Download JNTU Kakinada B.Tech 1-1 2013 Nov R10 MATHEMATICS I Question Paper

Download JNTUK (Jawaharlal Nehru Technological University Kakinada) B.Tech Supplementary 2013 Oct-Nov 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, Oct/Nov 2013
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 3y
1
+ xy = xy
2
(b) A body kept in air with temperature 25
0
C cools from 140
0
C to 80
0
C in 20
minutes. when will the body cools down to 35
0
C [8+7]
2. (a) Explain the procedure to nd Complete solution of second order non homo-
geneous dierential equation with constant coecients.
(b) Solve (D
2
4)y =x Sinx [8+7]
3. (a)A rectangular box open at the top is to have maximum capacity whose surface
area is 108 square ft. Find the dimensions of the rectangular box.
(b)Find the points on the ellipsoid
x
2
4
+
y
2
9
+
z
2
4
= 1 that are closest and farthest
from the point (2,0,-1). [8+7]
4. (a) Trace the curve x = a ( + sin), y = a (1 { cos).
(b) Trace the curve x
2=3
+y
2=3
=a
2=3
. [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
4
0
R
y
y
2
=4
y
x
2
+y
2
dx dy.
(b) Evaluate
R
1
0
R
)1x
2
0
R
)1x
2
y
2
0
xyz dzdy dx. [8+7]
7. (a)Prove that div(r/r) = 2/r.
(b) Show that A = (6xy+ z
3
)i +(3x
2
-z)j +(3x
2
z
2
-y)k is irrotational. Find  such
that A =r. Prove that div curl f = 0 [8+7]
8. (a)If f = y i + z j + x k, nd the circulation of f round the curve C, where C is the
circle x
2
+y
2
= 0, z = 0.
(b) If f = (x + y
2
)i { 2xj + 2yzk, evaluate
R
s
f:Nds where S is the surface of the
plane 2x + y + 2z = 6 in the rst octant. [8+7]
?????

FirstRanker.com - FirstRanker's Choice
Code No: R10102/R10 Set No. 1
I B.Tech I Semester Supplementary Examinations, Oct/Nov 2013
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 3y
1
+ xy = xy
2
(b) A body kept in air with temperature 25
0
C cools from 140
0
C to 80
0
C in 20
minutes. when will the body cools down to 35
0
C [8+7]
2. (a) Explain the procedure to nd Complete solution of second order non homo-
geneous dierential equation with constant coecients.
(b) Solve (D
2
4)y =x Sinx [8+7]
3. (a)A rectangular box open at the top is to have maximum capacity whose surface
area is 108 square ft. Find the dimensions of the rectangular box.
(b)Find the points on the ellipsoid
x
2
4
+
y
2
9
+
z
2
4
= 1 that are closest and farthest
from the point (2,0,-1). [8+7]
4. (a) Trace the curve x = a ( + sin), y = a (1 { cos).
(b) Trace the curve x
2=3
+y
2=3
=a
2=3
. [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
4
0
R
y
y
2
=4
y
x
2
+y
2
dx dy.
(b) Evaluate
R
1
0
R
)1x
2
0
R
)1x
2
y
2
0
xyz dzdy dx. [8+7]
7. (a)Prove that div(r/r) = 2/r.
(b) Show that A = (6xy+ z
3
)i +(3x
2
-z)j +(3x
2
z
2
-y)k is irrotational. Find  such
that A =r. Prove that div curl f = 0 [8+7]
8. (a)If f = y i + z j + x k, nd the circulation of f round the curve C, where C is the
circle x
2
+y
2
= 0, z = 0.
(b) If f = (x + y
2
)i { 2xj + 2yzk, evaluate
R
s
f:Nds where S is the surface of the
plane 2x + y + 2z = 6 in the rst octant. [8+7]
?????
Code No: R10102/R10 Set No. 2
I B.Tech I Semester Supplementary Examinations, Oct/Nov 2013
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 log x 2)ydx =x dy
(b) Find the orthogonal trajectory of the family of curves r =
2a
(1+Cos)
, where `a'
is a parameter [8+7]
2. (a) Solve (D
2
4D + 13)y =e
2x
(b) Solve (D
2
3D + 2)y =Coshx [8+7]
3. (a) Find the dimensions of a open rectangular tank of maximum capacity whose
surface area is 54 square feet.
(b) In a right angled triangle ABC with
6 B = 90

, nd the maximum of cosA cosB
cosC. [8+7]
4. (a) Trace the curve r
2
=a
2
cos 2.
(b) Trace the curve x = a ( + sin), y = a (1 + cos) . [8+7]
5. (a) Find the surface of the solid generated by the revolution of cardioid
r=a (1- cos) about the initial line.
(b) Find the surface of the solid generated by the revolution of the ellipsex
2
+4y
2
=
16 about its major axis . [8+7]
6. (a)Evaluate
RR
xydxdy over the positive Quadrant of the circle x
2
+y
2
=a
2
.
(b)Evaluate
R R R
V
(xy +yz +zx)dx dy dz where V is the region of space bound
by x = 0;x = 1;y = 0;y = 2;z = 0;z = 3. [8+7]
7. (a)Find the angle between the normals to the surface xy = z
2
at the points (4,1,2)
and (3,3,-3).
(b) Prove that div (r
n
r) = (n+3)r
n
[8+7]
8. (a)If f = 3xy i { y
2
j, evaluate
R
C
f:dr where C is the curve y =2x
2
, in xy plane
from (0, 0) to (1,2).
(b) Evaluate
R
s
f:Nds, where f = 18zi { 12j + 3yk and S is the part of the plane
2x + 3y + 6z = 12 located in rst octant. [8+7]
?????

FirstRanker.com - FirstRanker's Choice
Code No: R10102/R10 Set No. 1
I B.Tech I Semester Supplementary Examinations, Oct/Nov 2013
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 3y
1
+ xy = xy
2
(b) A body kept in air with temperature 25
0
C cools from 140
0
C to 80
0
C in 20
minutes. when will the body cools down to 35
0
C [8+7]
2. (a) Explain the procedure to nd Complete solution of second order non homo-
geneous dierential equation with constant coecients.
(b) Solve (D
2
4)y =x Sinx [8+7]
3. (a)A rectangular box open at the top is to have maximum capacity whose surface
area is 108 square ft. Find the dimensions of the rectangular box.
(b)Find the points on the ellipsoid
x
2
4
+
y
2
9
+
z
2
4
= 1 that are closest and farthest
from the point (2,0,-1). [8+7]
4. (a) Trace the curve x = a ( + sin), y = a (1 { cos).
(b) Trace the curve x
2=3
+y
2=3
=a
2=3
. [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
4
0
R
y
y
2
=4
y
x
2
+y
2
dx dy.
(b) Evaluate
R
1
0
R
)1x
2
0
R
)1x
2
y
2
0
xyz dzdy dx. [8+7]
7. (a)Prove that div(r/r) = 2/r.
(b) Show that A = (6xy+ z
3
)i +(3x
2
-z)j +(3x
2
z
2
-y)k is irrotational. Find  such
that A =r. Prove that div curl f = 0 [8+7]
8. (a)If f = y i + z j + x k, nd the circulation of f round the curve C, where C is the
circle x
2
+y
2
= 0, z = 0.
(b) If f = (x + y
2
)i { 2xj + 2yzk, evaluate
R
s
f:Nds where S is the surface of the
plane 2x + y + 2z = 6 in the rst octant. [8+7]
?????
Code No: R10102/R10 Set No. 2
I B.Tech I Semester Supplementary Examinations, Oct/Nov 2013
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 log x 2)ydx =x dy
(b) Find the orthogonal trajectory of the family of curves r =
2a
(1+Cos)
, where `a'
is a parameter [8+7]
2. (a) Solve (D
2
4D + 13)y =e
2x
(b) Solve (D
2
3D + 2)y =Coshx [8+7]
3. (a) Find the dimensions of a open rectangular tank of maximum capacity whose
surface area is 54 square feet.
(b) In a right angled triangle ABC with
6 B = 90

, nd the maximum of cosA cosB
cosC. [8+7]
4. (a) Trace the curve r
2
=a
2
cos 2.
(b) Trace the curve x = a ( + sin), y = a (1 + cos) . [8+7]
5. (a) Find the surface of the solid generated by the revolution of cardioid
r=a (1- cos) about the initial line.
(b) Find the surface of the solid generated by the revolution of the ellipsex
2
+4y
2
=
16 about its major axis . [8+7]
6. (a)Evaluate
RR
xydxdy over the positive Quadrant of the circle x
2
+y
2
=a
2
.
(b)Evaluate
R R R
V
(xy +yz +zx)dx dy dz where V is the region of space bound
by x = 0;x = 1;y = 0;y = 2;z = 0;z = 3. [8+7]
7. (a)Find the angle between the normals to the surface xy = z
2
at the points (4,1,2)
and (3,3,-3).
(b) Prove that div (r
n
r) = (n+3)r
n
[8+7]
8. (a)If f = 3xy i { y
2
j, evaluate
R
C
f:dr where C is the curve y =2x
2
, in xy plane
from (0, 0) to (1,2).
(b) Evaluate
R
s
f:Nds, where f = 18zi { 12j + 3yk and S is the part of the plane
2x + 3y + 6z = 12 located in rst octant. [8+7]
?????
Code No: R10102/R10 Set No. 3
I B.Tech I Semester Supplementary Examinations, Oct/Nov 2013
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 3y
1
+ xy = xy
2
(b) A body kept in air with temperature 25
0
C cools from 140
0
C to 80
0
C in 20
minutes. when will the body cools down to 35
0
C [8+7]
2. (a)Solve (D
2
+a
2
)y =Cosax
(b) Solve (D 2)
3
y =e
2x
[8+7]
3. (a)If u =e
xyz
then nd
@
3
u
@x@y@z
:
(b) Find the Jacobian of transformation from 3D-Cartesian coordinates to spherical
polar coordinates [8+7]
4. (a) Trace the curve x = a ( + sin), y = a (1 { cos).
(b) Trace the curve x
2=3
+y
2=3
=a
2=3
. [8+7]
5. (a) Find the volume of the solid generated by revolving one arc of the Cycloid x=a
(1-sin t), y=a(1-cos t) about its base.
(b) Prove that the curved surface area of a sphere of radius r intercepted between
two parallel planes at a distance a and b from the centre of the Sphere is 2r(b
a)when b> a: [8+7]
6. (a)Evaluate
RR
xydxdy over the positive Quadrant of the circle x
2
+y
2
=a
2
.
(b)Evaluate
R R R
V
(xy +yz +zx)dx dy dz where V is the region of space bound
by x = 0;x = 1;y = 0;y = 2;z = 0;z = 3. [8+7]
7. (a)Find the angle of intersection of the spheres x
2
+ y
2
+ z
2
= 4 and z = x
2
+ y
2
+ 3 at the point (2,-1,1).
(b) Prove that div grad r
n
= n(n+1)r
n2
. [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 (a)
R
v
r:fdV and
(b)
R
v
rfdV where V is the closed region bounded by x = 0, y = 0, z = 0, 2x
+ 2y +z = 4. [8+7]
?????

FirstRanker.com - FirstRanker's Choice
Code No: R10102/R10 Set No. 1
I B.Tech I Semester Supplementary Examinations, Oct/Nov 2013
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 3y
1
+ xy = xy
2
(b) A body kept in air with temperature 25
0
C cools from 140
0
C to 80
0
C in 20
minutes. when will the body cools down to 35
0
C [8+7]
2. (a) Explain the procedure to nd Complete solution of second order non homo-
geneous dierential equation with constant coecients.
(b) Solve (D
2
4)y =x Sinx [8+7]
3. (a)A rectangular box open at the top is to have maximum capacity whose surface
area is 108 square ft. Find the dimensions of the rectangular box.
(b)Find the points on the ellipsoid
x
2
4
+
y
2
9
+
z
2
4
= 1 that are closest and farthest
from the point (2,0,-1). [8+7]
4. (a) Trace the curve x = a ( + sin), y = a (1 { cos).
(b) Trace the curve x
2=3
+y
2=3
=a
2=3
. [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
4
0
R
y
y
2
=4
y
x
2
+y
2
dx dy.
(b) Evaluate
R
1
0
R
)1x
2
0
R
)1x
2
y
2
0
xyz dzdy dx. [8+7]
7. (a)Prove that div(r/r) = 2/r.
(b) Show that A = (6xy+ z
3
)i +(3x
2
-z)j +(3x
2
z
2
-y)k is irrotational. Find  such
that A =r. Prove that div curl f = 0 [8+7]
8. (a)If f = y i + z j + x k, nd the circulation of f round the curve C, where C is the
circle x
2
+y
2
= 0, z = 0.
(b) If f = (x + y
2
)i { 2xj + 2yzk, evaluate
R
s
f:Nds where S is the surface of the
plane 2x + y + 2z = 6 in the rst octant. [8+7]
?????
Code No: R10102/R10 Set No. 2
I B.Tech I Semester Supplementary Examinations, Oct/Nov 2013
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 log x 2)ydx =x dy
(b) Find the orthogonal trajectory of the family of curves r =
2a
(1+Cos)
, where `a'
is a parameter [8+7]
2. (a) Solve (D
2
4D + 13)y =e
2x
(b) Solve (D
2
3D + 2)y =Coshx [8+7]
3. (a) Find the dimensions of a open rectangular tank of maximum capacity whose
surface area is 54 square feet.
(b) In a right angled triangle ABC with
6 B = 90

, nd the maximum of cosA cosB
cosC. [8+7]
4. (a) Trace the curve r
2
=a
2
cos 2.
(b) Trace the curve x = a ( + sin), y = a (1 + cos) . [8+7]
5. (a) Find the surface of the solid generated by the revolution of cardioid
r=a (1- cos) about the initial line.
(b) Find the surface of the solid generated by the revolution of the ellipsex
2
+4y
2
=
16 about its major axis . [8+7]
6. (a)Evaluate
RR
xydxdy over the positive Quadrant of the circle x
2
+y
2
=a
2
.
(b)Evaluate
R R R
V
(xy +yz +zx)dx dy dz where V is the region of space bound
by x = 0;x = 1;y = 0;y = 2;z = 0;z = 3. [8+7]
7. (a)Find the angle between the normals to the surface xy = z
2
at the points (4,1,2)
and (3,3,-3).
(b) Prove that div (r
n
r) = (n+3)r
n
[8+7]
8. (a)If f = 3xy i { y
2
j, evaluate
R
C
f:dr where C is the curve y =2x
2
, in xy plane
from (0, 0) to (1,2).
(b) Evaluate
R
s
f:Nds, where f = 18zi { 12j + 3yk and S is the part of the plane
2x + 3y + 6z = 12 located in rst octant. [8+7]
?????
Code No: R10102/R10 Set No. 3
I B.Tech I Semester Supplementary Examinations, Oct/Nov 2013
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 3y
1
+ xy = xy
2
(b) A body kept in air with temperature 25
0
C cools from 140
0
C to 80
0
C in 20
minutes. when will the body cools down to 35
0
C [8+7]
2. (a)Solve (D
2
+a
2
)y =Cosax
(b) Solve (D 2)
3
y =e
2x
[8+7]
3. (a)If u =e
xyz
then nd
@
3
u
@x@y@z
:
(b) Find the Jacobian of transformation from 3D-Cartesian coordinates to spherical
polar coordinates [8+7]
4. (a) Trace the curve x = a ( + sin), y = a (1 { cos).
(b) Trace the curve x
2=3
+y
2=3
=a
2=3
. [8+7]
5. (a) Find the volume of the solid generated by revolving one arc of the Cycloid x=a
(1-sin t), y=a(1-cos t) about its base.
(b) Prove that the curved surface area of a sphere of radius r intercepted between
two parallel planes at a distance a and b from the centre of the Sphere is 2r(b
a)when b> a: [8+7]
6. (a)Evaluate
RR
xydxdy over the positive Quadrant of the circle x
2
+y
2
=a
2
.
(b)Evaluate
R R R
V
(xy +yz +zx)dx dy dz where V is the region of space bound
by x = 0;x = 1;y = 0;y = 2;z = 0;z = 3. [8+7]
7. (a)Find the angle of intersection of the spheres x
2
+ y
2
+ z
2
= 4 and z = x
2
+ y
2
+ 3 at the point (2,-1,1).
(b) Prove that div grad r
n
= n(n+1)r
n2
. [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 (a)
R
v
r:fdV and
(b)
R
v
rfdV where V is the closed region bounded by x = 0, y = 0, z = 0, 2x
+ 2y +z = 4. [8+7]
?????
Code No: R10102/R10 Set No. 4
I B.Tech I Semester Supplementary Examinations, Oct/Nov 2013
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(2x
2
y
2
+xy)dxx(x
2
y
2
xy) dy = 0
(b) The number N of bacteria in a culture grew at a rate proportional to N. The
number becomes 3 times of original (initial) in 2 hours. Then what is the time
required when the number reaches to 100 times of its original? [8+7]
2. (a) Solve y
11
+ 4y
1
12y =e
2x
3Sin 2x
(b) Solve (D
3
1)y = 0 [8+7]
3. (a)If u =e
xyz
then nd
@
3
u
@x@y@z
:
(b) Find the Jacobian of transformation from 3D-Cartesian coordinates to spherical
polar coordinates [8+7]
4. (a) Trace the curve x = 2 cos
3
t, y = 3 sin
3
t
(b) Trace the curve y
3
=x: [8+7]
5. (a) Find the surface of the solid generated by the revolution of cardioid r=a (1-
cos) about the initial line.
(b) Find the surface of the solid generated by the revolution of the ellipsex
2
+4y
2
=
16 about its Major axis . [8+7]
6. (a)Evaluate
RR
xydxdy over the positive Quadrant of the circle x
2
+y
2
=a
2
.
(b)Evaluate
R R R
V
(xy +yz +zx)dx dy dz where V is the region of space bound
by x = 0;x = 1;y = 0;y = 2;z = 0;z = 3. [8+7]
7. (a) Prove thatr(rA) =r(r.A) -r
2
A
(b) If f = x
2
yz, g = xy { 3z
2
then nd div(grad f grad g) [8+7]
8. Verify divergence theorem for F = x
2
i+y
2
j+z
2
k over the surface S of the solid cut
o by the plane x+y+z =a in the rst octant. [15]
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