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]



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]

*****






MATHEMATICS – I





- - -

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



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 =

*****






MATHEMATICS – I





- - -

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



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]

*****






MATHEMATICS – I





- - -

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∇ × = ∇ ∇ − ∇∇ × .



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]

*****

r07a1bs02 mathematics 1

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