uq-m2

Engineering Mathematics 2013

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 1

SUBJECT NAME : Engineering Mathematics - II

SUBJECT CODE : MA2161

MATERIAL NAME : University Questions

MATERIAL CODE : JM08AM1004

Name of the Student: Branch:

Unit – I (Ordinary Differential Equation)

Type – I to VI

1. Solve the equation 2 24 cos 2D y x x . (M/J 2009),(N/D 2011)

2. Solve the equation 23 2 2cos 2 3 2

xD D y x e . (N/D 2009)

3. Solve 2 316 cosD y x . (N/D 2010)

4. Solve : 2 23 2 sinD D y x x . (M/J 2011)

5. Solve the equation 25 4 sin 2

xD D y e x

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

6. Solve the equation 24 3 sin

xD D y e x

. (M/J 2010)

7. Solve: 24 3 cos 2

xD D y e x . (M/J 2012)

8. Solve 2 24 3 6 sin sin 2

xD D y e x x

. (N/D 2011)

Method of Variation of Parameters

1. Solve 2 2tanD a y ax by the method of variation of parameters. (M/J 2009)

2. Solve, 2

2

2tan

d ya y ax

dx by method of variation of parameters. (M/J 2011)

3. Apply method of variation of parameters to solve 24 cot 2D y x .

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



4. Solve 2 2secD a y ax using the method of variation of parameters.(M/J 2012)

5. Solve 2

2cos

d yy ecx

dx by the method of variation of parameters.

(A/M 2011),(ND 2012)

6. Solve 21 sinD y x x by the method of variation of parameters. (M/J 2010)

7. Using variation of parameters, solve 22 3 25

xD D y e

. (N/D 2011)

Cauchy and Legendre Equations

1. Solve the equation 2 23 5 cos logx D xD y x x . (M/J 2009)

2. Solve 2 2 23 4 cos logx D xD y x x . (N/D 2010)

3. Solve 2 2 24 sin logx D xD y x x . (M/J 2012),(N/D 2009)

4. Solve 2 2 22 4 2logx D xD y x x . (M/J 2010)

5. Solve the equation2

2 2

1 12logd y dy x

dx x dx x . (N/D 2012)

6. Solve 2

2 2

23 4 ln

d y dyx x y x x

dx dx . (N/D 2011)

7. Solve 2

2

2(1 ) (1 ) 2sin log(1 )

d y dyx x y x

dx dx . (A/M 2011)

8. Solve: d y dy

x x y xdx dx

2

2

2(1 ) (1 ) 4cos log(1 ) . (N/D 2011)

Simultaneous Differential Equations

1. Solve sin , cosdx dy

y t x tdt dt

given 2x and 0y at 0t . (M/J 2009)

2. Solve 2 sin 2 ,dx

y tdt

2 cos 2dy

x tdt

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



3. Solve 2 sin ,dx

y tdt

2 cosdy

x tdt

given 1x , 0y at 0t . (N/D 2010)

4. Solve dx

y tdt

and 2dyx t

dt . (A/M 2011)

5. Solve dx

y tdt

and 2dyx t

dt given x y (0) (0) 2 . (N/D 2011)

6. Solve tdxy e

dt ,

dyx t

dt . (N/D 2012)

7. Solve 22 3 2 ,

tdxx y e

dt 3 2 0.

dyx y

dt (M/J 2010)

8. Solve 2 3dx

x y tdt

and 23 2

tdyx y e

dt . (N/D 2011)

9. Solve for x from the equations 2 23

tD x y e , 2

3t

Dx Dy e . (M/J 2011)

Unit – II (Vector Calculus)

Simple problems on vector calculus

1. Find the directional derivative of 22xy z at the point 1, 1,3 in the direction of

2 2i j k . (M/J 2009)

2. Prove that 3 2 26 3 3F xy z i x z j xz y k is irrotational vector and

find the scalar potential such that F . (M/J 2010)

3. Show that 2 2 22 2 2 2F y xz i xy z j x z y z k is irrotational and

hence find its scalar potential. (M/J 2012)

4. Show that 2 2 22 2 2F xy z i x yz j y zx k is irrotational and

find its scalar potential. (N/D 2012)

5. Find the angle between the normals to the surface 3 24xy z at the

points 1, 1,2 and 4,1, 1 . (M/J 2009)



6. Find the angle between the normals to the surface 2xy z at the points 1,4,2 and

3, 3,3 . (A/M 2011)

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

23 (2 )F x i xz y j zk along the straight line from 0,0,0 to 2,1,3 .

(M/J 2012)

8. If r is the position vector of the point , ,x y z , Prove that 2 2( 1)

n nr n n r

.

(N/D 2010)

9. Determine ( )f r , where r xi yj zk , if ( )f r r is solenoidal and irrotational.

(N/D 2011)

10. If F is a vector point function, prove that 1 1 2curl curlF F F

.

(N/D 2011)(AUT)

11. Prove that curl div div u v v u u v u v v u . (N/D 2009)

12. Evaluate 2 2 2

C

x xy dx x y dy where C is the square bounded by the

lines 0, 1, 0 and 1x x y y . (N/D 2009),(N/D 2011)

13. Evaluate s

F n ds where 22F xyi yz j xzk and S is the surface of the

parallelepiped bounded by 0, 0, 0, 2, 1x y z x y and 3z . (M/J 2011)

Green’s Theorem

1. Verify Green’s theorem in a plane for 2 23 8 4 6

C

x y dx y xy dy , Where C is

the boundary of the region defined by the lines 0, 0x y and 1x y .

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

2. Verify Green’s theorem for 2 23 8 4 6

C

x y dx y xy dy where C is the boundary

of the region defined by 2 2, x y y x . (M/J 2010)



3. Verify Green’s theorem for 2 22V x y i xyj taken around the rectangle

bounded by the lines , 0x a y and y b . (N/D 2012)

Stoke’s Theorem

1. Verify Stoke’s theorem for 2F xyi yzj zxk where S is the open surface of the

rectangular parallelepiped formed by the planes 0, 1, 0, 2x x y y and 3z

above the XY plane. (M/J 2009)

2. Verify Stoke’s thorem for the vector ( )F y z i yzj xzk , where S is the surface

bounded by the planes 0, 0, 0, 1, 1, 1x y z x y z and C is the square

boundary on the xoy -plane. (N/D 2011)

3. Verify Stoke’s theorem when 2 2 22F xy x i x y j and C is the boundary of

the region enclosed by the parabolas 2y x and 2

x y . (N/D 2009)

4. Evaluate sin cos sinC

zdx xdy ydz by using Stoke’s theorem, where C is the

boundary of the rectangle defined by 0 , 0 1, 3x y z . (N/D 2009)

5. Using Stokes theorem, evaluate C

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

the boundary of the triangle with vertices at 0,0,0 , 1,0,0 , 1,1,0 .(M/J 2012)

6. Using Stoke’s theorem prove that curl grand 0 . (M/J 2011)

Gauss Divergence Theorem

1. Verify Gauss divergence theorem for 2 2 2F x i y j z k where S is the surface of

the cuboid formed by the planes 0, , 0, , 0x x a y y b z and z c .(M/J 2009)

2. Verify Gauss Divergence theorem for 24F xzi y j yzk over the cube bounded

by 0, 1, 0, 1, 0, 1x x y y z z . (N/D 2010),(A/M 2011),(N/D 2012)

3. Verify Gauss – divergence theorem for the vector function

3 22 2f x yz i x yj k over the cube bounded by 0, 0, 0x y z and

, ,x a y a z a . (M/J 2010),(N/D 2011)

4. Verify Gauss’s theorem for 2 2 2F x yz i y zx j z xy k over the

rectangular parallelepiped formed by 0 1,0 1x y and 0 1z .

(N/D 2011)(AUT)



Unit – III (Analytic Function)

Harmonic Function & Analytic Function

1. Verify that the families of curves 1

u c and 2

v c cut orthogonally, when 3u iv z .

(N/D 2009)

2. Prove that cosy

u e x and sin

xv e y

satisfy Laplace equations, but that u iv is

not an analytic function of z . (M/J 2011)

3. When the function ( )f z u iv is analytic, prove that the curves constantu and

constantv are orthogonal. (N/D 2009)

4. Show that the families of curves secn

r a n and cosn

r b ecn cut orthogonally.

(M/J 2011)

5. Show that 2 21log

2u x y is harmonic. Determine its analytic function. Find also

its conjugate. (A/M 2011)

6. Prove that 2 2u x y and

2 2

yv

x y

are harmonic but u iv is not regular.

(N/D 2010)

7. Prove that every analytic function w u iv can be expressed as a function z alone,

not as a function of z . (M/J 2010),(M/J 2012)

8. Find the analytic function ( )f z P iQ , if sin 2

cosh 2 cos 2

xP Q

y x

.(M/J 2009)

9. Determine the analytic function whose real part is sin 2

cosh 2 cos 2

x

y x. (N/D 2012)

10. If ( )w f z is analytic, prove thatdw w w

idz x y

. (A/M 2011)

11. Find the analytic function u iv , if 2 24u x y x xy y . Also find the

conjugate harmonic function v . (N/D 2009)

12. Find the analytic function w u iv when yv e y x x x

2cos 2 sin 2 and find u .

(N/D 2011)



13. Prove that ( cos sin )x

u e x y y y is harmonic and hence find the analytic function

( )f z u iv . (N/D 2010)

14. If ( )f z is a regular function of z , prove that 2 2

2 2

2 2( ) 4 ( )f z f z

x y

.

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

15. If ( )f z is an analytic function of z , prove that 2 2

2 2log ( ) 0f z

x y

. (M/J 2012)

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

2 22 22

2 2( ) ( ) ( )

p pf z p f z f z

x y

. (N/D 2011)(AUT)

Conformal Mapping

1. Find the image of the half plane x c , when 0c under the transformation 1

wz

.

Show the regions graphically. (M/J 2009),(N/D 2012)

2. Find the image of the circle 1 1z in the complex plane under the mapping 1

wz

.

(N/D 2009)

3. Find the image of the hyperbola 2 21x y under the transformation

1w

z .

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

4. Find the image of 2z under the mapping (1) 3 2w z i (2) 3w z .

(A/M 2011)

5. Prove that the transformation1

zw

z

maps the upper half of z - plane on to the

upper half of w - plane. What is the image of 1z under this transformation?

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

6. Show that the map wz

1

maps the totality of circles and straight lines as circles or

straight lines. (N/D 2011)



7. Prove that the transformation wz

1

maps the family of circles and straight lines into

the family of circles or straight lines. (N/D 2011)

8. Show that the transformation1

wz

transforms, in general, circles and straight lines

into circles and straight lines that are transformed into straight lines and circles

respectively. (N/D 2011)(AUT)

Bilinear Transformation

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

,1,0w i respectively. (M/J 2009)

2. Find the bilinear transformation which maps the points 0,1,z into

,1,w i i respectively. (M/J 2010),(M/J 2012)

3. Find the bilinear transformation that maps the points , , 0z i onto 0, ,w i respectively. (N/D 2012)

4. Find the bilinear transformation which maps the points 1, , 1z i into the points

, 0,w i i . Hence find the image of 1z . (M/J 2011)

5. 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

the unit circle of the z – plane on to upper half of the w – plane. (N/D 2010)

6. Find the Bilinear transformation that maps the points1 , , 2i i i of the z - plane

into the points 0,1, i of the w - plane. (N/D 2011)

Unit – IV (Complex Integration)

C.I.F and C.R.T

1. Evaluate

21 2c

zdz

z z where c is the circle

12

2z using Cauchy’s integral

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

2. Evaluate

22

( 1)

2 4C

zdz

z z

where C is 1 2z i using Cauchy’s integral formula.

(A/M 2011)



3. Evaluate2

4

2 5c

zdz

z z

, where C is the circle 1 2z i , using Cauchy’s integral

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

4. Using Cauchy’s integral formula evaluate2

1c

zdz

z , whereC is the circle 1z i .

(M/J 2011)

5. Using Cauchy’s integral formula, evaluate4 3

( 1)( 2)C

zdz

z z z

, Where ‘C ’ is the

circle3

2z . (M/J 2010)

6. Evaluate using Cauchy’s residue theorem,

C

z zdz

z z

2 2

sin cos

( 1)( 2), where C: z 3 .

(N/D 2011)

7. Evaluate2

1

( 1) ( 2)C

zdz

z z

, where C is the circle 2z i using Cauchy’s

residue theorem.

Contour Integral of Types – I ,II &III

1. Evaluate 2

02 cos

d

using contour integration.

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

2. Evaluate 2

0

0cos

da b

a b

, using contour integration. (N/D 2011)

3. Evaluate 2 2

0

sin, 0

cosd a b

a b

. (N/D 2012)

4. Evaluate 2

2

0

, 0 11 2 sin

dx

x x

. (M/J 2009)

5. Evaluate, by contour integration, 2

2

01 2 sin

d

a a

, 0 1a . (M/J 2011)



6. Evaluate

2

4 2

2

10 9

x xdx

x x

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

7. Evaluate

2

2 2 2 2

x dx

x a x b

, using contour integration, where 0a b .

(M/J 2009)

8. Evaluate 2 2

1 4

dx

x x

using contour integration. (N/D 2010)

9. Evaluate using contour integration

xdx

x

2

22

1. (N/D 2011)

10. Evaluate

3

2 20

dx

x a

, 0a using contour integration. (N/D 2009)

11. Evaluate2 2

0

cos mxdx

x a

, using contour integration. (M/J 2012)

12. Evaluate 2 2 2 2

cos x dx

x a x b

using contour integration, where 0a b .

(N/D 2011)

Taylor’s and Laurent’s Series

1. Expand 2

1( )

( 2)( 3)

zf z

z z

as a Laurent’s series in the region 2 3z .

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

2. Find the Laurent’s series of

2

2

1( )

5 6

zf z

z z valid in 2 3z .(M/J 2009)

3. Evaluate

1( )

1 3f z

z z

in Laurent series valid for the regions 3z and

1 3z . (N/D 2009),(M/J 2012)



4. Find the Laurants’s series expansion of

1( )

1f z

z z

valid in the regions

1 1, 1 1 2z z and 1 2z . (N/D 2011)

5. Find the Laurent’s series of 7 2

( )( 1)( 2)

zf z

z z z

in 1 1 3z . (M/J 2010)

6. Find the residues of

2

2 2( )

1 2

zf z

z z

at its isolated singularities using

Laurent’s series expansions. Also state the valid region. (N/D 2010),(N/D 2012)

Unit – V (Laplace Transform)

Laplace Transform of Periodic Function

1. Find the Laplace transform of, for 0

( )2 , for 2

t t af t

a t a t a

, ( 2 ) ( )f t a f t .

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

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

, 0( )

2 , 2

t tf t

t t

and ( 2 ) ( )f t f t . (M/J 2010),(M/J 2012)

3. Find the Laplace transform of , 0 1

( )0, 1 2

t tf t

t

and ( 2) ( )f t f t for 0t .

(N/D 2011)(AUT)

4. Find the Laplace transform of square wave function defined by

1, in 0( )

1, in 2

t af t

a t a

with period 2a . (N/D 2009)

5. Find the Laplace transform of

( ) , 0

, 2

f t t a

a t a

and ( 2 ) ( )f t a f t for all t . (N/D 2010)

6. Find the Laplace transform of a square wave function given by

aE t

f ta

E t a

for 02

( )

for 2

, and f t a f t ( ) ( ) . (N/D 2011)



7. Find the Laplace transform of the Half wave rectifier

sin , 0 /( )

0, / 2 /

t tf t

t

and ( 2 / ) ( )f t f t for all t .(N/D 2012)

Initial and Final Value Theorem& Other Simple Problems

1. Find the Laplace transform of 2cos 3

tte t

. (M/J 2009)

2. Verify initial and final value theorems for ( ) 1 (sin cos )t

f t e t t .

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

3. Find cos cosat bt

Lt

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

4. Find the Laplace transform ofat bt

e e

t

. (M/J 2012)

5. Find the Laplace transform of 4

0

sin 3

t

te t t dt

. (M/J 2009)

6. Evaluate tte t dt

2

0

cos using Laplace transforms. (N/D 2011),(M/J 2012)

7. Find the inverse Laplace transform of 2

1

1 4s s . (M/J 2009)

8. Find 2 2

1

2 2

1ln

s aL

s s b

. (N/D 2011)(AUT)

Laplace Transform Using Convolution Theorem

1. Using Convolution theorem

1 1L

s a s b

. (A/M 2011)

2. Apply convolution theorem to evaluate

1

22 2

sL

s a

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



3. Find

2

1

22

4

sL

s

using convolution theorem. (N/D 2012)

4. Find the inverse Laplace transform of

2

2 2 2 2

s

s a s b using convolution theorem.

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

5. Using convolution theorem find the inverse Laplace transform of 2

1

1 1s s .

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

6. Find

Ls s

1

2

1

4using convolution theorem. (N/D 2011)

Solving Differential Equation By Laplace Transform

1. Solve 2

23 2 2

d x dxx

dt dt , given 0x and 5

dx

dt for 0t using Laplace

transform method. (A/M 2011),(N/D 2012)

2. Solve the equation 9 cos 2 , (0) 1y y t y and 12

y

using Laplace

transform. (M/J 2009)

3. Solve the differential equation 2

2sin 2 ;

d yy t

dt (0) 0, (0) 0y y by using Laplace

transform method. (N/D 2009)

4. Using Laplace transform solve the differential equation 3 4 2t

y y y e with

(0) 1 (0)y y . (M/J 2010),(N/D 2010)

5. Solve the differential equation 2

23 2

td y dyy e

dt dt

with (0) 1y and (0) 0y ,

using Laplace transform. (M/J 2012)

6. Solve 23 2 4 , (0) 3, (0) 5

ty y y e y y , using Laplace transform.

(N/D 2011)(AUT)



7. Solve, by Laplace transform method, the equation 2

22 5 sin

td y dyy e t

dt dt

,

(0) 0, (0) 1y y . (M/J 2011)

8. Solve d y dy

y tdt dt

2

24 4 sin , if

dy

dt 0 and y 2 when 0t using Laplace

transforms. (N/D 2011)

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