Unit – I ( Vector Calculus - SNS Courseware

Engineering Mathematics

2016

SUBJECT NAME

: Mathematics - II

SUBJECT CODE

: MA6251

MATERIAL NAME

: University Questions

REGULATION

: R2013

UPDATED ON

: May-June 2016

TEXTBOOK FOR REFERENCE

: Sri Hariganesh Publications (Author: C. Ganesan)

To buy the book visit

www.hariganesh.com/textbook

(Scan the above QR code for the direct download of this material)

Unit – I (Vector Vector Calculus Calculus) • Simple problems on vector calculus 1.

Find the directional derivative of

i + 2 j + 2k . 2.

If

φ = 2xy + z 2 at the point ( 1, −1, 3 ) in the direction of (Textbook Page No.: 1.6)

∇φ = 2 xyz 3 i + x 2 z 3 j + 3 x 2 yz 2 k find φ ( x , y , z ) given that φ (1, −2, 2) = 4 .

Textbook Page No.: 1.1 16

3.

(M/J 2016)

(

Prove that F = 6 xy + z

) i + ( 3x

3

the scalar potential such that F

4.

(

Show that F = y + 2 xz 2

2

(

Show that F = 2 xy − z

2

find its scalar potential.

6.

(

Show that F = x + xy 2

2

= ∇ϕ .

− z ) j + ( 3 xz 2 − y ) k is irrotational vector and find (Textbook Page No.: 1.32) 2

(Textbook Page No.: 1.28)

)i +(x

2

)i +( y

2

(M/J 2012)

+ 2 yz ) j + ( y 2 − 2 zx ) k is irrotational and

(Textbook Page No.: 1.50) 2

(M/J 2010)

) i + ( 2 xy − z ) j + ( 2 x z − y + 2 z ) k is irrotational and

hence find its scalar alar potential.

5.

(M/J 2009)

(N/D 2012)

+ x 2 y ) j is irrotational and find its scalar potential.

Textbook ook Page No.: 1.50

Sri Hariganesh Publications (Ph: 9841168917, 8939331876)

(N/D 2013)

Page 1

Engineering Mathematics 7.

(


Find the angle between the normals to the surface

( 4,1, −1) . 9.

)

2 2 Prove that F = x − y + x iˆ − ( 2 xy + y ) ˆj is irrotational and hence find its scalar

potential.

8.

2016

xy 3 z 2 = 4 at the points ( −1, −1, 2 ) and


Find the angle between the normals to the surface

( −3, −3, 3 ) .

(N/D 2014)

(M/J 2009)

xy = z 2 at the points ( 1, 4, 2 ) and


(A/M 2011)

10. Find the angle between the normals to the surface x 2 = yz at the points ( 1,1,1) and

( 2, 4,1) .


(N/D 2014)

11. Find a and b so that the surfaces ax 3 − by 2 z − (a + 3) x 2 = 0 and 4 x 2 y − z 3 − 11 = 0 cut orthogonally at the point

( 2, −1, −3 ) .

(N/D 2013),(M/J 2016)

Textbook Page No.: 1.13

12. Find the value of n such that the vector r n r is both solenoidal and irrotational. Textbook Page No.: 1.44

(M/J 2014)

13. Find the work done in moving a particle in the force field given by

F = 3 x 2 i + (2 xz − y ) j + zk along the straight line from ( 0, 0, 0 ) to ( 2,1, 3 ) . Textbook Page No.: 1.55

(M/J 2012)

2 n n− 2 14. If r is the position vector of the point ( x , y , z ) , Prove that ∇ r = n( n + 1)r . Hence find

the value of

1 ∇2   . r


(N/D 2010),(M/J 2015)

15. Determine f ( r ) , where r = xi + yj + zk , if f ( r )r is solenoidal and irrotational. Textbook Page No.: 1.42

16. Prove that Curl Curl F = grad div F − ∇ 2 F . 17. Evaluate

∫(x

2

(N/D 2011) (M/J 2016)

+ xy ) dx + ( x 2 + y 2 ) dy where C is the square bounded by the

C

lines x = 0, x = 1, y = 0 and y = 1 .


(N/D 2009),(N/D 2011)

Page 2


2016


18. Evaluate

∫∫ F i n ds where F = 2 xyi + yz

2

j + xzk and S is the surface of the

s

parallelepiped bounded by x = 0, y = 0, z = 0, x = 2, y = 1 and z

= 3.

(M/J 2011)


• Green’s Theorem 1.

(

Verify Green’s theorem for V = x + y 2

2

) i − 2 xyj taken around the rectangle bounded by

the lines x = ± a , y = 0 and y = b .

(N/D 2012),(Jan 2016)


2.

Using Green’s theorem in a plane evaluate

∫  x (1 + y ) dx + ( x 2

3

+ y 3 ) dy  where C is

C

the square formed by x = ±1 and y = ±1 .

3.

Verify Green’s theorem in a plane for

∫ ( 3 x

(M/J 2016) 2

− 8 y 2 ) dx + ( 4 y − 6 xy ) dy  , Where C is the

C

boundary of the region defined by the lines x = 0, y = 0 and x + y = 1 . Textbook Page No.: 1.86

4.

(N/D 2010),(A/M 2011),(M/J 2011), (M/J 2012)

Verify Green’s theorem for

∫ (3x

2

− 8 y 2 ) dx + ( 4 y − 6 xy ) dy where C is the boundary of

C

the region defined by x

= y2 , y = x2 .

(M/J 2010)


5.

Using Green’s theorem, evaluate

∫ ( 3 x

2

− 8 y 2 ) dx + ( 4 y − 6 xy ) dy  , Where C is the

C

boundary of the triangle formed by the lines x = 0, y = 0 and x + y = 1 . (N/D 2014) Textbook Page No.: 1.148

6.

Using Green’s theorem, evaluate

∫ ( y − sin x ) dx + cos x dy where C is the triangle formed C

by

y = 0, x =

π 2

, y=

2x

π

.



(M/J 2015)

Page 3


2016

• Stoke’s Theorem 1.

(

Verify Stokes theorem for F = x − y 2

2

) i + 2 xyj

in the rectangular region of

bounded by the lines x = 0, y = 0, x = a and y = b .

z = 0 plane

(M/J 2014)


2.

(

Verify Stoke’s theorem for F = x + y 2

2

) i − 2 xyj taken around the rectangle formed by the

lines x = − a , x = a , y = 0 and y = b .

(N/D 2013)


3.

F = xyi − 2 yzj − zxk where S is the open surface of the rectangular parallelepiped formed by the planes x = 0, x = 1, y = 0, y = 2 and z = 3 Verify Stoke’s theorem for

above the XY plane.


(M/J 2009)

4.

= ( y − z )i + yzj − xzk , where S is the surface bounded by the planes x = 0, y = 0, z = 0, x = 1, y = 1, z = 1 and C is the square boundary (Textbook Page No.: 1.150) (N/D 2011) on the xoy -plane.

5.

Verify Stoke’s theorem when F = 2 xy − x

Verify Stoke’s thorem for the vector F

(

region enclosed by the parabolas

2

)i −(x

2

)

− y 2 j and C is the boundary of the

y 2 = x and x 2 = y .

(N/D 2009)


6.

Verify Stoke’s theorem for the vector field

F = (2 x − y )i − yz 2 j − y 2 zk over the upper half

surface x + y + z = 1 , bounded by its projection on the xy -plane. Textbook Page No.: 1.132 2

7.

Evaluate

2

2

(M/J 2013)

∫ ( sin zdx − cos xdy + sin ydz ) by using Stoke’s theorem, where C is the boundary C

of the rectangle defined by 0 ≤ x ≤ π , 0 ≤ y ≤ 1, z = 3 . Textbook Page No.: 1.139

8.

Using Stokes theorem, evaluate

∫ F idr , where F = y i + x 2

C

boundary of the triangle with vertices at

9.

(N/D 2009)

2

j − ( x + z )k and ‘C’ is the

( 0, 0, 0 ) , ( 1, 0, 0) , ( 1,1, 0) .

Textbook Page No.: 1.140 Using Stoke’s theorem prove that curl grand Textbook Page No.: 1.143

φ = 0.


(M/J 2012)

(M/J 2011)

Page 4


2016

• Gauss Divergence Theorem 1.

F = x 2 i + y 2 j + z 2 k where S is the surface of the cuboid formed by the planes x = 0, x = a , y = 0, y = b , z = 0 and z = c . (M/J 2009)

Verify Gauss divergence theorem for


2.

F = x 2 i + y 2 j + z 2 k taken over the cube bounded by the planes x = 0, y = 0, z = 0, x = 1, y = 1 and z = 1 . (M/J 2014)

Verify Gauss divergence theorem for


3.

F = 4 xzi − y 2 j + yzk over the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0, z = 1 .

Verify Gauss Divergence theorem for

(N/D 2010),(A/M 2011),(N/D 2012),(N/D 2013),(N/D 2014),(M/J 2015) Textbook Page No.: 1.93

4.

(

)

Verify Gauss – divergence theorem for the vector function f = x − yz i − 2 x yj + 2k 3

2

over the cube bounded by x = 0, y = 0, z = 0 and x = a , y = a , z = a . Textbook Page No.: 1.196

5.

(M/J 2010),(N/D 2011)

Verify divergence theorem for F = x i + zj + yzk over the cube formed by the planes 2

x = ± 1, y = ± 1, z = ± 1 . 6.

7.


(

) (

) (

(M/J 2013)

)

Verify Gauss’s theorem for F = x − yz i + y − zx j + z − xy k over the 2

2

2

rectangular parallelepiped bounded by x = 0, x = a , y = 0, y = b , z = 0 and

z = c.


(Jan 2016)

(

) (

) (

)

Verify Gauss’s theorem for F = x − yz i + y − zx j + z − xy k over the 2

2

rectangular parallelepiped formed by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and

2

0 ≤ z ≤ 1.


(N/D 2011)

Unit – II (Ordinary Differential Equation) • ODE with Constant Coefficients 1.

Solve the equation

(D

2

− 3 D + 2 ) y = 2cos ( 2 x + 3 ) + 2e x .

(N/D 2009)



Page 5


(

2

(

2

)

2016

2.

Solve D + 16 y = cos x .

3.

Solve D − 4 D + 3 y = cos 2 x + 2 x .

4.

Solve : D + 3 D + 2 y = sin x + x . (Textbook Page No.: 2.35)

5.

Solve the equation

(

3


)

2

)

2

(N/D 2010)

(D

2

(M/J 2014)

2

+ 5 D + 4 ) y = e − x sin 2 x .

(M/J 2011) (A/M 2011),(ND 2012)


(

)

−x

6.

Solve the equation D + 4 D + 3 y = e − sin x .

7.

Solve D − 4 D + 3 y = e cos 2 x .

8.

Solve D + 4 D + 3 y = 6e −

(

2

(

2

2

)

x

)

−2 x

(M/J 2010)


(M/J 2012)

sin x sin 2 x .

(N/D 2011)

+ sin 2 x . (Textbook Page No.: 2.56)

(M/J 2015)


(

2

(

2

(

3

)

9.

Solve D − 3 D + 2 y = xe

10.

Solve D + 2 D + 5 y = e − x .

11.

Solve D + 2 D + D y = e − + cos 2 x .

(Jan 2016)

12.

Solve

d2 y dy − 2 + y = 8 xe x sin x . 2 dx dx

(N/D 2013)

13.

Solve

(D

14.

Solve the equation

2

)

3x

−x

)

2

2

(N/D 2014)

−x

(Text Book Page No.: 2.53)

+ 2 D + 1) y = xe − x cos x .

(D

2

(M/J 2016)

+ 4 ) y = x 2 cos 2 x .

(M/J 2009),(N/D 2011)


• Method of Variation of Parameters 1.

Solve

d2 y + a 2 y = tan ax by method of variation of parameters. dx 2

Textbook Page No.: 2.110 2.

(M/J 2009),(M/J 2011),(M/J 2014)

Solve y ′′ + y = tan x using the method of variation of parameters.


(M/J 2016)

Page 6


Solve

2016

d2y + 4 y = tan 2 x by method of variation of parameters. dx 2

(N/D 2013),(N/D 2014)


(

)

Apply method of variation of parameters to solve D + 4 y = cot 2 x . 2


(

Solve D + a 2

2

(N/D 2009),(N/D 2011)

) y = sec ax using the method of variation of parameters.

(M/J 2012)


6.

Solve

d2 y + y = cos ecx by the method of variation of parameters. dx 2


(

(A/M 2011),(ND 2012)

)

Solve D + 1 y = x sin x by the method of variation of parameters. 2

(M/J 2010)


(

)

−x

Using variation of parameters, solve 2 D − D − 3 y = 25e − . 2

(N/D 2011)


9.

Solve

d2 y dy e− x + 2 + y = by the method of variation of parameters. dx 2 dx x2

(M/J 2013)


Solve, by the method of variation of parameters,

y′′ − 2 y′ + y = e x log x .(M/J 2015)


• Cauchy and Legendre Equations 1.

Solve

x2

d2y dy 1 + 4x + 2 y = x2 + 2 . 2 dx dx x

(M/J 2013)


(x

2

D 2 − xD + 1 ) y = sin ( log x ) .

2.

Solve

3.

Solve the equation

(x D 2

2

+ 3 xD + 5 ) y = x cos ( log x ) .


(N/D 2014) (M/J 2009)

Page 7


2016


Solve

(x D 2

2

− 3 xD + 4 ) y = x 2 cos ( log x ) .

(N/D 2010)


Solve

(x D 2

2

− xD + 4 ) y = x 2 sin ( log x ) .

(M/J 2012),(N/D 2009)


(x D 2

2

− xD − 2 ) y = x 2 log x .

6.

Solve

7.

Solve x D − 2 xD − 4 y = x + 2 log x .

(

2

)

2

2

(M/J 2016) (M/J 2010)


Solve

(x D 2

2

− xD + 1 ) y = log x + π .

(M/J 2015)

Textbook Page No.: 2.93 2

9.

 log x  Solve ( x D − xD + 1) y =   .  x 

10.

Solve the equation

2

2

d 2 y 1 dy 12 log x + . = dx 2 x dx x2

(M/J 2014)

(N/D 2012)


11.

Solve

x2

d2 y dy − 3x + 4 y = x 2 ln x . 2 dx dx

(N/D 2011)


d2y dy 12. Solve: (1 + x ) + (1 + x ) + y = 4cos [ log(1 + x )] . 2 dx dx 2

(N/D 2011)


d2y dy 13. Solve (1 + x ) + (1 + x ) + y = 2sin [ log(1 + x )] . 2 dx dx 2

(A/M 2011)



Page 8


Solve

( 3 x + 2)

2

d2y dy + 3 ( 3 x + 2 ) − 36 y = 3 x 2 + 4 x + 1 . 2 dx dx

2016

(M/J 2013)


Solve

( 2 x + 7)

2

y′′ − 6 ( 2 x + 7 ) y′ + 8 y = 8 x .

(Jan 2016)

• Simultaneous Differential Equations 1.

Solve

dx dy + 2 y = sin2t , −2x = cos2t . dt dt

(M/J 2012),(N/D 2009)


2.

Solve

dx dy + 2 y = − sin t , − 2 x = cos t . dt dt

(M/J 2014)


3.

Solve

dx dy + 2 y = − sin t , −2x = cost given x = 1 , y = 0 at t = 0 . dt dt

(N/D 2010)


4.

Solve

dx dy − y = t and + x = t2. dt dt

(A/M 2011),(M/J 2016)


5.

Solve

dx dy − y = t and + x = t 2 given x (0) = y (0) = 2 . dt dt

(N/D 2011)


6.

Solve

dx dy + y = et , x − =t. dt dt

(N/D 2012),(N/D 2014)


7.

Solve

dx dy + 2 x + 3 y = 2e 2 t , + 3 x + 2 y = 0. dt dt

(M/J 2010)



Page 9


Solve

2016

dx dy + 5x − 2 y = t, + 2 x + y = 0. dt dt

(M/J 2013)


9.

Solve

dy dx + 2 x − 3 y = t and − 3 x + 2 y = e 2t . dt dt

(N/D 2011)


10. Solve

dx dy + y = sin t , x + = cos t given x = 2 and y = 0 at t = 0 . dt dt

(M/J 2009)


11. Solve

dx dy + 4 x + 3 y = t and + 2 x + 5 y = e 2t . dt dt

(N/D 2013)


12. Solve the simultaneous differential equations:

dx dy dx + + 3 x = sin t , + y − x = cos t . dt dt dt


(M/J 2015)

13. Solve y′′ = x , x′′ = y .

(Jan 2016)

Unit – III (Laplace Transform) • Laplace Transform of Periodic Function 1.

for 0 < t < a t, , f ( t + 2a ) = f ( t ) .  2a − t , for a < t < 2a

Find the Laplace transform of f ( t ) =  Textbook Page No.: 3.54

2.

(M/J 2009),(N/D 2009),(A/M 2011),(N/D 2014),(M/J 2015)

Find the Laplace transform of the following triangular wave function given by

0≤ t ≤π t, and f ( t + 2π ) = f ( t ) . f (t ) =   2π − t , π ≤ t ≤ 2π

(M/J 2010),(M/J 2012)


3.

t, 0 < t < 1 and f ( t + 2) = f ( t ) for t > 0 .  0, 1 < t < 2

Find the Laplace transform of f ( t ) = 



(N/D 2011)(AUT)

Page 10


2016

Find the Laplace transform of square wave function defined by

1, in 0 < t < a with period 2a . f (t ) =   −1, in a < t < 2a

(N/D 2009)


Find the Laplace transform of square wave function (or Meoander function) of period

a  1, in 0 < t < 2 a as f ( t ) =  .  − 1, in a < t < a  2

(M/J 2013)


Find the Laplace transform of

f (t ) = E ,

0≤t ≤a

= − E , a ≤ t ≤ 2a

and f ( t + 2a ) = f ( t ) for all t .

(N/D 2010)


Find the Laplace transform of a square wave function given by

a   E for 0 ≤ t ≤ 2 , and f ( t + a ) = f ( t ) . f (t ) =   − E for a ≤ t ≤ a  2

(N/D 2011),(M/J 2016)


8.

 sin ω t , 0 < t < π / ω π / ω < t < 2π / ω  0,

Find the Laplace transform of the Half wave rectifier f ( t ) =  and f ( t + 2π / ω ) = f ( t ) for all t . Textbook Page No.: 3.51

(N/D 2012),(M/J 2014)

• Simple Problems and Initial & Final Value Theorem

(

−t

)

 e − t − cos t  L . t  

1.

Find L e − sin 3t

2.

Find the Laplace transform of e − t cos t .

2

and

−t

(Jan 2016)

(N/D 2014)



Page 11


2016 −2 t

Find the Laplace transform of te − cos 3 t .

(M/J 2009)



f ( t ) = te −3 t cos 2t .

(M/J 2014)

Textbook Page No.: 3.22 2 −3 t

5.

Find L  t e − sin 2t  .

6.

Find L  t e − cos t  .

7.

Verify initial and final value theorems for

(Text Book Page No.: 3.23)

2 −t

(M/J 2016)

f (t ) = 1 + e − t (sin t + cos t ) .


8.

(M/J 2013)

Find

(M/J 2010),(N/D 2010),(M/J 2012)

 cos at − cos bt  L . t  

(A/M 2011),(N/D 2012),(M/J 2015)


9.


1 − cos t . t


(N/D 2014)

10.


e at − e − bt . (Textbook Page No.: 3.25) t

(M/J 2012)

11.

Find the Laplace transform of e −

t

−4 t

∫ t sin 3t dt .

(M/J 2009)

0

Textbook Page No.: 3.32 ∞

12.

Evaluate

∫ te

−2 t

cos t dt using Laplace transforms.

(N/D 2011),(M/J 2012)

0


13.

Find the inverse Laplace transform of

1

( s + 1) ( s 2 + 4 )

.

(M/J 2009)



Page 12


2016

  s L−1  2 . 2  ( s + 1) ( s + 4 ) 

14.

Find


15.

Find the inverse Laplace transform of

 s+1 log  .  s −1

(M/J 2015)

(N/D 2013)


 1  s 2 + a 2   16. Find L  ln  2 . 2   s  s + b   −1

(N/D 2011)

 3 s 2 + 16 s + 26  . 2  s(s + 4 s + 13) 

17. Evaluate L−1 



18. Evaluate L−1  e −2 s

 

1

( s 2 + s + 1)

2

(N/D 2013)

 .  

(M/J 2016)

• Inverse Laplace Transform Using Convolution Theorem 1.



 .  ( s + a ) ( s + b ) 

Using Convolution theorem L− 1  

1

(A/M 2011)


2.

 s Apply convolution theorem to evaluate L− 1  2  s + a2 

(

)

2

 .  

(M/J 2010),(M/J 2012)


3.

 s2 Find L− 1   s2 + 4 

(

)

2

  using convolution theorem.  

(N/D 2012)


4.

Find the inverse Laplace transform of Textbook Page No.: 3.108

(s

s2

2

+ a 2 ) ( s 2 + b2 )

using convolution theorem.

(N/D 2010),(M/J 2011),(M/J 2014),(N/D 2014),(M/J 2016)


Page 13


2016

Using convolution theorem find the inverse Laplace transform of Textbook Page No.: 3.103

6.

Find

(s

1

2

+ 1) ( s + 1)

.

(N/D 2009),(N/D 2011)(AUT)

  1 L−1   using convolution theorem. 2  s ( s + 4 ) 

(N/D 2011)


7.

Using convolution theorem find the inverse Laplace transform of

(s


4

2

+ 2s + 5)

2

.

(M/J 2013)

• Solving Differential Equation By Laplace Transform 1.

d2x dx dx Solve − 3 + 2 x = 2 , given x = 0 and = 5 for t = 0 using Laplace transform 2 dt dt dt method.

2.


Solve the equation y ′′ + 9 y = cos 2 t , y (0) = 1 and

π  y   = −1 using Laplace transform.  2


3.

Using Laplace transform, solve

(M/J 2009)

d2 y + 4 y = sin 2t given y (0) = 3, y ′(0) = 4 . dt 2


4.

Solve the differential equation

(M/J 2014)

d2 y + y = sin 2t ; y (0) = 0, y ′(0) = 0 by using Laplace dt 2

transform method. 5.


Using Laplace transform solve the differential equation

y (0) = 1 = y ′(0) . 6.


(N/D 2009)

y′′ − 3 y′ − 4 y = 2e − t with


(M/J 2010),(N/D 2010)

d2 y dy − 3 + 2 y = e − t with y(0) = 1 and y′(0) = 0 , 2 dt dt

using Laplace transform. 7.

(A/M 2011),(N/D 2012)


(

)

Use Laplace transform to solve D − 3 D + 2 y = e 2

3t

(M/J 2012)

with y (0) = 1 and y ′(0) = 0 .



(N/D 2014)

Page 14


9.

Solve

2016

y′′ − 3 y′ + 2 y = 4e 2 t , y(0) = −3, y′(0) = 5 , using Laplace transform.


(N/D 2011)

Solve y ′′ + 5 y ′ + 6 y = 2, y (0) = 0, y ′(0) = 0 , using Laplace transform.

(M/J 2013)


10.

Solve, by Laplace transform method, the equation

y (0) = 0, y ′(0) = 1 .

d2 y dy + 2 + 5 y = e − t sin t , 2 dt dt


(M/J 2011),(Jan 2016)

d2y dy dy 11. Solve + 4 + 4 y = sin t , if = 0 and y = 2 when t = 0 using Laplace transforms. 2 dt dt dt Textbook Page No.: 3.133 12.

(N/D 2011)

y′′ + y′ = t 2 + 2t , y(0) = 4, y′(0) = −2 .

Using Laplace transforms, solve Textbook Page No.: 3.141

13.


(N/D 2013),(M/J 2016)

y′′ − 3 y′ + 2 y = 4t + e , where y (0) = 1, y ′(0) = −1 using

Laplace transforms.

3t


(M/J 2015)

Unit – IV (Analytic Function) • Harmonic Function & Analytic Function 1.

Prove that the real and imaginary parts of an analytic function are harmonic functions. Textbook Page No.: 4.23 (M/J 2014)

2.

Verify that the families of curves

u = c1 and v = c2 cut orthogonally, when u + iv = z 3 .


Prove that u = e

−y

cos x and v = e

an analytic function of z . 4.

Show that

u=


(M/J 2011)


(N/D 2009)

1 log ( x 2 + y 2 ) is harmonic. Determine its analytic function. Find also its 2

conjugate. 6.

sin y satisfy Laplace equations, but that u + iv is not

When the function f ( z ) = u + iv is analytic, prove that the curves u = constant and

v = constant are orthogonal. 5.

(N/D 2009) −x

(Textbook Page No.: 4.43) −2 xy

Prove that u = e −

the imaginary part.

(A/M 2011)

sin ( x 2 − y 2 ) is harmonic. Find the corresponding analytic function and (Textbook Page No.: 4.58)


(N/D 2013),(Jan 2016)

Page 15


Prove that

2016

u = x 2 − y 2 and v =

−y are harmonic but u + iv is not regular. x + y2 2


8.

Prove that

(N/D 2010)

u = x 2 − y 2 and v =

conjugates. 9.

−y are harmonic functions but not harmonic x + y2 2


(N/D 2014),(Jan 2016)

Prove that every analytic function w = u + iv can be expressed as a function z alone, not as a function of z . (Textbook Page No.: 4.19) (M/J 2010),(M/J 2012)

w=

z z where a ≠ 0 is analytic whereas w = is not analytic.(M/J 2016) z+a z +a

10.

Prove that

11.

Find the analytic function f ( z ) = P + iQ , if

P −Q =

sin 2 x . (M/J 2009) cosh 2 y − cos 2 x


12.

Determine the analytic function whose real part is

sin 2 x . cosh 2 y − cos 2 x


13.

(N/D 2012),(N/D 2014)

If w = f ( z ) is analytic, prove that

dw ∂w ∂w . = −i = dz ∂x ∂y

(A/M 2011)


(

Find the analytic function u + iv , if u = ( x − y ) x + 4 xy + y harmonic function v .

15.

2

2

) . Also find the conjugate


Determine the analytic function

w = u + iv if u = e 2 x ( x cos 2 y − y sin 2 y ) .


Find the analytic function

(M/J 2015)

w = u + iv when v = e −2 y ( y cos 2 x + x sin 2 x ) and find u .


Show that

Prove that

(N/D 2011)

v = e − x ( x cos y + y sin y ) is harmonic function. Hence find the analytic function

f ( z ) = u + iv . 18.

(N/D 2009)


(M/J 2014)

u = e x ( x cos y − y sin y ) is harmonic (satisfies Laplace’s equation) and hence find

the analytic function f ( z ) = u + iv .


(N/D 2010),(M/J 2013)

Page 16


2016


19.

If f ( z ) is a analytic function of Textbook Page No.: 4.27

 ∂2 ∂2  2 2 z , prove that  2 + 2  f ( z ) = 4 f ′( z ) . ∂y   ∂x (M/J 2009), (A/M 2011),(M/J 2013),(N/D 2014),(M/J 2016)

 ∂2 ∂2  20. If f ( z ) is an analytic function of z , prove that  + 2  log f ( z ) = 0 . (M/J 2012) 2 ∂y   ∂x Textbook Page No.: 4.28 21.

If f ( z ) is analytic function of

z

in any domain, prove that

 ∂2 ∂2  p 2 p− 2 2  2 + 2  f ( z ) = p f ′( z ) f ( z ) . ∂y   ∂x

(N/D 2011)(AUT)


• Conformal Mapping 1.

Find the image of the half plane

x > c , when c > 0 under the transformation w =

Show the regions graphically.

1 . z

(M/J 2009),(N/D 2012)


2.

Find the image of

1 z + 1 = 1 under the mapping w = . z

(M/J 2014)


3.

Find the image of the circle

1 z − 1 = 1 under the mapping w = . z

(N/D 2009)

1 z − 2i = 2 under the transformation w = . z

(M/J 2013)


4.

Find the image of the circle


5.

Find the image in the

1 w= . z

w -plane of the infinite strip

1 1 ≤ y ≤ under the transformation 4 2



(M/J 2015)

Page 17


2016

Find the image of the hyperbola

1 x 2 − y 2 = 1 under the transformation w = . z


(M/J 2010),(M/J 2012),(N/D 2012)

Find the image of z = 2 under the mapping (1)

w = z + 3 + 2i (2) w = 3 z .


8.

(A/M 2011)

Prove that the transformation w of

w - plane.

z maps the upper half of z - plane on to the upper half 1− z

=

z = 1 under this transformation?

What is the image of


9.

Prove that the transformation

(M/J 2010),(N/D 2012),(N/D 2013)

w=

of circles or straight lines.

10.

1 maps the family of circles and straight lines into the family z


Show that the transformation w

=

(N/D 2011)

1 transforms, in general, circles and straight lines into circles z

and straight lines that are transformed into straight lines and circles respectively. Textbook Page No.: 4.80 (N/D 2011)

• Bilinear Transformation 1.

Find the bilinear transformation which maps the points z = 0, − i , − 1 into w – plane

w = i ,1, 0 respectively. 2.


Find the bilinear transformation that transforms the points z = 1, i , − 1 of the z-plane into the points w = 2, i , − 2 of the w-plane.

3.

(M/J 2009)


(M/J 2016)

Find the bilinear transformation which maps the points z = 0,1, ∞ into

w = i ,1, − i respectively.

(M/J 2010),(M/J 2012),(M/J 2013)


4.

Find the bilinear transformation that maps the points z = ∞ , i , 0 onto w = 0, i , ∞ respectively. Textbook Page No.: 4.89

5.

(N/D 2012)

Find the bilinear map which maps the points z = 0, −1, i onto points w = i , 0, ∞ . Also find the image of the unit circle of the

z

plane.

(N/D 2013),(M/J 2015)



Page 18


Find the bilinear transformation that transforms 1, i and −1 of the z – plane onto 0, 1 and

∞ of the w – plane. 7.

2016


(M/J 2014)

Find the bilinear transformation that transforms 1, i and −1 of the z – plane onto 0, 1 and

∞ of the w – plane. Also show that the transformation maps interior of z – plane on to upper half of the w – plane.

the unit circle of the (N/D 2010)


8.

Find the bilinear transformation which maps the points z = 1, i , −1 into the points

w = i , 0, − i . Hence find the image of z < 1 .

(M/J 2011),(N/D 2014)


9.

Find the Bilinear transformation that maps the points 1 + i , − i , 2 − i of the points 0,1, i of the

w - plane.


z - plane

into the

(N/D 2011)

Unit – V (Complex Integration) • Cauchy Integral Formula and Cauchy Residue Theorem 1.

Using Cauchy’s integral formula, evaluate

4 − 3z

∫ z( z − 1)( z − 2) dz , Where ‘ C ’ is the C

circle

2.

z =

3 . 2


sin π z 2 + cos π z 2 Evaluate ∫ dz , where C is z = 3 . ( z − 1)( z − 2) C

(M/J 2010)

(N/D 2011),(M/J 2013)


3.

Using Cauchy’s integral formula evaluate

∫z c

2

z dz , where C is the circle z + i = 1 . +1


4.

Evaluate using Cauchy’s integral formula

(M/J 2011)

z +1

∫ ( z − 3 ) ( z − 1) dz where C

is the circle

z = 2.

C



(M/J 2016)

Page 19


Evaluate

∫z

2

c

2016

z+4 dz , where C is the circle z + 1 + i = 2 , using Cauchy’s integral + 2z + 5

formula.

(N/D 2010),(N/D 2011),(N/D 2012)


6.

Evaluate

∫z

2

c

z +1 dz , where C is the circle z + 1 + i = 2 , using Cauchy’s integral + 2z + 4

formula.

7.

Evaluate


z2 ∫C ( z − 1)2 ( z + 2) dz where C

is

(N/D 2014)

z = 3.

(M/J 2015)


8.

Evaluate

z −1 ∫C ( z + 1)2 ( z − 2) dz , where C is the circle z − i = 2 using Cauchy’s

residue theorem.

9.


Using Cauchy’s residue theorem evaluate

(M/J 2012)

z −1

∫ ( z − 1) ( z − 2) dz , where C is 2

z − i = 2.

C


10.

Evaluate

(M/J 2014)

zdz

∫ ( z − 1) ( z − 2 )

2

where

c is the circle z − 2 =

c


11.

Using Cauchy’s integral formula evaluate

1 using Cauchy’s integral formula. 2 (M/J 2009),(N/D 2009),(M/J 2012)

e 2z

∫ ( z + 1)

4

dz where C is z = 2 . (Jan 2016)

C

12.

Evaluate

∫ (z C

2

z +1 dz where C is z + 1 + i = 2 using Cauchy’s integral formula. + 2 z + 4)2


13.

Evaluate

(A/M 2011),(N/D 2013)

z 3 dz ∫C ( z − 1)4 ( z − 2)( z − 3) where C is z = 2.5 , using residue theorem.(Jan 2016)


Page 20


2016

• Contour Integral of Types – I ,II &III 2π

1.

Evaluate

dθ

∫ 2 + cos θ

using contour integration.


0

(N/D 2009), (M/J 2010), (N/D 2010) ,(A/M 2011) 2π

2.

Evaluate

dθ

∫ 13 + 5 cos θ


(N/D 2014)

0

Textbook Page No.: 5.155 2π

3.

Evaluate

dθ

∫ 13 + 12cos θ


(M/J 2016)

0

2π

4.

Evaluate

dθ

∫ 13 + 5 sin θ .


(M/J 2014)

0

2π

5.

Evaluate

dθ ∫ a + b cos θ ( a > b > 0 ) , using contour integration.

(N/D 2011)

0


6.

Evaluate

cos 2θ

∫ 5 + 4 cos θ dθ , using contour integration.

(N/D 2013)

0


7.

Evaluate

cos 3θ

∫ 5 − 4 cos θ dθ


(M/J 2013)

0


8.

Evaluate

∫ 0

sin 2 θ dθ , a > b > 0 . a + b cos θ

(N/D 2012)


9.

Evaluate

dθ

∫ 1 − 2 x sin θ + x

2

,

( 0 < x < 1) .

(M/J 2009)

0



Page 21


2016 2π

10.

Evaluate, by contour integration,

dθ

∫ 1 − 2a sin θ + a

2

, 0 < a < 1.

(M/J 2011)

0


11.

x2 − x + 2 dx Evaluate ∫ 4 x + 10 x 2 + 9 −∞



(M/J 2010),(A/M 2011),(N/D 2013)

∞

12.

Evaluate

x 2 dx ∫−∞ ( x 2 + a 2 ) ( x 2 + b2 ) , a > b > 0 .

(M/J 2009),(M/J 2013)


13.

Evaluate

x 2 dx using contour integration. ∫ 2 2 −∞ ( x + 9 ) ( x + 4 )

(N/D 2014)


14.

Evaluate

∫ (x

−∞

2

dx using contour integration. + 1) ( x 2 + 4 )

(N/D 2010)


15.

Evaluate by using contour integration

∫ 0

dx

.

(M/J 2014)

dx .

(N/D 2011)

(1 + x )

2 2


16.

Evaluate using contour integration

∫

−∞

x2

( x 2 + 1)

2


17.

Evaluate

∫ 0

dx

( x2 + a2 )

3

,

a > 0 using contour integration.

(N/D 2009)



Page 22

Engineering Mathematics ∞

18.

Evaluate

∫ 0

dx

( x2 + a2 )

2

2016

a > 0 using contour integration.

,

(M/J 2015)


19.

∫x

Evaluate

0

4

dx using contour integration. + a4

(Jan 2016)


20.

Evaluate

cos mx dx , using contour integration. 2 + a2

∫x 0

(M/J 2012)


21.

Evaluate

∫ 0

x sin mx dx where a > 0 , m > 0 . x2 + a2

(M/J 2016)


22.

∫ (x

Evaluate

−∞

cos x dx ,a > b > 0. 2 + a 2 ) ( x 2 + b2 )

(N/D 2011)


• Taylor’s and Laurent’s Series 1.

Expand

f (z) =

z > 3.

z2 − 1 as a Laurent’s series in the regions z < 2 , 2 < z < 3 and ( z + 2)( z + 3) (Textbook Page No.: 5.51)

(M/J 2009),(A/M 2011),(M/J 2011),(N/D 2011),(M/J 2013),(M/J 2014),(M/J 2015)

2.

Evaluate

f (z) =

1

( z + 1) ( z + 3 )

in Laurent series valid for the regions


3.

Expand as a Laurent’s series of the function (ii)

z > 3 and 1 < z < 3 .

(N/D 2009),(M/J 2012)

f (z) =

z in the region (i) z < 1 z − 3z + 2 2

1 < z < 2 (iii) z > 2 .


(M/J 2016)

Page 23


2016

0 < z −1 < 1.

1

f (z) =

Find the Laurent’s series expansion of

( z − 1) ( z − 2 )

valid in the regions


z > 2 and

(N/D 2014)

5.

z 2 − 4z + 2 Obtain the Laurent’s expansion of f ( z ) = 3 in 3 < z + 2 < 5 .(Jan 2016) z − 2z 2 − 5z + 6

6.

Find the Laurent’s series expansion of

1 valid in the regions z (1 − z )

f (z) =

z + 1 < 1, 1 < z + 1 < 2 and z + 1 > 2 .

(N/D 2011)


7.

Find the Laurent’s series of

f (z) =

7z − 2 in 1 < z + 1 < 3 . z ( z + 1)( z + 2)

(M/J 2010)


8.

Find the residues of f ( z ) =

z2

( z − 1) ( z + 2 ) 2

2

at its isolated singularities using Laurent’s

series expansions. Also state the valid region.

(N/D 2010),(N/D 2012)


9.

Find the residues of f ( z ) = expansion.

z2

( z + 2 ) ( z − 1)

2

at its isolated singularities using Laurent’s series


(N/D 2013)

Textbook for Reference: “ENGINEERING MATHEMATICS - II” Publication: Sri Hariganesh Publications

Author: C. Ganesan

Mobile: 9841168917, 8939331876 To buy the book visit

www.hariganesh.com/textbook

----All the Best---Sri Hariganesh Publications (Ph: 9841168917, 8939331876)

Page 24

Unit – I ( Vector Calculus - SNS Courseware

Recommend Documents