MA5-MATHEMATICS-II KINGS COLLEGE OF ENGINEERING Punalkulam DEPARTMENT OF MATHEMATICS ACADEMIC YEAR - ( Even Semese ) QUESTION BANK SUBJECT CODE: MA5 SUBJECT NAME: MATHEMATICS - II YEAR / SEM: I / II UNIT- I LAPLACE TRANSFORMS PART-A. Wie a funion fo whih Laplae ansfomaion doesno eis. Eplain why Laplae ansfom does no eis.. If L(f()) = F(s) wha is L(e -a f())? 3. Find he Laplae ansfom of e - (+). 4. Find he Laplae ansfom of e - sin. 5. Find L(sin). 6. sae he ondiion fo he eisene of Laplae Tansfom of f(). 7. Find he Laplae ansfom of. 8.Obain he Laplae ansfom of sin-os in he simplified fom. e 3 e 9. Find he Laplae ansfom of. s. If L(f()) = F(s), hen show ha L(f(a)) = F( ). a a. Veify he iniial value heoem fo f() = 5 + 4os.. Sae he final value heoem fo Laplae ansfoms. 3. If L{f()}=, find lim f(). s( s + ) 4. Find he invese Laplae ansfom of. 4 ( s + ) 5. Sae he onvoluion heoem fo Laplae ansfoms. 6. Find he invese Laplae Tansfom of. s 4s + 3 s + 7. Find he invese Laplae ansfom of. ( s + 4s + 5) s + 8. If L(f()) =. Find he value of s + 4 f ( ) d. KINGS COLLEGE OF ENGINEERING - PUNALKULAM
MA5-MATHEMATICS-II s + a 9. Find L ( log e ( )). s a. Find he Laplae ansfom of osha. PART-B 4 e sin. a) Find he Laplae ansfom of (i) e sin 3 and (ii). os b) Find (i) L and (ii) o s L. k. a) Find he Laplae ansfom of e 3 e (os) and. a b) Find L(. e.osb) and L [an ( s / a )]. 3. a) Veify he iniial and final value heoems fo he funion +e -. 4( s ) b) Find he invese Laplae ansfom of ( s + s + 7). 4. a) Find he invese Laplae ansfom of. s ( s + a ) s + 3 b) Find L - ( s + )( s )( s + ) s 5. a)using Convoluion heoem o find L - ( s + )( s + 4) s + b) Using onvoluion heoem find he invese Laplae ansfom of. ( s + 4s + 3) d y dy ' 6. a) Using Laplae ansfom, solve: 3 + y = 4, y() =, y () = 3. d d b)solve he diffeenial equaion, using Laplae ansfom y +5y +6y= given ha y()= and y ()=. 7. a) Using Laplae ansfom solve y +y -3y=3 given ha y()=4 and y ()= -7. d y dy b)using Laplae ansfom solve + = + given ha y=4 and d d y = - when =. 8. a) Using Laplae ansfom solve y -y +y=e given ha y()= and y ()=. d y dy 3 b) Solve using Laplae ansfom 3 + y = e give ha y()=. d d 9. a) Find he Laplae ansfom of f() = sin,<<, << and f(+) = f(). b) Find he Laplae ansfom of f() = E,<<a/ KINGS COLLEGE OF ENGINEERING - PUNALKULAM -E, a/<<a and f(+a) = f()
MA5-MATHEMATICS-II. a) Find he Laplae ansfom of f() =, << 4-, <<4, f(+4) = f(). b) Find he Laplae ansfom of f() = asinω, << /ω, /ω<</ω and f(+/ω) = f().. a) Find he Laplae ansfomof f() = k in, <<a -k in, a<<a and f(+a) = f(). b) Obain he Laplae ansfom of f() = os,<<, > and f(+) = f(). dy. a) Using Laplae ansfom, solve: + y + y( ) d =, y() =. d b) Solve using Laplae ansfom, PART A. If ( ) ( ) y + yd = e. UNIT- II VECTOR CALCULUS ( ), F = - y + i+ - y + y j + + k. Pove ha ( log ) = 3. Find n = n n-. Whee whee =I I = + y +. 4. Find. 5. Find he uni nomal o he sufae y= a (,,). 6. If f = + y + - 8, hen find gad f a (,,). 7. Define solenoidal veo and ioaional veo u 8. If V = ( + 3 y) i + ( y - ) j + ( + l ) k is solenoidal, find l. find gad ( div u F). k 9. Fo wha value of k is he veo solenoidal.. Find a,b, so ha he veo F = ( + y + a) i + ( b 3y ) j + ( 4 + y + )k is ioaional.. Is he veo i + y j + 3 k, Ioaional?. Show ha F = ( i + y j + ) k is a onsevaive veo field. 3. If F =y i + j +y k hen find F 4. If F = 5yi + y j, evaluae F.d Whee C is he pa of he uve y = beween = and =. 5. Find. ds Whee S is he sufae of he eahedon whose veies ae s (,,), (,,), (,,), (,,). 3 KINGS COLLEGE OF ENGINEERING - PUNALKULAM
MA5-MATHEMATICS-II 6. If S is any losed sufae enlosing a volume V and F = a i + by j + k, Pove ha F. nds =(a+b+) V ˆ S 7. Sae Geen s heoem in a plane. 8. Using Geen s heoem, Pove ha he aea enlosed by a simple losed uve C is ( dy yd) 9. Sae Gauss divegene heoem.. Sae Soke s heoem. PART - B. a If = i + y j + k and n ul ( ) =. b If = i + y j + k, div n =. Pove ha n ( ) = ( + 3) n n hen pove ha div gad( ) n( n ) - n and = +.Hene dedue ha divgad =. a Find he dieional deivaive of φ =3 +y-3 a (,,) in he dieion of b Find he angle beween he sufaes +y + =9 and = +y -3 a he poin (,-,). 3. a Find he values of a and b, if he sufaes a -by=(a+) and 4 y+ 3 =4 u ohogonally a he poin (,-,). 3 F = y os + i + y sin - 4 j + 3 k is ioaional and i + j - k b Pove ha ( ) ( ) find he sala poenial f. 3 4. a Find he wok done he foe F = 3yi y j moves a paile along he uve C: y= fom (,) o (,) in he y-plane. v b Evaluae f.. d Whee 3 f = ( y + ) i+ j + 3 k along he saigh line joining (, -,) and (3,,4) 5. a Evaluae f..ˆ nds f = ( + y ) i j + y and S is he sufae Whee k S of he + y + = 6 in he fis oan. b Using Geen s heoem in he plane evaluae (3 8y ) d + (4y 6y) dy Whee C is he bounday of he egion C enlosed by y= and y=. 6. a Apply Geen s heoem in he plane o evaluae (3 8y ) d + (4y 6y) dy Whee C is he bounday of he egion defined by by =, y= and + y=. C 4 KINGS COLLEGE OF ENGINEERING - PUNALKULAM
MA5-MATHEMATICS-II b Veify Geen s heoem in a plane fo ( - y)d + ( + y)dy whee C is he bounday of he ile +y = in he oy plane 7. a Veify Gauss s divegene heoem fo F = 4 i- y j + y k and C is is bounday ove he ube =, =, y =, y =, =, =. b Veify Gauss Divegene heoem fo F = i + y j + k Whee S is he sufae of he uboid fomed by he planes =, = a, y =, y = b, =, =. 8. a Veify Gauss s divegene heoem fo he funion F = y i + j + k ove he ylindial egion bounded by + y = 9, = and =. u b Veify soke s heoem fo F = ( - y ) i + y j in he eangula egion in he y plane bounded by he lines =, = a, y=, y= b. 9. a Veify soke s heoem fo F = ( - y) i - y j- y k Whee S is he uppe half of he sphee + y + = and C is is bounday b Evaluae ( e d + ydy d).using soke s heoem, whee C is he uve C y + = 4, =..a Veify soke s heoem fo F = - yi + y j + y k. Whee S is he uppe half of he sphee + y + =. u aken ound he eangle bounded by he b Veify soke s heoem fo F = ( + y ) i- y j lines = ± a, y= and y= b. UNIT- III ANALYTIC FUNCTIONS PART-A. Is f ( ) = 3 analyi? Jusify.. Pove ha is nowhee analyi. 3.Fo wha values of a,b and he funion f ( ) = ay + i( b y) is analyi. 4. If u+iv is analyi, show ha v iu &-v +iu ae also analyi. 5. Sae he ohogonal popey of an analyial funion. 6. Show ha he an analyi funion wih onsan eal pa is onsan. 7. Wie down he fomula fo finding an analyi funion f () = u + iv, wheneve he eal pa is given by using Milne Thomson mehod. 8. Find a so ha u (,y) = a y +y is hamoni. 9. Veify he funion u (,y) = log + y is hamoni o no.. Define he onfomal mapping.. Find he iial poins fo he ansfomaion W = ( α)( β ).. Find he image of he ile = unde he infomaion ω = 3. 3. Find he fied poins of he ansfomaion 3 w =. 4. Find he image of i = unde he mapping w =. 5. Define bilinea ansfomaion. 5 KINGS COLLEGE OF ENGINEERING - PUNALKULAM
MA5-MATHEMATICS-II 6 9 6. Find he fied poin of ansfomaions ω =. 4 6 7. Find he fied poins of he ansfomaion w= (3 4) 8. Find he fied poins of w =. ( ) + 9. Find he invaian poin of he bilinea ansfomaion w =. Wie he oss aio of he poins,, 3, 4. PART B y. a. If u= y and v= + y,pove ha u and v ae hamoni funions bu u+iv is no egula funion of. b.given ha f () =u+iv is an analyi funion and u+v = e (os y + sin y). find f().. a. Pove ha an analyi funion wih onsan modulus is onsan. ' b. If f ( ) is a egula funion of, pove ha + f ( ) = 4 f ( ) y 3. a. Show ha v = e ( osy ysiny ) is a hamoni funion. Find he oesponding analyi funion f (). b. Use Milne Thomson Mehod o find he hamoni onjugae u of v = e - ( yosy +(y )siny. Whee u+iv is he analyi funion. 4. a. Show ha he v = e ( yosy + siny )is hamoni and find he oesponding analyi funion f ( ) = u+iv. sin b. Find he analyi funion whose eal pa is (osh y os ) 5. a. Pove ha he funion u = 3 3y +3-3y + is hamoni. Find he onjugae hamoni funion V and he oesponding analyifunion f() b. Find he ohogonal ajeoies of he uves epesened by u(,y) = 3 y y 3 = 6. a. Disuss he onfomal mapping W=/ b. Find he image of he ile = in he omple plane unde he mapping ω = Show he egion gaphially. 7. a. Obain he image of i = unde he ansfomaionω =. b. Find he image of he ile = in he omple plane unde he mapping ω =. Show he egion gaphially. 8. a.find he image of +y= unde he ansfomaion w= b. Find he bilinea ansfomaions ha maps he poins = -,, in he plane on o he poins W =, i, 3i in he w plane. 9. a. Find he Bilinea ansfomaion ha maps he poins +i, -i, -i a he - plane ino he 6 KINGS COLLEGE OF ENGINEERING - PUNALKULAM.
MA5-MATHEMATICS-II poins,,i of he w-plane. b. Find he bilinea ansfomaion whih maps he poins =, i,- ino he poins w= i,,-i. Hene find he image of <..a. Find he bilinea ansfomaion whih maps he poins =,-I,- ino w = I,, espeively. b. Find he bilinea ansfomaion whih maps -,,i of he -plane ono -,-I, of he w-plane. Show ha unde his mapping he uppe half of he -plane maps ono he ineio of he uni ile w = PART - A y. Epess y ( + y ) 3 / 4 UNIT- IV MULTIPLE INTEGRALS d dy in pola o-odinaes.. Find he limis of inegaion in he double inegal f(,y) d dy whee R is in he fis R quadan and bounded by =, y =, y²= 4. 3. Skeh oughly he egion of inegaion fo he double inegal f(,y)dy d. a (a²-²) 4. Shade he egion of inegaion d dy (a-²) 5. Epess he egion, y,, ²+y²+² by iple inegaion. 6. Evaluae dd θ. 7. Find he aea of a ile of adius a by double inegaion in pola o-odinaes. a (a² - ²) 8. Evaluae (dy d). a b ddy 9. Evaluae y osθ. Evaluae dd θ... Epess f (, y) ddy in pola oodinae. +. Evaluae sin( y) ddy. 3 4 6 4 3. Daw he egion of inegaion fo ddy. 7 KINGS COLLEGE OF ENGINEERING - PUNALKULAM
MA5-MATHEMATICS-II y 4. Evaluae ddy ² + y² 5. Find d dy ove he egion bounded by, y, +y. a osθ 6. Evaluae dd θ 3 7. Evaluae ( y) ddy. 4 + 3 8. Evaluae y ddyd.. 9. Change he ode of inegaion f (, y) dyd. a b + y+. Evaluae e ddyd. PART -B. a Evaluae (+y)² d dy ove he aea bounded by ellipse ²/a² + y²/b² =. b Evaluae yddy whee R is he egion bounded by he paapola y = and he R lines y= and +y= lying in he fis quadan Π. a Evaluae sin θ d θd b Evaluae ddθ,ove he aea bounded beween he iles =osθand =4osθ. a a 3. a Change he ode of inegaion in a ydyd b Change he ode of inegaion and hene evaluae ( 4 y ) 4. a Evaluae ( + y) 3 and hen evaluae i. ( + y ) ddy ddy by hanging he ode of inegaion. b Change he ode of inegaion and evaluae he inegal + a a 8 KINGS COLLEGE OF ENGINEERING - PUNALKULAM ( y ) dyd 5. a Find by double inegaion of he aea beween he paabola y² = 4a and he line y =. b Find he aea of he egion bounded by y = ³, y = using double inegals 6. a Find he aea of he egion bounded by he paabolas y = ² and = y. b Find he aea enlosed by he wo paabolas y =4 and ²=4y
MA5-MATHEMATICS-II 7. a Evaluae a a ( + y ) y ddy by hanging ino pola o-odinaes. b Tansfom he inegal ino pola oodinaes and hene evaluae a a + y dyd.. 8. a By onveing ino pola o-odinaes evaluae, ( + y ) b Find he volume of he eahedon fomed by he planes =,y =, = and +y+=. ddy 9. a Evaluae yddyd aken ove he posiive oan of he sphee ²+y²+²=a. b Evaluae 4 a sin yddyd a. Epess he volume of he sphee ²+y²+² = a² as a volume inegal and hene evaluae i. b. Evaluae d dy d / a²-²-y²-² ove he fis oan of he sphee ²+y²+=a². PART A UNIT V COMPLEX INTEGRATION. Sae Cauhy s inegal fomula.. Evaluae d whee is he ile = 3 + 5 3. Evaluae d i 3 + 4. Evaluae d whee is = 4 whee is a ile = using Cauhy s inegal fomula. os 5. Evaluae d aound a eangle wih veies a ±i,-±i 6. Find he Lauen seies epansions of ( ) = e f 3 ( ) 7. Obain he epansion of log ( + ) when < 8. Epand a = in a Taylos seies 9. Epand os in a Taylo s seies a = / 4. Obain he Lauen epansion of he funion ( ) e abou =. in he neighbouhood of is 9 KINGS COLLEGE OF ENGINEERING - PUNALKULAM
MA5-MATHEMATICS-II. Evaluae d ( 3 ). Find he value of e d whee C is ile = 4 3. Deemine he esidues a poles of he funion ( ) = + f ( )( ) 4. Find he esidue a he pole of he funion + 3 f ( ) = ( + ) 5. Find he esidue of e a is singula poin. sin + os 6. Using Cauhy s inegal fomula evaluae 3 ( ( )) d whee is he ile =4 7. Sae Cauhy s Residue Theoem. dθ 8. Epess as a onou inegal aound he ile = + a osθ 9. Find he Residue of f ( ) = ( ) as is pole.. Find he esidue of o a he pole =. PART B e d.a. Using Cauhy s inegal fomula evaluae ( + )( + ) whee is = 3 + 4 b. Using Cauhy s inegal fomula evaluae d whee is + i = + + 5 +. a. By Cauhy s inegal fomula evaluae 4 3 4 + 4 b. Using Cauhy s inegal fomula evaluae ( )( ) whee is i = d whee is ile = 3. a. Epand f ( ) = d in Taylos seies if < ( + )( + 3) b. Epand a = in Taylos seies. 4.a. Find he Lauen seies epansion fo he funion 7 f ( ) = ( + ) ( )( ) in he annula egion < + < 3 b. Epand 3 + in Lauen seies valid in he egion (i) < < (ii) < < KINGS COLLEGE OF ENGINEERING - PUNALKULAM
MA5-MATHEMATICS-II 5.a. Obain he Lauen seies epansion fo he funion f ( ) = + 3 ( ) in he egion (i) <(ii)< <(iii) > 3 + b. Using esidue heoem o evaluae d ( )( 3) 6. a. Evaluae os + sin ) ( + )( + = 3 d b. Evaluae ( + 4) e 7. a. Evaluae ( + ) aound he ile = d Using Cauhy s esidue heoem. whee is he ile i = using Cauhy s esidue heoem d whee is he ile = 4 using Cauhy s esidue heoem dθ b. Evaluae by onou inegaion 5 sinθ 8. a. Using he mehod of onou inegaion evaluae ( + )( + d 4) dθ b. valuae,a>b>by onou inegaion. a + bosθ os θ 9.a. Evaluae dθ by onou inegaion 5 4osθ b. Pove ha d =, a, b > by using he mehod of onou ( + )( + ) a + b a b inegaion.. a. By he mehod of onou inegaion. Pove ha dθ =, if < a < aosϑ + a a b. Pove ha sin m + a ma d = e, m > ; a > KINGS COLLEGE OF ENGINEERING - PUNALKULAM