solve (x
2
+y
2
a
2
)xdx + (x
2
y
2
b
2
)ydy = 0. [7+8]
(b) If air is maintained at 20
0
Cand the temperature of the body cools from 80
0
C
to 60
0
C in 10 minutes, nd the temperature of the body after 30 minutes.
2. (a) Solve (D
2
+a
2
)y =Secax
(b) Solve (D
2
+ 4)y =e
x
+Sin 2x [8+7]
3. (a) If V = log (x
2
+y
2
) +x 2y nd
@V
@x
:,
@V
@y
;
@
2
V
@x
2
:
@
2
V
@y
2
:
(b) If U = xe
xy
where x
2
+y
2
+ 2xy = 1; nd
@
2
U
@x
2
: [8+7]
4. (a) Trace the curve r = 2 + 3 sin.
(b) Trace the curve y
2
(2ax) =x
3
. [8+7]
5. (a) Find the surface of the solid generated by revolution of the lemniscate r
2
=
a
2
cos
2
 about the initial line.
(b) Show that the whole length of the curve x
2
(a
2
x
2
) = 8a
2
y
2
isa
p
2. [8+7]
6. (a) Show that
R
4a
0
R
y
y
2
4a
x
2
y
2
x
2
+y
2
dx dy = 8a
2


2

5
3


.
(b) Evaluate
RR
R
ydxdy where R is the domain bounded by y-axis, the curve
y=x
2
and the line x +y = 2 in the rst quadrants . [8+7]
7. (a) If V= e
xyz
(i+j+k), nd curl V.
(b) Find the constants a and b so that the surface ax
2
-byz = (a+2)x will be
orthogonal to the surface 4x
2
y +z
3
=4 at the point (1,-1,2) [8+7]
8. (a) Show that the area of the ellipse x
2
/a
2
+ y
2
/b
2
= 1 is ab
(b) If f = (2x
2
{ 3z)i { 2xyj { 4xzk, evaluate
(i)
R
v
r:fdV and
(ii)
R
v
rfdV where V is the closed region bounded by x = 0, y = 0, z = 0,
2x + 2y +z = 4. [8+7]
?????
1 of 1
Code No: R10102/R10 Set No. 2
I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014
MATHEMATICS-I
( Common to Civil Engineering, Electrical & Electronics Engineering,
Mechanical Engineering, Electronics & Communication Engineering,
Computer Science & Engineering, Chemical Engineering, Electronics &
Instrumentation Engineering, Bio-Medical Engineering, Information
Technology, Electronics & Computer Engineering, Aeronautical
Engineering, Bio-Technology, Automobile Engineering, Mining and
Petroliem Technology)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
?????
1. (a) Solve e
y

1+
dy
dx


= e
x
(b) Show that the family of curves
x
2
a
2
+?
+
y
2
a
2
+?
= 1, where ??? is a parameter is self
orthogonal. [8+7]
2. (a) Solve (D
2
+9)y = 2Cos
2
x. (b) Solve
d
2
y
dx
2
+4y = 2e
x
Sin
2
x. [8+7]
3. (a) Calculate the approximate value of
?
10 to four decimal places using Taylor?s
theorem.
(b) Find 3 positive numbers whose sum is 600 and whose product is maximum.
[8+7]
4. (a) Trace the curve y = x
2
(x
2
?4). (b)Trace the curve r = cos?. [8+7]
5. (a) The ?gure bounded by a parabola and the tangents at the extremities of its
latusrectum revolves about the axis of the parabola, Find the volume of the
solid thus generated. [8+7]
(b) The segment of the parabola y
2
=4ax which is cuto? by the latus rectum
revolves about the directrix.Find the volume of rotation of the annular region.
6. (a) Evaluate
R R
(x+y)
2
dx dy. over the area bounded by the ellipse
x
2
a
2
+
y
2
b
2
= 1.
(b) TransformthefollowingtoCartesianformandhenceevaluate
R
?
0
R
a
0
r
3
sin?drd?.
[8+7]
7. (a) Prove that?r = r/r
(b) Find the angle between the surfaces x
2
+ y
2
+ z
2
= 9 and z=x
2
+ y
2
?3 at the
point (2,-1,2). [8+7]
8. (a) Evaluate
RR
S
(yzi+zxj+xyk).dS whereSisthesurfaceofthespherex
2
+y
2
+z
2
=a
2
in the ?rst octant.
(b) Evaluate
H
c
(x
2
? 2xy)dx + (x
2
y + 3)dy around the boundary of the region
de?ned by y
2
=8x and x=2. [8+7]
?????
1 of 1

FirstRanker.com - FirstRanker's Choice
Code No: R10102/R10 Set No. 1
I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014
MATHEMATICS-I
( Common to Civil Engineering, Electrical & Electronics Engineering,
Mechanical Engineering, Electronics & Communication Engineering,
Computer Science & Engineering, Chemical Engineering, Electronics &
Instrumentation Engineering, Bio-Medical Engineering, Information
Technology, Electronics & Computer Engineering, Aeronautical
Engineering, Bio-Technology, Automobile Engineering, Mining and
Petroliem Technology)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
?????
1. (a) Solve (x
2
+y
2
a
2
)xdx + (x
2
y
2
b
2
)ydy = 0. [7+8]
(b) If air is maintained at 20
0
Cand the temperature of the body cools from 80
0
C
to 60
0
C in 10 minutes, nd the temperature of the body after 30 minutes.
2. (a) Solve (D
2
+a
2
)y =Secax
(b) Solve (D
2
+ 4)y =e
x
+Sin 2x [8+7]
3. (a) If V = log (x
2
+y
2
) +x 2y nd
@V
@x
:,
@V
@y
;
@
2
V
@x
2
:
@
2
V
@y
2
:
(b) If U = xe
xy
where x
2
+y
2
+ 2xy = 1; nd
@
2
U
@x
2
: [8+7]
4. (a) Trace the curve r = 2 + 3 sin.
(b) Trace the curve y
2
(2ax) =x
3
. [8+7]
5. (a) Find the surface of the solid generated by revolution of the lemniscate r
2
=
a
2
cos
2
 about the initial line.
(b) Show that the whole length of the curve x
2
(a
2
x
2
) = 8a
2
y
2
isa
p
2. [8+7]
6. (a) Show that
R
4a
0
R
y
y
2
4a
x
2
y
2
x
2
+y
2
dx dy = 8a
2


2

5
3


.
(b) Evaluate
RR
R
ydxdy where R is the domain bounded by y-axis, the curve
y=x
2
and the line x +y = 2 in the rst quadrants . [8+7]
7. (a) If V= e
xyz
(i+j+k), nd curl V.
(b) Find the constants a and b so that the surface ax
2
-byz = (a+2)x will be
orthogonal to the surface 4x
2
y +z
3
=4 at the point (1,-1,2) [8+7]
8. (a) Show that the area of the ellipse x
2
/a
2
+ y
2
/b
2
= 1 is ab
(b) If f = (2x
2
{ 3z)i { 2xyj { 4xzk, evaluate
(i)
R
v
r:fdV and
(ii)
R
v
rfdV where V is the closed region bounded by x = 0, y = 0, z = 0,
2x + 2y +z = 4. [8+7]
?????
1 of 1
Code No: R10102/R10 Set No. 2
I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014
MATHEMATICS-I
( Common to Civil Engineering, Electrical & Electronics Engineering,
Mechanical Engineering, Electronics & Communication Engineering,
Computer Science & Engineering, Chemical Engineering, Electronics &
Instrumentation Engineering, Bio-Medical Engineering, Information
Technology, Electronics & Computer Engineering, Aeronautical
Engineering, Bio-Technology, Automobile Engineering, Mining and
Petroliem Technology)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
?????
1. (a) Solve e
y

1+
dy
dx


= e
x
(b) Show that the family of curves
x
2
a
2
+?
+
y
2
a
2
+?
= 1, where ??? is a parameter is self
orthogonal. [8+7]
2. (a) Solve (D
2
+9)y = 2Cos
2
x. (b) Solve
d
2
y
dx
2
+4y = 2e
x
Sin
2
x. [8+7]
3. (a) Calculate the approximate value of
?
10 to four decimal places using Taylor?s
theorem.
(b) Find 3 positive numbers whose sum is 600 and whose product is maximum.
[8+7]
4. (a) Trace the curve y = x
2
(x
2
?4). (b)Trace the curve r = cos?. [8+7]
5. (a) The ?gure bounded by a parabola and the tangents at the extremities of its
latusrectum revolves about the axis of the parabola, Find the volume of the
solid thus generated. [8+7]
(b) The segment of the parabola y
2
=4ax which is cuto? by the latus rectum
revolves about the directrix.Find the volume of rotation of the annular region.
6. (a) Evaluate
R R
(x+y)
2
dx dy. over the area bounded by the ellipse
x
2
a
2
+
y
2
b
2
= 1.
(b) TransformthefollowingtoCartesianformandhenceevaluate
R
?
0
R
a
0
r
3
sin?drd?.
[8+7]
7. (a) Prove that?r = r/r
(b) Find the angle between the surfaces x
2
+ y
2
+ z
2
= 9 and z=x
2
+ y
2
?3 at the
point (2,-1,2). [8+7]
8. (a) Evaluate
RR
S
(yzi+zxj+xyk).dS whereSisthesurfaceofthespherex
2
+y
2
+z
2
=a
2
in the ?rst octant.
(b) Evaluate
H
c
(x
2
? 2xy)dx + (x
2
y + 3)dy around the boundary of the region
de?ned by y
2
=8x and x=2. [8+7]
?????
1 of 1
Code No: R10102/R10 Set No. 3
I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014
MATHEMATICS-I
( Common to Civil Engineering, Electrical & Electronics Engineering,
Mechanical Engineering, Electronics & Communication Engineering,
Computer Science & Engineering, Chemical Engineering, Electronics &
Instrumentation Engineering, Bio-Medical Engineering, Information
Technology, Electronics & Computer Engineering, Aeronautical
Engineering, Bio-Technology, Automobile Engineering, Mining and
Petroliem Technology)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
?????
1. (a) Solve y(Sinx?y)dx = Cosx dy
(b) If the temperature of air is maintained at 20
0
Cand the temperature of the
body cools from 100
0
Cto80
0
C in 10 minutes, ?nd the temperature of the
body after 20 minutes. [8+7]
2. (a) Solve (D
2
?4D+13)y = e
2x
(b) Solve (D
2
?3D+2)y = Coshx [8+7]
3. (a) If r + s + t = x, s + t = xy, t = xyz, ?nd
?(r,s,t)
?(x,y,z)
.
(b) Find the extreme points of f(x,y) = xy +
8
x
+
8
y
. [8+7]
4. (a) Trace the curve y = 5cosh

x
5


.
(b) Trace the curve y
2
= (4?x)(3?x
2
).. [8+7]
5. (a) Find the length of the arc of the curve y =log (secx) from x = o to
?
3
.
(b) Find the perimeter of the loop of the curve 3ay
2
=x(x-a)
2.
[8+7]
6. (a) Evaluate
R R
rdrd? over the region bounded by the cardioid r=a(1+cos?) and
out side the circle r=a .
(b) Change the order of Integration & evaluate
Z
4a
0
Z
2
?
ax
x
2
4a
dydx [8+7]
7. (a) Prove that (F??)?r = -2F
(b) Determine the constant a so that the vector V = (x+3y)i+(y-z)j+(x+az)k is
solenoidal. [8+7]
8. Apply Stokes theorem, to evaluate
H
c
ydx + zdy + xdzwhere C is the curve of
intersection of the sphere x
2
+ y
2
+ z
2
= a
2
and x + z = a. [15]
?????
1 of 1

FirstRanker.com - FirstRanker's Choice
Code No: R10102/R10 Set No. 1
I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014
MATHEMATICS-I
( Common to Civil Engineering, Electrical & Electronics Engineering,
Mechanical Engineering, Electronics & Communication Engineering,
Computer Science & Engineering, Chemical Engineering, Electronics &
Instrumentation Engineering, Bio-Medical Engineering, Information
Technology, Electronics & Computer Engineering, Aeronautical
Engineering, Bio-Technology, Automobile Engineering, Mining and
Petroliem Technology)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
?????
1. (a) Solve (x
2
+y
2
a
2
)xdx + (x
2
y
2
b
2
)ydy = 0. [7+8]
(b) If air is maintained at 20
0
Cand the temperature of the body cools from 80
0
C
to 60
0
C in 10 minutes, nd the temperature of the body after 30 minutes.
2. (a) Solve (D
2
+a
2
)y =Secax
(b) Solve (D
2
+ 4)y =e
x
+Sin 2x [8+7]
3. (a) If V = log (x
2
+y
2
) +x 2y nd
@V
@x
:,
@V
@y
;
@
2
V
@x
2
:
@
2
V
@y
2
:
(b) If U = xe
xy
where x
2
+y
2
+ 2xy = 1; nd
@
2
U
@x
2
: [8+7]
4. (a) Trace the curve r = 2 + 3 sin.
(b) Trace the curve y
2
(2ax) =x
3
. [8+7]
5. (a) Find the surface of the solid generated by revolution of the lemniscate r
2
=
a
2
cos
2
 about the initial line.
(b) Show that the whole length of the curve x
2
(a
2
x
2
) = 8a
2
y
2
isa
p
2. [8+7]
6. (a) Show that
R
4a
0
R
y
y
2
4a
x
2
y
2
x
2
+y
2
dx dy = 8a
2


2

5
3


.
(b) Evaluate
RR
R
ydxdy where R is the domain bounded by y-axis, the curve
y=x
2
and the line x +y = 2 in the rst quadrants . [8+7]
7. (a) If V= e
xyz
(i+j+k), nd curl V.
(b) Find the constants a and b so that the surface ax
2
-byz = (a+2)x will be
orthogonal to the surface 4x
2
y +z
3
=4 at the point (1,-1,2) [8+7]
8. (a) Show that the area of the ellipse x
2
/a
2
+ y
2
/b
2
= 1 is ab
(b) If f = (2x
2
{ 3z)i { 2xyj { 4xzk, evaluate
(i)
R
v
r:fdV and
(ii)
R
v
rfdV where V is the closed region bounded by x = 0, y = 0, z = 0,
2x + 2y +z = 4. [8+7]
?????
1 of 1
Code No: R10102/R10 Set No. 2
I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014
MATHEMATICS-I
( Common to Civil Engineering, Electrical & Electronics Engineering,
Mechanical Engineering, Electronics & Communication Engineering,
Computer Science & Engineering, Chemical Engineering, Electronics &
Instrumentation Engineering, Bio-Medical Engineering, Information
Technology, Electronics & Computer Engineering, Aeronautical
Engineering, Bio-Technology, Automobile Engineering, Mining and
Petroliem Technology)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
?????
1. (a) Solve e
y

1+
dy
dx


= e
x
(b) Show that the family of curves
x
2
a
2
+?
+
y
2
a
2
+?
= 1, where ??? is a parameter is self
orthogonal. [8+7]
2. (a) Solve (D
2
+9)y = 2Cos
2
x. (b) Solve
d
2
y
dx
2
+4y = 2e
x
Sin
2
x. [8+7]
3. (a) Calculate the approximate value of
?
10 to four decimal places using Taylor?s
theorem.
(b) Find 3 positive numbers whose sum is 600 and whose product is maximum.
[8+7]
4. (a) Trace the curve y = x
2
(x
2
?4). (b)Trace the curve r = cos?. [8+7]
5. (a) The ?gure bounded by a parabola and the tangents at the extremities of its
latusrectum revolves about the axis of the parabola, Find the volume of the
solid thus generated. [8+7]
(b) The segment of the parabola y
2
=4ax which is cuto? by the latus rectum
revolves about the directrix.Find the volume of rotation of the annular region.
6. (a) Evaluate
R R
(x+y)
2
dx dy. over the area bounded by the ellipse
x
2
a
2
+
y
2
b
2
= 1.
(b) TransformthefollowingtoCartesianformandhenceevaluate
R
?
0
R
a
0
r
3
sin?drd?.
[8+7]
7. (a) Prove that?r = r/r
(b) Find the angle between the surfaces x
2
+ y
2
+ z
2
= 9 and z=x
2
+ y
2
?3 at the
point (2,-1,2). [8+7]
8. (a) Evaluate
RR
S
(yzi+zxj+xyk).dS whereSisthesurfaceofthespherex
2
+y
2
+z
2
=a
2
in the ?rst octant.
(b) Evaluate
H
c
(x
2
? 2xy)dx + (x
2
y + 3)dy around the boundary of the region
de?ned by y
2
=8x and x=2. [8+7]
?????
1 of 1
Code No: R10102/R10 Set No. 3
I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014
MATHEMATICS-I
( Common to Civil Engineering, Electrical & Electronics Engineering,
Mechanical Engineering, Electronics & Communication Engineering,
Computer Science & Engineering, Chemical Engineering, Electronics &
Instrumentation Engineering, Bio-Medical Engineering, Information
Technology, Electronics & Computer Engineering, Aeronautical
Engineering, Bio-Technology, Automobile Engineering, Mining and
Petroliem Technology)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
?????
1. (a) Solve y(Sinx?y)dx = Cosx dy
(b) If the temperature of air is maintained at 20
0
Cand the temperature of the
body cools from 100
0
Cto80
0
C in 10 minutes, ?nd the temperature of the
body after 20 minutes. [8+7]
2. (a) Solve (D
2
?4D+13)y = e
2x
(b) Solve (D
2
?3D+2)y = Coshx [8+7]
3. (a) If r + s + t = x, s + t = xy, t = xyz, ?nd
?(r,s,t)
?(x,y,z)
.
(b) Find the extreme points of f(x,y) = xy +
8
x
+
8
y
. [8+7]
4. (a) Trace the curve y = 5cosh

x
5


.
(b) Trace the curve y
2
= (4?x)(3?x
2
).. [8+7]
5. (a) Find the length of the arc of the curve y =log (secx) from x = o to
?
3
.
(b) Find the perimeter of the loop of the curve 3ay
2
=x(x-a)
2.
[8+7]
6. (a) Evaluate
R R
rdrd? over the region bounded by the cardioid r=a(1+cos?) and
out side the circle r=a .
(b) Change the order of Integration & evaluate
Z
4a
0
Z
2
?
ax
x
2
4a
dydx [8+7]
7. (a) Prove that (F??)?r = -2F
(b) Determine the constant a so that the vector V = (x+3y)i+(y-z)j+(x+az)k is
solenoidal. [8+7]
8. Apply Stokes theorem, to evaluate
H
c
ydx + zdy + xdzwhere C is the curve of
intersection of the sphere x
2
+ y
2
+ z
2
= a
2
and x + z = a. [15]
?????
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Code No: R10102/R10 Set No. 4
I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014
MATHEMATICS-I
( Common to Civil Engineering, Electrical & Electronics Engineering,
Mechanical Engineering, Electronics & Communication Engineering,
Computer Science & Engineering, Chemical Engineering, Electronics &
Instrumentation Engineering, Bio-Medical Engineering, Information
Technology, Electronics & Computer Engineering, Aeronautical
Engineering, Bio-Technology, Automobile Engineering, Mining and
Petroliem Technology)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
?????
1. (a) Solve (x+1)
dy
dx
?y = e
3x
(x+1)
2
(b) Find the orthogonal trajectory of the family of curves x
2/3
+y
2/3
= a
2/3
, where
?a? is a parameter [8+7]
2. (a) Solve (D
3
?6D
2
+11D?6)y = e
?2x
+e
?3x
(b) Solve
d
2
y
dx
2
?8
dy
dx
+15y = 0 [8+7]
3. (a) If a =
yz
x
, b =
xz
y
, c =
xy
z
, ?nd
?(x,y,z)
?(a,b,c)
.
(b) Find the minimum value of x
2
+y
2
+z
2
, give that xyz = a
3
[8+7]
4. (a) Trace the curve r = cos 4?.
(b) Trace the curvey
2
(1?x) = x
2
(1+x).. [8+7]
5. Prove that the volume of the solid generated by the revolution about the x?axis
of the loop of the curve x = t
2
,y = t?
1
3
t
3
is
3?
4
. [8+7]
6. (a) By changing the order of integration evaluate
Z
1
0
Z
)2?x
2
0
x
)x
2
+y
2
dydx.
(b) Evaluate
Z
a
0
Z
)a
2
?x
2
a?x
y dx dy by using change of order of integration . [8+7]
7. (a) If V= e
xyz
(i+j+k), ?nd curl V.
(b) Find the constants a and b so that the surface ax
2
-byz = (a+2)x will be
orthogonal to the surface 4x
2
y +z
3
=4 at the point (1,-1,2) [8+7]
8. (a) Use divergence theorem to evaluate
RR
S
(x
3
i + y
3
j + z
3
k).Nds, and S is the
surface of the sphere x
2
+y
2
+z
2
=r
2
.
(b) Using Green?s theorem, Find the area bounded by the hypocycloid x
2/3
+y
2/3
=
a
2/3
, a>0. Given that the parametric equations are x =a cos
3
?, y =a sin
3
?.
[8+7]
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