# JNTU Kakinada B-Tech 1-2 R161203 MATHEMATICS-III R16 May 2018 Question Paper

JNTU Kakinada (JNTUK) B-Tech First Year Second Semester (1-2) MATHEMATICS-III Common to to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE R16 Regulation May 2018 Question Paper

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE)Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to reduce the given matrix into Echelon form. (2M)
b)Find the Eigen value of the matrix A =
4 2
1 5
? ?
? ?
? ?
. (2M)
c) Find the point of the curve r = a (1 + cos ?) where tangent coincide with the radius
vector. (2M)
d) Evaluate
??
+
2
1
4
3
y
dxdy e xy ) ( (2M)
e) Show that ( 1) ( ) n n n G + = G for ` 0 n > (2M)
3 3
3 x y xyz f = + + at (1,1, 2) – (2M)
g) Find the work done in moving particle in the force field
2
3 F x i j zk = + + along the
straight line (0, 0, 0) to (2, 1, 3). (2M)
PART -B

2.

a) Reduce the matrix
3 2 0 1
0 2 2 1
1 2 3 2
0 1 2 1
A
– – ? ?
? ?
? ?
=
? ? – –
? ?
? ?
in to normal form hence find the rank.
(7M)
b) If consistent, solve the system of equations.
x + y + z + t = 4
x ? z + 2t = 2
y + z ? 3t = ?1
x + 2y ? z + t = 3. (7M)
3. a) Determine the diagonal matrix orthogonally similar to the matrix.
A =
6 2 2
2 3 1
2 1 3
– ? ?
? ?
– –
? ?
? ? –
? ?(7M)
b) Find the Nature , index and signature of the quadratic form
2 2 2
10 2 5 4 10 6 x y z xy xz yz + + – – +

(7M)
SET – 1
R16
1 of 2

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE)Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to reduce the given matrix into Echelon form. (2M)
b)Find the Eigen value of the matrix A =
4 2
1 5
? ?
? ?
? ?
. (2M)
c) Find the point of the curve r = a (1 + cos ?) where tangent coincide with the radius
vector. (2M)
d) Evaluate
??
+
2
1
4
3
y
dxdy e xy ) ( (2M)
e) Show that ( 1) ( ) n n n G + = G for ` 0 n > (2M)
3 3
3 x y xyz f = + + at (1,1, 2) – (2M)
g) Find the work done in moving particle in the force field
2
3 F x i j zk = + + along the
straight line (0, 0, 0) to (2, 1, 3). (2M)
PART -B

2.

a) Reduce the matrix
3 2 0 1
0 2 2 1
1 2 3 2
0 1 2 1
A
– – ? ?
? ?
? ?
=
? ? – –
? ?
? ?
in to normal form hence find the rank.
(7M)
b) If consistent, solve the system of equations.
x + y + z + t = 4
x ? z + 2t = 2
y + z ? 3t = ?1
x + 2y ? z + t = 3. (7M)
3. a) Determine the diagonal matrix orthogonally similar to the matrix.
A =
6 2 2
2 3 1
2 1 3
– ? ?
? ?
– –
? ?
? ? –
? ?(7M)
b) Find the Nature , index and signature of the quadratic form
2 2 2
10 2 5 4 10 6 x y z xy xz yz + + – – +

(7M)
SET – 1
R16
1 of 2

Code No: R161203

4. a)By change of order of integration evaluate
2 2
0( )
a a
x
x y dy dx +
? ?(7M)

b)Evaluate
2 2
sin? ( )/ p/2
0 0 0
?
a a r a
r dr d dz

? ? ?

(7M)
5.
a)Evaluate
2
4
0
3
x
dx
8

?
(7M)

b)Show that
? ?
8 8
p
= =
0 0
2 2
2 2
1
dx x dx x cos sin
(7M)
6.
a)Show that ( )
n
f r a r = ? is solenoidal where =

?+

?+

and
? =
?+ ?+

(7M)

b)Prove that
3 4
1 3
r
r r
? ? ? ?
? ? =
? ? ? ?
? ? ? ?

(7M)
7. a) Verify stoke?s theorem for F y i z j x k = + + for the upper part of the sphere
x
2
+ y
2
+ z
2
= 1. (7M)
b)Verify Green?s theorem in the plane for
?
+ +
c
2 2
dy x dx y xy ) ( . Where c is the
closed curve of the region bounded by y=x & y=x
2 (7M)

2 of 2
SET – 1
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE)Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to reduce the given matrix into Echelon form. (2M)
b)Find the Eigen value of the matrix A =
4 2
1 5
? ?
? ?
? ?
. (2M)
c) Find the point of the curve r = a (1 + cos ?) where tangent coincide with the radius
vector. (2M)
d) Evaluate
??
+
2
1
4
3
y
dxdy e xy ) ( (2M)
e) Show that ( 1) ( ) n n n G + = G for ` 0 n > (2M)
3 3
3 x y xyz f = + + at (1,1, 2) – (2M)
g) Find the work done in moving particle in the force field
2
3 F x i j zk = + + along the
straight line (0, 0, 0) to (2, 1, 3). (2M)
PART -B

2.

a) Reduce the matrix
3 2 0 1
0 2 2 1
1 2 3 2
0 1 2 1
A
– – ? ?
? ?
? ?
=
? ? – –
? ?
? ?
in to normal form hence find the rank.
(7M)
b) If consistent, solve the system of equations.
x + y + z + t = 4
x ? z + 2t = 2
y + z ? 3t = ?1
x + 2y ? z + t = 3. (7M)
3. a) Determine the diagonal matrix orthogonally similar to the matrix.
A =
6 2 2
2 3 1
2 1 3
– ? ?
? ?
– –
? ?
? ? –
? ?(7M)
b) Find the Nature , index and signature of the quadratic form
2 2 2
10 2 5 4 10 6 x y z xy xz yz + + – – +

(7M)
SET – 1
R16
1 of 2

Code No: R161203

4. a)By change of order of integration evaluate
2 2
0( )
a a
x
x y dy dx +
? ?(7M)

b)Evaluate
2 2
sin? ( )/ p/2
0 0 0
?
a a r a
r dr d dz

? ? ?

(7M)
5.
a)Evaluate
2
4
0
3
x
dx
8

?
(7M)

b)Show that
? ?
8 8
p
= =
0 0
2 2
2 2
1
dx x dx x cos sin
(7M)
6.
a)Show that ( )
n
f r a r = ? is solenoidal where =

?+

?+

and
? =
?+ ?+

(7M)

b)Prove that
3 4
1 3
r
r r
? ? ? ?
? ? =
? ? ? ?
? ? ? ?

(7M)
7. a) Verify stoke?s theorem for F y i z j x k = + + for the upper part of the sphere
x
2
+ y
2
+ z
2
= 1. (7M)
b)Verify Green?s theorem in the plane for
?
+ +
c
2 2
dy x dx y xy ) ( . Where c is the
closed curve of the region bounded by y=x & y=x
2 (7M)

2 of 2
SET – 1
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE) Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to reduce the given matrix into Normal form. (2M)
2 2 2
3 3 2 x y z yz + + – (2M)
c) Write the tangents at the origin of the curve a
2
y
2
= x
2(a
2
? x
2
). (2M)
d)Evaluate
1 1 1
0 0 0
dx dy dz
???(2M)
e)Prove that
0
2
2 1 2 1
sin cos ( , )
m n
d m n
p
? ? ? ?
– –
=
?(2M)
f)Find the maximum value of the directional derivative of
2 4
2x y z f = – – at (2, 1,1) – (2M)
g) Write Stoke?s theorem. (2M)
PART -B

2.

a) For what value of k the matrix A =
4 4 3 1
1 1 1 0
2 2 2
9 9 3
k
k
– ? ?
? ?

? ?
? ?
? ?
? ?
has rank 3.
(7M)

b)Solve the following system of equations
8 3 2 20
4 11 33
6 3 12 35
x y z
x y z
x y z
– + =
+ – =
+ + =
by using.
Gauss ? Seidel method.
(7M)
3. a) Determine the characteristic roots and the corresponding characteristic vectors of
the matrix.
A =
3 10 5
2 3 4
3 5 7
? ?
? ?
– – –
? ?
? ?
? ?(7M)
b) Find the Nature , index and signature of the quadratic form
2 2 2
4 3 8 4 6 x y z xy xz yz + + – + –

(7M)
SET – 2
R16
1 of 2

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE)Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to reduce the given matrix into Echelon form. (2M)
b)Find the Eigen value of the matrix A =
4 2
1 5
? ?
? ?
? ?
. (2M)
c) Find the point of the curve r = a (1 + cos ?) where tangent coincide with the radius
vector. (2M)
d) Evaluate
??
+
2
1
4
3
y
dxdy e xy ) ( (2M)
e) Show that ( 1) ( ) n n n G + = G for ` 0 n > (2M)
3 3
3 x y xyz f = + + at (1,1, 2) – (2M)
g) Find the work done in moving particle in the force field
2
3 F x i j zk = + + along the
straight line (0, 0, 0) to (2, 1, 3). (2M)
PART -B

2.

a) Reduce the matrix
3 2 0 1
0 2 2 1
1 2 3 2
0 1 2 1
A
– – ? ?
? ?
? ?
=
? ? – –
? ?
? ?
in to normal form hence find the rank.
(7M)
b) If consistent, solve the system of equations.
x + y + z + t = 4
x ? z + 2t = 2
y + z ? 3t = ?1
x + 2y ? z + t = 3. (7M)
3. a) Determine the diagonal matrix orthogonally similar to the matrix.
A =
6 2 2
2 3 1
2 1 3
– ? ?
? ?
– –
? ?
? ? –
? ?(7M)
b) Find the Nature , index and signature of the quadratic form
2 2 2
10 2 5 4 10 6 x y z xy xz yz + + – – +

(7M)
SET – 1
R16
1 of 2

Code No: R161203

4. a)By change of order of integration evaluate
2 2
0( )
a a
x
x y dy dx +
? ?(7M)

b)Evaluate
2 2
sin? ( )/ p/2
0 0 0
?
a a r a
r dr d dz

? ? ?

(7M)
5.
a)Evaluate
2
4
0
3
x
dx
8

?
(7M)

b)Show that
? ?
8 8
p
= =
0 0
2 2
2 2
1
dx x dx x cos sin
(7M)
6.
a)Show that ( )
n
f r a r = ? is solenoidal where =

?+

?+

and
? =
?+ ?+

(7M)

b)Prove that
3 4
1 3
r
r r
? ? ? ?
? ? =
? ? ? ?
? ? ? ?

(7M)
7. a) Verify stoke?s theorem for F y i z j x k = + + for the upper part of the sphere
x
2
+ y
2
+ z
2
= 1. (7M)
b)Verify Green?s theorem in the plane for
?
+ +
c
2 2
dy x dx y xy ) ( . Where c is the
closed curve of the region bounded by y=x & y=x
2 (7M)

2 of 2
SET – 1
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE) Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to reduce the given matrix into Normal form. (2M)
2 2 2
3 3 2 x y z yz + + – (2M)
c) Write the tangents at the origin of the curve a
2
y
2
= x
2(a
2
? x
2
). (2M)
d)Evaluate
1 1 1
0 0 0
dx dy dz
???(2M)
e)Prove that
0
2
2 1 2 1
sin cos ( , )
m n
d m n
p
? ? ? ?
– –
=
?(2M)
f)Find the maximum value of the directional derivative of
2 4
2x y z f = – – at (2, 1,1) – (2M)
g) Write Stoke?s theorem. (2M)
PART -B

2.

a) For what value of k the matrix A =
4 4 3 1
1 1 1 0
2 2 2
9 9 3
k
k
– ? ?
? ?

? ?
? ?
? ?
? ?
has rank 3.
(7M)

b)Solve the following system of equations
8 3 2 20
4 11 33
6 3 12 35
x y z
x y z
x y z
– + =
+ – =
+ + =
by using.
Gauss ? Seidel method.
(7M)
3. a) Determine the characteristic roots and the corresponding characteristic vectors of
the matrix.
A =
3 10 5
2 3 4
3 5 7
? ?
? ?
– – –
? ?
? ?
? ?(7M)
b) Find the Nature , index and signature of the quadratic form
2 2 2
4 3 8 4 6 x y z xy xz yz + + – + –

(7M)
SET – 2
R16
1 of 2

Code No: R161203

4. a) Trace the curve r
2
= a
2
cos 2 ? (7M)

b)Evaluate
2 2( ) x y dx dy +
? ?
over the area bounded by the Ellipse
2 2
2 2
1
x y
a b
+ =
(7M)
5.
a)Evaluate
2
0
0, 1
bx
a dx b a
8

> >
?

(7M)

b)Show that ( )
1
1
0( 1) !
log( 1)
n
n
m
n
n
x x dx
m
+

=
+
?
(7M)
6. a)Find the constants ?a? and ?b? such that the surfaces 5x
2
-2yz-9x=0 and ax
2
y+bz
3
=4
cuts orthogonally at (1,-1,2)
(7M)
b)Show that the vector ( ) ( ) ( )
2 2 2
x yz i y zx j z xy k – + – + – is irrotational and find
its scalar potential.
(7M)
7. a) If f
%
=
2(3 2 ) 4 5 x z i xy j xk – – – Evaluate
V
Cur F dv
?
, where v is volume bounded by
the planes x = 0; y = 0; z = 0 and 3x + 2y ? 3z = 6. (7M)
b)Evaluate cos (1 sin ) over
c
y dx x y dy a + –
?
closed curve c given by x
2
+ y
2
= 1; z = 0
using Green?s theorem. (7M)

2 of 2
SET – 2
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE)Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to reduce the given matrix into Echelon form. (2M)
b)Find the Eigen value of the matrix A =
4 2
1 5
? ?
? ?
? ?
. (2M)
c) Find the point of the curve r = a (1 + cos ?) where tangent coincide with the radius
vector. (2M)
d) Evaluate
??
+
2
1
4
3
y
dxdy e xy ) ( (2M)
e) Show that ( 1) ( ) n n n G + = G for ` 0 n > (2M)
3 3
3 x y xyz f = + + at (1,1, 2) – (2M)
g) Find the work done in moving particle in the force field
2
3 F x i j zk = + + along the
straight line (0, 0, 0) to (2, 1, 3). (2M)
PART -B

2.

a) Reduce the matrix
3 2 0 1
0 2 2 1
1 2 3 2
0 1 2 1
A
– – ? ?
? ?
? ?
=
? ? – –
? ?
? ?
in to normal form hence find the rank.
(7M)
b) If consistent, solve the system of equations.
x + y + z + t = 4
x ? z + 2t = 2
y + z ? 3t = ?1
x + 2y ? z + t = 3. (7M)
3. a) Determine the diagonal matrix orthogonally similar to the matrix.
A =
6 2 2
2 3 1
2 1 3
– ? ?
? ?
– –
? ?
? ? –
? ?(7M)
b) Find the Nature , index and signature of the quadratic form
2 2 2
10 2 5 4 10 6 x y z xy xz yz + + – – +

(7M)
SET – 1
R16
1 of 2

Code No: R161203

4. a)By change of order of integration evaluate
2 2
0( )
a a
x
x y dy dx +
? ?(7M)

b)Evaluate
2 2
sin? ( )/ p/2
0 0 0
?
a a r a
r dr d dz

? ? ?

(7M)
5.
a)Evaluate
2
4
0
3
x
dx
8

?
(7M)

b)Show that
? ?
8 8
p
= =
0 0
2 2
2 2
1
dx x dx x cos sin
(7M)
6.
a)Show that ( )
n
f r a r = ? is solenoidal where =

?+

?+

and
? =
?+ ?+

(7M)

b)Prove that
3 4
1 3
r
r r
? ? ? ?
? ? =
? ? ? ?
? ? ? ?

(7M)
7. a) Verify stoke?s theorem for F y i z j x k = + + for the upper part of the sphere
x
2
+ y
2
+ z
2
= 1. (7M)
b)Verify Green?s theorem in the plane for
?
+ +
c
2 2
dy x dx y xy ) ( . Where c is the
closed curve of the region bounded by y=x & y=x
2 (7M)

2 of 2
SET – 1
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE) Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to reduce the given matrix into Normal form. (2M)
2 2 2
3 3 2 x y z yz + + – (2M)
c) Write the tangents at the origin of the curve a
2
y
2
= x
2(a
2
? x
2
). (2M)
d)Evaluate
1 1 1
0 0 0
dx dy dz
???(2M)
e)Prove that
0
2
2 1 2 1
sin cos ( , )
m n
d m n
p
? ? ? ?
– –
=
?(2M)
f)Find the maximum value of the directional derivative of
2 4
2x y z f = – – at (2, 1,1) – (2M)
g) Write Stoke?s theorem. (2M)
PART -B

2.

a) For what value of k the matrix A =
4 4 3 1
1 1 1 0
2 2 2
9 9 3
k
k
– ? ?
? ?

? ?
? ?
? ?
? ?
has rank 3.
(7M)

b)Solve the following system of equations
8 3 2 20
4 11 33
6 3 12 35
x y z
x y z
x y z
– + =
+ – =
+ + =
by using.
Gauss ? Seidel method.
(7M)
3. a) Determine the characteristic roots and the corresponding characteristic vectors of
the matrix.
A =
3 10 5
2 3 4
3 5 7
? ?
? ?
– – –
? ?
? ?
? ?(7M)
b) Find the Nature , index and signature of the quadratic form
2 2 2
4 3 8 4 6 x y z xy xz yz + + – + –

(7M)
SET – 2
R16
1 of 2

Code No: R161203

4. a) Trace the curve r
2
= a
2
cos 2 ? (7M)

b)Evaluate
2 2( ) x y dx dy +
? ?
over the area bounded by the Ellipse
2 2
2 2
1
x y
a b
+ =
(7M)
5.
a)Evaluate
2
0
0, 1
bx
a dx b a
8

> >
?

(7M)

b)Show that ( )
1
1
0( 1) !
log( 1)
n
n
m
n
n
x x dx
m
+

=
+
?
(7M)
6. a)Find the constants ?a? and ?b? such that the surfaces 5x
2
-2yz-9x=0 and ax
2
y+bz
3
=4
cuts orthogonally at (1,-1,2)
(7M)
b)Show that the vector ( ) ( ) ( )
2 2 2
x yz i y zx j z xy k – + – + – is irrotational and find
its scalar potential.
(7M)
7. a) If f
%
=
2(3 2 ) 4 5 x z i xy j xk – – – Evaluate
V
Cur F dv
?
, where v is volume bounded by
the planes x = 0; y = 0; z = 0 and 3x + 2y ? 3z = 6. (7M)
b)Evaluate cos (1 sin ) over
c
y dx x y dy a + –
?
closed curve c given by x
2
+ y
2
= 1; z = 0
using Green?s theorem. (7M)

2 of 2
SET – 2
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE)Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to find the inverse of the given matrix by Jordan
method. (2M)
b) Find the Eigen value of Adj A if the ??? is the Eigen value of A. (2M)
c) Write the symmetry of the curve y
2
(2a ? x) = x
3
(2M)

d)Evaluate
? ?

3
0
x
x
dy dx xy (2M)
e)Find the value of
1 1
,
2 2
?
? ?
? ?
? ?(2M)
f)Find the angle between the surfaces
2 2 2
9 x y z + + = and
2 2
3 z x y = + – at the
point( ) 2, 1, 2 – . (2M)
g) Write the physical interpretation of Gauss divergence theorem. (2M)
PART -B

2.

a) Reduce the matrix to Echelon form and find its rank
2 1 3 4
0 3 4 1
2 3 7 5
2 5 11 6
– ? ?
? ?
? ?
? ?
? ?
? ?
(7M)

b)Solve the equations
10 12,
2 10 13,
5 7.
x y z
x y z
x y z
+ + =
+ + =
+ + =
by Gauss ? Jordan method.
(7M)
3. a) Find the Natural frequencies and normal modes of vibrating system for which
mass
1 0
0 2
M
? ?
=
? ?
? ?
and stiffness
2 1
1 3
K
? ?
=
? ?
? ?(7M)

b)Verify Cayley-Hamilton theorem for the matrix A =
1 2 3
2 4 5
3 5 6
? ?
? ?
? ?
? ?
? ?
. Hence find A
?1

(7M)
SET – 3
R16
1 of 2

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE)Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to reduce the given matrix into Echelon form. (2M)
b)Find the Eigen value of the matrix A =
4 2
1 5
? ?
? ?
? ?
. (2M)
c) Find the point of the curve r = a (1 + cos ?) where tangent coincide with the radius
vector. (2M)
d) Evaluate
??
+
2
1
4
3
y
dxdy e xy ) ( (2M)
e) Show that ( 1) ( ) n n n G + = G for ` 0 n > (2M)
3 3
3 x y xyz f = + + at (1,1, 2) – (2M)
g) Find the work done in moving particle in the force field
2
3 F x i j zk = + + along the
straight line (0, 0, 0) to (2, 1, 3). (2M)
PART -B

2.

a) Reduce the matrix
3 2 0 1
0 2 2 1
1 2 3 2
0 1 2 1
A
– – ? ?
? ?
? ?
=
? ? – –
? ?
? ?
in to normal form hence find the rank.
(7M)
b) If consistent, solve the system of equations.
x + y + z + t = 4
x ? z + 2t = 2
y + z ? 3t = ?1
x + 2y ? z + t = 3. (7M)
3. a) Determine the diagonal matrix orthogonally similar to the matrix.
A =
6 2 2
2 3 1
2 1 3
– ? ?
? ?
– –
? ?
? ? –
? ?(7M)
b) Find the Nature , index and signature of the quadratic form
2 2 2
10 2 5 4 10 6 x y z xy xz yz + + – – +

(7M)
SET – 1
R16
1 of 2

Code No: R161203

4. a)By change of order of integration evaluate
2 2
0( )
a a
x
x y dy dx +
? ?(7M)

b)Evaluate
2 2
sin? ( )/ p/2
0 0 0
?
a a r a
r dr d dz

? ? ?

(7M)
5.
a)Evaluate
2
4
0
3
x
dx
8

?
(7M)

b)Show that
? ?
8 8
p
= =
0 0
2 2
2 2
1
dx x dx x cos sin
(7M)
6.
a)Show that ( )
n
f r a r = ? is solenoidal where =

?+

?+

and
? =
?+ ?+

(7M)

b)Prove that
3 4
1 3
r
r r
? ? ? ?
? ? =
? ? ? ?
? ? ? ?

(7M)
7. a) Verify stoke?s theorem for F y i z j x k = + + for the upper part of the sphere
x
2
+ y
2
+ z
2
= 1. (7M)
b)Verify Green?s theorem in the plane for
?
+ +
c
2 2
dy x dx y xy ) ( . Where c is the
closed curve of the region bounded by y=x & y=x
2 (7M)

2 of 2
SET – 1
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE) Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to reduce the given matrix into Normal form. (2M)
2 2 2
3 3 2 x y z yz + + – (2M)
c) Write the tangents at the origin of the curve a
2
y
2
= x
2(a
2
? x
2
). (2M)
d)Evaluate
1 1 1
0 0 0
dx dy dz
???(2M)
e)Prove that
0
2
2 1 2 1
sin cos ( , )
m n
d m n
p
? ? ? ?
– –
=
?(2M)
f)Find the maximum value of the directional derivative of
2 4
2x y z f = – – at (2, 1,1) – (2M)
g) Write Stoke?s theorem. (2M)
PART -B

2.

a) For what value of k the matrix A =
4 4 3 1
1 1 1 0
2 2 2
9 9 3
k
k
– ? ?
? ?

? ?
? ?
? ?
? ?
has rank 3.
(7M)

b)Solve the following system of equations
8 3 2 20
4 11 33
6 3 12 35
x y z
x y z
x y z
– + =
+ – =
+ + =
by using.
Gauss ? Seidel method.
(7M)
3. a) Determine the characteristic roots and the corresponding characteristic vectors of
the matrix.
A =
3 10 5
2 3 4
3 5 7
? ?
? ?
– – –
? ?
? ?
? ?(7M)
b) Find the Nature , index and signature of the quadratic form
2 2 2
4 3 8 4 6 x y z xy xz yz + + – + –

(7M)
SET – 2
R16
1 of 2

Code No: R161203

4. a) Trace the curve r
2
= a
2
cos 2 ? (7M)

b)Evaluate
2 2( ) x y dx dy +
? ?
over the area bounded by the Ellipse
2 2
2 2
1
x y
a b
+ =
(7M)
5.
a)Evaluate
2
0
0, 1
bx
a dx b a
8

> >
?

(7M)

b)Show that ( )
1
1
0( 1) !
log( 1)
n
n
m
n
n
x x dx
m
+

=
+
?
(7M)
6. a)Find the constants ?a? and ?b? such that the surfaces 5x
2
-2yz-9x=0 and ax
2
y+bz
3
=4
cuts orthogonally at (1,-1,2)
(7M)
b)Show that the vector ( ) ( ) ( )
2 2 2
x yz i y zx j z xy k – + – + – is irrotational and find
its scalar potential.
(7M)
7. a) If f
%
=
2(3 2 ) 4 5 x z i xy j xk – – – Evaluate
V
Cur F dv
?
, where v is volume bounded by
the planes x = 0; y = 0; z = 0 and 3x + 2y ? 3z = 6. (7M)
b)Evaluate cos (1 sin ) over
c
y dx x y dy a + –
?
closed curve c given by x
2
+ y
2
= 1; z = 0
using Green?s theorem. (7M)

2 of 2
SET – 2
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE)Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to find the inverse of the given matrix by Jordan
method. (2M)
b) Find the Eigen value of Adj A if the ??? is the Eigen value of A. (2M)
c) Write the symmetry of the curve y
2
(2a ? x) = x
3
(2M)

d)Evaluate
? ?

3
0
x
x
dy dx xy (2M)
e)Find the value of
1 1
,
2 2
?
? ?
? ?
? ?(2M)
f)Find the angle between the surfaces
2 2 2
9 x y z + + = and
2 2
3 z x y = + – at the
point( ) 2, 1, 2 – . (2M)
g) Write the physical interpretation of Gauss divergence theorem. (2M)
PART -B

2.

a) Reduce the matrix to Echelon form and find its rank
2 1 3 4
0 3 4 1
2 3 7 5
2 5 11 6
– ? ?
? ?
? ?
? ?
? ?
? ?
(7M)

b)Solve the equations
10 12,
2 10 13,
5 7.
x y z
x y z
x y z
+ + =
+ + =
+ + =
by Gauss ? Jordan method.
(7M)
3. a) Find the Natural frequencies and normal modes of vibrating system for which
mass
1 0
0 2
M
? ?
=
? ?
? ?
and stiffness
2 1
1 3
K
? ?
=
? ?
? ?(7M)

b)Verify Cayley-Hamilton theorem for the matrix A =
1 2 3
2 4 5
3 5 6
? ?
? ?
? ?
? ?
? ?
. Hence find A
?1

(7M)
SET – 3
R16
1 of 2

Code No: R161203

4. a) Find the volume of region bounded by the surface z = x
2
+ y
2
and z = 2x. (7M)

b) Evaluate
2 2
2 2
0 0
a x
a
x y dy dx

+
? ?
by changing in to polar co-ordinates.
(7M)
5.
a)Show that
1 1 1( ) ( ) ( ) ( , ) 0, 0
b
m n m n
a
x a b x dx b a m n m n ?
– – + –
– – = – > >
?
(7M)

b)Evaluate ( )
4
1
0
log x x dx
?

(7M)
6. a) Find the directional derivative of xyz f = at (1,-1, 1) along the direction which
makes equal angles with the positive direction of x , y , z axes (7M)
b)Prove that div 0 curl f =
(7M)
7. a)Verify Green?s theorem for
2 2(3 8 ) (4 6 )
c
x y dx y xy dy – + –
?
where c is the boundary of
the region enclosed by the lines. x = 0 y = 0 x + y = 1. (7M)
b) Find the flux of vector function ( 2 ) ( 3 ) (5 ) F x z i x y j x y k = – + + + through the upper
side of the triangle ABC with vertices (1,0,0), (0,1,0), (0,0,1). (7M)

2 of 2
SET – 3
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE)Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to reduce the given matrix into Echelon form. (2M)
b)Find the Eigen value of the matrix A =
4 2
1 5
? ?
? ?
? ?
. (2M)
c) Find the point of the curve r = a (1 + cos ?) where tangent coincide with the radius
vector. (2M)
d) Evaluate
??
+
2
1
4
3
y
dxdy e xy ) ( (2M)
e) Show that ( 1) ( ) n n n G + = G for ` 0 n > (2M)
3 3
3 x y xyz f = + + at (1,1, 2) – (2M)
g) Find the work done in moving particle in the force field
2
3 F x i j zk = + + along the
straight line (0, 0, 0) to (2, 1, 3). (2M)
PART -B

2.

a) Reduce the matrix
3 2 0 1
0 2 2 1
1 2 3 2
0 1 2 1
A
– – ? ?
? ?
? ?
=
? ? – –
? ?
? ?
in to normal form hence find the rank.
(7M)
b) If consistent, solve the system of equations.
x + y + z + t = 4
x ? z + 2t = 2
y + z ? 3t = ?1
x + 2y ? z + t = 3. (7M)
3. a) Determine the diagonal matrix orthogonally similar to the matrix.
A =
6 2 2
2 3 1
2 1 3
– ? ?
? ?
– –
? ?
? ? –
? ?(7M)
b) Find the Nature , index and signature of the quadratic form
2 2 2
10 2 5 4 10 6 x y z xy xz yz + + – – +

(7M)
SET – 1
R16
1 of 2

Code No: R161203

4. a)By change of order of integration evaluate
2 2
0( )
a a
x
x y dy dx +
? ?(7M)

b)Evaluate
2 2
sin? ( )/ p/2
0 0 0
?
a a r a
r dr d dz

? ? ?

(7M)
5.
a)Evaluate
2
4
0
3
x
dx
8

?
(7M)

b)Show that
? ?
8 8
p
= =
0 0
2 2
2 2
1
dx x dx x cos sin
(7M)
6.
a)Show that ( )
n
f r a r = ? is solenoidal where =

?+

?+

and
? =
?+ ?+

(7M)

b)Prove that
3 4
1 3
r
r r
? ? ? ?
? ? =
? ? ? ?
? ? ? ?

(7M)
7. a) Verify stoke?s theorem for F y i z j x k = + + for the upper part of the sphere
x
2
+ y
2
+ z
2
= 1. (7M)
b)Verify Green?s theorem in the plane for
?
+ +
c
2 2
dy x dx y xy ) ( . Where c is the
closed curve of the region bounded by y=x & y=x
2 (7M)

2 of 2
SET – 1
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE) Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to reduce the given matrix into Normal form. (2M)
2 2 2
3 3 2 x y z yz + + – (2M)
c) Write the tangents at the origin of the curve a
2
y
2
= x
2(a
2
? x
2
). (2M)
d)Evaluate
1 1 1
0 0 0
dx dy dz
???(2M)
e)Prove that
0
2
2 1 2 1
sin cos ( , )
m n
d m n
p
? ? ? ?
– –
=
?(2M)
f)Find the maximum value of the directional derivative of
2 4
2x y z f = – – at (2, 1,1) – (2M)
g) Write Stoke?s theorem. (2M)
PART -B

2.

a) For what value of k the matrix A =
4 4 3 1
1 1 1 0
2 2 2
9 9 3
k
k
– ? ?
? ?

? ?
? ?
? ?
? ?
has rank 3.
(7M)

b)Solve the following system of equations
8 3 2 20
4 11 33
6 3 12 35
x y z
x y z
x y z
– + =
+ – =
+ + =
by using.
Gauss ? Seidel method.
(7M)
3. a) Determine the characteristic roots and the corresponding characteristic vectors of
the matrix.
A =
3 10 5
2 3 4
3 5 7
? ?
? ?
– – –
? ?
? ?
? ?(7M)
b) Find the Nature , index and signature of the quadratic form
2 2 2
4 3 8 4 6 x y z xy xz yz + + – + –

(7M)
SET – 2
R16
1 of 2

Code No: R161203

4. a) Trace the curve r
2
= a
2
cos 2 ? (7M)

b)Evaluate
2 2( ) x y dx dy +
? ?
over the area bounded by the Ellipse
2 2
2 2
1
x y
a b
+ =
(7M)
5.
a)Evaluate
2
0
0, 1
bx
a dx b a
8

> >
?

(7M)

b)Show that ( )
1
1
0( 1) !
log( 1)
n
n
m
n
n
x x dx
m
+

=
+
?
(7M)
6. a)Find the constants ?a? and ?b? such that the surfaces 5x
2
-2yz-9x=0 and ax
2
y+bz
3
=4
cuts orthogonally at (1,-1,2)
(7M)
b)Show that the vector ( ) ( ) ( )
2 2 2
x yz i y zx j z xy k – + – + – is irrotational and find
its scalar potential.
(7M)
7. a) If f
%
=
2(3 2 ) 4 5 x z i xy j xk – – – Evaluate
V
Cur F dv
?
, where v is volume bounded by
the planes x = 0; y = 0; z = 0 and 3x + 2y ? 3z = 6. (7M)
b)Evaluate cos (1 sin ) over
c
y dx x y dy a + –
?
closed curve c given by x
2
+ y
2
= 1; z = 0
using Green?s theorem. (7M)

2 of 2
SET – 2
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE)Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to find the inverse of the given matrix by Jordan
method. (2M)
b) Find the Eigen value of Adj A if the ??? is the Eigen value of A. (2M)
c) Write the symmetry of the curve y
2
(2a ? x) = x
3
(2M)

d)Evaluate
? ?

3
0
x
x
dy dx xy (2M)
e)Find the value of
1 1
,
2 2
?
? ?
? ?
? ?(2M)
f)Find the angle between the surfaces
2 2 2
9 x y z + + = and
2 2
3 z x y = + – at the
point( ) 2, 1, 2 – . (2M)
g) Write the physical interpretation of Gauss divergence theorem. (2M)
PART -B

2.

a) Reduce the matrix to Echelon form and find its rank
2 1 3 4
0 3 4 1
2 3 7 5
2 5 11 6
– ? ?
? ?
? ?
? ?
? ?
? ?
(7M)

b)Solve the equations
10 12,
2 10 13,
5 7.
x y z
x y z
x y z
+ + =
+ + =
+ + =
by Gauss ? Jordan method.
(7M)
3. a) Find the Natural frequencies and normal modes of vibrating system for which
mass
1 0
0 2
M
? ?
=
? ?
? ?
and stiffness
2 1
1 3
K
? ?
=
? ?
? ?(7M)

b)Verify Cayley-Hamilton theorem for the matrix A =
1 2 3
2 4 5
3 5 6
? ?
? ?
? ?
? ?
? ?
. Hence find A
?1

(7M)
SET – 3
R16
1 of 2

Code No: R161203

4. a) Find the volume of region bounded by the surface z = x
2
+ y
2
and z = 2x. (7M)

b) Evaluate
2 2
2 2
0 0
a x
a
x y dy dx

+
? ?
by changing in to polar co-ordinates.
(7M)
5.
a)Show that
1 1 1( ) ( ) ( ) ( , ) 0, 0
b
m n m n
a
x a b x dx b a m n m n ?
– – + –
– – = – > >
?
(7M)

b)Evaluate ( )
4
1
0
log x x dx
?

(7M)
6. a) Find the directional derivative of xyz f = at (1,-1, 1) along the direction which
makes equal angles with the positive direction of x , y , z axes (7M)
b)Prove that div 0 curl f =
(7M)
7. a)Verify Green?s theorem for
2 2(3 8 ) (4 6 )
c
x y dx y xy dy – + –
?
where c is the boundary of
the region enclosed by the lines. x = 0 y = 0 x + y = 1. (7M)
b) Find the flux of vector function ( 2 ) ( 3 ) (5 ) F x z i x y j x y k = – + + + through the upper
side of the triangle ABC with vertices (1,0,0), (0,1,0), (0,0,1). (7M)

2 of 2
SET – 3
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE)Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Find the Rank of the matrix
1 1 1
1 1 1
1 1 1
? ?
? ?
? ?
? ?
? ?
(2M)
b) Prove the AB and BA has same Eigen values. (2M)
c)Write the Asymptote of the curve y=
2
2
1
? 1
x
x
+
(2M)
d)Evaluate
3 2
0 1( ) xy x y dx dy +
? ?(2M)
e)Show that
1
2
p
? ?
G =
? ?
? ?(2M)
f)Show that ( )
2
2 r r ? =
(2M)
g) Write Green?s theorem. (2M)
PART -B
2.
a)Reduce the matrix
1 2 3 2
2 2 1 3
3 0 4 1
A
– ? ?
? ?
= –
? ?
? ?
? ?
into PAQ form and hence find the rank
of the matrix. (7M)

b)Solve the equations
8,
2 3 2 19
4 2 3 23
x y z
x y z
x y z
+ + =
+ + =
+ + =
by Gauss ? Elimination method.
(7M)
3.
a)Diagonalize the matrix A =
1 2 3
0 3 1
0 0 1
? ?
? ?
? ?
? ?
? ?
if possible. (7M)
b) Find the Nature, index and signature of the quadratic form
2 2 2
3 5 3 2 2 2 x y z xy xz yz + + – + – by orthogonal reduction.

(7M)
SET – 4
R16
1 of 2

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE)Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to reduce the given matrix into Echelon form. (2M)
b)Find the Eigen value of the matrix A =
4 2
1 5
? ?
? ?
? ?
. (2M)
c) Find the point of the curve r = a (1 + cos ?) where tangent coincide with the radius
vector. (2M)
d) Evaluate
??
+
2
1
4
3
y
dxdy e xy ) ( (2M)
e) Show that ( 1) ( ) n n n G + = G for ` 0 n > (2M)
3 3
3 x y xyz f = + + at (1,1, 2) – (2M)
g) Find the work done in moving particle in the force field
2
3 F x i j zk = + + along the
straight line (0, 0, 0) to (2, 1, 3). (2M)
PART -B

2.

a) Reduce the matrix
3 2 0 1
0 2 2 1
1 2 3 2
0 1 2 1
A
– – ? ?
? ?
? ?
=
? ? – –
? ?
? ?
in to normal form hence find the rank.
(7M)
b) If consistent, solve the system of equations.
x + y + z + t = 4
x ? z + 2t = 2
y + z ? 3t = ?1
x + 2y ? z + t = 3. (7M)
3. a) Determine the diagonal matrix orthogonally similar to the matrix.
A =
6 2 2
2 3 1
2 1 3
– ? ?
? ?
– –
? ?
? ? –
? ?(7M)
b) Find the Nature , index and signature of the quadratic form
2 2 2
10 2 5 4 10 6 x y z xy xz yz + + – – +

(7M)
SET – 1
R16
1 of 2

Code No: R161203

4. a)By change of order of integration evaluate
2 2
0( )
a a
x
x y dy dx +
? ?(7M)

b)Evaluate
2 2
sin? ( )/ p/2
0 0 0
?
a a r a
r dr d dz

? ? ?

(7M)
5.
a)Evaluate
2
4
0
3
x
dx
8

?
(7M)

b)Show that
? ?
8 8
p
= =
0 0
2 2
2 2
1
dx x dx x cos sin
(7M)
6.
a)Show that ( )
n
f r a r = ? is solenoidal where =

?+

?+

and
? =
?+ ?+

(7M)

b)Prove that
3 4
1 3
r
r r
? ? ? ?
? ? =
? ? ? ?
? ? ? ?

(7M)
7. a) Verify stoke?s theorem for F y i z j x k = + + for the upper part of the sphere
x
2
+ y
2
+ z
2
= 1. (7M)
b)Verify Green?s theorem in the plane for
?
+ +
c
2 2
dy x dx y xy ) ( . Where c is the
closed curve of the region bounded by y=x & y=x
2 (7M)

2 of 2
SET – 1
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE) Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to reduce the given matrix into Normal form. (2M)
2 2 2
3 3 2 x y z yz + + – (2M)
c) Write the tangents at the origin of the curve a
2
y
2
= x
2(a
2
? x
2
). (2M)
d)Evaluate
1 1 1
0 0 0
dx dy dz
???(2M)
e)Prove that
0
2
2 1 2 1
sin cos ( , )
m n
d m n
p
? ? ? ?
– –
=
?(2M)
f)Find the maximum value of the directional derivative of
2 4
2x y z f = – – at (2, 1,1) – (2M)
g) Write Stoke?s theorem. (2M)
PART -B

2.

a) For what value of k the matrix A =
4 4 3 1
1 1 1 0
2 2 2
9 9 3
k
k
– ? ?
? ?

? ?
? ?
? ?
? ?
has rank 3.
(7M)

b)Solve the following system of equations
8 3 2 20
4 11 33
6 3 12 35
x y z
x y z
x y z
– + =
+ – =
+ + =
by using.
Gauss ? Seidel method.
(7M)
3. a) Determine the characteristic roots and the corresponding characteristic vectors of
the matrix.
A =
3 10 5
2 3 4
3 5 7
? ?
? ?
– – –
? ?
? ?
? ?(7M)
b) Find the Nature , index and signature of the quadratic form
2 2 2
4 3 8 4 6 x y z xy xz yz + + – + –

(7M)
SET – 2
R16
1 of 2

Code No: R161203

4. a) Trace the curve r
2
= a
2
cos 2 ? (7M)

b)Evaluate
2 2( ) x y dx dy +
? ?
over the area bounded by the Ellipse
2 2
2 2
1
x y
a b
+ =
(7M)
5.
a)Evaluate
2
0
0, 1
bx
a dx b a
8

> >
?

(7M)

b)Show that ( )
1
1
0( 1) !
log( 1)
n
n
m
n
n
x x dx
m
+

=
+
?
(7M)
6. a)Find the constants ?a? and ?b? such that the surfaces 5x
2
-2yz-9x=0 and ax
2
y+bz
3
=4
cuts orthogonally at (1,-1,2)
(7M)
b)Show that the vector ( ) ( ) ( )
2 2 2
x yz i y zx j z xy k – + – + – is irrotational and find
its scalar potential.
(7M)
7. a) If f
%
=
2(3 2 ) 4 5 x z i xy j xk – – – Evaluate
V
Cur F dv
?
, where v is volume bounded by
the planes x = 0; y = 0; z = 0 and 3x + 2y ? 3z = 6. (7M)
b)Evaluate cos (1 sin ) over
c
y dx x y dy a + –
?
closed curve c given by x
2
+ y
2
= 1; z = 0
using Green?s theorem. (7M)

2 of 2
SET – 2
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE)Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Write the working procedure to find the inverse of the given matrix by Jordan
method. (2M)
b) Find the Eigen value of Adj A if the ??? is the Eigen value of A. (2M)
c) Write the symmetry of the curve y
2
(2a ? x) = x
3
(2M)

d)Evaluate
? ?

3
0
x
x
dy dx xy (2M)
e)Find the value of
1 1
,
2 2
?
? ?
? ?
? ?(2M)
f)Find the angle between the surfaces
2 2 2
9 x y z + + = and
2 2
3 z x y = + – at the
point( ) 2, 1, 2 – . (2M)
g) Write the physical interpretation of Gauss divergence theorem. (2M)
PART -B

2.

a) Reduce the matrix to Echelon form and find its rank
2 1 3 4
0 3 4 1
2 3 7 5
2 5 11 6
– ? ?
? ?
? ?
? ?
? ?
? ?
(7M)

b)Solve the equations
10 12,
2 10 13,
5 7.
x y z
x y z
x y z
+ + =
+ + =
+ + =
by Gauss ? Jordan method.
(7M)
3. a) Find the Natural frequencies and normal modes of vibrating system for which
mass
1 0
0 2
M
? ?
=
? ?
? ?
and stiffness
2 1
1 3
K
? ?
=
? ?
? ?(7M)

b)Verify Cayley-Hamilton theorem for the matrix A =
1 2 3
2 4 5
3 5 6
? ?
? ?
? ?
? ?
? ?
. Hence find A
?1

(7M)
SET – 3
R16
1 of 2

Code No: R161203

4. a) Find the volume of region bounded by the surface z = x
2
+ y
2
and z = 2x. (7M)

b) Evaluate
2 2
2 2
0 0
a x
a
x y dy dx

+
? ?
by changing in to polar co-ordinates.
(7M)
5.
a)Show that
1 1 1( ) ( ) ( ) ( , ) 0, 0
b
m n m n
a
x a b x dx b a m n m n ?
– – + –
– – = – > >
?
(7M)

b)Evaluate ( )
4
1
0
log x x dx
?

(7M)
6. a) Find the directional derivative of xyz f = at (1,-1, 1) along the direction which
makes equal angles with the positive direction of x , y , z axes (7M)
b)Prove that div 0 curl f =
(7M)
7. a)Verify Green?s theorem for
2 2(3 8 ) (4 6 )
c
x y dx y xy dy – + –
?
where c is the boundary of
the region enclosed by the lines. x = 0 y = 0 x + y = 1. (7M)
b) Find the flux of vector function ( 2 ) ( 3 ) (5 ) F x z i x y j x y k = – + + + through the upper
side of the triangle ABC with vertices (1,0,0), (0,1,0), (0,0,1). (7M)

2 of 2
SET – 3
R16

Code No: R161203
I B. Tech II Semester Regular/Supplementary Examinations, April/May – 2018
MATHEMATICS-III (Com. to CE,CSE,IT,AE,AME,EIE,EEE,ME,ECE,Metal E,Min E,E Com E,Agri E,Chem E,PCE,PE)Time: 3 hours Max. Marks: 70
Note: 1. Question Paper consists of two parts (Part-A and Part-B) 2. Answer ALL the question in Part-A
3. Answer any FOUR Questions from Part-B
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
PART ?A
1. a) Find the Rank of the matrix
1 1 1
1 1 1
1 1 1
? ?
? ?
? ?
? ?
? ?
(2M)
b) Prove the AB and BA has same Eigen values. (2M)
c)Write the Asymptote of the curve y=
2
2
1
? 1
x
x
+
(2M)
d)Evaluate
3 2
0 1( ) xy x y dx dy +
? ?(2M)
e)Show that
1
2
p
? ?
G =
? ?
? ?(2M)
f)Show that ( )
2
2 r r ? =
(2M)
g) Write Green?s theorem. (2M)
PART -B
2.
a)Reduce the matrix
1 2 3 2
2 2 1 3
3 0 4 1
A
– ? ?
? ?
= –
? ?
? ?
? ?
into PAQ form and hence find the rank
of the matrix. (7M)

b)Solve the equations
8,
2 3 2 19
4 2 3 23
x y z
x y z
x y z
+ + =
+ + =
+ + =
by Gauss ? Elimination method.
(7M)
3.
a)Diagonalize the matrix A =
1 2 3
0 3 1
0 0 1
? ?
? ?
? ?
? ?
? ?
if possible. (7M)
b) Find the Nature, index and signature of the quadratic form
2 2 2
3 5 3 2 2 2 x y z xy xz yz + + – + – by orthogonal reduction.

(7M)
SET – 4
R16
1 of 2

Code No: R161203

4. a)Trace the curve x = a cost+
2
a
log tan
2
t/2, y = a sint (7M)
b) Find the area between the circles r = a cos? and r = 2a cos?.
(7M)
5. a)Prove that( ) ( )
1
0
1 1(1 ) ( , )
m n m
n
m n
x x
dx
a bx a b a
m n ?
+
– –

=
+ +
?(7M)

b)Evaluate
?
8

0
4 x
dx x e
6

(7M)
6. a)Find the directional derivative of the function
2
cos
x
e yz
at the origin in the
direction to the tangent to the curve sin , cos ,
4
x a t y a t z at at t
p
= = = = (7M)
b)Show that ( ) ( )( ) . . curl curl f f f f =?? ?? =? ? – ??
if ( ) , , f x y z is vector
point function.
(7M)
7. a) Verify Gauss Divergence theorem for
2 2 2( ) ( ) ( ) F x yz i y zx j z xy k = – + – + – taken
over the rectangular parallelepiped 0 ; 0 ; 0 x a y b z c = = = = = = . (7M)
b)Evaluate ( ).
s
F n ds ??
??
where
2 2( 4) 3 (2 ) F x y i xy j xy z k = + – + + + and s in the
surface of the paraboloid z = 4 ? x
2
? y
2
above the xy plane. (7M)

2 of 2
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R16

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