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Code No: R10102/R10

I B.Tech I Semester Regular Examinations, January / February 2011

MATHEMATICS - I

(Common to All Branches)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

*********

1 (a) Solve 2 23 2 1dy

x x xydx

= − +

(b) If 30% of a radioactive substance disappears in 10 days, how long will it take for

90% of it to disappear?

[8M + 7M]

2 (a) Solve 3 2 3( 3 4) (1 )xD D y e−− + = +

(b) Solve 2( 5 6) sin 4 sinD D y x x+ − =

[8M + 7M]

3 (a) Expand 1tan x− in a series of powers of (x-1) up to the term contains the fourth

degree

(b) If u x y z= + + ,uv y z= + ,uvw z= show that 2( , , )

( , , )

x y zu z

u v w

∂=

∂

[8M + 7M]

4 (a) Trace the curve 2 2( ) (3 )y a x x a x+ = −

(b) Trace the curve cosr a b θ= + ,( a b< )

[8M + 7M]

5 (a) Find the perimeter of the loop of the curve 2 29 ( 3 )a y x x a= −

(b) A sector of a circle of radius a and angle 060 rotates about its middle radius.

Find the volume of the solid formed.

[8M + 7M]

6 (a) Evaluate 21 1

2 20 0 (1 )

x dydx

x y

+

+ +∫ ∫

(b) Evaluate 22 2

2 20 0

x x xdydx

x y

−

+∫ ∫ by changing into polar coordinates.

[8M + 7M]

7 (a) Show that 2

(log )r

rr

∇ =

(b) Find the value of a, b and c such that

( ) ( 2 ) ( 2 )x y az i bx y z j x c y z k+ + + + − + − + + is irrotational.

[8M + 7M]

Page 1 of 2

Set No. 1

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Code No: R10102/R10

8 (a) By using Greens theorem find the areas bounded by one arc of the cycloid

( sin )x a θ θ= − , (1 cos )y a θ= − , a>0, 0 2θ π< <

(b) Evaluate by using divergence theorem, .Su n ds∫ ∫ where u r xi y j zk= = + + and

S is the surface of the sphere 2 2 2 9x y z+ + = .

[8M + 7M]

Page 2 of 2

Set No. 1

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JNTUWORLD

Code No: R10102/R10

I B.Tech I Semester Regular Examinations, January / February 2011

MATHEMATICS - I

(Common to All Branches)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

********

1 (a) Solve sin 2 tandy

x y xdx

− =

(b) If the surroundings are maintained at 030 c and the temperature of body cools from 080 c

to 060 c in 12 mins.,find the temperature of body after 24 mints.

[8M + 7M]

2 (a) Solve 3 2( 3 4) sinh 2 7D D y x+ − = +

(b) Solve 2( 4 3)

xe

D D y e+ + =

[8M + 7M]

3 (a) Obtained the Taylor’s series exapansion of sinx in powers of 4

xπ

−

(b) If yz

ux

= ,zx

vy

= ,xy

wz

= show that ( , , )

4( , , )

u v w

x y z

∂=

∂

[8M + 7M]

4 (a) Trace the curve 2 3(2 )y a x x− =

(b) Trace the curve 3cosx θ= , 3siny θ=

[8M + 7M]

5 (a) Find the perimeter of the loop of the curve 2 29 ( 3 )a y x x a= −

(b) A basin is formed by the revolution of the curve 3 64 ( 0)x y y= > about the y-axis. If the

depth of the basin is 8 cm, how many cubic cm of water will it hold?

[8M + 7M]

6 (a) Evaluate by changing the order of integration 2

4 2

04

a ax

x

a

d ydx∫ ∫

(b) Evaluate2 20

a a

y

xdydx

x y+∫ ∫ by changing into polar coordinates.

[8M + 7M]

7 (a) The temperature at a point ( , , )x y z is given by 2 2( , , )T x y z x y z= + − a mosquito located

at (1,1, 2) desires to fly in such a direction that it will get worm as soon as possible. In

what direction should it fly?

(b) If ( , , )x y zφ is a solution of Laplace equation, then show that Gradφ is both solenoidal

and irrotational.

[8M + 7M]

Page 1 of 2

Set No. 2

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Code No: R10102/R10

8 (a) If 2 2( 4) 3 (2 )f x y i x y j xz z k= + − + + + and S is the upper half of the sphere

2 2 2 16x y z+ + = . Show by using Stokes theorem that 3. 2SCurl f nds aπ=∫ .

(b) If S is the surface of the tetrahedron bounded by the planes 0x = , 0y = , 0z = and

1ax by cz+ + = . Show that 1

.2S

r ndsabc

=∫ .

[8M + 7M]

Page 2 of 2

Set No. 2

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JNTUWORLD

Code No: R10102/R10

I B.Tech I Semester Regular Examinations, January / February 2011

MATHEMATICS - I

(Common to All Branches)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

*********

1 (a) Solve 32 sec tandy

y x y xdx

− =

(b) Find the orthogonal trajectories of the family of circle 2 2 2 1 0x y fy+ + + =

[8M + 7M]

2 (a) Solve 2( 4) sinh 54 8D y x x x− = + +

(b) Solve 2( 2 1) sinxD D y xe x− + =

[8M + 7M]

3 (a) Find the minimum value of 2 2 2x y z+ + when ax by cz p+ + =

(b) If 2 2x y

zx y

+=

+,then prove that

2 2

22

z z zx y

x x y x

∂ ∂ ∂+ =

∂ ∂ ∂ ∂

[8M + 7M]

4 (a) Trace the curve 3 3 3x y axy+ = , 0a>

(b) Trace the curve 2cos log tan2 2

a tx a t= + , siny a t=

[8M + 7M]

5 (a) Find the area of the surface generated by revolving the loop of 2 29 ( 3)y x x= − about the

x axis− .

(b) Find the volume of the solid generated by the revolving one arc of the cycloid

( sin ), (1 cos )x a y aθ θ θ= + = + about the x axis−

[8M + 7M]

6 (a) Evaluate 2 2 2( )z x y z dxdydz+ +∫ ∫ ∫ through the volume of the cylinder 2 2 2x y a+ =

intercepted by the planes 0z = and z h= .

(b) Evaluate 3r drdθ∫ ∫ over area bounded by the circles 2 cosr θ= and 4cosr θ=

[8M + 7M]

7 (a) Find the directional derivative of the function 2 3x y yzφ = + at the point (2, 1,1)− in the

direction of the normal to the surface 2log 4 0z yx − + = at ( 1,2,1)−

(b) Show that the vector field 2 2 2 22 ( cos ) (2 cos )f x y z i x z z yz j x yz y yz k= + + + + is

irrotational. Find the potential function.

[8M + 7M]

Page 1 of 2

Set No. 3

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Code No: R10102/R10

8 (a) Verify Greens theorem for the functions ( , ) ( sin )M x y y x= − and ( , ) cosN x y x= in the

region bounded by the triangle in the xy-plane bounded by the lines 0,2

y xπ

= = and

2xy

π= .

(b) Using Divergence theorem evaluate S

xdydz ydzdx zdxdy+ +∫ ∫ , where

2 2 2 2:S x y z a+ + = .

[8M + 7M]

Page 2 of 2

Set No. 3

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JNTUWORLD

Code No: R10102/R10

I B.Tech I Semester Regular Examinations, January / February 2011

MATHEMATICS - I

(Common to All Branches)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

*********

1 (a) Solve 2 2 2 2 2 3( ) (3 2 ) 0xy x dx x y x y x dy− + + − =

(b) Find the orthogonal trajectories of the family of circle 2 2 2 1 0x y fy+ + + =

[8M + 7M]

2 (a) Solve 2( 1) cosD x t t+ = given x=0, 0dx

dt= at t =0

(b) Solve 2 2( ) tanD a y ax+ =

[8M + 7M]

3 (a) Trace the curve 3 3 3x y axy+ = , 0a>

(b) Trace the curve 2cos log tan2 2

a tx a t= + , siny a t=

[8M + 7M]

4 (a) Find the minimum value of 2 3 4x y z subject to the condition 2 3 4x y z a+ + =

(b) If (1 )x u v= + (1 )y v u= + then prove that ( , )

1( , )

x yu v

u v

∂= + +

∂

[8M + 7M]

5 (a) Find the volume of the spindle shaped solid formed by revolving the asteroid 2 2 2

3 3 3x y a+ = about x axis− .

(b) Find the surface area generated by the revolution of one arc of a catenary

cosh( / )y c x c= about x-axis.

[8M + 7M]

6 (a) Evaluate 2

4 2 20

a a

ax

y dydx

y a x−∫ ∫

(b) Evaluate 2 2

r dr d

a r

θ

+∫ ∫ over one loop of the lemniscates of Bernoulli 2 2 cos 2r a θ=

[8M + 7M]

Page 1 of 2

Set No. 4

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Code No: R10102/R10

7 (a) Find the constants a and b so that the surface 2 ( 2)ax b yz a x− = + will be orthogonal to

the surface 2 34 4x y z+ = at the point ( ), ,−1 1 2 .

(b) Determine the constant a so that the vector ( 3 ) ( 2 ) ( )u x y i y z j x a y k= + + − + + is

solenoidal.

[8M + 7M]

8 Verify Gauss divergence theorem for 3 ( ) ( )F xzi y zx j x yz z k= + + + + , where V is the

volume bounded by the coordinate planes and the plane 2 3 6 12x y z+ + = in the first

octant.

[15M]

Page 2 of 2

Set No. 4

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