r07a1bs02 Mathematics 1
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R07 SET-1Code.No: 41007
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
I B.TECH SUPPLEMENTARY EXAMINATIONS FEBRUARY - 2010
MATHEMATICS – I
(COMMON TO CE, EEE, ME, ECE, CSE, CHEM, EIE, BME, IT, E.CON.E, MCT, CSS,
ETM, MMT, ECC, MEP, AE, ICE, AME, BT)
Time: 3hours Max.Marks:80 Answer any FIVE questions
All questions carry equal marks
- - -
1.a) Solve 1 1
x x
y y xe dx e dy
y
⎛ ⎞ ⎛ ⎞+ + − 0=⎜ ⎟
⎝ ⎠⎝ ⎠⎜ ⎟⎜ ⎟
.
b) Find the orthogonal trajectories of the family of curves r n
= an
cos nθ. [8+8]
2.a) Solve ( D2+4 ) y= e
x+sin 2x+cos 2x
b) Solve2
2cosd y y x x
dx+ = by the method of variation of parameters. [8+8]
3.a) Verify Rolle’s theorem for the function log2
( )
x ab
x a b
⎡ ⎤+⎢ ⎥+⎣ ⎦
in [ a, b ] where a > 0, b > 0.
b) Find the minimum values of x2+y
2+z
2given that xyz = a
3[8+8]
4.a) Find the radius of curvature of any point on the cure y = cosx
c h c
b) Find the envelope of the family of straight lines y = 2 2 2mx a m b+ + where m is the
parameter.
5.a) Find the perimeter of the curve x2+y
2= r
2?
b) Evaluate
2 22 11 1
0 0 0
x y x
xyz dz dydx
− −−
∫ ∫ . [8+8]∫
6.a) Test for convergence of ( )2 21 1n n+ − −∑
b) Test for convergence of 2 33 4 5
2 (
2 3 4
x x x x+ + + + − − − − − > 0) [8+8]
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7.a) Evaluate3
.r
r
⎛ ⎞∇ ⎜
⎝ ⎠⎟ where r xi y j zk = + + and r r =
b) Use divergence theorem to evaluate
.S
F d s∫ ∫ where 3 3 3F x i y j z k = + + and ‘s’ is the surface of the sphere x
2+y
2+z
2=a
2. [8+8]
8.a) Show that L1 1
t sπ
⎧ ⎫=⎨ ⎬
⎩ ⎭
b) If 3 2
12
t L
sπ
⎧ ⎫⎪ ⎪=⎨ ⎬
⎪ ⎪⎩ ⎭show that
1 2
1 1 L
st π
⎧ ⎫=⎨ ⎬
⎩ ⎭.
c) Using Laplace transform, solve 2 3dx
x ydt
0− + = and 2dy
x ydt
0+ − = given
x = 8, y = 3 when t = 0. [5+5+6]
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R07 SET-2Code.No: 41007
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
I B.TECH SUPPLEMENTARY EXAMINATIONS FEBRUARY - 2010
MATHEMATICS – I
(COMMON TO CE, EEE, ME, ECE, CSE, CHEM, EIE, BME, IT, E.CON.E, MCT, CSS,
ETM, MMT, ECC, MEP, AE, ICE, AME, BT)
Time: 3hours Max.Marks:80 Answer any FIVE questions
All questions carry equal marks
- - -
1.a) Solve 2 2 x dy y dx x y dx− = +
b) Prove that the system of parabolas 2 4 ( ) y a x a= + is self orthogonal. [8+8]
2.a) Solve ( D2- 4D + 3 ) y = sin 3x cos 2x
b) Solve the method of variation of parameters. [8+8]2( 1) cos D y+ = x by
3.a) Find C of the Lagrange’s theorem of
f (x) = ( x-1) (x-2) (x-3) on [ 0, 4]
b) Show that the function u = xy + yz + zx, v = x2
+ y2
+ z2
and w = x + y + z are
functionally related. Find the relation between them. [8+8]
4.a) Find the radius of curvature ρ at any point of the cylinder x = a(θ + sin θ) ,
y = a(1-cos θ) .
b) Find the coordinates of the centre of curvature at any point of the rectangle hyperbola
x y = c2. [8+8]
5.a) Find the length of the arc of the parabola y2
= 4 ax measured from the vertex to bothextremities of the latus-rectum.
b) Evaluate1
1 0
( )
z x z
x z
x y z dx dy dz
+
− −
+ +∫ ∫ ∫ [8+8]
6.a) Test for convergence of ( )3 3
1
1n
n n∞
=
+ −∑ .
b) Find the nature of the series3 3.6 3.6.9
4 4.7 4.7.10+ + + − − − − − + ∞ [8+8]
7.a) Show that the vector ( ) ( ) ( )2 2 2 x yz i y zx j z xy k − + − + − is irrotational and find its scalar
potential.
b) Using Divergence theorem evaluate ( )S
x dy d z y d z dx z dx dy+ +∫ ∫ where
. [8+8]2 2 2:s x y z a+ + = 2
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8.a) Find the Laplace transform of 2 sin3t
t e t
b) Find 1
2 2
s L
s a
− ⎛ ⎞⎜ ⎟−⎝ ⎠
c) Solve the differential equation using Laplace transform2
23 2 t d x dx
x e
dt dt
−+ + = where
x(0) = 0, . [5+5+6]1(0) 1 x =
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R07 SET-3Code.No: 41007
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
I B.TECH SUPPLEMENTARY EXAMINATIONS FEBRUARY - 2010
MATHEMATICS – I
(COMMON TO CE, EEE, ME, ECE, CSE, CHEM, EIE, BME, IT, E.CON.E, MCT, CSS,
ETM, MMT, ECC, MEP, AE, ICE, AME, BT)
Time: 3hours Max.Marks:80 Answer any FIVE questions
All questions carry equal marks
- - -
1.a) Solve ( ) ( )2 2 2 24 2 4 2 x xy y dx y xy x dy− − + − − = 0
b) Show that the system of confocal conics2 2
2 21
x y
a bλ λ + =
+ +where λ is a parameter, is
self orthogonal. [8+8]
2.a) Solve ( )2 23 2 2 D D y x− + =
xb) Solve ( ) by the method of variation of parameters. [8+8]2 2 tan D a y a+ =
3.a) Show that for 0 < a < b < 1,( )
( )
1 1
2 2
1 1
1 1
Tan b Tan a
a b a
− −−> >
b+ − +
b) Find whether the following functions are functionally dependent or not. If they arefunctionally dependent, find the relation between them
(i) µ = sin , x
e y cos x
v e y=
(ii) x y
µ = , x y x y
µ +=−
[8+8]
4.a) Find the radius of curvature at the point3 3
,2 2
a a⎛ ⎜ of the curve
⎞⎟
⎝ ⎠
3 3 3 x y ax+ = y
b) Find the coordinates of the centre of curvature at any point of the parabola
Y2= 4ax. Hence prove that its evolute is 27ay
2= 4( x-2a )
3. [8+8]
5.a) Find the length of the catenary coshx
y c
c
⎛ ⎞= ⎜ ⎟
⎝ ⎠
measured from the vertex to any point
( x, y ) on it.
b) Evaluate∫ ∫ [8+8]2 2( )
0 0
x ye dx
∞ ∞− +
dy
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6.a) Test for convergence of the series3 5 71.3 1.3.5
1 2.3 2.4.5 2.4.6.7
x x x x+ + + + − − − − −
b) Examine the following series for absolute and conditional convergence
1 1 1 1( 1)
5 2 5 3 5 4 5
n
n− + − − − − − + − + − − − [8+8]
7.a) Prove that curl ( ) ( . ) ( . )a b a div b bdiv a b a a b× = − + ∇ − ∇
b) By transforming into triple integral, evaluate 3 2 2( ) x d y dz x y d z dx x z dx dy+ +∫ ∫ where s
is the closed surface consisting of the cylinder 2 2 2 x y a+ = and the circular discs
z = 0, z = b. [8+8]
8.a) Find the Laplace transformer of t ( 3 sin 2t – 2 cos 2t )
b) Find ( )1
22
3
6 13
s
Ls s
−⎛ ⎞
+
⎜ ⎟⎜ ⎟+ +⎝ ⎠
c) Solve given x = 0,2( 1) cos D x t + = 2t 0dx
dt = at t = 0. [5+5+6]
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R07 SET-4Code.No: 41007
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
I B.TECH SUPPLEMENTARY EXAMINATIONS FEBRUARY - 2010
MATHEMATICS – I
(COMMON TO CE, EEE, ME, ECE, CSE, CHEM, EIE, BME, IT, E.CON.E, MCT, CSS,
ETM, MMT, ECC, MEP, AE, ICE, AME, BT)
Time: 3hours Max.Marks:80 Answer any FIVE questions
All questions carry equal marks
- - -
1.a) Solvedy y
dx x xy=
+.
b) Solve 3 2sin 2 cosdy
x y x ydx
+ = . [8+8]
2.a) Solve
2
32 6 13 8 sin 2 xd y dy y e
dx dx− + = x .
b) Apply the method of variation of parameters to solve2
2cos
d y y ec
dx+ = x [8+8]
3.a) Find C of Cauchy’s mean value theorem for ( ) f x = x and1
( )g x x
= in [ a, b ]?
b) If 1
x y
xyµ
+=
−and v find1 1
Tan x Tan y− −= +
( , )
( , )
v
x y
µ ∂
∂. Hence prove that µ and are
functionally dependent. Find the functional relation between them. [8+8]
v
4.a) Prove that the radius of curvature of the curve 2 3 3 xy a x= − at the point ( a, 0 ) is3
2
a.
b) Show that the evolute of the ellipse cos , sin x a y bθ θ = = is ( )2 2 2
3 3 32 2( ) ( )ax by a b+ = − .
[8+8]
5.a) Find the perimeter of the loop of the curve 3ay2
= x( x-a )2
.
b) Evaluate1 1
2 20 0 (1 )(1 )
dxdy
x y− −∫ ∫ . [8+8]
6.a) Test for convergence of the series4.7..........(3 1)
1.2.3.................
nn x
n
+∑
b) Test for convergence of the series2 4 6
1 . [8+8]...........2! 4! 6!
x x x− + − +
7.a) Prove that 2( ) ( . )a a a∇ × = ∇ ∇ − ∇∇ × .
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b) Compute over the surface of the sphere2 2 2(ax by cz ds+ +∫ )2 2 2 1 x y z+ + = . [8+8]
8.a) Find the Laplace transform of .3 2 sint t e t
b) Find the inverse Laplace transform of
( )
4
1 ( 2)s s+ +
.
c) Find 1
2
3
4 13
s L
s s
− −⎧ ⎫⎨ ⎬
+ +⎩ ⎭. [5+5+6]
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