Download JNTUK (Jawaharlal Nehru Technological University Kakinada) B.Tech Supplementary 2014 Feb-March R10 I Semester (1st Year 1st Sem) MATHEMATICS I Question Paper.

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICS-I

( Common to Civil Engineering, Electrical & Electronics Engineering,

Mechanical Engineering, Electronics & Communication Engineering,

Computer Science & Engineering, Chemical Engineering, Electronics &

Instrumentation Engineering, Bio-Medical Engineering, Information

Technology, Electronics & Computer Engineering, Aeronautical

Engineering, Bio-Technology, Automobile Engineering, Mining and

Petroliem Technology)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Solve (x

2

+y

2

a

2

)xdx + (x

2

y

2

b

2

)ydy = 0. [7+8]

(b) If air is maintained at 20

0

Cand the temperature of the body cools from 80

0

C

to 60

0

C in 10 minutes, nd the temperature of the body after 30 minutes.

2. (a) Solve (D

2

+a

2

)y =Secax

(b) Solve (D

2

+ 4)y =e

x

+Sin 2x [8+7]

3. (a) If V = log (x

2

+y

2

) +x 2y nd

@V

@x

:,

@V

@y

;

@

2

V

@x

2

:

@

2

V

@y

2

:

(b) If U = xe

xy

where x

2

+y

2

+ 2xy = 1; nd

@

2

U

@x

2

: [8+7]

4. (a) Trace the curve r = 2 + 3 sin.

(b) Trace the curve y

2

(2ax) =x

3

. [8+7]

5. (a) Find the surface of the solid generated by revolution of the lemniscate r

2

=

a

2

cos

2

about the initial line.

(b) Show that the whole length of the curve x

2

(a

2

x

2

) = 8a

2

y

2

isa

p

2. [8+7]

6. (a) Show that

R

4a

0

R

y

y

2

4a

x

2

y

2

x

2

+y

2

dx dy = 8a

2

2

5

3

.

(b) Evaluate

RR

R

ydxdy where R is the domain bounded by y-axis, the curve

y=x

2

and the line x +y = 2 in the rst quadrants . [8+7]

7. (a) If V= e

xyz

(i+j+k), nd curl V.

(b) Find the constants a and b so that the surface ax

2

-byz = (a+2)x will be

orthogonal to the surface 4x

2

y +z

3

=4 at the point (1,-1,2) [8+7]

8. (a) Show that the area of the ellipse x

2

/a

2

+ y

2

/b

2

= 1 is ab

(b) If f = (2x

2

{ 3z)i { 2xyj { 4xzk, evaluate

(i)

R

v

r:fdV and

(ii)

R

v

rfdV where V is the closed region bounded by x = 0, y = 0, z = 0,

2x + 2y +z = 4. [8+7]

?????

1 of 1

FirstRanker.com - FirstRanker's Choice

Code No: R10102/R10 Set No. 1

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICS-I

( Common to Civil Engineering, Electrical & Electronics Engineering,

Mechanical Engineering, Electronics & Communication Engineering,

Computer Science & Engineering, Chemical Engineering, Electronics &

Instrumentation Engineering, Bio-Medical Engineering, Information

Technology, Electronics & Computer Engineering, Aeronautical

Engineering, Bio-Technology, Automobile Engineering, Mining and

Petroliem Technology)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Solve (x

2

+y

2

a

2

)xdx + (x

2

y

2

b

2

)ydy = 0. [7+8]

(b) If air is maintained at 20

0

Cand the temperature of the body cools from 80

0

C

to 60

0

C in 10 minutes, nd the temperature of the body after 30 minutes.

2. (a) Solve (D

2

+a

2

)y =Secax

(b) Solve (D

2

+ 4)y =e

x

+Sin 2x [8+7]

3. (a) If V = log (x

2

+y

2

) +x 2y nd

@V

@x

:,

@V

@y

;

@

2

V

@x

2

:

@

2

V

@y

2

:

(b) If U = xe

xy

where x

2

+y

2

+ 2xy = 1; nd

@

2

U

@x

2

: [8+7]

4. (a) Trace the curve r = 2 + 3 sin.

(b) Trace the curve y

2

(2ax) =x

3

. [8+7]

5. (a) Find the surface of the solid generated by revolution of the lemniscate r

2

=

a

2

cos

2

about the initial line.

(b) Show that the whole length of the curve x

2

(a

2

x

2

) = 8a

2

y

2

isa

p

2. [8+7]

6. (a) Show that

R

4a

0

R

y

y

2

4a

x

2

y

2

x

2

+y

2

dx dy = 8a

2

2

5

3

.

(b) Evaluate

RR

R

ydxdy where R is the domain bounded by y-axis, the curve

y=x

2

and the line x +y = 2 in the rst quadrants . [8+7]

7. (a) If V= e

xyz

(i+j+k), nd curl V.

(b) Find the constants a and b so that the surface ax

2

-byz = (a+2)x will be

orthogonal to the surface 4x

2

y +z

3

=4 at the point (1,-1,2) [8+7]

8. (a) Show that the area of the ellipse x

2

/a

2

+ y

2

/b

2

= 1 is ab

(b) If f = (2x

2

{ 3z)i { 2xyj { 4xzk, evaluate

(i)

R

v

r:fdV and

(ii)

R

v

rfdV where V is the closed region bounded by x = 0, y = 0, z = 0,

2x + 2y +z = 4. [8+7]

?????

1 of 1

Code No: R10102/R10 Set No. 2

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICS-I

( Common to Civil Engineering, Electrical & Electronics Engineering,

Mechanical Engineering, Electronics & Communication Engineering,

Computer Science & Engineering, Chemical Engineering, Electronics &

Instrumentation Engineering, Bio-Medical Engineering, Information

Technology, Electronics & Computer Engineering, Aeronautical

Engineering, Bio-Technology, Automobile Engineering, Mining and

Petroliem Technology)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Solve e

y

1+

dy

dx

= e

x

(b) Show that the family of curves

x

2

a

2

+?

+

y

2

a

2

+?

= 1, where ??? is a parameter is self

orthogonal. [8+7]

2. (a) Solve (D

2

+9)y = 2Cos

2

x. (b) Solve

d

2

y

dx

2

+4y = 2e

x

Sin

2

x. [8+7]

3. (a) Calculate the approximate value of

?

10 to four decimal places using Taylor?s

theorem.

(b) Find 3 positive numbers whose sum is 600 and whose product is maximum.

[8+7]

4. (a) Trace the curve y = x

2

(x

2

?4). (b)Trace the curve r = cos?. [8+7]

5. (a) The ?gure bounded by a parabola and the tangents at the extremities of its

latusrectum revolves about the axis of the parabola, Find the volume of the

solid thus generated. [8+7]

(b) The segment of the parabola y

2

=4ax which is cuto? by the latus rectum

revolves about the directrix.Find the volume of rotation of the annular region.

6. (a) Evaluate

R R

(x+y)

2

dx dy. over the area bounded by the ellipse

x

2

a

2

+

y

2

b

2

= 1.

(b) TransformthefollowingtoCartesianformandhenceevaluate

R

?

0

R

a

0

r

3

sin?drd?.

[8+7]

7. (a) Prove that?r = r/r

(b) Find the angle between the surfaces x

2

+ y

2

+ z

2

= 9 and z=x

2

+ y

2

?3 at the

point (2,-1,2). [8+7]

8. (a) Evaluate

RR

S

(yzi+zxj+xyk).dS whereSisthesurfaceofthespherex

2

+y

2

+z

2

=a

2

in the ?rst octant.

(b) Evaluate

H

c

(x

2

? 2xy)dx + (x

2

y + 3)dy around the boundary of the region

de?ned by y

2

=8x and x=2. [8+7]

?????

1 of 1

FirstRanker.com - FirstRanker's Choice

Code No: R10102/R10 Set No. 1

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICS-I

( Common to Civil Engineering, Electrical & Electronics Engineering,

Mechanical Engineering, Electronics & Communication Engineering,

Computer Science & Engineering, Chemical Engineering, Electronics &

Instrumentation Engineering, Bio-Medical Engineering, Information

Technology, Electronics & Computer Engineering, Aeronautical

Engineering, Bio-Technology, Automobile Engineering, Mining and

Petroliem Technology)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Solve (x

2

+y

2

a

2

)xdx + (x

2

y

2

b

2

)ydy = 0. [7+8]

(b) If air is maintained at 20

0

Cand the temperature of the body cools from 80

0

C

to 60

0

C in 10 minutes, nd the temperature of the body after 30 minutes.

2. (a) Solve (D

2

+a

2

)y =Secax

(b) Solve (D

2

+ 4)y =e

x

+Sin 2x [8+7]

3. (a) If V = log (x

2

+y

2

) +x 2y nd

@V

@x

:,

@V

@y

;

@

2

V

@x

2

:

@

2

V

@y

2

:

(b) If U = xe

xy

where x

2

+y

2

+ 2xy = 1; nd

@

2

U

@x

2

: [8+7]

4. (a) Trace the curve r = 2 + 3 sin.

(b) Trace the curve y

2

(2ax) =x

3

. [8+7]

5. (a) Find the surface of the solid generated by revolution of the lemniscate r

2

=

a

2

cos

2

about the initial line.

(b) Show that the whole length of the curve x

2

(a

2

x

2

) = 8a

2

y

2

isa

p

2. [8+7]

6. (a) Show that

R

4a

0

R

y

y

2

4a

x

2

y

2

x

2

+y

2

dx dy = 8a

2

2

5

3

.

(b) Evaluate

RR

R

ydxdy where R is the domain bounded by y-axis, the curve

y=x

2

and the line x +y = 2 in the rst quadrants . [8+7]

7. (a) If V= e

xyz

(i+j+k), nd curl V.

(b) Find the constants a and b so that the surface ax

2

-byz = (a+2)x will be

orthogonal to the surface 4x

2

y +z

3

=4 at the point (1,-1,2) [8+7]

8. (a) Show that the area of the ellipse x

2

/a

2

+ y

2

/b

2

= 1 is ab

(b) If f = (2x

2

{ 3z)i { 2xyj { 4xzk, evaluate

(i)

R

v

r:fdV and

(ii)

R

v

rfdV where V is the closed region bounded by x = 0, y = 0, z = 0,

2x + 2y +z = 4. [8+7]

?????

1 of 1

Code No: R10102/R10 Set No. 2

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICS-I

( Common to Civil Engineering, Electrical & Electronics Engineering,

Mechanical Engineering, Electronics & Communication Engineering,

Computer Science & Engineering, Chemical Engineering, Electronics &

Instrumentation Engineering, Bio-Medical Engineering, Information

Technology, Electronics & Computer Engineering, Aeronautical

Engineering, Bio-Technology, Automobile Engineering, Mining and

Petroliem Technology)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Solve e

y

1+

dy

dx

= e

x

(b) Show that the family of curves

x

2

a

2

+?

+

y

2

a

2

+?

= 1, where ??? is a parameter is self

orthogonal. [8+7]

2. (a) Solve (D

2

+9)y = 2Cos

2

x. (b) Solve

d

2

y

dx

2

+4y = 2e

x

Sin

2

x. [8+7]

3. (a) Calculate the approximate value of

?

10 to four decimal places using Taylor?s

theorem.

(b) Find 3 positive numbers whose sum is 600 and whose product is maximum.

[8+7]

4. (a) Trace the curve y = x

2

(x

2

?4). (b)Trace the curve r = cos?. [8+7]

5. (a) The ?gure bounded by a parabola and the tangents at the extremities of its

latusrectum revolves about the axis of the parabola, Find the volume of the

solid thus generated. [8+7]

(b) The segment of the parabola y

2

=4ax which is cuto? by the latus rectum

revolves about the directrix.Find the volume of rotation of the annular region.

6. (a) Evaluate

R R

(x+y)

2

dx dy. over the area bounded by the ellipse

x

2

a

2

+

y

2

b

2

= 1.

(b) TransformthefollowingtoCartesianformandhenceevaluate

R

?

0

R

a

0

r

3

sin?drd?.

[8+7]

7. (a) Prove that?r = r/r

(b) Find the angle between the surfaces x

2

+ y

2

+ z

2

= 9 and z=x

2

+ y

2

?3 at the

point (2,-1,2). [8+7]

8. (a) Evaluate

RR

S

(yzi+zxj+xyk).dS whereSisthesurfaceofthespherex

2

+y

2

+z

2

=a

2

in the ?rst octant.

(b) Evaluate

H

c

(x

2

? 2xy)dx + (x

2

y + 3)dy around the boundary of the region

de?ned by y

2

=8x and x=2. [8+7]

?????

1 of 1

Code No: R10102/R10 Set No. 3

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICS-I

( Common to Civil Engineering, Electrical & Electronics Engineering,

Mechanical Engineering, Electronics & Communication Engineering,

Computer Science & Engineering, Chemical Engineering, Electronics &

Instrumentation Engineering, Bio-Medical Engineering, Information

Technology, Electronics & Computer Engineering, Aeronautical

Engineering, Bio-Technology, Automobile Engineering, Mining and

Petroliem Technology)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Solve y(Sinx?y)dx = Cosx dy

(b) If the temperature of air is maintained at 20

0

Cand the temperature of the

body cools from 100

0

Cto80

0

C in 10 minutes, ?nd the temperature of the

body after 20 minutes. [8+7]

2. (a) Solve (D

2

?4D+13)y = e

2x

(b) Solve (D

2

?3D+2)y = Coshx [8+7]

3. (a) If r + s + t = x, s + t = xy, t = xyz, ?nd

?(r,s,t)

?(x,y,z)

.

(b) Find the extreme points of f(x,y) = xy +

8

x

+

8

y

. [8+7]

4. (a) Trace the curve y = 5cosh

x

5

.

(b) Trace the curve y

2

= (4?x)(3?x

2

).. [8+7]

5. (a) Find the length of the arc of the curve y =log (secx) from x = o to

?

3

.

(b) Find the perimeter of the loop of the curve 3ay

2

=x(x-a)

2.

[8+7]

6. (a) Evaluate

R R

rdrd? over the region bounded by the cardioid r=a(1+cos?) and

out side the circle r=a .

(b) Change the order of Integration & evaluate

Z

4a

0

Z

2

?

ax

x

2

4a

dydx [8+7]

7. (a) Prove that (F??)?r = -2F

(b) Determine the constant a so that the vector V = (x+3y)i+(y-z)j+(x+az)k is

solenoidal. [8+7]

8. Apply Stokes theorem, to evaluate

H

c

ydx + zdy + xdzwhere C is the curve of

intersection of the sphere x

2

+ y

2

+ z

2

= a

2

and x + z = a. [15]

?????

1 of 1

FirstRanker.com - FirstRanker's Choice

Code No: R10102/R10 Set No. 1

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICS-I

( Common to Civil Engineering, Electrical & Electronics Engineering,

Mechanical Engineering, Electronics & Communication Engineering,

Computer Science & Engineering, Chemical Engineering, Electronics &

Instrumentation Engineering, Bio-Medical Engineering, Information

Technology, Electronics & Computer Engineering, Aeronautical

Engineering, Bio-Technology, Automobile Engineering, Mining and

Petroliem Technology)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Solve (x

2

+y

2

a

2

)xdx + (x

2

y

2

b

2

)ydy = 0. [7+8]

(b) If air is maintained at 20

0

Cand the temperature of the body cools from 80

0

C

to 60

0

C in 10 minutes, nd the temperature of the body after 30 minutes.

2. (a) Solve (D

2

+a

2

)y =Secax

(b) Solve (D

2

+ 4)y =e

x

+Sin 2x [8+7]

3. (a) If V = log (x

2

+y

2

) +x 2y nd

@V

@x

:,

@V

@y

;

@

2

V

@x

2

:

@

2

V

@y

2

:

(b) If U = xe

xy

where x

2

+y

2

+ 2xy = 1; nd

@

2

U

@x

2

: [8+7]

4. (a) Trace the curve r = 2 + 3 sin.

(b) Trace the curve y

2

(2ax) =x

3

. [8+7]

5. (a) Find the surface of the solid generated by revolution of the lemniscate r

2

=

a

2

cos

2

about the initial line.

(b) Show that the whole length of the curve x

2

(a

2

x

2

) = 8a

2

y

2

isa

p

2. [8+7]

6. (a) Show that

R

4a

0

R

y

y

2

4a

x

2

y

2

x

2

+y

2

dx dy = 8a

2

2

5

3

.

(b) Evaluate

RR

R

ydxdy where R is the domain bounded by y-axis, the curve

y=x

2

and the line x +y = 2 in the rst quadrants . [8+7]

7. (a) If V= e

xyz

(i+j+k), nd curl V.

(b) Find the constants a and b so that the surface ax

2

-byz = (a+2)x will be

orthogonal to the surface 4x

2

y +z

3

=4 at the point (1,-1,2) [8+7]

8. (a) Show that the area of the ellipse x

2

/a

2

+ y

2

/b

2

= 1 is ab

(b) If f = (2x

2

{ 3z)i { 2xyj { 4xzk, evaluate

(i)

R

v

r:fdV and

(ii)

R

v

rfdV where V is the closed region bounded by x = 0, y = 0, z = 0,

2x + 2y +z = 4. [8+7]

?????

1 of 1

Code No: R10102/R10 Set No. 2

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICS-I

( Common to Civil Engineering, Electrical & Electronics Engineering,

Mechanical Engineering, Electronics & Communication Engineering,

Computer Science & Engineering, Chemical Engineering, Electronics &

Instrumentation Engineering, Bio-Medical Engineering, Information

Technology, Electronics & Computer Engineering, Aeronautical

Engineering, Bio-Technology, Automobile Engineering, Mining and

Petroliem Technology)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Solve e

y

1+

dy

dx

= e

x

(b) Show that the family of curves

x

2

a

2

+?

+

y

2

a

2

+?

= 1, where ??? is a parameter is self

orthogonal. [8+7]

2. (a) Solve (D

2

+9)y = 2Cos

2

x. (b) Solve

d

2

y

dx

2

+4y = 2e

x

Sin

2

x. [8+7]

3. (a) Calculate the approximate value of

?

10 to four decimal places using Taylor?s

theorem.

(b) Find 3 positive numbers whose sum is 600 and whose product is maximum.

[8+7]

4. (a) Trace the curve y = x

2

(x

2

?4). (b)Trace the curve r = cos?. [8+7]

5. (a) The ?gure bounded by a parabola and the tangents at the extremities of its

latusrectum revolves about the axis of the parabola, Find the volume of the

solid thus generated. [8+7]

(b) The segment of the parabola y

2

=4ax which is cuto? by the latus rectum

revolves about the directrix.Find the volume of rotation of the annular region.

6. (a) Evaluate

R R

(x+y)

2

dx dy. over the area bounded by the ellipse

x

2

a

2

+

y

2

b

2

= 1.

(b) TransformthefollowingtoCartesianformandhenceevaluate

R

?

0

R

a

0

r

3

sin?drd?.

[8+7]

7. (a) Prove that?r = r/r

(b) Find the angle between the surfaces x

2

+ y

2

+ z

2

= 9 and z=x

2

+ y

2

?3 at the

point (2,-1,2). [8+7]

8. (a) Evaluate

RR

S

(yzi+zxj+xyk).dS whereSisthesurfaceofthespherex

2

+y

2

+z

2

=a

2

in the ?rst octant.

(b) Evaluate

H

c

(x

2

? 2xy)dx + (x

2

y + 3)dy around the boundary of the region

de?ned by y

2

=8x and x=2. [8+7]

?????

1 of 1

Code No: R10102/R10 Set No. 3

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICS-I

( Common to Civil Engineering, Electrical & Electronics Engineering,

Mechanical Engineering, Electronics & Communication Engineering,

Computer Science & Engineering, Chemical Engineering, Electronics &

Instrumentation Engineering, Bio-Medical Engineering, Information

Technology, Electronics & Computer Engineering, Aeronautical

Engineering, Bio-Technology, Automobile Engineering, Mining and

Petroliem Technology)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Solve y(Sinx?y)dx = Cosx dy

(b) If the temperature of air is maintained at 20

0

Cand the temperature of the

body cools from 100

0

Cto80

0

C in 10 minutes, ?nd the temperature of the

body after 20 minutes. [8+7]

2. (a) Solve (D

2

?4D+13)y = e

2x

(b) Solve (D

2

?3D+2)y = Coshx [8+7]

3. (a) If r + s + t = x, s + t = xy, t = xyz, ?nd

?(r,s,t)

?(x,y,z)

.

(b) Find the extreme points of f(x,y) = xy +

8

x

+

8

y

. [8+7]

4. (a) Trace the curve y = 5cosh

x

5

.

(b) Trace the curve y

2

= (4?x)(3?x

2

).. [8+7]

5. (a) Find the length of the arc of the curve y =log (secx) from x = o to

?

3

.

(b) Find the perimeter of the loop of the curve 3ay

2

=x(x-a)

2.

[8+7]

6. (a) Evaluate

R R

rdrd? over the region bounded by the cardioid r=a(1+cos?) and

out side the circle r=a .

(b) Change the order of Integration & evaluate

Z

4a

0

Z

2

?

ax

x

2

4a

dydx [8+7]

7. (a) Prove that (F??)?r = -2F

(b) Determine the constant a so that the vector V = (x+3y)i+(y-z)j+(x+az)k is

solenoidal. [8+7]

8. Apply Stokes theorem, to evaluate

H

c

ydx + zdy + xdzwhere C is the curve of

intersection of the sphere x

2

+ y

2

+ z

2

= a

2

and x + z = a. [15]

?????

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Code No: R10102/R10 Set No. 4

I B.Tech I Semester Supplementary Examinations, Feb/Mar 2014

MATHEMATICS-I

( Common to Civil Engineering, Electrical & Electronics Engineering,

Mechanical Engineering, Electronics & Communication Engineering,

Computer Science & Engineering, Chemical Engineering, Electronics &

Instrumentation Engineering, Bio-Medical Engineering, Information

Technology, Electronics & Computer Engineering, Aeronautical

Engineering, Bio-Technology, Automobile Engineering, Mining and

Petroliem Technology)

Time: 3 hours Max Marks: 75

Answer any FIVE Questions

All Questions carry equal marks

?????

1. (a) Solve (x+1)

dy

dx

?y = e

3x

(x+1)

2

(b) Find the orthogonal trajectory of the family of curves x

2/3

+y

2/3

= a

2/3

, where

?a? is a parameter [8+7]

2. (a) Solve (D

3

?6D

2

+11D?6)y = e

?2x

+e

?3x

(b) Solve

d

2

y

dx

2

?8

dy

dx

+15y = 0 [8+7]

3. (a) If a =

yz

x

, b =

xz

y

, c =

xy

z

, ?nd

?(x,y,z)

?(a,b,c)

.

(b) Find the minimum value of x

2

+y

2

+z

2

, give that xyz = a

3

[8+7]

4. (a) Trace the curve r = cos 4?.

(b) Trace the curvey

2

(1?x) = x

2

(1+x).. [8+7]

5. Prove that the volume of the solid generated by the revolution about the x?axis

of the loop of the curve x = t

2

,y = t?

1

3

t

3

is

3?

4

. [8+7]

6. (a) By changing the order of integration evaluate

Z

1

0

Z

)2?x

2

0

x

)x

2

+y

2

dydx.

(b) Evaluate

Z

a

0

Z

)a

2

?x

2

a?x

y dx dy by using change of order of integration . [8+7]

7. (a) If V= e

xyz

(i+j+k), ?nd curl V.

(b) Find the constants a and b so that the surface ax

2

-byz = (a+2)x will be

orthogonal to the surface 4x

2

y +z

3

=4 at the point (1,-1,2) [8+7]

8. (a) Use divergence theorem to evaluate

RR

S

(x

3

i + y

3

j + z

3

k).Nds, and S is the

surface of the sphere x

2

+y

2

+z

2

=r

2

.

(b) Using Green?s theorem, Find the area bounded by the hypocycloid x

2/3

+y

2/3

=

a

2/3

, a>0. Given that the parametric equations are x =a cos

3

?, y =a sin

3

?.

[8+7]

?????

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This post was last modified on 03 December 2019