Engineering Mathematics Material - Hariganesh.com

Engineering Mathematics

2016

SUBJECT NAME

: Mathematics - II

SUBJECT CODE

: MA6251

MATERIAL NAME

: University Questions

REGULATION

: R2013

UPDATED ON

: Nov-Dec 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 Calculus)  Simple problems on vector calculus 1.

2.

Find the directional derivative of 

 2xy  z 2 at the point  1, 1, 3  in the direction of

i  2 j  2k .

(Textbook Page No.: 1.6)

Find the directional derivative of 

 4xz 2  x 2 yz at  1, 2,1 in the direction of

2i  3 j  4k . 3.

If

(N/D 2016)

  2 xyz 3 i  x 2 z 3 j  3 x 2 yz 2 k find  ( x, y, z ) given that  (1, 2, 2)  4 .

Textbook Page No.: 1.14

4.

Prove that

Show that

Show that

  .


(M/J 2010)

F   y 2  2 xz 2  i   2 xy  z  j   2 x 2 z  y  2z  k is irrotational and

hence find its scalar potential.

6.

(M/J 2016)

F   6 xy  z 3  i   3 x 2  z  j   3 xz 2  y  k is irrotational vector and find

the scalar potential such that F

5.

(M/J 2009)


(M/J 2012)

F   2 xy  z 2  i   x 2  2 yz  j   y 2  2zx  k is irrotational and

find its scalar potential.


Sri Hariganesh Publications (Ph: 9841168917, 8939331876)

(N/D 2012)

Page 1

Engineering Mathematics 7.

Show that

2016

F   x 2  xy 2  i   y 2  x 2 y  j is irrotational and find its scalar potential.


8.

Prove that

F   x 2  y 2  x  iˆ   2 xy  y  ˆj is irrotational and hence find its scalar

potential.

9.

Prove that



Find the angle between the normals to the surface

 4,1, 1 . 11.


 2, 4,1 . 13.

Find the angle between the surfaces

Find

xy  z 2 at the points  1, 4, 2  and (A/M 2011)

x 2  yz at the points  1,1,1 and


 2, 1, 2  . 14.

(M/J 2009)



(N/D 2016)

xy 3 z 2  4 at the points  1, 1, 2  and


 3, 3, 3 . 12.

(N/D 2014)

F   y 2 cos x  z 3  iˆ   2 y sin x  4  ˆj  3 xz 2 kˆ is irrotational and find its

scalar potential.

10.

(N/D 2013)

(N/D 2014)

x 2  y 2  z 2  9 and z  x 2  y 2  3 at the point


(N/D 2016)

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

orthogonally at the point

 2, 1, 3 .

cut

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


15.

Find the value of

n such that the vector r n r

is both solenoidal and irrotational.


16.

(M/J 2014)

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.: 1.51

17.

If

r is the position vector of the point  x , y, z  , Prove that  2 r n  n(n  1)r n 2 . Hence find

the value of

18.

(M/J 2012)

Determine

1 2   . r


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

f ( r ) , where r  xi  yj  zk , if f ( r )r is solenoidal and irrotational.


Page 2


2016


(N/D 2011)

19.

Prove that Curl Curl F  graddiv F   2 F .

(M/J 2016)

20.

Evaluate

x

2

 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)


21.

Evaluate

 F

n ds where F  2 xyi  yz 2 j  xzk and S is the surface of the

s

x  0, y  0, z  0, x  2, y  1 and z  3 .

parallelepiped bounded by

(M/J 2011)


 Green’s Theorem 1.

Verify Green’s theorem for the lines

V   x 2  y 2  i  2 xyj taken around the rectangle bounded by

x   a, y  0 and y  b .

(N/D 2012),(Jan 2016)


2.

  x 1  y  dx   x 2

Using Green’s theorem in a plane evaluate

C

the square formed by

3

 y 3  dy  where C is

x  1 and y  1 .

(M/J 2016)


3.

Verify Green’s theorem in a plane for

  3 x C

4.

2

 8 y 2  dx   4 y  6 xy  dy  , Where C is the

boundary of the region defined by the lines

x  0, y  0 and x  y  1 .


(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 C

2

 8 y 2  dx   4 y  6 xy  dy  , Where C is the

boundary of the triangle formed by the lines

x  0, y  0 and x  y  1 . (N/D 2014)


Page 3


2016


6.

Using Green’s theorem, evaluate

  y  sin x  dx  cos x dy where C

is the triangle formed

C

by

y  0, x 

 2

, y

2x



.


(M/J 2015)

 Stoke’s Theorem 1.

Verify Stokes theorem for bounded by the lines

F   x 2  y 2  i  2 xyj in the rectangular region of z  0 plane

x  0, y  0, x  a and y  b .

(M/J 2014)


2.

Verify Stoke’s theorem for lines

F   x 2  y 2  i  2 xyj taken around the rectangle formed by the

x  a, x  a, y  0 and y  b .

(N/D 2013)


3.

Verify Stoke’s theorem for

F  xyi  2 yzj  zxk where S is the open surface of the

rectangular parallelepiped formed by the planes above the XY plane.

4.

on the

5.


Verify Stoke’s thorem for the vector F bounded by the planes

x  0, x  1, y  0, y  2 and z  3 (M/J 2009)

 ( y  z )i  yzj  xzk , where S is the surface

x  0, y  0, z  0, x  1, y  1, z  1 and C is the square boundary

xoy -plane.


Verify Stoke’s theorem when

(N/D 2011)

F   2 xy  x 2  i   x 2  y 2  j and C is the boundary of the

region enclosed by the parabolas

y 2  x and x 2  y .

(N/D 2009)


6.

Verify Stoke’s theorem for the vector field surface

F  (2 x  y )i  yz 2 j  y 2 zk over the upper half

x 2  y 2  z 2  1 , bounded by its projection on the xy -plane.


7.

Evaluate

(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   , 0  y  1, z  3 .

(N/D 2009)



Page 4


2016

Using Stokes theorem, evaluate

F

dr , where F  y 2 i  x 2 j  ( x  z )k and ‘C’ is the

C

boundary of the triangle with vertices at

 0,0,0 , 1,0,0  , 1,1,0  .

(M/J 2012)


9.

Using Stoke’s theorem prove that curl grand 

 0.

(M/J 2011)


 Gauss Divergence Theorem 1.

Verify Gauss Divergence theorem for

F  4 xzi  y 2 j  yzk over the cube bounded by

x  0, x  1, y  0, y  1, z  0, z  1 . (N/D 2010),(A/M 2011),(N/D 2012),(N/D 2013),(N/D 2014),(M/J 2015) Textbook Page No.: 1.87

2.

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


3.

Verify Gauss divergence theorem for the planes

F  x 2 i  y 2 j  z 2 k taken over the cube bounded by

x  0, y  0, z  0, x  1, y  1 and z  1 .

(M/J 2014)


4.

Verify Gauss – divergence theorem for the vector function over the cube bounded by

f   x 3  yz  i  2 x 2 yj  2k

x  0, y  0, z  0 and x  a, y  a, z  a .


5.

Verify divergence theorem for

x  1, y  1, z  1 . 6.

Verify Gauss’s theorem for F

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

F  x 2 i  zj  yzk over the cube formed by the planes (Textbook Page No.: 1.100)

(M/J 2013)

  x 2  yz  i   y 2  zx  j   z 2  xy  k over the

rectangular parallelepiped bounded by

x  0, x  a, y  0, y  b, z  0 and z  c .


7.

Verify Gauss’s theorem for F

(Jan 2016)

  x 2  yz  i   y 2  zx  j   z 2  xy  k over the

rectangular parallelepiped formed by

0  x  1,0  y  1 and 0  z  1 .



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

Page 5


2016

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

Solve

D

2

 2 D  2  y  e 2 x  cos 2 x .

(N/D 2016)

Textbook Page No.: 2.31 2.

Solve

D

3

 2 D2  D  y  e  x  cos 2 x .

(Jan 2016)


Solve the equation

D

2

 3 D  2  y  2cos  2 x  3   2e x .

(N/D 2009)


Solve

D

2

 16  y  cos3 x .

5.

Solve

D

2

 4 D  3  y  cos 2 x  2 x 2 .

6.

Solve :

7.

Solve the equation

D

2


(N/D 2010) (M/J 2014)

 3 D  2  y  sin x  x 2 . (Textbook Page No.: 2.37)

D

2

 5 D  4  y  e  x sin 2 x .

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


Solve the equation

D

2

 4 D  3  y  e  x sin x .

(M/J 2010)


Solve

D

2

 4 D  3  y  e x cos 2 x .

10.

Solve

D

2

 4 D  3  y  6e 2 x sin x sin 2 x .


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


Solve

D

2

 3 D  2  y  xe 3 x  sin 2 x . (Textbook Page No.: 2.59)

(M/J 2015)

12.

Solve

D

2

 2D  5 y  e  x x 2 .

(N/D 2014)


Page 6


2016

13.

Solve

d2 y dy  2  y  8 xe x sin x . 2 dx dx

(Text Book Page No.: 2.56)

(N/D 2013)

14.

Solve

D


(M/J 2016)

15.

Solve the equation

2

 2 D  1 y  xe  x cos x .

D

2

 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. 2 dx


Solve

y  y  tan x

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

using the method of variation of parameters.

(M/J 2016)


d2 y  4 y  tan 2 x by method of variation of parameters. dx 2

3.

Solve

4.

Apply method of variation of parameters to solve

D

2

 4  y  cot 2 x .


Solve

D

2

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

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

 a 2  y  sec ax using the method of variation of parameters.

(M/J 2012)


6.

Using method of variation of parameters, Solve

d2 y  y  sec x . dx 2

(N/D 2016)


7.

d2 y  y  cos ecx by the method of variation of parameters. Solve dx 2 Textbook Page No.: 2.128

8.

Solve

D

2

(A/M 2011),(ND 2012)

 1 y  cos ecx cot x using the method of variation of parameters.


Solve

D

2

 1 y  x sin x by the method of variation of parameters.


(A/M 2015) (M/J 2010)

Page 7


2016


Using variation of parameters, solve

 2D

2

 D  3  y  25e  x .

(N/D 2011)


11.

Solve

d2 y dy e x by the method of variation of parameters.  2  y  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.

d2 y dy 1 Solve x  4x  2 y  x2  2 . 2 dx dx x 2

(M/J 2013)


Solve

x D

3.

Solve

x D

2

2

2

2

 xD  1 y  sin  log x  .

(N/D 2014)

 2 xD  4  y  x 2  2log x .

(M/J 2010)


4.

d2 y dy Solve x x  y  log x . 2 dx dx 2

(N/D 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)


Solve the equation

x D 2

2

 3 xD  5  y  x cos  log x  .

(M/J 2009)

Textbook Page No.: 2.


Page 8


Solve

x D 2

2

2016

 xD  2  y  x 2 log x .

(M/J 2016)

Textbook Page No.: 2.80 2

9.

 log x  Solve  x D  xD  1 y    . (Textbook Page No.: 2.82)  x 

10.

Solve the equation

2

2

d 2 y 1 dy 12log 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)


12.

Solve:

d2 y dy  (1  x )  y  4cos  log(1  x ) . 2 dx dx

(1  x )2

(N/D 2011)


13.

Solve

(1  x )2

d2 y dy  (1  x )  y  2sin  log(1  x ) . 2 dx dx

(A/M 2011)


14.

Solve

 3 x  2

2

d2 y dy  3  3 x  2   36 y  3 x 2  4 x  1 . 2 dx dx

(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 y  x, x  y .

2.

Solve

( Textbook Page No.: 2.102 )

dx dy  2 y  sin 2t ,  2 x  cos 2t . dt dt

(Jan 2016)

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



Page 9


Solve

2016

dx dy  2 y   sin t ,  2 x  cos t . dt dt

(M/J 2014)


4.

Solve

dx dy  2 y   sin t ,  2 x  cos t given x  1 , y  0 at t  0 . dt dt

(N/D 2010)


5.

Solve

dx dy  y  t and  x  t2. dt dt

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


6.

Solve

dx dy  y  t and  x  t 2 given x(0)  y(0)  2 . dt dt

(N/D 2011)


7.

Solve

dx dy  y  et , x  t. dt dt

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


8.

Solve

dx dy  2 x  3 y  2e 2 t ,  3 x  2 y  0. dt dt

(M/J 2010)


9.

Solve

dx dy  5x  2 y  t,  2 x  y  0. dt dt

(M/J 2013)


10.

Solve

dx dy  2 x  3 y  0 and 3 x   2 y  2e 2 t . dt dt

(N/D 2016)


11.

Solve

dx dy  2 x  3 y  t and  3 x  2 y  e 2t . dt dt

(N/D 2011)


12.

Solve

dx dy  y  sin t , x   cos t given x  2 and y  0 at t  0 . dt dt


(M/J 2009)

Page 10


2016


13.

Solve

dx dy  4 x  3 y  t and  2 x  5 y  e 2t . dt dt

(N/D 2013)


14.

Solve the simultaneous differential equations:

dx dy dx   3 x  sin t ,  y  x  cos t . dt dt dt


(M/J 2015)

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

Find the Laplace transform of Textbook Page No.: 3.51

2.

for 0  t  a t , , f (t  2a )  f (t ) . f (t )    2a  t , for a  t  2a (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  t , and f (t  2 )  f (t ) . f (t )    2  t ,   t  2

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


3.

Find the Laplace transform of

t , 0  t  1 and f ( t  2)  f ( t ) for t  0 . f (t )    0, 1  t  2


(N/D 2011)(AUT)

Find the Laplace transform of square wave function defined by

1, in 0  t  a f (t )   with period 2a .  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)



Page 11


2016


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 )   a   E for ta  2

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


8.

Find the Laplace transform of the Half wave rectifier and

sin  t , 0  t   /  f (t )    /   t  2 /   0,

f (t  2 /  )  f (t ) for all t .


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

 Simple Problems and Initial & Final Value Theorem 1.

Find

 e  t  cos t  L  e  t sin2 3t  and L  . t  

(Jan 2016)


Find the Laplace transform of e

t

t cos t .

(N/D 2014)

te 2t cos 3t .

(M/J 2009)

f (t )  te 3 t cos 2t .

(M/J 2014)



4.


5.

Find

L  t 2e 3 t sin 2t  .

(Text Book Page No.: 3.21)

(M/J 2013)

6.

Find

L  t 2e  t cos t  .


(M/J 2016)


Page 12


2016

Verify initial and final value theorems for

f (t )  1  e  t (sin t  cos t ) .


8.

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)



1  cos t . t


(N/D 2014)

10.


e at  e  bt . (Textbook Page No.: 3.24) t

(M/J 2012)

t

11.


e

4 t

 t sin 3t dt .

(M/J 2009)

0

Textbook Page No.: 3.31 

12.

Evaluate

 te

2 t

cos t dt using Laplace transforms.

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

0


Find the inverse Laplace transform of

1

 s  1  s 2  4 

.

(M/J 2009)


  s  1  L 14. Find  2 . 2   s  1 s  4    


(M/J 2015)

  s2 1  . L 15. Find 2 2 2 2   s  a  s  b  


(A/M 2015)

16.

Find the inverse Laplace transform of

 s 1 log  .  s 1

(N/D 2013)



Page 13


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 

  s2  a2  

2

 .  

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




3.

  using convolution theorem.   s2  42   

Find L1 

s2

(N/D 2012)


4.



  using convolution theorem. 2 2   s  4  s  9   s

Using convolution theorem, find L1 

Textbook Page No.: 3.105 (similar problem)

5.

Find the inverse Laplace transform of Textbook Page No.: 3.103

6.

s

(N/D 2016)

s2

2

 a 2  s 2  b 2 

using convolution theorem.

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

Using convolution theorem find the inverse Laplace transform of

1 .  s  1  s  1 2


7.

Find

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

  1   L1   using convolution theorem. 2 s s  4      

(N/D 2011)


8.

Using convolution theorem find the inverse Laplace transform of

s



4

2

 2s  5

2

.

(M/J 2013)

Page 14


2016

 Solving Differential Equation By Laplace Transform 1.

Solve

dx d2x dx  5 for t  0 using Laplace transform  3  2 x  2 , given x  0 and 2 dt dt dt

method.

2.

Solve the equation


  y  9 y  cos 2t , y(0)  1 and 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.


Using Laplace transform, solve

Use Laplace transform to solve


D

2

Solve

(N/D 2016)

D

2

 3 D  2  y  e 3 t with y(0)  1 and y(0)  0 .

Solve

(N/D 2014)

y  3 y  2 y  4e 2t , y(0)  3, y(0)  5 , using Laplace transform.


(M/J 2012)

 3 D  2  y  e 3 t given y(0)  1 and y(0)  1 .


(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


(N/D 2009)

y  3 y  4 y  2e  t with


using Laplace transform. 7.

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

(N/D 2011)

y  5 y  6 y  2, y(0)  0, y(0)  0 , using Laplace transform. (M/J 2013)


d2 y dy  2  5 y  e  t sin t , 11. Solve, by Laplace transform method, the equation 2 dt dt (Textbook Page No.: 3.132) (M/J 2011),(Jan 2016) y(0)  0, y(0)  1 . Sri Hariganesh Publications (Ph: 9841168917, 8939331876)

Page 15


Solve

2016

dy d2 y dy  0 and y  2 when t  0 using Laplace transforms.  4  4 y  sin t , if 2 dt dt dt


(N/D 2011)

y  y  t 2  2t , y(0)  4, y(0)  2 .

Using Laplace transforms, solve Textbook Page No.: 3.136

14.


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

y  3 y  2 y  4t  e , where y(0)  1, y(0)  1 using 3t

Laplace transforms.


(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.: 4.21

2.

Verify that the families of curves

(M/J 2014)

u  c1 and v  c2 cut orthogonally, when u  iv  z 3 .


Prove that

4.

When the function orthogonal.

5.

Show that

u

x

f ( z )  u  iv is analytic, prove that the curves u  c1 and v  c2 are


Prove that


Prove that

u  x 2  y 2 and v 


Prove that

u  x 2  y 2 and v 

conjugates.

(N/D 2013),(Jan 2016)

y are harmonic but u  iv is not regular. x  y2 2


(A/M 2011)

u  e 2 xy sin  x 2  y 2  is harmonic. Find the corresponding analytic function and

the imaginary part. 7.

(N/D 2009),(N/D 2016)

1 log  x 2  y 2  is harmonic. Determine its analytic function. Find also its 2

conjugate. 6.

(N/D 2009)

cos x and v  e sin y satisfy Laplace equations, but that u  iv is not an analytic function of z . (Textbook Page No.: 4.29) (M/J 2011) ue

y

(N/D 2010)

y are harmonic functions but not harmonic x  y2 2



(N/D 2014),(Jan 2016)

Page 16


2016

9.

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.: 4.18) (M/J 2010),(M/J 2012)

10.

Prove that

11.

Find the analytic function

w

z z where a  0 is analytic whereas w  is not analytic.(M/J 2016) z a za f ( z )  P  iQ , if P  Q 

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


Determine the analytic function whose real part is

sin 2 x . cosh 2 y  cos 2 x


If

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

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  harmonic function

15.

v.

 x  y   x 2  4 xy  y 2  . Also find the conjugate


Determine the analytic function

w  u  iv

if

(N/D 2009)

u  e 2 x ( x cos 2 y  y sin 2 y ) .



(M/J 2015)

w  u  iv when v  e 2 y  y cos 2 x  x sin 2 x  and find u .


Show 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.



f ( z )  u  iv

whose real part is

(M/J 2014)

u  e x ( x cos y  y sin y ) . Find

also the conjugate harmonic u .

(N/D 2016)


Prove that

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 17


If

2016

 2 2  2 2 f ( z ) is a analytic function of z , prove that  2  2  f ( z )  4 f ( z ) . y   x


21.

If

(M/J 2009), (A/M 2011),(M/J 2013),(N/D 2014),(M/J 2016)

 2 2  f ( z ) is an analytic function of z , prove that  2  2  log f ( z )  0 . (M/J 2012) y   x


If

f ( z ) is analytic function of z in any domain, prove that

 2 2  p 2 p 2  f ( z )  p2 f ( z ) f ( z ) .  2 2  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

z  1  1 under the mapping w 

1 . z

(M/J 2014)


3.

Find the image of the circle

z  1  1 under the mapping w 

1 . z

(N/D 2009)


4.

Find the image of the circle

z  2i  2 under the transformation w 

1 . z

(M/J 2013)


Find the image in the

w

1 . z

w -plane of the infinite strip

1 1  y  under the transformation 4 2



(M/J 2015)

Page 18


2016 x 2  y 2  1 under the transformation w 

Find the image of the hyperbola Textbook Page No.: 4.70

7.

Find the image of

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

z  2 under the mapping (1) w  z  3  2i (2) w  3z .


(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


Prove that the transformation

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

w

of circles or straight lines. 10.

1 . z

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


Show that the transformation w



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

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

w  i ,1,0 respectively. 2.


Find the bilinear transformation that transforms the points points

3.

z  0,  i , 1 into w – plane

w  2, i ,  2 of the w-plane.

(M/J 2009)

z  1, i ,  1 of the z-plane into the



w  i ,1, 1 respectively.

(M/J 2016)

z  0,1,  into (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.


5.

(N/D 2012)

Find the bilinear map which maps the points image of the unit circle of the

z

z  0, 1, i onto points w  i ,0,  . Also find the

plane.

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



Page 19


2016

Find the bilinear transform which maps

i ,  i ,1 in z-plane into 0,1, 

of w-plane respectively.


7.

(Jan 2016)


z  0, 1,  1 onto the points

w   1, 0,  . Find also the invariant points of the transformation.

(N/D 2016)


8.

Find the bilinear transformation that transforms

 of the w – plane. 9.

1, i and 1 of the z – plane onto 0, 1 and


Find the bilinear transformation that transforms

(M/J 2014)

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)


10.


z  1, i , 1 into the points

w  i ,0,  i . Hence find the image of z  1 .

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


11.

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  z 2  cos  z 2 dz , where C is z  3 . Evaluate  ( z  1)( z  2) C

(M/J 2010)

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



Page 20


2016

Using Cauchy’s integral formula evaluate

z

2

c

z dz , where C is the circle z  i  1 . 1


(M/J 2011)

Evaluate using Cauchy’s integral formula

z 1

  z  3 z  1 dz where C

is the circle

z 2.

C


Evaluate

z

2

c

(M/J 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)


Evaluate

z

2

c

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

formula.

7.


Using Cauchy’s integral formula, evaluate

z  1  1. 8.

(N/D 2014)

z C ( z  1) ( z  2) dz , where is the circle 2


z2 Evaluate C ( z  1)2 ( z  2) dz where C

is

(N/D 2016)

z  3.

(M/J 2015)


Evaluate

zdz

  z  1 z  2

2

where

c is the circle z  2 

c


10.

Using Cauchy’s integral formula evaluate

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

e2z

  z  1

4

dz where C is z  2 . (Jan 2016)

C


Evaluate

 (z C

2

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



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

Page 21


Evaluate

2016

z 1

 ( z  1) ( z  2) dz , where C is the circle z  i  2 using Cauchy’s 2

C

residue theorem.

13.


Using Cauchy’s residue theorem evaluate

(M/J 2012)

z 1 C ( z  1)2 ( z  2) dz , where C is z  i  2 .


Evaluate

 (z C

2

(M/J 2014)

z dz , where C is the circle z  i  1 , using Cauchy’s residue theorem.  1)2


(N/D 2016)

3

15.

Evaluate

z dz

 ( z  1) ( z  2)( z  3) where C 4

is

z  2.5 , using residue theorem.(Jan 2016)

C

 Contour Integral of Types – I ,II &III 2

1.

Evaluate

d

 2  cos 

using contour integration.


0

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

2.

Evaluate

d

 13  5cos


(N/D 2014)

0

Textbook Page No.: 5.154 2

3.

Evaluate

d

 13  12cos


(M/J 2016)

0


4.

Evaluate

d

 13  5sin  .


(M/J 2014)

0

2

5.

Evaluate

d  a  b cos  a  b  0 , using contour integration.

(N/D 2011)

0


6.

Evaluate

cos 2

 5  4cos  d , using contour integration.

(N/D 2013)

0


Page 22


2016


7.

Evaluate

cos 3

 5  4cos d


(M/J 2013)

0


8.

Evaluate

 0

sin 2  d , a  b  0 . a  b cos 

(N/D 2012)


9.

Evaluate

d

 1  2 x sin   x

2

,

 0  x  1 .

(M/J 2009)

0


10.

Evaluate, by contour integration,

d

 1  2a sin  a

2

,0  a  1.

(M/J 2011)

0


x 2dx 11. Evaluate  , a  b  0. 2 2 2 2   x  a  x  b 

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


x 2 dx 12. Evaluate  using contour integration. 2 2   x  9  x  4 

(N/D 2014)


13.

Evaluate

 x



2

dx using contour integration.  1 x 2  4 

(N/D 2010)


14.

Evaluate

x2  x  2  x 4  10 x 2  9 dx


using contour integration. (M/J 2010),(A/M 2011),(N/D 2013)


Page 23


2016 

15.

Evaluate by using contour integration

 0

dx

.

(M/J 2014)

dx .

(N/D 2011)

1  x2 

2


16.

Evaluate using contour integration





x

x2 2

 1

2


17.

Evaluate

 0

x

dx 2

 a2 

3

,

a  0 using contour integration.

(N/D 2009)

a  0 using contour integration.

(M/J 2015)


18.

Evaluate

 0

dx

 x2  a2 

2

,


19.

Evaluate

x 0

dx using contour integration.  a4

4

(Jan 2016)

Textbook Page No.: 5.138 (similar problem) 

20.

Evaluate

cos mx dx , using contour integration. 2  a2

x 0

(M/J 2012),(N/D 2016)


21.

Evaluate

 0

x sin mx dx where a  0 , m  0 . x2  a2

(M/J 2016)


22.

Evaluate

 x



2

cos x dx ,a  b  0.  a 2  x 2  b 2 

(N/D 2011)



Page 24


2016

 Taylor’s and Laurent’s Series 1.

f (z) 

Expand

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.: 5.53)

(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.

4.

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

Expand as a Laurent’s series of the function (ii) 1 

z  3 and 1  z  3 .

z  2 (iii) z  2 .

f (z) 

z in the region (i) z  1 z  3z  2 2


Find the Laurent’s series expansion of

f (z) 

(M/J 2016)

1 valid in the region 1  z  1  2 . z  5z  6 2


5.


0  z  1  1. 6.

(N/D 2016)

f (z) 

1

 z  1 z  2 

valid in the regions


Obtain the Laurent’s expansion of

f (z) 

z  2 and

(N/D 2014)

z 2  4z  2 in 3  z  2  5 .(Jan 2016) z 3  2z 2  5z  6


7.


f (z) 

1 valid in the regions z 1  z 

z  1  1, 1  z  1  2 and z  1  2 . Textbook Page No.: 5.58

8.

Find the Laurent’s series of

f (z) 

(N/D 2011)

7z  2 in 1  z  1  3 . z( z  1)( z  2)

(M/J 2010)


9.

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)

Page 25


2016


10.

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” Edition: 2nd Edition Publication: Sri Hariganesh Publications

Author: C. Ganesan

Mobile: 9841168917, 8939331876 To buy the book visit

www.hariganesh.com/textbook

----All the Best ----


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