Download JNTUK (Jawaharlal Nehru Technological University Kakinada) B.Tech Supplementary-Regular 2012 January R10 I Semester (1st Year 1st Sem) Maths Question Paper.
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Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the differential equations of all parabolas with x-axis as its axis and ( ? , 0) as its
focus.
(b) Find the orthogonal trajectories of coaxial circles x
2
+ y
2
+ 2 ? y + c = 2 , where ? is
the parameter.
[7M + 8M]
2.(a)
Solve
( D
2
? 2) y = e
?
2x
+ cos x
d
2
y
+ 4
dy
+ 5 y = 2 sin hx
dy
? 1
Solve
dx
2
dx subject to y=o and
dx
(b) at x=0.
[7M + 8M]
3.(a) If u = xy + yz + zx , v = x
2
+ y
2
+ z
2
and w = x + y + z , verify whether there exists a
possible relationship in between u, v and w. If so find the relation.
(b) Find the minimum value of x
2
+ y
2
+ z
2
on the plane x ? y ? z ? 3 a
4.(a) Trace the curve
x
(
x
2
+
y
2
)
=
(b) Trace the polar curve r ? 2 ? ?
[7M + 8M]
4 ( x
2
? y
2
)
3 cos ? .
[7M + 8M]
5.(a) Find the perimeter of one loop of the curve 3a y
2
= x ( x ? a)
2
.
(b) Find the volume generated by revolving the area bounded by one loop of the
curve r ? a (1 ? cos ? ) about the initial line.
[7M + 8M]
6.(a)
Evaluate
?
?
? ?
e
? y
dx.dy by changing the order of integration.
y
0 x
x
(b) Evaluate
?
2
?
2 x ? x
2
by changing into polar coordinates.
dy dx
0 0
x
2
+ y
2
[7M + 8M]
7.(a) Find the directional derivative of ? ( x, y , z ) = x y
2
+ y z
3
at the point (2,-1,1) in the
direction of the vector i ? 2 j ? 2k .
(b)
Find
curl [ r f ( r )]
where
r ? xi ? yj ? zk
,
r ?| r |
[7M + 8M]
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the differential equations of all parabolas with x-axis as its axis and ( ? , 0) as its
focus.
(b) Find the orthogonal trajectories of coaxial circles x
2
+ y
2
+ 2 ? y + c = 2 , where ? is
the parameter.
[7M + 8M]
2.(a)
Solve
( D
2
? 2) y = e
?
2x
+ cos x
d
2
y
+ 4
dy
+ 5 y = 2 sin hx
dy
? 1
Solve
dx
2
dx subject to y=o and
dx
(b) at x=0.
[7M + 8M]
3.(a) If u = xy + yz + zx , v = x
2
+ y
2
+ z
2
and w = x + y + z , verify whether there exists a
possible relationship in between u, v and w. If so find the relation.
(b) Find the minimum value of x
2
+ y
2
+ z
2
on the plane x ? y ? z ? 3 a
4.(a) Trace the curve
x
(
x
2
+
y
2
)
=
(b) Trace the polar curve r ? 2 ? ?
[7M + 8M]
4 ( x
2
? y
2
)
3 cos ? .
[7M + 8M]
5.(a) Find the perimeter of one loop of the curve 3a y
2
= x ( x ? a)
2
.
(b) Find the volume generated by revolving the area bounded by one loop of the
curve r ? a (1 ? cos ? ) about the initial line.
[7M + 8M]
6.(a)
Evaluate
?
?
? ?
e
? y
dx.dy by changing the order of integration.
y
0 x
x
(b) Evaluate
?
2
?
2 x ? x
2
by changing into polar coordinates.
dy dx
0 0
x
2
+ y
2
[7M + 8M]
7.(a) Find the directional derivative of ? ( x, y , z ) = x y
2
+ y z
3
at the point (2,-1,1) in the
direction of the vector i ? 2 j ? 2k .
(b)
Find
curl [ r f ( r )]
where
r ? xi ? yj ? zk
,
r ?| r |
[7M + 8M]
Page 1 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the differential equations of all parabolas with x-axis as its axis and ( ? , 0) as its
focus.
(b) Find the orthogonal trajectories of coaxial circles x
2
+ y
2
+ 2 ? y + c = 2 , where ? is
the parameter.
[7M + 8M]
2.(a)
Solve
( D
2
? 2) y = e
?
2x
+ cos x
d
2
y
+ 4
dy
+ 5 y = 2 sin hx
dy
? 1
Solve
dx
2
dx subject to y=o and
dx
(b) at x=0.
[7M + 8M]
3.(a) If u = xy + yz + zx , v = x
2
+ y
2
+ z
2
and w = x + y + z , verify whether there exists a
possible relationship in between u, v and w. If so find the relation.
(b) Find the minimum value of x
2
+ y
2
+ z
2
on the plane x ? y ? z ? 3 a
4.(a) Trace the curve
x
(
x
2
+
y
2
)
=
(b) Trace the polar curve r ? 2 ? ?
[7M + 8M]
4 ( x
2
? y
2
)
3 cos ? .
[7M + 8M]
5.(a) Find the perimeter of one loop of the curve 3a y
2
= x ( x ? a)
2
.
(b) Find the volume generated by revolving the area bounded by one loop of the
curve r ? a (1 ? cos ? ) about the initial line.
[7M + 8M]
6.(a)
Evaluate
?
?
? ?
e
? y
dx.dy by changing the order of integration.
y
0 x
x
(b) Evaluate
?
2
?
2 x ? x
2
by changing into polar coordinates.
dy dx
0 0
x
2
+ y
2
[7M + 8M]
7.(a) Find the directional derivative of ? ( x, y , z ) = x y
2
+ y z
3
at the point (2,-1,1) in the
direction of the vector i ? 2 j ? 2k .
(b)
Find
curl [ r f ( r )]
where
r ? xi ? yj ? zk
,
r ?| r |
[7M + 8M]
Page 1 of 2
Set No. 1
Code No: R10102 / R10
8.(a) Compute the line integral ?
( y
2
dx ? x
2
dy)
round the triangle whose vertices are
(1,0),(0,1) and (-1,0) in the xy-plane.
(b) Evaluate the integral I = ? ?
S
x
3
dydz + x
2
y dzdx + x
2
z dxdy using divergence theorem,
where S is the surface consisting of the cylinder x
2
? y
2
? a
2
(0 ? z ? b) and the
circular disks $z=0$ and z ? b ( x
2
? y
2
? a
2
) .
[7M + 8M]
Page 2 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the differential equations of all parabolas with x-axis as its axis and ( ? , 0) as its
focus.
(b) Find the orthogonal trajectories of coaxial circles x
2
+ y
2
+ 2 ? y + c = 2 , where ? is
the parameter.
[7M + 8M]
2.(a)
Solve
( D
2
? 2) y = e
?
2x
+ cos x
d
2
y
+ 4
dy
+ 5 y = 2 sin hx
dy
? 1
Solve
dx
2
dx subject to y=o and
dx
(b) at x=0.
[7M + 8M]
3.(a) If u = xy + yz + zx , v = x
2
+ y
2
+ z
2
and w = x + y + z , verify whether there exists a
possible relationship in between u, v and w. If so find the relation.
(b) Find the minimum value of x
2
+ y
2
+ z
2
on the plane x ? y ? z ? 3 a
4.(a) Trace the curve
x
(
x
2
+
y
2
)
=
(b) Trace the polar curve r ? 2 ? ?
[7M + 8M]
4 ( x
2
? y
2
)
3 cos ? .
[7M + 8M]
5.(a) Find the perimeter of one loop of the curve 3a y
2
= x ( x ? a)
2
.
(b) Find the volume generated by revolving the area bounded by one loop of the
curve r ? a (1 ? cos ? ) about the initial line.
[7M + 8M]
6.(a)
Evaluate
?
?
? ?
e
? y
dx.dy by changing the order of integration.
y
0 x
x
(b) Evaluate
?
2
?
2 x ? x
2
by changing into polar coordinates.
dy dx
0 0
x
2
+ y
2
[7M + 8M]
7.(a) Find the directional derivative of ? ( x, y , z ) = x y
2
+ y z
3
at the point (2,-1,1) in the
direction of the vector i ? 2 j ? 2k .
(b)
Find
curl [ r f ( r )]
where
r ? xi ? yj ? zk
,
r ?| r |
[7M + 8M]
Page 1 of 2
Set No. 1
Code No: R10102 / R10
8.(a) Compute the line integral ?
( y
2
dx ? x
2
dy)
round the triangle whose vertices are
(1,0),(0,1) and (-1,0) in the xy-plane.
(b) Evaluate the integral I = ? ?
S
x
3
dydz + x
2
y dzdx + x
2
z dxdy using divergence theorem,
where S is the surface consisting of the cylinder x
2
? y
2
? a
2
(0 ? z ? b) and the
circular disks $z=0$ and z ? b ( x
2
? y
2
? a
2
) .
[7M + 8M]
Page 2 of 2
Set No. 2
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the solution of the differential equation
dy
=
xe
y ? x
2
and y(0) = 0 .
dx
(b) A body initially at 80
0
C cools down to 50
0
C in 10 minutes, the temperature of the
air being 40
0
C. What will be the temperature of the body after 20 minutes?
[7M + 8M]
d
2
y
+ 9 y = e
2
x
x
2
2.(a)
Solve
dx
2
(b) Find the general solution of
d
2
y
? 2
dy
+ y = e
x
sin 2x
dx
2
dx
[7M + 8M]
3.(a) Verify whether the functions u = sin
?1
x + sin
?1
y and v = x 1 ? y
2
+ y 1 ? x
2
are
functionally dependent. If so, find the relation between them.[7 M+8 M]
(b) Prove that the rectangular solid of maximum volume that can be inscribed into a
sphere of radius ?a ? is a cube.
[7M + 8M]
4.(a) Trace the parametric curve x = a (cos ? +
1
log tan
2
(
t
) and y = a sin t .
2 2
(b) Trace the lemniscate r
2
= a
2
cos 2 ? .
[7M + 8M]
5.(a) Find the surface area generated by revolving the arc of the curve y = a cosh ( x / c)
from x=0 to x=c about the x-axis.
(b) Find the total length of the lamniscate r
2
= a
2
cos 2? .
[7M + 8M]
6.(a) Find the area of the region which is outside the circle r=1 and inside the cordioid
r = (1 + cos ? )
(b) Evaluate the followingintegralbychanginginto polarcoordinates
? ?
1 ? ( x
+
y
2
)
dx dy over the positive coordinate of the circle x
2
+ y
2
= 1
1 + x
2
+ y
2
[7M + 8M]
Page 1 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the differential equations of all parabolas with x-axis as its axis and ( ? , 0) as its
focus.
(b) Find the orthogonal trajectories of coaxial circles x
2
+ y
2
+ 2 ? y + c = 2 , where ? is
the parameter.
[7M + 8M]
2.(a)
Solve
( D
2
? 2) y = e
?
2x
+ cos x
d
2
y
+ 4
dy
+ 5 y = 2 sin hx
dy
? 1
Solve
dx
2
dx subject to y=o and
dx
(b) at x=0.
[7M + 8M]
3.(a) If u = xy + yz + zx , v = x
2
+ y
2
+ z
2
and w = x + y + z , verify whether there exists a
possible relationship in between u, v and w. If so find the relation.
(b) Find the minimum value of x
2
+ y
2
+ z
2
on the plane x ? y ? z ? 3 a
4.(a) Trace the curve
x
(
x
2
+
y
2
)
=
(b) Trace the polar curve r ? 2 ? ?
[7M + 8M]
4 ( x
2
? y
2
)
3 cos ? .
[7M + 8M]
5.(a) Find the perimeter of one loop of the curve 3a y
2
= x ( x ? a)
2
.
(b) Find the volume generated by revolving the area bounded by one loop of the
curve r ? a (1 ? cos ? ) about the initial line.
[7M + 8M]
6.(a)
Evaluate
?
?
? ?
e
? y
dx.dy by changing the order of integration.
y
0 x
x
(b) Evaluate
?
2
?
2 x ? x
2
by changing into polar coordinates.
dy dx
0 0
x
2
+ y
2
[7M + 8M]
7.(a) Find the directional derivative of ? ( x, y , z ) = x y
2
+ y z
3
at the point (2,-1,1) in the
direction of the vector i ? 2 j ? 2k .
(b)
Find
curl [ r f ( r )]
where
r ? xi ? yj ? zk
,
r ?| r |
[7M + 8M]
Page 1 of 2
Set No. 1
Code No: R10102 / R10
8.(a) Compute the line integral ?
( y
2
dx ? x
2
dy)
round the triangle whose vertices are
(1,0),(0,1) and (-1,0) in the xy-plane.
(b) Evaluate the integral I = ? ?
S
x
3
dydz + x
2
y dzdx + x
2
z dxdy using divergence theorem,
where S is the surface consisting of the cylinder x
2
? y
2
? a
2
(0 ? z ? b) and the
circular disks $z=0$ and z ? b ( x
2
? y
2
? a
2
) .
[7M + 8M]
Page 2 of 2
Set No. 2
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the solution of the differential equation
dy
=
xe
y ? x
2
and y(0) = 0 .
dx
(b) A body initially at 80
0
C cools down to 50
0
C in 10 minutes, the temperature of the
air being 40
0
C. What will be the temperature of the body after 20 minutes?
[7M + 8M]
d
2
y
+ 9 y = e
2
x
x
2
2.(a)
Solve
dx
2
(b) Find the general solution of
d
2
y
? 2
dy
+ y = e
x
sin 2x
dx
2
dx
[7M + 8M]
3.(a) Verify whether the functions u = sin
?1
x + sin
?1
y and v = x 1 ? y
2
+ y 1 ? x
2
are
functionally dependent. If so, find the relation between them.[7 M+8 M]
(b) Prove that the rectangular solid of maximum volume that can be inscribed into a
sphere of radius ?a ? is a cube.
[7M + 8M]
4.(a) Trace the parametric curve x = a (cos ? +
1
log tan
2
(
t
) and y = a sin t .
2 2
(b) Trace the lemniscate r
2
= a
2
cos 2 ? .
[7M + 8M]
5.(a) Find the surface area generated by revolving the arc of the curve y = a cosh ( x / c)
from x=0 to x=c about the x-axis.
(b) Find the total length of the lamniscate r
2
= a
2
cos 2? .
[7M + 8M]
6.(a) Find the area of the region which is outside the circle r=1 and inside the cordioid
r = (1 + cos ? )
(b) Evaluate the followingintegralbychanginginto polarcoordinates
? ?
1 ? ( x
+
y
2
)
dx dy over the positive coordinate of the circle x
2
+ y
2
= 1
1 + x
2
+ y
2
[7M + 8M]
Page 1 of 2
Set No. 2
Code No: R10102 / R10
7.(a) Find the directional derivative of the divergence of F = x yi + y z j + z
2
k at the point
(2,1,2) in the direction of the outer normal to the sphere x
2
+ y
2
+ z
2
= 9 .
(b) Find the value of a,b and c such that ( x + y + a z ) i + ( b x + 2 y ? z ) j + ( ? x + c y + 2 z )k is
irrotational.
[7M + 8M]
8.(a) If f = ( x
2
+ y ? 4) i + 3 x y j + (2 xz + z
2
)k and S is the upper half of the sphere
x
2
+ y
2
+ z
2
= 16 . Show by using Stokes theorem that
?
Curl f .n ds = 2 ? a
3
.
S
(b) If S is the surface of the tetrahedron bounded by the planes x = 0 , y = 0 , z = 0 and
ax + by + cz = 1. Show that
?
r .nds =
1
.
S 2abc
[7M + 8M]
Page 2 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the differential equations of all parabolas with x-axis as its axis and ( ? , 0) as its
focus.
(b) Find the orthogonal trajectories of coaxial circles x
2
+ y
2
+ 2 ? y + c = 2 , where ? is
the parameter.
[7M + 8M]
2.(a)
Solve
( D
2
? 2) y = e
?
2x
+ cos x
d
2
y
+ 4
dy
+ 5 y = 2 sin hx
dy
? 1
Solve
dx
2
dx subject to y=o and
dx
(b) at x=0.
[7M + 8M]
3.(a) If u = xy + yz + zx , v = x
2
+ y
2
+ z
2
and w = x + y + z , verify whether there exists a
possible relationship in between u, v and w. If so find the relation.
(b) Find the minimum value of x
2
+ y
2
+ z
2
on the plane x ? y ? z ? 3 a
4.(a) Trace the curve
x
(
x
2
+
y
2
)
=
(b) Trace the polar curve r ? 2 ? ?
[7M + 8M]
4 ( x
2
? y
2
)
3 cos ? .
[7M + 8M]
5.(a) Find the perimeter of one loop of the curve 3a y
2
= x ( x ? a)
2
.
(b) Find the volume generated by revolving the area bounded by one loop of the
curve r ? a (1 ? cos ? ) about the initial line.
[7M + 8M]
6.(a)
Evaluate
?
?
? ?
e
? y
dx.dy by changing the order of integration.
y
0 x
x
(b) Evaluate
?
2
?
2 x ? x
2
by changing into polar coordinates.
dy dx
0 0
x
2
+ y
2
[7M + 8M]
7.(a) Find the directional derivative of ? ( x, y , z ) = x y
2
+ y z
3
at the point (2,-1,1) in the
direction of the vector i ? 2 j ? 2k .
(b)
Find
curl [ r f ( r )]
where
r ? xi ? yj ? zk
,
r ?| r |
[7M + 8M]
Page 1 of 2
Set No. 1
Code No: R10102 / R10
8.(a) Compute the line integral ?
( y
2
dx ? x
2
dy)
round the triangle whose vertices are
(1,0),(0,1) and (-1,0) in the xy-plane.
(b) Evaluate the integral I = ? ?
S
x
3
dydz + x
2
y dzdx + x
2
z dxdy using divergence theorem,
where S is the surface consisting of the cylinder x
2
? y
2
? a
2
(0 ? z ? b) and the
circular disks $z=0$ and z ? b ( x
2
? y
2
? a
2
) .
[7M + 8M]
Page 2 of 2
Set No. 2
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the solution of the differential equation
dy
=
xe
y ? x
2
and y(0) = 0 .
dx
(b) A body initially at 80
0
C cools down to 50
0
C in 10 minutes, the temperature of the
air being 40
0
C. What will be the temperature of the body after 20 minutes?
[7M + 8M]
d
2
y
+ 9 y = e
2
x
x
2
2.(a)
Solve
dx
2
(b) Find the general solution of
d
2
y
? 2
dy
+ y = e
x
sin 2x
dx
2
dx
[7M + 8M]
3.(a) Verify whether the functions u = sin
?1
x + sin
?1
y and v = x 1 ? y
2
+ y 1 ? x
2
are
functionally dependent. If so, find the relation between them.[7 M+8 M]
(b) Prove that the rectangular solid of maximum volume that can be inscribed into a
sphere of radius ?a ? is a cube.
[7M + 8M]
4.(a) Trace the parametric curve x = a (cos ? +
1
log tan
2
(
t
) and y = a sin t .
2 2
(b) Trace the lemniscate r
2
= a
2
cos 2 ? .
[7M + 8M]
5.(a) Find the surface area generated by revolving the arc of the curve y = a cosh ( x / c)
from x=0 to x=c about the x-axis.
(b) Find the total length of the lamniscate r
2
= a
2
cos 2? .
[7M + 8M]
6.(a) Find the area of the region which is outside the circle r=1 and inside the cordioid
r = (1 + cos ? )
(b) Evaluate the followingintegralbychanginginto polarcoordinates
? ?
1 ? ( x
+
y
2
)
dx dy over the positive coordinate of the circle x
2
+ y
2
= 1
1 + x
2
+ y
2
[7M + 8M]
Page 1 of 2
Set No. 2
Code No: R10102 / R10
7.(a) Find the directional derivative of the divergence of F = x yi + y z j + z
2
k at the point
(2,1,2) in the direction of the outer normal to the sphere x
2
+ y
2
+ z
2
= 9 .
(b) Find the value of a,b and c such that ( x + y + a z ) i + ( b x + 2 y ? z ) j + ( ? x + c y + 2 z )k is
irrotational.
[7M + 8M]
8.(a) If f = ( x
2
+ y ? 4) i + 3 x y j + (2 xz + z
2
)k and S is the upper half of the sphere
x
2
+ y
2
+ z
2
= 16 . Show by using Stokes theorem that
?
Curl f .n ds = 2 ? a
3
.
S
(b) If S is the surface of the tetrahedron bounded by the planes x = 0 , y = 0 , z = 0 and
ax + by + cz = 1. Show that
?
r .nds =
1
.
S 2abc
[7M + 8M]
Page 2 of 2
Set No. 3
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
( x
2
+ y
2
)
dy
= xy
1.(a) Solve
dx
.
(b) A colony of bacteria is grown under ideal condition in laboratory so that the
population increases exponentially with time. At the end of 3 hours there are 10000
bacteria. At the end of 5 hours there are 40000. How many bacteria were present
initially?
[7M + 8M]
2.(a)
Solve
( D
3
? 6 D
2
+ 11 D ? 6) y = e
?2
x
+ x
3
(b)
Solve
( D
2
+ 1) y = x
2
e
2
x
+ x cos x
.
[7M + 8M]
3.(a) If
u = x + y + z
,
u
2
v = y + z
and
u
3
w
=
z
, then find
?(u
,
v
,
w)
.
?(
x, y , z)
(b) Find the minimum and maximum distances of a point on the curve 2 x
2
+ 4 xy + 4 y
2
? 8 = 0 .
4.(a) Trace the parametric curve
x
=
a
(
t
?
sin
t
)
(b) Trace the curve y
2
( x ? a ) = x
2
( x + a) and
[7M + 8M]
and
y = a (1 + cos t )
a > 0
[7M +
8M] 5.(a) Find the volume of the solid formed by revolving the area bounded by the curve
27 a y
2
= 4 ( x ? 2 a)
3
about x-axis
(b) Find the length of the loop of the curve r = a (1 ? cos ? ) .
[7M + 8M]
6.(a) Find the area of the loop of the curve x
3
+ y
3
= 3a x y , by transforming it into polar
coordinates.
(b) Change the order of integration and evaluate I =
?
0
1
?
x
x
x y dy dx.
[7M + 8M]
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the differential equations of all parabolas with x-axis as its axis and ( ? , 0) as its
focus.
(b) Find the orthogonal trajectories of coaxial circles x
2
+ y
2
+ 2 ? y + c = 2 , where ? is
the parameter.
[7M + 8M]
2.(a)
Solve
( D
2
? 2) y = e
?
2x
+ cos x
d
2
y
+ 4
dy
+ 5 y = 2 sin hx
dy
? 1
Solve
dx
2
dx subject to y=o and
dx
(b) at x=0.
[7M + 8M]
3.(a) If u = xy + yz + zx , v = x
2
+ y
2
+ z
2
and w = x + y + z , verify whether there exists a
possible relationship in between u, v and w. If so find the relation.
(b) Find the minimum value of x
2
+ y
2
+ z
2
on the plane x ? y ? z ? 3 a
4.(a) Trace the curve
x
(
x
2
+
y
2
)
=
(b) Trace the polar curve r ? 2 ? ?
[7M + 8M]
4 ( x
2
? y
2
)
3 cos ? .
[7M + 8M]
5.(a) Find the perimeter of one loop of the curve 3a y
2
= x ( x ? a)
2
.
(b) Find the volume generated by revolving the area bounded by one loop of the
curve r ? a (1 ? cos ? ) about the initial line.
[7M + 8M]
6.(a)
Evaluate
?
?
? ?
e
? y
dx.dy by changing the order of integration.
y
0 x
x
(b) Evaluate
?
2
?
2 x ? x
2
by changing into polar coordinates.
dy dx
0 0
x
2
+ y
2
[7M + 8M]
7.(a) Find the directional derivative of ? ( x, y , z ) = x y
2
+ y z
3
at the point (2,-1,1) in the
direction of the vector i ? 2 j ? 2k .
(b)
Find
curl [ r f ( r )]
where
r ? xi ? yj ? zk
,
r ?| r |
[7M + 8M]
Page 1 of 2
Set No. 1
Code No: R10102 / R10
8.(a) Compute the line integral ?
( y
2
dx ? x
2
dy)
round the triangle whose vertices are
(1,0),(0,1) and (-1,0) in the xy-plane.
(b) Evaluate the integral I = ? ?
S
x
3
dydz + x
2
y dzdx + x
2
z dxdy using divergence theorem,
where S is the surface consisting of the cylinder x
2
? y
2
? a
2
(0 ? z ? b) and the
circular disks $z=0$ and z ? b ( x
2
? y
2
? a
2
) .
[7M + 8M]
Page 2 of 2
Set No. 2
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the solution of the differential equation
dy
=
xe
y ? x
2
and y(0) = 0 .
dx
(b) A body initially at 80
0
C cools down to 50
0
C in 10 minutes, the temperature of the
air being 40
0
C. What will be the temperature of the body after 20 minutes?
[7M + 8M]
d
2
y
+ 9 y = e
2
x
x
2
2.(a)
Solve
dx
2
(b) Find the general solution of
d
2
y
? 2
dy
+ y = e
x
sin 2x
dx
2
dx
[7M + 8M]
3.(a) Verify whether the functions u = sin
?1
x + sin
?1
y and v = x 1 ? y
2
+ y 1 ? x
2
are
functionally dependent. If so, find the relation between them.[7 M+8 M]
(b) Prove that the rectangular solid of maximum volume that can be inscribed into a
sphere of radius ?a ? is a cube.
[7M + 8M]
4.(a) Trace the parametric curve x = a (cos ? +
1
log tan
2
(
t
) and y = a sin t .
2 2
(b) Trace the lemniscate r
2
= a
2
cos 2 ? .
[7M + 8M]
5.(a) Find the surface area generated by revolving the arc of the curve y = a cosh ( x / c)
from x=0 to x=c about the x-axis.
(b) Find the total length of the lamniscate r
2
= a
2
cos 2? .
[7M + 8M]
6.(a) Find the area of the region which is outside the circle r=1 and inside the cordioid
r = (1 + cos ? )
(b) Evaluate the followingintegralbychanginginto polarcoordinates
? ?
1 ? ( x
+
y
2
)
dx dy over the positive coordinate of the circle x
2
+ y
2
= 1
1 + x
2
+ y
2
[7M + 8M]
Page 1 of 2
Set No. 2
Code No: R10102 / R10
7.(a) Find the directional derivative of the divergence of F = x yi + y z j + z
2
k at the point
(2,1,2) in the direction of the outer normal to the sphere x
2
+ y
2
+ z
2
= 9 .
(b) Find the value of a,b and c such that ( x + y + a z ) i + ( b x + 2 y ? z ) j + ( ? x + c y + 2 z )k is
irrotational.
[7M + 8M]
8.(a) If f = ( x
2
+ y ? 4) i + 3 x y j + (2 xz + z
2
)k and S is the upper half of the sphere
x
2
+ y
2
+ z
2
= 16 . Show by using Stokes theorem that
?
Curl f .n ds = 2 ? a
3
.
S
(b) If S is the surface of the tetrahedron bounded by the planes x = 0 , y = 0 , z = 0 and
ax + by + cz = 1. Show that
?
r .nds =
1
.
S 2abc
[7M + 8M]
Page 2 of 2
Set No. 3
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
( x
2
+ y
2
)
dy
= xy
1.(a) Solve
dx
.
(b) A colony of bacteria is grown under ideal condition in laboratory so that the
population increases exponentially with time. At the end of 3 hours there are 10000
bacteria. At the end of 5 hours there are 40000. How many bacteria were present
initially?
[7M + 8M]
2.(a)
Solve
( D
3
? 6 D
2
+ 11 D ? 6) y = e
?2
x
+ x
3
(b)
Solve
( D
2
+ 1) y = x
2
e
2
x
+ x cos x
.
[7M + 8M]
3.(a) If
u = x + y + z
,
u
2
v = y + z
and
u
3
w
=
z
, then find
?(u
,
v
,
w)
.
?(
x, y , z)
(b) Find the minimum and maximum distances of a point on the curve 2 x
2
+ 4 xy + 4 y
2
? 8 = 0 .
4.(a) Trace the parametric curve
x
=
a
(
t
?
sin
t
)
(b) Trace the curve y
2
( x ? a ) = x
2
( x + a) and
[7M + 8M]
and
y = a (1 + cos t )
a > 0
[7M +
8M] 5.(a) Find the volume of the solid formed by revolving the area bounded by the curve
27 a y
2
= 4 ( x ? 2 a)
3
about x-axis
(b) Find the length of the loop of the curve r = a (1 ? cos ? ) .
[7M + 8M]
6.(a) Find the area of the loop of the curve x
3
+ y
3
= 3a x y , by transforming it into polar
coordinates.
(b) Change the order of integration and evaluate I =
?
0
1
?
x
x
x y dy dx.
[7M + 8M]
Page 1 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the differential equations of all parabolas with x-axis as its axis and ( ? , 0) as its
focus.
(b) Find the orthogonal trajectories of coaxial circles x
2
+ y
2
+ 2 ? y + c = 2 , where ? is
the parameter.
[7M + 8M]
2.(a)
Solve
( D
2
? 2) y = e
?
2x
+ cos x
d
2
y
+ 4
dy
+ 5 y = 2 sin hx
dy
? 1
Solve
dx
2
dx subject to y=o and
dx
(b) at x=0.
[7M + 8M]
3.(a) If u = xy + yz + zx , v = x
2
+ y
2
+ z
2
and w = x + y + z , verify whether there exists a
possible relationship in between u, v and w. If so find the relation.
(b) Find the minimum value of x
2
+ y
2
+ z
2
on the plane x ? y ? z ? 3 a
4.(a) Trace the curve
x
(
x
2
+
y
2
)
=
(b) Trace the polar curve r ? 2 ? ?
[7M + 8M]
4 ( x
2
? y
2
)
3 cos ? .
[7M + 8M]
5.(a) Find the perimeter of one loop of the curve 3a y
2
= x ( x ? a)
2
.
(b) Find the volume generated by revolving the area bounded by one loop of the
curve r ? a (1 ? cos ? ) about the initial line.
[7M + 8M]
6.(a)
Evaluate
?
?
? ?
e
? y
dx.dy by changing the order of integration.
y
0 x
x
(b) Evaluate
?
2
?
2 x ? x
2
by changing into polar coordinates.
dy dx
0 0
x
2
+ y
2
[7M + 8M]
7.(a) Find the directional derivative of ? ( x, y , z ) = x y
2
+ y z
3
at the point (2,-1,1) in the
direction of the vector i ? 2 j ? 2k .
(b)
Find
curl [ r f ( r )]
where
r ? xi ? yj ? zk
,
r ?| r |
[7M + 8M]
Page 1 of 2
Set No. 1
Code No: R10102 / R10
8.(a) Compute the line integral ?
( y
2
dx ? x
2
dy)
round the triangle whose vertices are
(1,0),(0,1) and (-1,0) in the xy-plane.
(b) Evaluate the integral I = ? ?
S
x
3
dydz + x
2
y dzdx + x
2
z dxdy using divergence theorem,
where S is the surface consisting of the cylinder x
2
? y
2
? a
2
(0 ? z ? b) and the
circular disks $z=0$ and z ? b ( x
2
? y
2
? a
2
) .
[7M + 8M]
Page 2 of 2
Set No. 2
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the solution of the differential equation
dy
=
xe
y ? x
2
and y(0) = 0 .
dx
(b) A body initially at 80
0
C cools down to 50
0
C in 10 minutes, the temperature of the
air being 40
0
C. What will be the temperature of the body after 20 minutes?
[7M + 8M]
d
2
y
+ 9 y = e
2
x
x
2
2.(a)
Solve
dx
2
(b) Find the general solution of
d
2
y
? 2
dy
+ y = e
x
sin 2x
dx
2
dx
[7M + 8M]
3.(a) Verify whether the functions u = sin
?1
x + sin
?1
y and v = x 1 ? y
2
+ y 1 ? x
2
are
functionally dependent. If so, find the relation between them.[7 M+8 M]
(b) Prove that the rectangular solid of maximum volume that can be inscribed into a
sphere of radius ?a ? is a cube.
[7M + 8M]
4.(a) Trace the parametric curve x = a (cos ? +
1
log tan
2
(
t
) and y = a sin t .
2 2
(b) Trace the lemniscate r
2
= a
2
cos 2 ? .
[7M + 8M]
5.(a) Find the surface area generated by revolving the arc of the curve y = a cosh ( x / c)
from x=0 to x=c about the x-axis.
(b) Find the total length of the lamniscate r
2
= a
2
cos 2? .
[7M + 8M]
6.(a) Find the area of the region which is outside the circle r=1 and inside the cordioid
r = (1 + cos ? )
(b) Evaluate the followingintegralbychanginginto polarcoordinates
? ?
1 ? ( x
+
y
2
)
dx dy over the positive coordinate of the circle x
2
+ y
2
= 1
1 + x
2
+ y
2
[7M + 8M]
Page 1 of 2
Set No. 2
Code No: R10102 / R10
7.(a) Find the directional derivative of the divergence of F = x yi + y z j + z
2
k at the point
(2,1,2) in the direction of the outer normal to the sphere x
2
+ y
2
+ z
2
= 9 .
(b) Find the value of a,b and c such that ( x + y + a z ) i + ( b x + 2 y ? z ) j + ( ? x + c y + 2 z )k is
irrotational.
[7M + 8M]
8.(a) If f = ( x
2
+ y ? 4) i + 3 x y j + (2 xz + z
2
)k and S is the upper half of the sphere
x
2
+ y
2
+ z
2
= 16 . Show by using Stokes theorem that
?
Curl f .n ds = 2 ? a
3
.
S
(b) If S is the surface of the tetrahedron bounded by the planes x = 0 , y = 0 , z = 0 and
ax + by + cz = 1. Show that
?
r .nds =
1
.
S 2abc
[7M + 8M]
Page 2 of 2
Set No. 3
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
( x
2
+ y
2
)
dy
= xy
1.(a) Solve
dx
.
(b) A colony of bacteria is grown under ideal condition in laboratory so that the
population increases exponentially with time. At the end of 3 hours there are 10000
bacteria. At the end of 5 hours there are 40000. How many bacteria were present
initially?
[7M + 8M]
2.(a)
Solve
( D
3
? 6 D
2
+ 11 D ? 6) y = e
?2
x
+ x
3
(b)
Solve
( D
2
+ 1) y = x
2
e
2
x
+ x cos x
.
[7M + 8M]
3.(a) If
u = x + y + z
,
u
2
v = y + z
and
u
3
w
=
z
, then find
?(u
,
v
,
w)
.
?(
x, y , z)
(b) Find the minimum and maximum distances of a point on the curve 2 x
2
+ 4 xy + 4 y
2
? 8 = 0 .
4.(a) Trace the parametric curve
x
=
a
(
t
?
sin
t
)
(b) Trace the curve y
2
( x ? a ) = x
2
( x + a) and
[7M + 8M]
and
y = a (1 + cos t )
a > 0
[7M +
8M] 5.(a) Find the volume of the solid formed by revolving the area bounded by the curve
27 a y
2
= 4 ( x ? 2 a)
3
about x-axis
(b) Find the length of the loop of the curve r = a (1 ? cos ? ) .
[7M + 8M]
6.(a) Find the area of the loop of the curve x
3
+ y
3
= 3a x y , by transforming it into polar
coordinates.
(b) Change the order of integration and evaluate I =
?
0
1
?
x
x
x y dy dx.
[7M + 8M]
Page 1 of 2
Set No. 3
Code No: R10102 / R10
7.(a) In what direction from the point (1, 3, 2) is the directional derivative of ? = 2 x z ? y
2
is maximum and what is its magnitude.
= ( y
2
cos x + z
3
)i + (2 y sin x ? 4) j + (3 x z
2
+ 2)k
(b) Show that F is a conservative force
field and find its scalar potential.
[7M + 8M]
8.(a) Show that F = (2 xy + z
3
) i + x
2
j + 3xz
2
k is a conservative force field. Find the scalar
potential and the work done in moving an object in this field from (1,-2,1) to (3,1,4).
(b)
Verify Green's theorem ,if
Mdx
?
Ndy
is
( xy + y
2
)dx + x
2
dy
with c: closed curve of
the region bounded by
y
?
x
and
y
=
x
2
.
[7M + 8M]
Page 2 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the differential equations of all parabolas with x-axis as its axis and ( ? , 0) as its
focus.
(b) Find the orthogonal trajectories of coaxial circles x
2
+ y
2
+ 2 ? y + c = 2 , where ? is
the parameter.
[7M + 8M]
2.(a)
Solve
( D
2
? 2) y = e
?
2x
+ cos x
d
2
y
+ 4
dy
+ 5 y = 2 sin hx
dy
? 1
Solve
dx
2
dx subject to y=o and
dx
(b) at x=0.
[7M + 8M]
3.(a) If u = xy + yz + zx , v = x
2
+ y
2
+ z
2
and w = x + y + z , verify whether there exists a
possible relationship in between u, v and w. If so find the relation.
(b) Find the minimum value of x
2
+ y
2
+ z
2
on the plane x ? y ? z ? 3 a
4.(a) Trace the curve
x
(
x
2
+
y
2
)
=
(b) Trace the polar curve r ? 2 ? ?
[7M + 8M]
4 ( x
2
? y
2
)
3 cos ? .
[7M + 8M]
5.(a) Find the perimeter of one loop of the curve 3a y
2
= x ( x ? a)
2
.
(b) Find the volume generated by revolving the area bounded by one loop of the
curve r ? a (1 ? cos ? ) about the initial line.
[7M + 8M]
6.(a)
Evaluate
?
?
? ?
e
? y
dx.dy by changing the order of integration.
y
0 x
x
(b) Evaluate
?
2
?
2 x ? x
2
by changing into polar coordinates.
dy dx
0 0
x
2
+ y
2
[7M + 8M]
7.(a) Find the directional derivative of ? ( x, y , z ) = x y
2
+ y z
3
at the point (2,-1,1) in the
direction of the vector i ? 2 j ? 2k .
(b)
Find
curl [ r f ( r )]
where
r ? xi ? yj ? zk
,
r ?| r |
[7M + 8M]
Page 1 of 2
Set No. 1
Code No: R10102 / R10
8.(a) Compute the line integral ?
( y
2
dx ? x
2
dy)
round the triangle whose vertices are
(1,0),(0,1) and (-1,0) in the xy-plane.
(b) Evaluate the integral I = ? ?
S
x
3
dydz + x
2
y dzdx + x
2
z dxdy using divergence theorem,
where S is the surface consisting of the cylinder x
2
? y
2
? a
2
(0 ? z ? b) and the
circular disks $z=0$ and z ? b ( x
2
? y
2
? a
2
) .
[7M + 8M]
Page 2 of 2
Set No. 2
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the solution of the differential equation
dy
=
xe
y ? x
2
and y(0) = 0 .
dx
(b) A body initially at 80
0
C cools down to 50
0
C in 10 minutes, the temperature of the
air being 40
0
C. What will be the temperature of the body after 20 minutes?
[7M + 8M]
d
2
y
+ 9 y = e
2
x
x
2
2.(a)
Solve
dx
2
(b) Find the general solution of
d
2
y
? 2
dy
+ y = e
x
sin 2x
dx
2
dx
[7M + 8M]
3.(a) Verify whether the functions u = sin
?1
x + sin
?1
y and v = x 1 ? y
2
+ y 1 ? x
2
are
functionally dependent. If so, find the relation between them.[7 M+8 M]
(b) Prove that the rectangular solid of maximum volume that can be inscribed into a
sphere of radius ?a ? is a cube.
[7M + 8M]
4.(a) Trace the parametric curve x = a (cos ? +
1
log tan
2
(
t
) and y = a sin t .
2 2
(b) Trace the lemniscate r
2
= a
2
cos 2 ? .
[7M + 8M]
5.(a) Find the surface area generated by revolving the arc of the curve y = a cosh ( x / c)
from x=0 to x=c about the x-axis.
(b) Find the total length of the lamniscate r
2
= a
2
cos 2? .
[7M + 8M]
6.(a) Find the area of the region which is outside the circle r=1 and inside the cordioid
r = (1 + cos ? )
(b) Evaluate the followingintegralbychanginginto polarcoordinates
? ?
1 ? ( x
+
y
2
)
dx dy over the positive coordinate of the circle x
2
+ y
2
= 1
1 + x
2
+ y
2
[7M + 8M]
Page 1 of 2
Set No. 2
Code No: R10102 / R10
7.(a) Find the directional derivative of the divergence of F = x yi + y z j + z
2
k at the point
(2,1,2) in the direction of the outer normal to the sphere x
2
+ y
2
+ z
2
= 9 .
(b) Find the value of a,b and c such that ( x + y + a z ) i + ( b x + 2 y ? z ) j + ( ? x + c y + 2 z )k is
irrotational.
[7M + 8M]
8.(a) If f = ( x
2
+ y ? 4) i + 3 x y j + (2 xz + z
2
)k and S is the upper half of the sphere
x
2
+ y
2
+ z
2
= 16 . Show by using Stokes theorem that
?
Curl f .n ds = 2 ? a
3
.
S
(b) If S is the surface of the tetrahedron bounded by the planes x = 0 , y = 0 , z = 0 and
ax + by + cz = 1. Show that
?
r .nds =
1
.
S 2abc
[7M + 8M]
Page 2 of 2
Set No. 3
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
( x
2
+ y
2
)
dy
= xy
1.(a) Solve
dx
.
(b) A colony of bacteria is grown under ideal condition in laboratory so that the
population increases exponentially with time. At the end of 3 hours there are 10000
bacteria. At the end of 5 hours there are 40000. How many bacteria were present
initially?
[7M + 8M]
2.(a)
Solve
( D
3
? 6 D
2
+ 11 D ? 6) y = e
?2
x
+ x
3
(b)
Solve
( D
2
+ 1) y = x
2
e
2
x
+ x cos x
.
[7M + 8M]
3.(a) If
u = x + y + z
,
u
2
v = y + z
and
u
3
w
=
z
, then find
?(u
,
v
,
w)
.
?(
x, y , z)
(b) Find the minimum and maximum distances of a point on the curve 2 x
2
+ 4 xy + 4 y
2
? 8 = 0 .
4.(a) Trace the parametric curve
x
=
a
(
t
?
sin
t
)
(b) Trace the curve y
2
( x ? a ) = x
2
( x + a) and
[7M + 8M]
and
y = a (1 + cos t )
a > 0
[7M +
8M] 5.(a) Find the volume of the solid formed by revolving the area bounded by the curve
27 a y
2
= 4 ( x ? 2 a)
3
about x-axis
(b) Find the length of the loop of the curve r = a (1 ? cos ? ) .
[7M + 8M]
6.(a) Find the area of the loop of the curve x
3
+ y
3
= 3a x y , by transforming it into polar
coordinates.
(b) Change the order of integration and evaluate I =
?
0
1
?
x
x
x y dy dx.
[7M + 8M]
Page 1 of 2
Set No. 3
Code No: R10102 / R10
7.(a) In what direction from the point (1, 3, 2) is the directional derivative of ? = 2 x z ? y
2
is maximum and what is its magnitude.
= ( y
2
cos x + z
3
)i + (2 y sin x ? 4) j + (3 x z
2
+ 2)k
(b) Show that F is a conservative force
field and find its scalar potential.
[7M + 8M]
8.(a) Show that F = (2 xy + z
3
) i + x
2
j + 3xz
2
k is a conservative force field. Find the scalar
potential and the work done in moving an object in this field from (1,-2,1) to (3,1,4).
(b)
Verify Green's theorem ,if
Mdx
?
Ndy
is
( xy + y
2
)dx + x
2
dy
with c: closed curve of
the region bounded by
y
?
x
and
y
=
x
2
.
[7M + 8M]
Page 2 of 2
Set No. 4
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
x
dy
? y = x x
2
+ y
2
1.(a) Solve
dx
(b) A body is heated to 110
0
C is placed in air at 10
0
C. After 1 hour its temperature is
80
0
C. When will the temperature be 30
0
C?
[7M + 8M]
2.(a)
Solve
( D
2
+ 3 D + 2) y = sin x sin 2x
(b)
Solve
( D
2
+ 2 D ? 3) y = x
3
e
?2
x
.
[7M + 8M]
3.(a) Verify whether the functions u =
x ? y
and v =
x + z
are functionally dependent. If
x + z y + z
so, find the relation in between them.
(b) The temperature T at any point ( x , y , z ) in the space is given as T = 400 x
2
y z . Find
the highest temperature on the surface of the sphere x
2
+ y
2
+ z
2
= 1
[7M + 8M].
4.(a)
Trace the curve
x
3
+ y
3
= 3a x y
(b) Trace the polar curve r = a (1 ? sin ? ) .
[7M + 8M]
5 (a) Find the surface area generated by revolving the arc x
2 /3
+ y
2 /3
= a
2 /3
about x-axis.
(b) Find the volume of the solid generated by revolving the cardioid r = a (1 + cos ? )
about the initial line.
[7M + 8M]
6.(a) Find the area of a plate in the form of a quadrant of an ellipse x
2
/ a
2
+ y
2
/ b
2
= 1 by
changing into polar coordinates.
4 a 2
By changing the order of integration, evaluate the integral
?
a y
(b) ?y
2
d x dy .
0
4a
[7M + 8M]
Page 1 of 2
FirstRanker.com - FirstRanker's Choice
Set No. 1
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the differential equations of all parabolas with x-axis as its axis and ( ? , 0) as its
focus.
(b) Find the orthogonal trajectories of coaxial circles x
2
+ y
2
+ 2 ? y + c = 2 , where ? is
the parameter.
[7M + 8M]
2.(a)
Solve
( D
2
? 2) y = e
?
2x
+ cos x
d
2
y
+ 4
dy
+ 5 y = 2 sin hx
dy
? 1
Solve
dx
2
dx subject to y=o and
dx
(b) at x=0.
[7M + 8M]
3.(a) If u = xy + yz + zx , v = x
2
+ y
2
+ z
2
and w = x + y + z , verify whether there exists a
possible relationship in between u, v and w. If so find the relation.
(b) Find the minimum value of x
2
+ y
2
+ z
2
on the plane x ? y ? z ? 3 a
4.(a) Trace the curve
x
(
x
2
+
y
2
)
=
(b) Trace the polar curve r ? 2 ? ?
[7M + 8M]
4 ( x
2
? y
2
)
3 cos ? .
[7M + 8M]
5.(a) Find the perimeter of one loop of the curve 3a y
2
= x ( x ? a)
2
.
(b) Find the volume generated by revolving the area bounded by one loop of the
curve r ? a (1 ? cos ? ) about the initial line.
[7M + 8M]
6.(a)
Evaluate
?
?
? ?
e
? y
dx.dy by changing the order of integration.
y
0 x
x
(b) Evaluate
?
2
?
2 x ? x
2
by changing into polar coordinates.
dy dx
0 0
x
2
+ y
2
[7M + 8M]
7.(a) Find the directional derivative of ? ( x, y , z ) = x y
2
+ y z
3
at the point (2,-1,1) in the
direction of the vector i ? 2 j ? 2k .
(b)
Find
curl [ r f ( r )]
where
r ? xi ? yj ? zk
,
r ?| r |
[7M + 8M]
Page 1 of 2
Set No. 1
Code No: R10102 / R10
8.(a) Compute the line integral ?
( y
2
dx ? x
2
dy)
round the triangle whose vertices are
(1,0),(0,1) and (-1,0) in the xy-plane.
(b) Evaluate the integral I = ? ?
S
x
3
dydz + x
2
y dzdx + x
2
z dxdy using divergence theorem,
where S is the surface consisting of the cylinder x
2
? y
2
? a
2
(0 ? z ? b) and the
circular disks $z=0$ and z ? b ( x
2
? y
2
? a
2
) .
[7M + 8M]
Page 2 of 2
Set No. 2
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
1.(a) Find the solution of the differential equation
dy
=
xe
y ? x
2
and y(0) = 0 .
dx
(b) A body initially at 80
0
C cools down to 50
0
C in 10 minutes, the temperature of the
air being 40
0
C. What will be the temperature of the body after 20 minutes?
[7M + 8M]
d
2
y
+ 9 y = e
2
x
x
2
2.(a)
Solve
dx
2
(b) Find the general solution of
d
2
y
? 2
dy
+ y = e
x
sin 2x
dx
2
dx
[7M + 8M]
3.(a) Verify whether the functions u = sin
?1
x + sin
?1
y and v = x 1 ? y
2
+ y 1 ? x
2
are
functionally dependent. If so, find the relation between them.[7 M+8 M]
(b) Prove that the rectangular solid of maximum volume that can be inscribed into a
sphere of radius ?a ? is a cube.
[7M + 8M]
4.(a) Trace the parametric curve x = a (cos ? +
1
log tan
2
(
t
) and y = a sin t .
2 2
(b) Trace the lemniscate r
2
= a
2
cos 2 ? .
[7M + 8M]
5.(a) Find the surface area generated by revolving the arc of the curve y = a cosh ( x / c)
from x=0 to x=c about the x-axis.
(b) Find the total length of the lamniscate r
2
= a
2
cos 2? .
[7M + 8M]
6.(a) Find the area of the region which is outside the circle r=1 and inside the cordioid
r = (1 + cos ? )
(b) Evaluate the followingintegralbychanginginto polarcoordinates
? ?
1 ? ( x
+
y
2
)
dx dy over the positive coordinate of the circle x
2
+ y
2
= 1
1 + x
2
+ y
2
[7M + 8M]
Page 1 of 2
Set No. 2
Code No: R10102 / R10
7.(a) Find the directional derivative of the divergence of F = x yi + y z j + z
2
k at the point
(2,1,2) in the direction of the outer normal to the sphere x
2
+ y
2
+ z
2
= 9 .
(b) Find the value of a,b and c such that ( x + y + a z ) i + ( b x + 2 y ? z ) j + ( ? x + c y + 2 z )k is
irrotational.
[7M + 8M]
8.(a) If f = ( x
2
+ y ? 4) i + 3 x y j + (2 xz + z
2
)k and S is the upper half of the sphere
x
2
+ y
2
+ z
2
= 16 . Show by using Stokes theorem that
?
Curl f .n ds = 2 ? a
3
.
S
(b) If S is the surface of the tetrahedron bounded by the planes x = 0 , y = 0 , z = 0 and
ax + by + cz = 1. Show that
?
r .nds =
1
.
S 2abc
[7M + 8M]
Page 2 of 2
Set No. 3
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
( x
2
+ y
2
)
dy
= xy
1.(a) Solve
dx
.
(b) A colony of bacteria is grown under ideal condition in laboratory so that the
population increases exponentially with time. At the end of 3 hours there are 10000
bacteria. At the end of 5 hours there are 40000. How many bacteria were present
initially?
[7M + 8M]
2.(a)
Solve
( D
3
? 6 D
2
+ 11 D ? 6) y = e
?2
x
+ x
3
(b)
Solve
( D
2
+ 1) y = x
2
e
2
x
+ x cos x
.
[7M + 8M]
3.(a) If
u = x + y + z
,
u
2
v = y + z
and
u
3
w
=
z
, then find
?(u
,
v
,
w)
.
?(
x, y , z)
(b) Find the minimum and maximum distances of a point on the curve 2 x
2
+ 4 xy + 4 y
2
? 8 = 0 .
4.(a) Trace the parametric curve
x
=
a
(
t
?
sin
t
)
(b) Trace the curve y
2
( x ? a ) = x
2
( x + a) and
[7M + 8M]
and
y = a (1 + cos t )
a > 0
[7M +
8M] 5.(a) Find the volume of the solid formed by revolving the area bounded by the curve
27 a y
2
= 4 ( x ? 2 a)
3
about x-axis
(b) Find the length of the loop of the curve r = a (1 ? cos ? ) .
[7M + 8M]
6.(a) Find the area of the loop of the curve x
3
+ y
3
= 3a x y , by transforming it into polar
coordinates.
(b) Change the order of integration and evaluate I =
?
0
1
?
x
x
x y dy dx.
[7M + 8M]
Page 1 of 2
Set No. 3
Code No: R10102 / R10
7.(a) In what direction from the point (1, 3, 2) is the directional derivative of ? = 2 x z ? y
2
is maximum and what is its magnitude.
= ( y
2
cos x + z
3
)i + (2 y sin x ? 4) j + (3 x z
2
+ 2)k
(b) Show that F is a conservative force
field and find its scalar potential.
[7M + 8M]
8.(a) Show that F = (2 xy + z
3
) i + x
2
j + 3xz
2
k is a conservative force field. Find the scalar
potential and the work done in moving an object in this field from (1,-2,1) to (3,1,4).
(b)
Verify Green's theorem ,if
Mdx
?
Ndy
is
( xy + y
2
)dx + x
2
dy
with c: closed curve of
the region bounded by
y
?
x
and
y
=
x
2
.
[7M + 8M]
Page 2 of 2
Set No. 4
Code No: R10102 / R10
I B.Tech I Semester Regular/Supplementary Examinations January 2012
MATHEMATICS - I
(Common to all branches)
Time: 3 hours Max Marks: 75
Answer any FIVE Questions
All Questions carry equal marks
*********
x
dy
? y = x x
2
+ y
2
1.(a) Solve
dx
(b) A body is heated to 110
0
C is placed in air at 10
0
C. After 1 hour its temperature is
80
0
C. When will the temperature be 30
0
C?
[7M + 8M]
2.(a)
Solve
( D
2
+ 3 D + 2) y = sin x sin 2x
(b)
Solve
( D
2
+ 2 D ? 3) y = x
3
e
?2
x
.
[7M + 8M]
3.(a) Verify whether the functions u =
x ? y
and v =
x + z
are functionally dependent. If
x + z y + z
so, find the relation in between them.
(b) The temperature T at any point ( x , y , z ) in the space is given as T = 400 x
2
y z . Find
the highest temperature on the surface of the sphere x
2
+ y
2
+ z
2
= 1
[7M + 8M].
4.(a)
Trace the curve
x
3
+ y
3
= 3a x y
(b) Trace the polar curve r = a (1 ? sin ? ) .
[7M + 8M]
5 (a) Find the surface area generated by revolving the arc x
2 /3
+ y
2 /3
= a
2 /3
about x-axis.
(b) Find the volume of the solid generated by revolving the cardioid r = a (1 + cos ? )
about the initial line.
[7M + 8M]
6.(a) Find the area of a plate in the form of a quadrant of an ellipse x
2
/ a
2
+ y
2
/ b
2
= 1 by
changing into polar coordinates.
4 a 2
By changing the order of integration, evaluate the integral
?
a y
(b) ?y
2
d x dy .
0
4a
[7M + 8M]
Page 1 of 2
Set No. 4
Code No: R10102 / R10
7.(a) Find the constants a and b so that the surface a x
2
? b y z = ( a + 2) x will be
orthogonal to the surface 4 x
2
y + z
3
= 4 at the point (1, ?1, 2) .
(b) Determine the constant b such that A = (b x
2
y + y z )i + ( x y
2
? x z
2
) j + (2 x y z
? 2 x
2
y
2
)k has zero divergence.
[7M + 8M]
Evaluate
?
8.(a) f
where f = x
2
i + y
2
j and curve c is the arc of the parabola $y=x^2$ .dr
c
in the xy-plane from (0,0) to (1,1).
(b) Evaluate by Stokes theorem
?
( x + y ) dx + (2 x ? z ) dy + ( y + z )dz , where C is the
C
boundary of the triangle vertices (0,0,0), (1,0,0) and (1,1,0).
[7M + 8M]
Page 2 of 2
FirstRanker.com - FirstRanker's Choice
This post was last modified on 03 December 2019