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THIRD SEMESTER B.TECH. [ENGINEERING] (14 SCHEME) DEGREE EXAMINATION, NOVEMBER 2015 EN 14 301-ENGINEERING MATHEMATIC~III (Common for all Branches) Time : Three Hours

Maximum: 100 Marks Part A

Answer any eight questions. Each question carries 5 marks. 1. Prove that w

= sin z is an entire function. If so find

dw .

dz

2. Show that ex (x cosy - y sin y) is a harmonic function.

3. Find and graph the

i~age

of -1 ~ x ~ 1, -

J, · m ) 4. Provethat'f.(z-z dz= {21ti 0

c

0

if . 1f

1t

< y < 1t under the mapping w

m=-1

m ;e -1

5. Using Cauchy's integral formula, evaluate

andinteger.

J

z

c(z-1){z-2)

.

.

.

= ez.

2

dz where C is

!z- 21 = Ji.

9z+i z+z

6. Fmd the poles and residues of - -3 • 7. Express v = (1, -2, 5) in R3 as a linear combination of the vectors u1 = (1, 1, 1), u2

= (1, 2, 3)

and

u3 = ( 2, -1, 1). 8. Let_W be the subspace ofR5 generated by the vectors u = {1, 2, 3,-:-1, 2) an.d v = (2, 4, 7, 2, -1). Find a basis ofthe orthogonal complement

wJ..

9. Find the Fourier sine. integral representation of

ofW.

f (t) = e-at , 0 < t < oo, a > 0.

10. Find the Fourier cosine transform of the function

.

·

f ( x) = {cos x. .

0

' 0 < x a (8 x 5

= 40 marks) Turn over

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.1:

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Part B · ·• il~·;

Answer all questions. Each question carries 15 marks. sin 2x

. fi~twn . wh ose reaI part IS . . d the anaIytlc 11 . ( a ) F m

·

cosh 2y +cos 2x

(b) Prove that an analytic function with constant modulus is a constant.

Or 12. (a) Discqss the tt~sfor:Uation w = z +.!. What are its fixed points. What are the critical points? z Show that the transformation maps the circle

lzl = c

into an ellipse. Discuss the case when

c = 1. (b) Find the Mobius transformation that maps the points ( 2, i, .

.

13. (a) Using Cauchy's residue theorem evaluate~

•

2

2}

into the points ( l, i,

-1}.

2

7

.

sm 1t z cos 1t z dz where Cis the circle lzl = 2. c (z-1} (z-2}

(b) Expand

z(z-1

)(

z-2

} in the region (i)

lzl > 2;

(ii)

lz I< 1 ; (iii) 1
Or 14. (a) Evaluate

2 Jlt 0

sin

e

3 + COS 8

dS ; (b) Evaluate

<X>J

_

00

x

( X

2

2

) (

+ 4 x2 +9

)

dx.

15. (a) Find an orthonormal basis for the subspace spanned by

(1, 1, 1, 1), (1, 2, 4, 5}

and

(1,.:...3,-4,-2) inR4. (b )F Find a, b,

c such that ( 2, 1, -1 ~, (a, 1, -1}

and _( b, 3, c} form an orthogonal basis of R3 .

Or 16. (a) Define an inner product space. Let x =(Xp x2 } and y

=(YP y 2 }.

Determine whether

(b) State Schwartz's Inequality and triangle Inequality. Using the standard inner product verify them for the vectors x = ( -2, 3, 1} and y = (3, -4, -1} in R3 •

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17. (a) Find a Fourier cosine and sine integral representation of the function

f(t)=

(b)

cost { 0

,

0~t ~~

'

t>~.

If~ {f(t)} = F( w) then show that ~ {f(t-t0 )} = e-lwt.

F( w).

Or

:'

18. (a) Find the Fourier integral repre~entation of

""ssin X0

X

X

3

{1f (t):::;

t

0

2

,

I1 I> 1 . Hence evaluate 1 1 t 1<

COS X cos (X) dx. 2

(b) Find the Fourier sine and cosine transform of e-ltl.

(4 x 15 = 60 marks)

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