Anna University Coimbatore 2009 B.E Electrical and Electronics Engineering Engineering Mathematics 3 - Question Paper

9.

10.

mm. life# %

w..

v--^!?

Vs/

-

*

* f

Find z[e^at+b]

Prove initial value theorem on z-transform.

PART -B

(5 x 16 = 80)

11.    (a) Form the partial differential equation by eliminating f from the relation (8)

f (xy-z², x²-y-z)=0

(b) Solve [D² -DD' -2D'²]z = 2x + 3y + e^3x+4y    (8)

(OR)

12.    (a) Solve x(z²-y²) p + y(x²-z²)q = z(y²-x²)    (8) (b) Solve [D² + 3DD'-4D^/2]z=cos(2x + y) + xy (8)

2 fcaa

13. (a)

2    Q.CODE: 081001

(8)

0    , 71 < x < 0

7tX _n

-. 0 < X < 71

1    4

and deduce

Find the Fourier expansion of f(x) =

1

(8)

(b)

Find the half range cosine series of x - x + in (0,1)

(OR)

14. (a) If f(x) = x - x² in the range (0, {), find the half - range sine series of f(x). (8)

1111 Deduce that -_T-_r + -_T- + 1³ 3 5 7

71

32

.00 =

(b) Find the Fourier series y = f(x) upto second harmonics from the following data.

(8)

O 71/

/3

0.50

⁴%

1.30

5 y. 1.76

271

1.80

71

2.16

y=f(x): 1.80 0.30

15 (a)

' Find the Fourier transform of e

(b) _x2 *

Using transform method, find =- dx , a > 0

₀^J (x +a^z)

-X²/

^/2 and xe

(8)

(8)

(OR)

a-|xj, for Ixl < a

16. (a)

(8)

Find the Fourier transform of f(x) =

and deduce

0 , for Ixl > a , a > 0

sin⁴1

r bir

J ~

dt

(b)     x²

Find finite Fourier sine and cosine transform of x + in 0<x< n

3 27i

(8)

A string is stretched between two fixed points x=0 and x= I and released at (16)

2Kx

, when 0 < x

rest from the initial deflection given by f(x) =

2K

(-x), when <x<

Find the deflection of the string at any time.

19. (a)

(8)

1

Find z

and z[coshat sinbt]

n(n + 1)

(b) Solve y_n+2 + 2y_n+i + y_n = n given y₀ = 0 and yi=0 using z-transform.

(OR)

(8)

20. (a)

(8)

z² + z

z² +z (z-1)²

-1

and z ¹

Find z

[_(z 1)(z² +1)

(b)

(8)

Using convolution theorem, evaluate z

(z-1)(z-3)

*

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