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### Transcript of 23421332 Engineering Mathematics III Important University Questions Unit i Fourier Series Two Marks

ENGINEERING MATHEMATICS-IIIIMPORTANT UNIVERSITY QUESTIONSUNIT-IFOURIER SERIESTWO MARKS1)Determine nb in the Fourier series expansion of ( ) ( ) x x f 21in 2 0 < < xwith period 2 . (May/June 2007)2)Define root mean square value of ( ) x f inb x a < < . (M/J07)3)If ( ) 20, 5 0, c o s< xe . (Nov/Dec 2005)14) Find the Fourier cosine integral representation of( )'01x f

1 1 0>< 0.(May 2006)17) If( ) { } ( ) s f x f F then givethe value of ( ) { } ax f F. (May 2006)18) Find the Fourier transform of( )'01x f 11>< xx. Hence evaluate (i)

,_

032coscos sindxxxx x x(ii) ,_

02315cos sin dsss s s. (April2005) (Dec 2008)17) Find the Fourier Sine transform of( ) 0 >axeax. (Nov/Dec 2006)18) Find the Fourier Sine and cosine transform of xe2 . Hence find the value of the following integrals (i)( )+ 0224 x dx(ii) ( )+ 02224dxx x. (A.U.Model Qu)19) Evaluate(i) ( )( )+ +02 2 2 2b x a xdx(ii) ( )( )+ +02 24 1 x xdxusing Fourier transform.(Nov/Dec 2008)20) Find the Fourier Sine and cosine transform of 1 nx .(May 2006)21) Using Parsevals identity for Fourier cosine transform of axe evaluate ( )+ 022 2x adx .(Nov/Dec 2007)822) Find the Fourier Sine transform of( ) 0 , >a eax. Hence find[ ]axS xe F. Hence deduce the inversion formula. (May/June 2007)23) Find the Fourier Sine transform off(x) defined as

( )'0sin xx f wherewhere a xa x>< < 0. (Dec 2008) 24) Findthe Fourier transform of( )'0 x 1x f

otherwisefor 1 x. Hence find the values of(i)

,_

0dt4t t sinand (ii)

,_

0dx2xx sin(Dec 2008)25) Find the finite sine and cosine transform of( )2x1 x f ,_

in the interval ( ) , 0. (Dec 2008)26) Find the Fourier transform of ( )' 0 x ax f ,, a xa x> > 0. (April 2003)3) A tightly stretched string with fixed end points x=0 and x=l is initially in a position given by

,_

lxy x y30 sin ) 0 , (. It is released from rest from this position. Find the displacement at anytime t . (Nov 2004)4) A tightly stretched string of length 2l has its ends fastened at x=0 , x=2l. The midpoint of the string is then taken to height b and then released from rest in that position. Find the lateral displacement of a point of the string at time t from yhe instant of release.(May 2005)5) A string of length l has its ends x=0 , x=lfixed. The point where 3lx is drawn aside a small distance h ,the displacement ) , ( t x ysatisfies .22222xyat y Find) , ( t x yat any time t .6) An elastic string of length 2l fixed at both ends is disturbed from its equilibrium position by imparting to each point an initial velocity of magnitude ). 2 (2x lx k Find the displacement function ) , ( t x y. (May 06)7) A uniform string is stretched and fastened to two points l apart. Motion is started by displacing the string into the form of the curve ), ( x l kx y and then releasing it from this position at time t=0. Find the displacement 17of the point of the string at a distance x from one end at time t . (A.U.Tri.Nov/Dec 2008)(Dec 2008)(May/June 2009)8) If a string of length l is initially at rest in its equilibrium position and each of its points is given a velocity v such that ') ( x l c cxv forfor l xllx