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

Subject Code: R13102/R13
I B. Tech I Semester Regular Examinations Feb./Mar. - 2014
MATHEMATICS-I
(Common to All Branches)
Time: 3 hours Max. Marks: 70
Question Paper Consists of Part-A and Part-B
Answering the question in Part-A is Compulsory,
Three Questions should be answered from Part-B
*****
PART-A
1.(i) Find the orthogonal trajectories of the curve   1  cos
.
(ii) If  
,   
,    
, find
,,
,,
, given that
,,
,,
 


.
(iii) Find the Laplace transform of !  "
!, 0 \$ ! \$ 1
0, ! % 1
& using Heaviside function.
(iv) Let the heat conduction in a thin metallic bar of length L is governed by the equation

'
(
 


)
'

)
, t > 0. If both ends of the bar are held at constant temperature zero and the bar
is initially has temperature f(x), find the temperature u(x,t).
(v) Solve p
2
+pq = z
2
.
(vi) Find
*
+
)
,-+.-

 . [4+4+4+4+3+3]
PART- B

2.(a) Solve 2

0   1 1   0  1  0
(b) Find the complete solution of 
??
 2 

2
3
 2

cos 2 [8+8]

3.(a) Solve
4
4
 2 
3



(b) Find the solution of
4
)

4
)
 4  6 3  cos 2 . [8+8]

4.(a) Find the Laplace transform of ! 
89: ;(,89: <(
(
.
(b) If  ?>?,   ?@? ,   ?@> and
@  
, >   
 1 ?  
, find
,,
,,
. [8+8]

5.(a) Expand  ,   2

ln1  in powers of x and y using MacLaurin?s Series
(b) Solve 
??
0 8
?
 15   9!2
(
, 0  5 1 
?
0  10 using Laplace transforms
[8+8]

6.(a) Solve    F 0    G 

0 

.
(b) Solve the partial differential equation px+qy =1. [8+8]

7.(a) Find the partial differential equation of all spheres whose centers lie on z- axis.
(b) Find the solution of the wave equation

)
'
(
)
 


)
'

)
, if the initial deflection is
  H
I
J
 0 \$ \$ K/2
I
J
K 0 
J

\$ \$ K
& and initial velocity equal to 0. [8+8]

Page 1 of 1

Set No - 1

FirstRanker.com - FirstRanker's Choice
Subject Code: R13102/R13
I B. Tech I Semester Regular Examinations Feb./Mar. - 2014
MATHEMATICS-I
(Common to All Branches)
Time: 3 hours Max. Marks: 70
Question Paper Consists of Part-A and Part-B
Answering the question in Part-A is Compulsory,
Three Questions should be answered from Part-B
*****
PART-A
1.(i) Find the orthogonal trajectories of the curve   1  cos
.
(ii) If  
,   
,    
, find
,,
,,
, given that
,,
,,
 


.
(iii) Find the Laplace transform of !  "
!, 0 \$ ! \$ 1
0, ! % 1
& using Heaviside function.
(iv) Let the heat conduction in a thin metallic bar of length L is governed by the equation

'
(
 


)
'

)
, t > 0. If both ends of the bar are held at constant temperature zero and the bar
is initially has temperature f(x), find the temperature u(x,t).
(v) Solve p
2
+pq = z
2
.
(vi) Find
*
+
)
,-+.-

 . [4+4+4+4+3+3]
PART- B

2.(a) Solve 2

0   1 1   0  1  0
(b) Find the complete solution of 
??
 2 

2
3
 2

cos 2 [8+8]

3.(a) Solve
4
4
 2 
3



(b) Find the solution of
4
)

4
)
 4  6 3  cos 2 . [8+8]

4.(a) Find the Laplace transform of ! 
89: ;(,89: <(
(
.
(b) If  ?>?,   ?@? ,   ?@> and
@  
, >   
 1 ?  
, find
,,
,,
. [8+8]

5.(a) Expand  ,   2

ln1  in powers of x and y using MacLaurin?s Series
(b) Solve 
??
0 8
?
 15   9!2
(
, 0  5 1 
?
0  10 using Laplace transforms
[8+8]

6.(a) Solve    F 0    G 

0 

.
(b) Solve the partial differential equation px+qy =1. [8+8]

7.(a) Find the partial differential equation of all spheres whose centers lie on z- axis.
(b) Find the solution of the wave equation

)
'
(
)
 


)
'

)
, if the initial deflection is
  H
I
J
 0 \$ \$ K/2
I
J
K 0 
J

\$ \$ K
& and initial velocity equal to 0. [8+8]

Page 1 of 1

Set No - 1
Subject Code: R13102/R13
I B. Tech I Semester Regular Examinations Feb./Mar. - 2014
MATHEMATICS-I
(Common to All Branches)
Time: 3 hours Max. Marks: 70
Question Paper Consists of Part-A and Part-B
Answering the question in Part-A is Compulsory,
Three Questions should be answered from Part-B
*****
PART-A
1.(i) Find the complete solution of M
-
 16   0.
(ii) If  
,   
,   , find
,,
,,
, given that
,,
,,
 .
(iii) Solve

F

 

G

 

.
(iv) Find the solution, by Laplace transform method, of the integro-differential equation

?
 3  2 O ! 1!  !
(
P

(v) Find the differential equation of the orthogonal trajectories for the family of parabola
through the origin and foci on y-axis.
(vi) Find the solution of wave equation in one dimension using the method of separation of
variables.
[3+3+4+4+4+4]
PART-B
2.(a) Solve 

0 2

1  2

0

1  0
(b) Find the complete solution of 
??
 5
?
0 6  4  .
[8+8]
3.(a) Solve cos x dy  ysinx 0 y 1 .
(b) Find the solution of
4
)

4
)
0 4
4
4
 3  2 2
3
 32

cos 2 .
[8+8]
4.(a) Find the Laplace transform of !  O 2
,'
 @ 1@
(
P
.
(b) Find the shortest distance from origin to the surface 

 2.
[8+8]
5.(a) Find
',U
,
if @  2  1 >  

0 

,where  
1   
.
(b) Solve 
??
0 8
?
 15   9!2
(
, 0  5 1 
?
0  10 using Laplace transforms
[8+8]
6.(a) Form the partial differential equation by eliminating the arbitrary function
from        .
(b) Find the solution of VM

0 MM
W
0 2M
W

X   0 1 2

, where M 


and M
W



.
[8+8]
7.(a) Solve the partial differential equation F  G  .
(b) Find the temperature in a bar of length l which is perfectly insulated laterally and whose
ends O and A are kept at 0
o
C, given that the initial temperature at any point P of the rod is
given by f(x).
[8+8]
Page 1 of 1

Set No - 2

FirstRanker.com - FirstRanker's Choice
Subject Code: R13102/R13
I B. Tech I Semester Regular Examinations Feb./Mar. - 2014
MATHEMATICS-I
(Common to All Branches)
Time: 3 hours Max. Marks: 70
Question Paper Consists of Part-A and Part-B
Answering the question in Part-A is Compulsory,
Three Questions should be answered from Part-B
*****
PART-A
1.(i) Find the orthogonal trajectories of the curve   1  cos
.
(ii) If  
,   
,    
, find
,,
,,
, given that
,,
,,
 


.
(iii) Find the Laplace transform of !  "
!, 0 \$ ! \$ 1
0, ! % 1
& using Heaviside function.
(iv) Let the heat conduction in a thin metallic bar of length L is governed by the equation

'
(
 


)
'

)
, t > 0. If both ends of the bar are held at constant temperature zero and the bar
is initially has temperature f(x), find the temperature u(x,t).
(v) Solve p
2
+pq = z
2
.
(vi) Find
*
+
)
,-+.-

 . [4+4+4+4+3+3]
PART- B

2.(a) Solve 2

0   1 1   0  1  0
(b) Find the complete solution of 
??
 2 

2
3
 2

cos 2 [8+8]

3.(a) Solve
4
4
 2 
3



(b) Find the solution of
4
)

4
)
 4  6 3  cos 2 . [8+8]

4.(a) Find the Laplace transform of ! 
89: ;(,89: <(
(
.
(b) If  ?>?,   ?@? ,   ?@> and
@  
, >   
 1 ?  
, find
,,
,,
. [8+8]

5.(a) Expand  ,   2

ln1  in powers of x and y using MacLaurin?s Series
(b) Solve 
??
0 8
?
 15   9!2
(
, 0  5 1 
?
0  10 using Laplace transforms
[8+8]

6.(a) Solve    F 0    G 

0 

.
(b) Solve the partial differential equation px+qy =1. [8+8]

7.(a) Find the partial differential equation of all spheres whose centers lie on z- axis.
(b) Find the solution of the wave equation

)
'
(
)
 


)
'

)
, if the initial deflection is
  H
I
J
 0 \$ \$ K/2
I
J
K 0 
J

\$ \$ K
& and initial velocity equal to 0. [8+8]

Page 1 of 1

Set No - 1
Subject Code: R13102/R13
I B. Tech I Semester Regular Examinations Feb./Mar. - 2014
MATHEMATICS-I
(Common to All Branches)
Time: 3 hours Max. Marks: 70
Question Paper Consists of Part-A and Part-B
Answering the question in Part-A is Compulsory,
Three Questions should be answered from Part-B
*****
PART-A
1.(i) Find the complete solution of M
-
 16   0.
(ii) If  
,   
,   , find
,,
,,
, given that
,,
,,
 .
(iii) Solve

F

 

G

 

.
(iv) Find the solution, by Laplace transform method, of the integro-differential equation

?
 3  2 O ! 1!  !
(
P

(v) Find the differential equation of the orthogonal trajectories for the family of parabola
through the origin and foci on y-axis.
(vi) Find the solution of wave equation in one dimension using the method of separation of
variables.
[3+3+4+4+4+4]
PART-B
2.(a) Solve 

0 2

1  2

0

1  0
(b) Find the complete solution of 
??
 5
?
0 6  4  .
[8+8]
3.(a) Solve cos x dy  ysinx 0 y 1 .
(b) Find the solution of
4
)

4
)
0 4
4
4
 3  2 2
3
 32

cos 2 .
[8+8]
4.(a) Find the Laplace transform of !  O 2
,'
 @ 1@
(
P
.
(b) Find the shortest distance from origin to the surface 

 2.
[8+8]
5.(a) Find
',U
,
if @  2  1 >  

0 

,where  
1   
.
(b) Solve 
??
0 8
?
 15   9!2
(
, 0  5 1 
?
0  10 using Laplace transforms
[8+8]
6.(a) Form the partial differential equation by eliminating the arbitrary function
from        .
(b) Find the solution of VM

0 MM
W
0 2M
W

X   0 1 2

, where M 


and M
W



.
[8+8]
7.(a) Solve the partial differential equation F  G  .
(b) Find the temperature in a bar of length l which is perfectly insulated laterally and whose
ends O and A are kept at 0
o
C, given that the initial temperature at any point P of the rod is
given by f(x).
[8+8]
Page 1 of 1

Set No - 2
Subject Code: R13102/R13
I B. Tech I Semester Regular Examinations Feb./Mar. - 2014
MATHEMATICS-I
(Common to All Branches)
Time: 3 hours Max. Marks: 70
Question Paper Consists of Part-A and Part-B
Answering the question in Part-A is Compulsory,
Three Questions should be answered from Part-B
*****
PART-A
1.(i) Find the dimensions of rectangular box of maximum capacity whose surface area is S.
(ii) Find the orthogonal trajectories of the family of curves
/3
 
/3
 
/3
.
(iii) A generator having emf 100 volts is connected in series with a 10 ohm resistor and an
inductor of 2 henries. If the switch is closed at a time t =0, find the current at time t>0.
(iv) Find the Laplace transform of !  "
!, 0 \$ ! \$ 1
0, ! % 1
& using Heaviside function.
(v) Solve pq+qx = y.
(vi) Find the solution of 2


0 3


 0 by the method of separation of variables.
[4+4+4+4+3+3]
PART- B
2.(a) Solve 1   1  1 0  1  0
(b) Find the complete solution of 
??
 4  2



.
[8+8]
3.(a) Solve 2x y
?
 y 
Y
)
Z
[
, y1  2.
(b) Find the solution of
4
)

4
)
0 4
4
4
0 5  2

 3 cos4  3 .
[8+8]
4.(a) Find the Laplace transform of !  !2
,(
 !.
(b) Find the maxima and minima of
3
 3 

0 15

0 15 

 72 .
[8+8]
5.(a) Expand  ,   2

in powers of (x-1) and (y-1).
(b) Solve 
??
 7
?
 10   42
,3(
, 0  0 1 
?
0  01 using Laplace transforms.
[8+8]
6.(a) Form the partial differential equation by eliminating the arbitrary constants ?a? and ?b?
from 2 

)
;
)


)
<
)
.
(b) Find the solution of V4M

 12MM
W
 9M
W

X  2
3,
, where M 


and M
W



.
[8+8]
7.(a) Solve the partial differential equation F tan  G tan   tan .
(b) A tightly stretched string with fixed end points x=0 and x=l is initially in a position given
by   
P

3
_
J
. If it is released from rest from this position, find the displacement
 , ! .
[8+8]

Page 1 of 1

Set No - 3

FirstRanker.com - FirstRanker's Choice
Subject Code: R13102/R13
I B. Tech I Semester Regular Examinations Feb./Mar. - 2014
MATHEMATICS-I
(Common to All Branches)
Time: 3 hours Max. Marks: 70
Question Paper Consists of Part-A and Part-B
Answering the question in Part-A is Compulsory,
Three Questions should be answered from Part-B
*****
PART-A
1.(i) Find the orthogonal trajectories of the curve   1  cos
.
(ii) If  
,   
,    
, find
,,
,,
, given that
,,
,,
 


.
(iii) Find the Laplace transform of !  "
!, 0 \$ ! \$ 1
0, ! % 1
& using Heaviside function.
(iv) Let the heat conduction in a thin metallic bar of length L is governed by the equation

'
(
 


)
'

)
, t > 0. If both ends of the bar are held at constant temperature zero and the bar
is initially has temperature f(x), find the temperature u(x,t).
(v) Solve p
2
+pq = z
2
.
(vi) Find
*
+
)
,-+.-

 . [4+4+4+4+3+3]
PART- B

2.(a) Solve 2

0   1 1   0  1  0
(b) Find the complete solution of 
??
 2 

2
3
 2

cos 2 [8+8]

3.(a) Solve
4
4
 2 
3



(b) Find the solution of
4
)

4
)
 4  6 3  cos 2 . [8+8]

4.(a) Find the Laplace transform of ! 
89: ;(,89: <(
(
.
(b) If  ?>?,   ?@? ,   ?@> and
@  
, >   
 1 ?  
, find
,,
,,
. [8+8]

5.(a) Expand  ,   2

ln1  in powers of x and y using MacLaurin?s Series
(b) Solve 
??
0 8
?
 15   9!2
(
, 0  5 1 
?
0  10 using Laplace transforms
[8+8]

6.(a) Solve    F 0    G 

0 

.
(b) Solve the partial differential equation px+qy =1. [8+8]

7.(a) Find the partial differential equation of all spheres whose centers lie on z- axis.
(b) Find the solution of the wave equation

)
'
(
)
 


)
'

)
, if the initial deflection is
  H
I
J
 0 \$ \$ K/2
I
J
K 0 
J

\$ \$ K
& and initial velocity equal to 0. [8+8]

Page 1 of 1

Set No - 1
Subject Code: R13102/R13
I B. Tech I Semester Regular Examinations Feb./Mar. - 2014
MATHEMATICS-I
(Common to All Branches)
Time: 3 hours Max. Marks: 70
Question Paper Consists of Part-A and Part-B
Answering the question in Part-A is Compulsory,
Three Questions should be answered from Part-B
*****
PART-A
1.(i) Find the complete solution of M
-
 16   0.
(ii) If  
,   
,   , find
,,
,,
, given that
,,
,,
 .
(iii) Solve

F

 

G

 

.
(iv) Find the solution, by Laplace transform method, of the integro-differential equation

?
 3  2 O ! 1!  !
(
P

(v) Find the differential equation of the orthogonal trajectories for the family of parabola
through the origin and foci on y-axis.
(vi) Find the solution of wave equation in one dimension using the method of separation of
variables.
[3+3+4+4+4+4]
PART-B
2.(a) Solve 

0 2

1  2

0

1  0
(b) Find the complete solution of 
??
 5
?
0 6  4  .
[8+8]
3.(a) Solve cos x dy  ysinx 0 y 1 .
(b) Find the solution of
4
)

4
)
0 4
4
4
 3  2 2
3
 32

cos 2 .
[8+8]
4.(a) Find the Laplace transform of !  O 2
,'
 @ 1@
(
P
.
(b) Find the shortest distance from origin to the surface 

 2.
[8+8]
5.(a) Find
',U
,
if @  2  1 >  

0 

,where  
1   
.
(b) Solve 
??
0 8
?
 15   9!2
(
, 0  5 1 
?
0  10 using Laplace transforms
[8+8]
6.(a) Form the partial differential equation by eliminating the arbitrary function
from        .
(b) Find the solution of VM

0 MM
W
0 2M
W

X   0 1 2

, where M 


and M
W



.
[8+8]
7.(a) Solve the partial differential equation F  G  .
(b) Find the temperature in a bar of length l which is perfectly insulated laterally and whose
ends O and A are kept at 0
o
C, given that the initial temperature at any point P of the rod is
given by f(x).
[8+8]
Page 1 of 1

Set No - 2
Subject Code: R13102/R13
I B. Tech I Semester Regular Examinations Feb./Mar. - 2014
MATHEMATICS-I
(Common to All Branches)
Time: 3 hours Max. Marks: 70
Question Paper Consists of Part-A and Part-B
Answering the question in Part-A is Compulsory,
Three Questions should be answered from Part-B
*****
PART-A
1.(i) Find the dimensions of rectangular box of maximum capacity whose surface area is S.
(ii) Find the orthogonal trajectories of the family of curves
/3
 
/3
 
/3
.
(iii) A generator having emf 100 volts is connected in series with a 10 ohm resistor and an
inductor of 2 henries. If the switch is closed at a time t =0, find the current at time t>0.
(iv) Find the Laplace transform of !  "
!, 0 \$ ! \$ 1
0, ! % 1
& using Heaviside function.
(v) Solve pq+qx = y.
(vi) Find the solution of 2


0 3


 0 by the method of separation of variables.
[4+4+4+4+3+3]
PART- B
2.(a) Solve 1   1  1 0  1  0
(b) Find the complete solution of 
??
 4  2



.
[8+8]
3.(a) Solve 2x y
?
 y 
Y
)
Z
[
, y1  2.
(b) Find the solution of
4
)

4
)
0 4
4
4
0 5  2

 3 cos4  3 .
[8+8]
4.(a) Find the Laplace transform of !  !2
,(
 !.
(b) Find the maxima and minima of
3
 3 

0 15

0 15 

 72 .
[8+8]
5.(a) Expand  ,   2

in powers of (x-1) and (y-1).
(b) Solve 
??
 7
?
 10   42
,3(
, 0  0 1 
?
0  01 using Laplace transforms.
[8+8]
6.(a) Form the partial differential equation by eliminating the arbitrary constants ?a? and ?b?
from 2 

)
;
)


)
<
)
.
(b) Find the solution of V4M

 12MM
W
 9M
W

X  2
3,
, where M 


and M
W



.
[8+8]
7.(a) Solve the partial differential equation F tan  G tan   tan .
(b) A tightly stretched string with fixed end points x=0 and x=l is initially in a position given
by   
P

3
_
J
. If it is released from rest from this position, find the displacement
 , ! .
[8+8]

Page 1 of 1

Set No - 3
Subject Code: R13102/R13
I B. Tech I Semester Regular Examinations Feb./Mar. - 2014
MATHEMATICS-I
(Common to All Branches)
Time: 3 hours Max. Marks: 70
Question Paper Consists of Part-A and Part-B
Answering the question in Part-A is Compulsory,
Three Questions should be answered from Part-B
*****
PART-A
1.(i) Find the distance from the centre at which the velocity in simple harmonic motion
will be 1/3rd of the maximum.
(ii) Find a point with in a triangle such that the sum of the squares of its distances from the
three vertices is minimum.
(iii) Find the solution, by Laplace transform method, of the integro-differential
equation 
?
 4  O ! 1!
(
P
, 0  0.
(iv) Uranium disintegrates at a rate proportional to the amount present at that time. If M and N
grams of Uranium that rae present at times T
1
and T
2
respectively, find the half life of
Uranium.
(v) Find the complete solution of

M
3
0 3M
2
M
W
 3 MM
W 2
0 M`
3
  0.
(vi) Solve 

 1  F

 G

.
[4+4+4+4+3+3]
PART- B
2.(a) Solve 3

 4  0 1    2 1  0
(b) Find the solution of
4
)

4
)
 5
4
4
0 6  4  .
[8+8]
3.(a) Find the complete solution of "  2 

2
3
 2

cos 2 .
(b) Solve x z?  zlogz  zlogz

.
[8+8]
4.(a) Find the Laplace transform of !  !2
(
 2!.
(b) If @  
,*


[
.
[
? .
?

, prove that @

 @


d

tan @.
[8+8]
5.(a) If ?   0   0  0  , find the value of
e


e


e

.
(b) Solve 
??
 2
?
 5   2
,(
sin ! , 0  0 1 
?
0  1 using Laplace transforms.
[8+8]
6.(a) Form the partial differential equation by eliminating the arbitrary constants ?a? and ?b?
from     f  

 f

.
(b) Using method of separation of variables, solve @
(
 2
,(
 with @ , 0  @0, !  0.
[8+8]
7.(a) Find the temperature in a thin metal rod of length L, with both ends insulated and with
initial temperature in the rod is 
_
g
.
(b) Solve the partial differential equation F x

 qy

 z

.
[8+8]
Page 1 of 1
Set No - 4

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