TWO MARKS WITH ANSWER
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- Walter James
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1 TWO MARKS WITH ANSWER MA65/TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS REGULATION: UNIT I PARTIAL DIFFERENTIAL EQUATIONS Formtio o rti dirti utio Sigur itgr -- Soutio o tdrd t o irt ordr rti dirti utio - Lgrg ir utio -- Lir rti dirti utio o cod d highr ordr with cott coicit o oth homogou d ohomogou t. Form th PDE imitig d rom N/D 4,A/M A:... Sutitut d i, w gt. 4. Fid th PDE o th mi o hr hvig thir ctr o th i = =.N/D A : Th utio o uch hr i r Prti dirtitig with rct to d, w gt... Simir...
2 From d. Form th rti dirti utio r M/J A:... r Prti dirtitig with rct to d, w gt Sutitut d i, w gt r r [ r i rticur cott ] 4. Fid th Prti dirti utio o hr who ctr i o th i. N/D A: Eutio o uch hr... r c Prti dirtitig with rct to d, w gt c c c d c c c From d, 5. Form th rti dirti utio imitig th ritrr uctio rom. N/D, M/J A:
3 Giv rio c writt,... Lt u, v i o th orm u, v... Th imitio o φ rom giv u u v v... u u v v... 4 Uig 4 i, w gt 6. Fid th rti dirti utio o cuttig u itrct rom th d. A: Th utio o uch i... Prti dirtitig with rct to, w gt... Prti dirtitig with rct to, w gt
4 From d, 7. Form PDE imitig th uctio rom th rtio. N/D4,N/D A: Prti dirtitig with rct to d, w gt,... ' '... ' ' gt w d From 8.Form th PDE rom M/J 4,N/D A: Sutitut d i, w gt 9. Di ordr o.d.. N/D A: Th ordr o.d. i th ordr o th hight rti drivtiv occurrig i it.
5 . Form th.d. o th orm t g t N/D,N/D A: '' '' ' ' '' ''..... ' ' t g t t t g t t t g t r t g t t g t From d 4 t. Form th rti dirti utio imitig rom, M/J 4 A:...,,...., v u orm th i o v u Lt Th imitio o rom giv..... v u v u v u v u... 4 Uig 4 i, w gt......form th PDE rom = +. M/J 4,N/D A:
6 Sutitut d i, w gt.form th rti dirti utio imitig th ritrr uctio rom,. N/D 4 A:,.... Lt u, v i o th orm u, v... Th imitio o rom giv u u v v... u u Uig 4 i, w gt v v Sov A: M/J 4
7 Giv 4... g g 5. Sov =. M/J4 A: Thi i C.I. Thr i o S.I
8 6. Fid th comt outio o + = N/D4 A:. Thi i o th t Thror th tri outio i. To id th comt itgr C.I Su vu i Su i, w gt 7.Fid th igur itgr o th rti dirti utio i = A: N/D,N/D 9 Thi i o th t Thi C.I i To id th igur itgr Su & i,w gt which i th S.I. 8. Fid th C.I o M/J,N/D A: Thi i CLAIRAUT S t.thror th comt itgr C.I i
9 9.Fid th comt itgr o =. N/D,N/D,A/M 5 A: =. Lt which i C.I..Fid th comt itgr o + = M/J,M/J,N/D A:. Thi i o th t Thror th tri outio i To id th comt itgr C.I. Su vu i Su i, w gt.sov th rti dirti utio = A/M A:
10 . Sov N/D A: Giv Rcd Th A.E i Th outio.sov th utio D D' A/M,N/D A: Giv D D' Rcd Th A.E i Th outio 4. Sov D 7DD' 6D' =. M/J A: Giv D 7DD' 6D' =. Rcd Th A.E i Th outio
11 5.Sov A : Giv D D D Rcd Th A.E i Th outio 6. Sov D D' A: 4 4 A/M,M/J 4 A.E i 7.Sov D D D. N/D A: Giv D D D Rcd Th A.E i Th outio 8. Fid th P.I o D DD' D' =. N/D A : Giv D DD' D' =. Rcd D= d
12 9.Fid th rticur itgr o D D' DD ' N/D A : Giv D D' DD' Rcd, d.form th PDE imitig d rom N/D, M/J, A/M,M/J A :... Sutitut d i, w gt. 4 UNIT II FOURIER SERIES Diricht coditio Gr Fourir ri Odd d v uctio H rg i ri H rg coi ri Com orm o Fourir ri Prv idtit Hrmoic i.. Writ th Diricht coditio or uctio to dd Fourir ri or Stt th uicit coditio or uctio to rd Fourir ri. M/J4,N/D4,M/J,M/J,N/D,N/D,N/D A: A uctio did i, c dd iiit trigoomtric ri o th orm = co i rovidd
13 i i did d ig vud ct oi t iit umr o oit i,. ii i riodic i,. iii d r icwi cotiuou i,. iv h o or iit umr o mim or miim i,.. Writ th Fourir coicit i, c c. A: Th Fourir ri or th uctio i th itrv c, c i giv = co i c d c c co d c c i d c whr Th vu, d r kow Eur ormu or Fourir coicit.. Giv th rio or th Fourir ri co-icit or th uctio did i,.a/m,n/d. i A: Giv = i - = -i- = --i = i - = = i i v uctio.. 4. Fid th vu o i th Fourir ri io o i,.n/d A: Giv d
14 d = = =. 5. Fid th vu o i th ri o i,. N/D A: Giv i, coidr,=, = = d d * = d = d 6. I, th id. N/D, A: 7. I A: = i oit o dicotiuit um o th Fourir ri = 4 = = co dduc tht.... N/D4, A/M 6 Hr w coidr th itrv,. i oit o cotiuit LHS = RHS = 4 co coidr = &u = co co co = 4...
15 co co co = 4... = 4... = 4... = Oti th irt trm o th Fourir ri or th uctio,.n/d 9 A: Giv, Firt trm o th Fourir ri d d [, = d i v uctio = = 9. Fid th cott trm i th Fourir io o co i, M/J,N/D,SEP9 A: Cott trm = = co i v uctio d co d co d
16 co d d =. rm : Cott t. I Fid th vu o i th Fourir io o. A: Giv,, Itrv=, =, d i d i i i d d i i d d co co = co. Fid th um o th Fourir ri or,, t =. N/D A: Giv,, t =. Hr = i oit o dicotiuit.
17 Sum = = =. I co 4 i, th dduc tht vu o N/D4 A: co i Giv 4 = i oit o dicotiuit coidr = LHS = RHS = co = Writ dow th orm o th Fourir ri o odd uctio i, d ocitd Eur ormu or Fourir coicit. N/D A: Th Fourir ri o odd uctio i, i i Whr,, i d. 4. I i v uctio i th itrv,, wht i th vu o?. MAY 4 A: I i v uctio i th itrv,,th vu o. 5. Fid i = i dd Fourir ri i,. MAY4,SEP9 A: Giv i,,
18 d i v uctio = d = d i,, = d =. 6. Fid th h rg i ri or k i, MAY 4 A: Giv = k i, Th h rg i ri i, i i i d k i d k k co, 4k, i i i v i odd,,5... 4k i 7. Fid th h rg i ri or i th itrv MAY4,N/D,SEP9 A: Giv = i, Th h rg i ri i, i i
19 i d i d co, 4, i i i v i odd,, i 8. Fid th h rg i ri or i th itrv A: Giv = i, Th h rg i ri i, i i d i Su i d 4 co co 4 co 4 =, 8, i i i v i odd 8 i,, Fid th h rg i ri io o = i, N/D A: Giv = i, Th h rg i ri i, i i
20 d i Su i d co co co =, i 4, i i v i odd 4 i,,5.... Th coi ri or = i or < < i giv i = co dduc tht... = N/D A: Giv i co co i, Hr i oit o cotiuit coidr = i i co co co co co co co 4 co Prov tht h rg coi ri o i, A: h th vu.
21 Giv i, d d = = 6. Th h rg i ri 8k k i, i i. Dduc th vu o,, A: Giv = k i, Th H rg i ri i,,5... 8k i k = 8k i,,5... i, i,,i Su k k 8 i 8k 8k 5 + i + i k k i 8k + i 8k 5 + i... 5 k 4 8k + 8k 8 + k... 5 k 4 8k... 5 k 4 8k I th h rg i io o co i, id th vu o. A:
22 Giv = co i, d i d co i d co i i = ico d i = co co =,co = = co co 4. Stt Prv Idtit Prv Thorm. N/D4,N/D,SEP9 A: I th Fourir ri corrodig to covrg uiorm to i, th d 5. Di Root M Sur vu o uctio ovr th itrv,. M/J,N/D,N/D,M/J,SEP9 A: Root m ur vu o th uctio = ovr th itrv, i did RMS = d or d. 6. Fid th R.M.S. Vu o i th itrv -, A: Giv = i -, d
23 d d 4 = = 5 5 = Fid th R.M.S. Vu o i th itrv, N/D4,N/D A: i, RMS = d = d = 5 5 = 5 5 = 5 8. Fid th R.M.S. Vu o i th itrv, N/D4,N/D,N/D A: Giv = i, RMS = d = d = = = 9. Writ th com orm o th Fourir ri o or.writ dow th com Fourir ri, ttig th ormu or coicit C. M/J A: Com orm o Fourir ri o i, c c i i C
24 whr C C C i d. Di Hrmoic i. M/J, M/J A : Th roc o idig th Fourir ri or uctio giv umric vu i kow hrmoic i.. Stt th cod hrmoic i Fourir ri io. N/D A : Scod hrmoic i Fourir ri io i th itrv, i giv ow: co i co i Whr [m vu o i, ] [m vu o co i, ] [m vu o i i, ] [m vu o co i, ] [m vu o i i, ]
25 UNIT III APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS Ciictio o PDE Mthod o rtio o vri Soutio o o dimio wv utio O dimio ht coductio Std tt outio o two dimio utio o ht coductio cudig iutd dg. u u. Ci th rti dirti utio 4. [Nov/Dc 9] t A: Th d u u u u u ordr.d. i A, B, C,,, u,, u u 4 A 4, B & C t B - 4AC Th utio i roic. u u. Ci th rti dirti utio [Nov/Dc ] t A: u u A, B & C t B - 4AC Th utio i roic.. Ci th rti dirti utio u u. [M/Ju ] A: A, B & C B - 4AC -4 For >, B 4AC, th utio i itic For <, B 4AC, th utio i hroic. 4. Ci th rti dirti utio A:. [Nov/Dc 4] A -, B - & C B - 4AC 4 4
26 For, =, B 4AC, th utio i itic For, > or, <, B 4AC, th utio i hroic. 5. Ci th rti dirti utio or,. A: [Nov/Dc 4] Th d u u u u u ordr.d. i A, B, C,,, u,, B 4AC 4 4 i w oitiv i. I, i gtiv. B B 4AC v 4AC. Thror th utio i itic u u u 6. Ci [Nov/Dc ] A: A, B & C B - 4AC 4 B 4AC Th utio i roic. u u u 7. Ci 8 [Nov/Dc ] A: A 8, B & C - B B - 4AC 4AC Th utio i hroic 4 96
27 8. Wht r th umtio md to driv o dimio wv utio? [M/Ju 4] A: Th m o th trig r uit gth i cott. Th trig i rct tic d do ot or ritc to dig. c Th tio cud trtchig th trig or iig it t th d oit i o rg tht th ctio o th grvittio orc o th trig c gctd. d Th trig rorm m trvr motio i vrtic, tht i vr rtic o th trig rmi m i out vu. 9. Wht r th w umd to driv o dimio ht utio? [M/Ju 4] A: Ht ow rom highr tmrtur to owr tmrtur. Th mout o ht ruird to roduc giv tmrtur chg i od i roortio to th m o th od d to th tmrtur chg. c Th rt t which ht ow through r i roortio to th r d to th tmrtur grdit orm to th r.. Writ th iiti coditio o th wv utio i th trig h iiti dicmt ut o iiti vocit. [Ari/M ] A: i, t or t ii, t or t, iii t iv, k. Writ dow oi outio o o dimio wv utio A : t 4 i, t c c c c [Nov/Dc 9, Nov/Dc, M/Ju 4, Nov/Dc 4] t ii, t c5 co c6 i c7 co t c8 i t iii, t c c c t 9 c. I th d o trig o gth r id d th midoit o th trig i drw id through hight h d th trig i rd rom rt, tt th iiti d oudr coditio. [M/Ju ] A : i, t or t ii, t or t
28 , iii t iv,,,. I td tt coditio driv th outio o o dimio ht ow utio. [Nov/Dc 4] A: Wh td tt coditio it th ht ow utio i iddt o tim t. u t u d u Th ht ow utio com or d d d u d d d du d du d du c du c d d du c d u = c c 4. Writ dow th thr oi outio o o dimio ht utio. [Nov/Dc, Nov/Dc 4, M/Ju, Nov/Dc ] A: Th thr oi outio o o dimio ht utio r u, t c c c u, t c4 c5 c6 t t u, t c7 c8 co c9 i 5. Wht i th ic dirc tw th outio o o dimio wv utio d o dimio ht utio with rct to th tim? [M/Ju ]
29 A: Soutio o th o dimio wv utio i o riodic i tur. But outio o th o dimio ht utio i ot o riodic i tur. 6. I th wv utio A: c t, wht do c td or? [Nov/Dc ] c T m m Tio r uit gth o th trig 7. I th o dimio ht utio ut c u,wht i A: Wht i i c?or [M/Ju ] u u? [M/Ju 4] t K i kow diuivit o th mtri o th r 8. A tight trtchd trig with id d oit = d = iiti i oitio giv, i.i it i rd rom rt i thi oitio, writ th oudr coditio. [Ari/M ] A: Boudr coditio r i, t or t ii, t or t Iiti coditio r, iii t iv, i 9. Mod th oudr vu rom: A uiorm tic trig o gth 6 cm i ujctd to cott tio o kg. Th d id d th iiti dicmt i 6, < < 6,whi th iiti vocit i ro. A: [M/Ju ] Th wv utio i t c Boudr coditio r i, t or t ii 6, t or t
30 Iiti coditio r, iii t iv, 6, 6. A rod 4 cm og with iutd id h it d A d B kt t ⁰C d 6⁰C rctiv. Fid th td tt tmrtur t octio 5 cm rom A. A: Wh th td tt coditio it th ht ow utio i [Ar/M ] u u = c c Th oudr coditio r u = u4 = 6 Aig i, w gt u = c = Sutitutig c = i, w gt u = c Aig i, w gt u 4 = c 4 6 c c 4 4 Sutitutig c i, w gt u Th td tt tmrtur t octio 5 cm rom A u O d o th rod o gth cm i kt t ⁰C d othr d o th rod i kt t 5⁰C uti td coditio rvi. Fid th td tt tmrtur. [M/Ju 4] A: Wh th td tt coditio it th ht ow utio i
31 u u = c c Th oudr coditio r u = u = 5 Aig i, w gt u = c = Sutitutig c = i, w gt u = c Aig i, w gt u = c 5 c 5 c Sutitutig c i, w gt u. Wh th d o rod o gth cm r mitid t th tmrtur o ⁰C d ⁰C rctiv uti td tt coditio rvi. Fid th td tt tmrtur o th rod. [Nov/Dc, Stmr 9] A: Wh th td tt coditio it th ht ow utio i u u = c c Th oudr coditio r u = u = Aig i, w gt u = c = Sutitutig c = i, w gt u = c
32 Aig i, w gt u = c c c Sutitutig c i, w gt u. A iutd rod o gth h it d A d B kt t ⁰C d 8⁰C rctiv. Fid th td tt outio o th rod. [Nov/Dc ] A: Wh th td tt coditio it th ht ow utio i u Th oudr coditio r u = c c u = u = 8 Aig i, w gt u = c = Sutitutig c = i, w gt u = Aig i, w gt c u = c 8 c 8 c 8 Sutitutig c 8 8 i, w gt u 4. Di td tt coditio o ht ow. [Nov/Dc ] A: Th tt i which th tmrtur do ot vr with rct to tim t i cd td tt. Thror wh td tt coditio it, u, t com u.
33 5. Wht i th td tt ht utio i two dimio Crti orm? [M/Ju, M/Ju, Nov/Dc ] A: u u 6. Writ th oudr coditio or -D ht utio i td tt coditio. [Nov/Dc ] A: Th oudr coditio r i u, or ii u, or iii iv u, or d u, or 7. Writ thr oi outio o td tt two dimio ht utio. A: i u, c c c co c i 4 ii u, c co c c c 5 6 i 7 8 iii u, c c c c 9 [Ari/M, Nov/Dc ] 8. Writ dow th thr oi outio o Lc utio i two dimio. A: i u, c c c co c i 4 ii u, c co c c c 5 6 i 7 8 iii u, c c c c 9 [M/Ju, Ari/M ] 9. A t i oudd th i =,=,= d =. It c r iutd. Th dg coicidig with - i i kt t C. Th dg coicidig with - i i kt t 5 C. Th othr two dg r kt t C. Writ th oudr coditio tht r dd or ovig two dimio ht ow utio. [Nov/Dc, Nov/Dc ] A: Th oudr coditio r i u, or
34 ii iii iv u, or u, 5 or d u, or. Writ th oudr coditio or th oowig rom. A rctgur t i oudd th i =, =, = d =. It urc r iutd. Th tmrtur og = d = r kt t C d th othr t C. [Nov/Dc ] A : Th oudr coditio r i u, or ii iii iv u, or u, or d u, or UNIT IV FOURIER TRANSFORMS Sttmt o Fourir itgr thorm Fourir trorm ir Fourir i d coi trorm Prorti Trorm o im uctio Covoutio thorm Prv idtit.. Stt Fourir itgr thorm. M/J,N/D,N/D4,M/J 4,N/D,M/J A: Fourir itgr ormu i, t co t dt d. Stt th coditio or th itc o Fourir trorm o uctio.m/j A: i w did d ig vud ct t iit umr o oit i,. i riodic i,., d r icwi cotiuou i,. d covrg.. Writ th Fourir trorm ir. N/D,N/D. A: Th Fourir trorm o i did
35 F[ ] i d Th ivr Fourir trorm dotd F F F d i Th utio d r cd Fourir trorm ir.. i did 4. Writ th Fourir coi trorm ir. A: Th iiit Fourir coi trorm o i did F c [ ] co d Th ivr Fourir coi trorm dotd F F. i did F c co d Th utio d r cd Fourir coi trorm ir. F c 5. Prov tht F [ co ] c A: Th Fourir coi trorm o i did F c c F c [ ] co d F c [ co ] co co d co co d co d Fc [ co ] Fc Fc 6.Writ th Fourir i trorm ir. A: co d Th iiit Fourir i trorm o i did F [ ] i d Th ivr Fourir i trorm dotd F F. i did F i d Th utio d r cd Fourir i trorm ir.
36 7.Di Covoutio thorm o Fourir Trorm? N/D,MAY,APR9, SEP 9 A: I F d G r th Fourir trorm o d g rctiv th th Fourir trorm o th covoutio o d g i th roduct o thir Fourir trorm. i.., F[ * g ] F. G 8. Di rciroc with rct to Fourir trorm. N/D A: I trorm o uctio i u to th th uctio i cd rciroc. Em: Th Fourir trorm o rciroc with rct to Fourir trorm. i.hc i cd 9. I F i th Fourir trorm o, id th Fourir trorm o F whr. OR Stt d rov Chg o c o rort o Fourir trorm. M/J 4, M/J,A/M A: Th Fourir trorm o i did F[ ] F[ ] Put d i d i d d d d = F F [ ] = F,, i d i d,
37 .Wht i th Fourir trorm o, i th Fourir trorm o i F? orstt hitig thorm o Fourir trorm. N/D,A/M 5, N/D,N/D4, M/J,A/M A: Th Fourir trorm o i did F[ ] i d F[ ] Put d d F[ ] = i d i i,, i F i i d i d d.i F i th Fourir trorm o, how tht th Fourir trorm o i i F. or Stt hitig thorm o Fourir trorm. N/D 4,N/D A: Th Fourir trorm o i did F[ ] i d F[ i ] i F[ i ] = F. i d i i d.i F c i th Fourir coi trorm, rov tht th Fourir Coi Trorm c i F. A/M A:
38 F [ ] co d F c [ ] co d Put d d d d,, F [ ] c d co co d F c Fc [ ] Fc. Prov tht coi trorm i ir i tur. A: To rov : F [ g ] F G C C Th Fourir coi trorm o i did F c [ ] co d C F C [ g ] g co d co d g co d F c G c 4. Stt Prv idtit or Fourir trorm?n/d,m/j 4,N/D,M/J A:,N/D Lt F th Fourir trorm o. Th d F d
39 5. Stt Prv idtit or Fourir i d coi trorm. A: Prv idtit or Fourir i trorm d F d Prv idtit or Fourir coi trorm d F C d 6. Stt th Modutio thorm.orstt d rov modutio thorm or Fourir Trorm.or I F{} =, id F{ co }. N/D,N/D 4,M/J A: Th Fourir trorm o i did F[ ] i d F[ co ] i co d i i i d F[ co ] F F i d i d 7. I F i th Fourir Trorm o, th rov tht F. N/D A: Th Fourir trorm o i did F[ ] i d Tkig com cojuct o oth id w gt
40 i F[ ] d ut : d d : i F[ ] d F[ ] F i d i d 8.Prov tht FS d F C d. A/M A: Th Fourir coi trorm o i did Hc F C d d F S [ ] F C co d d d d d co d co d i d F C co d i d F S Show tht FC d F S d. 9.Fid th i trorm o -, >. N/D 4, M/J,M/J,N/D,M/J A: Th Fourir i trorm o i did
41 i ] [ i i ] [ d F d d F.Fid th i trorm o -. M/J,MAY8 A: 9 i ] [ i ] [ d F d F. Fid th Fourir Si Trorm o = /. N/D,M/J 4,A/M 5 A:. i ] [ i ] [ d F d F. Fid th Fourir coi Trorm o -,. A/M A: co ] [ co ] [ d FC d F C. Fid th Fourir coi Trorm o -. N/D,D9 A: co ] [ co ] [ d FC d F C 4.Fid th Fourir coi Trorm o -. N/D A: 4 co ] [ co ] [ d F C d F C 5.Fid th Fourir coi Trorm o 5 5. N/D. A :
42 co co 5 co co 5 co 5 ] [5 co ] [ d d d d d F d F C C Fid th Fourir trorm o,. N/D A: d F i ] [ ] [ d d F i i Hr = i i d d i i d d i i i i i i i i i, i i ] [ F
43 7. Fid th Fourir trorm o ik &,,. N/D,MAY A: d F i ] [ i ik d d k i k i k i k i k i k i i i k i i k i k i ] [ F k i k i k i 8. I F F, rov tht F d d F. A : d F i ] [ d i d d d d F d d i i i d i d i d i d d d F d d d F d d i i i
44 d d i F F i d d or F F d 9.Fid th Fourir trorm o A :,,. N/D Th Fourir trorm o i did F[ ] i d i i i d d d i d i d i d i i d i i i i i i i i F[ ].Fid th i trorm o -. A: F [ ] F [ ] i d i d 4. Fid th Fourir coi trorm o i co A: F C [ ] co d APRIL APRIL = co d co d
45 = co co d.co d = co co d = i i UNIT V - TRANSFORMS AND DIFFERENCE EQUATIONS - trorm - Emtr rorti Ivr - trorm uig rti rctio d ridu Covoutio thorm - Formtio o dirc utio Soutio o dirc utio uig trorm. Di -Trorm. [ N/D 9, 8] A: Lt uc did or =,,, th -trorm i did =, com umr =F Thi i cd two idd or itr -trorm... Prov -Trorm i ir trorm. [M/J ] A: g = g g g = = g g = g. I id [ ] A: F [A/M 5] 4. Stt th iiti vu thorm o -Trorm. [N/D 4, M/J ]
46 A: I F, th imf 5. Stt th i vu thorm o -Trorm N/D A: I t F th im t = im F t. 6. Stt th covoutio thorm o -trormtio. [M/J,M/J 4,A/M 5] A: I d G i * g F. G ii t * gt F. G F r th -Trorm o d g th 7. Fid A: i or k othrwi k. k 8. Fid [N/D 4, N/D, N/D ] A: = og
47 og og og 9. Fid [ ]. M/J4,A/M,N/D,SEP9 A: =.Fid i [ N/D 8,A/M ] A: co i W Kow tht i ut co i i =.Fid th - trorm o othrwi or! [ M/J,N/D ] A:!!
48 = +!! = +!! =. Fid th - trorm o! [ N/D ] A:!!! = + = +!! =.Fid th - trorm o. N/D,M/J 4 A: d X.. 4 d d d. 4 4.Fid. [N/D 4,M/J ] A:
49 = =. 5.Fid N/D 4 A: 6. Fid co θ N/D A: W. K. T. Put = iθ i i
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