# Download JNTUA B.Tech 1-1 R19 2020 January 19A54101 Algebra And Calculus Question Paper

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Code: 19A54101

B.Tech I Year I Semester (R19) Regular Examinations January 2020
ALGEBRA & CALCULUS
(Common to all branches)

Time: 3 hours Max. Marks: 70

PART ? A
(Compulsory Question)

*****
1 Answer the following: (10 X 02 = 20 Marks)

(a) Find the rank of the matrix ?? = ?
8
0
?8

1
3
?1

3
2
?3

6
2
4
?.
(b)
If ? is an Eigen value of a matrix A then prove that
m
? is an Eigen value of
m
A . ( m being a
positive integer)

(c) Discuss the application of Rolle?s theorem to the function ( ) [ ]
sec 0,2 f x x in ? = .
(d) State Maclaurin?s theorem with Lagrange?s form of remainder.

(e)
Evaluate
z
x
?
?
and
z
y
?
?
, if
( )
22
log z xy = + .

(f)
If
2 2 2 2
2, 2 u x y v x y =?=? find
( )
( )
,
,
uv
J
xy
?
=
?
.

(g) Evaluate
1
00
x
x
y
e dydx
??
.

(h) Evaluate
( )
2 22
000
ab c
x y z dzdydx ++
???
.

(i) Show that ( ) ( )
2
21
0
2 ,0
x n
n e x dx n
?
? ?
?= >
?
.

(j)
Express the integral
2
0
cot d
?
??
?
in terms of beta function.

PART ? B
(Answer all five units, 5 X 10 = 50 Marks)

UNIT ? I

2 (a) Reduce the matrix ?? = ?
?2
1
1
0

?1
2
0
1

?3
3
1
1

?1
?1
1
?1
? to Echelon form and hence find its rank.
(b) Find the Eigen values and Eigen vectors of the matrix ?? = ?
2 0 1
0 2 0
1 0 2
?.
OR
3 (a) Test for consistency the following equations and solve them if consistent :

5 3 7 4,
3 26 2 9 ,
7 2 10 5.
xy z
x yz
xy z
++ =
+ +=
+ + =

(b) Verify Cayley-Hamilton theorem for the matrix ?? = ?
1 3 7
4 2 3
1 2 1
? and hence find its inverse.
Contd. in page 2

Page 1 of 2

R19
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Code: 19A54101

B.Tech I Year I Semester (R19) Regular Examinations January 2020
ALGEBRA & CALCULUS
(Common to all branches)

Time: 3 hours Max. Marks: 70

PART ? A
(Compulsory Question)

*****
1 Answer the following: (10 X 02 = 20 Marks)

(a) Find the rank of the matrix ?? = ?
8
0
?8

1
3
?1

3
2
?3

6
2
4
?.
(b)
If ? is an Eigen value of a matrix A then prove that
m
? is an Eigen value of
m
A . ( m being a
positive integer)

(c) Discuss the application of Rolle?s theorem to the function ( ) [ ]
sec 0,2 f x x in ? = .
(d) State Maclaurin?s theorem with Lagrange?s form of remainder.

(e)
Evaluate
z
x
?
?
and
z
y
?
?
, if
( )
22
log z xy = + .

(f)
If
2 2 2 2
2, 2 u x y v x y =?=? find
( )
( )
,
,
uv
J
xy
?
=
?
.

(g) Evaluate
1
00
x
x
y
e dydx
??
.

(h) Evaluate
( )
2 22
000
ab c
x y z dzdydx ++
???
.

(i) Show that ( ) ( )
2
21
0
2 ,0
x n
n e x dx n
?
? ?
?= >
?
.

(j)
Express the integral
2
0
cot d
?
??
?
in terms of beta function.

PART ? B
(Answer all five units, 5 X 10 = 50 Marks)

UNIT ? I

2 (a) Reduce the matrix ?? = ?
?2
1
1
0

?1
2
0
1

?3
3
1
1

?1
?1
1
?1
? to Echelon form and hence find its rank.
(b) Find the Eigen values and Eigen vectors of the matrix ?? = ?
2 0 1
0 2 0
1 0 2
?.
OR
3 (a) Test for consistency the following equations and solve them if consistent :

5 3 7 4,
3 26 2 9 ,
7 2 10 5.
xy z
x yz
xy z
++ =
+ +=
+ + =

(b) Verify Cayley-Hamilton theorem for the matrix ?? = ?
1 3 7
4 2 3
1 2 1
? and hence find its inverse.
Contd. in page 2

Page 1 of 2

R19

Code: 19A54101

UNIT ? II

4 (a) Verify Rolle?s theorem for
21 2
( ) ( ) in (0, ).
mn
fx x a x a
?
= ?
(b)
Verify Taylor?s theorem for ?? ( ?? ) = (1 ? ?? )
5
2
with Lagrange?s form of remainder up to two terms in the
interval ] 1 , 0 [ .
OR
5 (a) Verify Lagrange?s Mean value theorem for ( ) ( ) ( ) 3 2 1 ) ( ? ? ? = x x x x f in ] 4 , 0 [ .
(b) Verify Cauchy?s mean value theorem for ( ) sin fx x = and ( ) cos gx x = in the interval
[ ]
, . ab

UNIT ? III

6 (a)
If .
(, , )
, ,,
( , , )
x yz
u x y z uv y z uvw z Find
uv w
?
= ++ = + =
?

(b) Discuss the maxima and minima of
32
. ( , ) (1 ) f xy x y x y = ? ?
OR
7 (a) Determine whether the following functions are functionally dependent or not. If functionally
dependent, find the functional relation between them: ?? = ?? 2
+ ?? 2
+ 2 ?? ?? + 2 ?? + 2 ?? , ?? = ?? ?? ?? ?? .

(b) Find the maximum and minimum distances of the point ( ) 3, 4, 12 from the sphere
2 22
. 1 x y z + +=

UNIT ? IV

8

(a) Change the order of integration and hence evaluate
( )
2
4 22
0
.
aa
ax
y dy dx
I
y ax
=
?
??

(b) Compute the volume of the sphere
2 2 2 2
a z y x = + + , using spherical coordinates.
OR
9 (a) Evaluate the double integral
2 2
( )
00
xy
e dx dy
??
?+
??
, using polar coordinates.
(b)
Evaluate
00 0
.
x yz
xy ax
e dz dy dx
++
+
?? ?

UNIT ? V

10 (a)
Prove that
( )
21
1
,2 ,
2
m
m mm ??
?
??
=
??
??
.

(b) Show that ( )
( )
( )
1
1
0
1!
log
1
n
n
m
n
n
x x dx
m
+
?
=
+
?
, where n is a positive integer and 1 m >? . Hence evaluate
( )
1
3
0
log x x dx
?
.
OR
11 (a) Express the following integrals in terms of gamma functions: (i)
0
c
x
x
dx
c
?
?
. (ii)
2
0
bx
a dx
?
?
?
.

(b) Express
1
0
(1 )
m np
x x dx ?
?
in terms of Gamma function and hence evaluate
( )
1
0 1
n
dx
x ?
?
.
*****
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