Mathematics Iii May2004 Or 220556

OR

Code No: 220556 II-B.Tech. II-Semester Supplementary Examinations, April/May-2004 MATHEMATICS-III (Common to Electronics and Electrical Engineering, Electronics and Communication Engineering, Electronics and Instrumentation Engineering, BioMedical Engineering, Metallurgy and Material Technology, Electronics and Control Engineering and Electronics and Telematics) Time: 3 Hours Max. Marks: 70 Answer any FIVE questions All questions carry equal marks --1

1.a)

Show that

∫ x ( log x ) m

n

dx

( − 1) n n! (m + 1) n +1

=

0

where n is a positive

integer and m > -1 b)

2.a) b)

∞

y n −1

∫ (1 + y )

Show that β(m,n) =

m+ n

dy

0

Establish the formula

Pn' +1 ( x ) −Pn' −1 ( x ) =( 2n +1) Pn ( x )

When n is a +ve integer prove that Pn(x) =

1

π

(x ± π∫

x 2 − 1 cosθ) n

dθ

0

3.a) b) 4.a)

Prove that

 ∂2 ∂2  +  ∂x 2 ∂y 2 

  | Re al f ( z ) | 2 = 2 | f ' ( z ) | 2  

where w =f(z) is analytic.

Find k such that f(x,y) = x3 + 3kxy2 may be harmonic and find its conjugate. Evaluate using Cauchy’s Integral Formula ( z − 2) dz

∫ ( z − 1) ( z + 2) 3

3

C

with c : | Z | = 3 b)

Evaluate c∫ ( y2 + z2 ) dx +( z2 + x2)dy + ( x2 + y2)dz from ( 0,0,0 ) to ( 1,1,1 ) where C is the curve x = t, y=t2 , z=t3 in the parameter form.

5.a)

Locate and classify the singularities of f ( z) =

4z + 3 z ( z 2 +1)( z −1) 3

b) Obtain the Taylor Expansion of e(1+z) in power of (z-1) Contd…2

Code No.:220556 6.

b) 8.a) b)

OR

x2dx (x2+1)(x2+4) Show that the polynomial Z5 + Z3 + 2 Z + 3 has just one zero in the first quadrant of the complex plane. Show that the equation Z4 + 4( 1+i) Z + 1 = 0 has one root in each quadrant. Evaluate

7.a)

-2-

-∞

∫

Determine the bilinear transformation that maps the points (1-2i, 2+i, 2+3i) into the points (2+i, 1+3i, 4). Prove that the transformation w=sin z maps the families of lines x=constant and y=constant in to two families of confocal central conics. ^^^