Download JNTUA B.Tech 1-1 R13 2019 Dec 13A54101 Mathematics I Question Paper

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Code: 13A54101


B.Tech I Year (R13) Supplementary Examinations December 2019
MATHEMATICS ? I
(Common to all branches)

Time: 3 hours Max. Marks: 70

PART ? A
(Compulsory Question)

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

(a) Solve
22
( ) 2 x y dx xydy ?? .

(b) Solve the differential equation :
4 3 2
( 2 2 2 1) 0 D D D D y ? ? ? ? ?

(c) Solve the differential equation:
22
( 3 1) 0 x D xD y ? ? ? .

(d)
If
3 3 3
log(x y z 3xyz) then find
UUU
U
x y z
???
? ? ? ? ? ?
? ? ?
.

(e)
If
( , )
cos , sin then find ?
( , )
xy
x r y r
r
??
?
?
??
?


(f) If
3 2 2 2
( , ) 3 3 3 4 f x y x xy x y ? ? ? ? ? then find critical points.

(g)
Evaluate
? ? ? ?
11
22
00
1
11
dxdy
xy ??
??
.

(h) Evaluate
0 0 0
xy ax
x y z
e dxdydz
?
??
? ? ?
.

(i) Find Curl F for F zi xj yk ? ? ? .
(j) State Gauss divergence theorem.

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

UNIT ? I

2 (a) Solve (D
4
+ 2D
2
+ 1)y = e
x
cos x.

(b)
Find the orthogonal trajectories of the family of confocal conics
22
22
1
xy
ab ?
??
?
, where ?? is a
parameter.
OR
3 (a) Solve (D
2
- 2D + 1)y = x e
x
sin x.
(b) Using method of variation of parameter Solve (D
2
+ 4)y = tan 2x.

UNIT ? II

4 (a) Verify Rolle?s theorem for the function f(x) =
? ?
sin
0,
x
x
in
e
? .
(b) Find the coordinates of the centre of curvature at (?? ?? 2
, 2???? ) on the parabola ?? 2
= 4???? .
OR
5 (a) Find the maximum and minimum values of ?? 3
+ 3?? ?? 2
? 15?? 2
? 15?? 2
+ 72?? .
(b) Find C of Cauchy?s Mean Value theorem for ?? (?? ) = sin?? , ?? (?? ) = cos?? ???? ?? , ?? .

Contd. in page 2


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Code: 13A54101


B.Tech I Year (R13) Supplementary Examinations December 2019
MATHEMATICS ? I
(Common to all branches)

Time: 3 hours Max. Marks: 70

PART ? A
(Compulsory Question)

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

(a) Solve
22
( ) 2 x y dx xydy ?? .

(b) Solve the differential equation :
4 3 2
( 2 2 2 1) 0 D D D D y ? ? ? ? ?

(c) Solve the differential equation:
22
( 3 1) 0 x D xD y ? ? ? .

(d)
If
3 3 3
log(x y z 3xyz) then find
UUU
U
x y z
???
? ? ? ? ? ?
? ? ?
.

(e)
If
( , )
cos , sin then find ?
( , )
xy
x r y r
r
??
?
?
??
?


(f) If
3 2 2 2
( , ) 3 3 3 4 f x y x xy x y ? ? ? ? ? then find critical points.

(g)
Evaluate
? ? ? ?
11
22
00
1
11
dxdy
xy ??
??
.

(h) Evaluate
0 0 0
xy ax
x y z
e dxdydz
?
??
? ? ?
.

(i) Find Curl F for F zi xj yk ? ? ? .
(j) State Gauss divergence theorem.

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

UNIT ? I

2 (a) Solve (D
4
+ 2D
2
+ 1)y = e
x
cos x.

(b)
Find the orthogonal trajectories of the family of confocal conics
22
22
1
xy
ab ?
??
?
, where ?? is a
parameter.
OR
3 (a) Solve (D
2
- 2D + 1)y = x e
x
sin x.
(b) Using method of variation of parameter Solve (D
2
+ 4)y = tan 2x.

UNIT ? II

4 (a) Verify Rolle?s theorem for the function f(x) =
? ?
sin
0,
x
x
in
e
? .
(b) Find the coordinates of the centre of curvature at (?? ?? 2
, 2???? ) on the parabola ?? 2
= 4???? .
OR
5 (a) Find the maximum and minimum values of ?? 3
+ 3?? ?? 2
? 15?? 2
? 15?? 2
+ 72?? .
(b) Find C of Cauchy?s Mean Value theorem for ?? (?? ) = sin?? , ?? (?? ) = cos?? ???? ?? , ?? .

Contd. in page 2


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Code: 13A54101




UNIT ? III

6 (a) Trace the curve y
2
(a - x) = x
2
(a + x).
(b)
Find the surface area of the solid of revolution of one loop of the curve
22
cos 2 ra ? ? about the initial
line.
OR
7 (a) Evaluate ? ?
???????? 1+?? 2
+?? 2
?1+?? 2
0
1
0
.
(b)
By changing the order of integration , evaluate
2
2 4
0
4
.
ax a
x
a
dydx
??


UNIT ? IV

8 (a) Find the Laplace transform of: (i) {
sin 3?? . cos ?? ?? }. (ii) {?? 2
sin2?? }.
(b) Apply Convolution theorem to find ?? ?1
{
1
(?? +1)(?? 2
+1)
}.
OR
9 (a) Find the Laplace transform of e
-3t
(2 cos 5t ? 3 sin 5t ).
(b) Solve the D.E ?? ?? ? 2?? ? ? 8?? = 0, ?? (0) = 3, ?? ?(0) = 6. Using Laplace transform.

UNIT ? V

10 (a) Find the directional derivative of xyz
2
+ xz at (1, 1, 1) in the direction of 2 3 . i j k ??

(b) Verify Green?s theorem for ?
[(???? + ?? 2
)???? + ?? 2
???? ]
?? where C is bounded by ?? = ?? and ?? = ?? 2
.
OR
11
Verify Gauss divergence theorem for
2
44 F xyi y j zk ? ? ? where S is the surface of the cube
bounded by x = 0, x = 1; y = 0, y = 1; z = 0, z = 1.

*****
























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