Engineering materials and hand notes, Summaries of Engineering Mathematics

Download Engineering materials and hand notes and more Summaries Engineering Mathematics in PDF only on Docsity! VVHOUHOVELYUVYUYUUYVUUUUUUUUUUUYUYEYEYUYUYYEEeyY penn Ond- \- Matrices wes IN le aT | * ° ‘ Marty, A Matix & arn array m,n clomente “Tranged th o™ horizontal us And wn Werthical | Colocci. Such a Moatst. will be denoted as A and tH ob dina ads mn AW Ma Ay He hy p. [Meese 4m Aro Qin, a —- » ame Who Aur a ao we the | Stmenle ath “Such a Motte wor<ll have Sine mxn Sead ele: Stenexally Matic anu donolect Zé Gpital | 2 Theeg Sioart System J inclusive ~the nein bere | Const tating 2 -maline whch & Common th we CjyeC jel \- Viens wsiloes te wl reteufast' | I A Mottrix Connet be gttduceol t 4 Sire Myr hos Whoucasr ~the da teouminant an he | aduucact a Single <apapben | Scanned by CamScanner Scanned with CamScanner : | ugith non — vange ting Te ame “{ Colonie th ediices =o hut = thy “Hips cy Ta 2 | MNugt he equa to ‘ho. Colounas a | Jofominant { ° An interchanging { Rows 8% — Glownne a qjivuen wi I Fmdanl will be | “Kank 4 a Matity t— The Kank x! 4 a Mix BP w& a east +e angen Sy" > 4 ctkast One Y- Yourd Square matt | determinan - Whee Ge \). and rnalot have seul dondid a Move youd Square wll dilemfrant - Jt & * Renenlary Gang omndign:- | The Elanuntany Trang owns n which & alto Knovon ae | An tale, ierchonding “the given 2 R. Hull ying with a non yes “pubes Tbe ny YOu oy ; | Colpuimns 7 “the guicen rely 3: Multiply wilh @ Non - Ze ‘umber fe “he q es Ma rot be ental te an |e Trangormalion 4 tu belongs to =e 7 = 0 (1) t S999) Le 4s ig ? as. seks 2 fh hf leaf Scanned by CamScanner Scanned with CamScanner yr Kk, Ay [s 6 + 8 Et g 4 (Je ty ty fc 4 ow (4 g 4 Voom pw 6 UF te 49 er WY -K-K, Ss + Fw Ss € Dw 5S 6 AB A \ | Ty (WY WAY Ge Ld WW Is ho, | Nw NA ‘\S GG 38 LAL va, 1% fe tA WAN Ko- QK-SK, j Xy-7 Kau Ky But Ryo Av (oy \ ‘ © 4 2 3 6 «a ® w e 0 0 Which & ch Echelon fe | esr. | Scanned by CamScanner Scanned with CamScanner Scanned by CamScanner D i rata | Yy- vu ¥ ae E xvas ¥° eo Vel yp 2 ——e, 0 © ow 5 wo o ~ F, Whith w& Pr Echelon en" y= Be BS Y -¢ | }Sdke- tt Q- \' 2 -5 | ay Lam & > SN HS Jax Ky -> Ex - 2K) Ra Ky- Ry) hy? Ry SK i Scanned with CamScanner 9 OA es PORREELER OREO ROO IEE DOLE ee coe ‘De ee et rr ie — ee SA A Ti EE — } ea { . Nd C ., ny e000 ; ——> of L 7— TN nm es 0 | — oy Bt pe 26 NG al at | [An ai? -oN + BL Tas T ° ay 4 _ ~~ i ‘ i Zz _— 9 =e, | ~_ aq Ne a“ ht — 4 | J _ Oo ! l —~- oN = Ww a SE 4 oT! an a = a0 _ Le? - dm TT | w ar aT d . i a7 T par me Ly © oO ! = - Oo? lg ei O = 3 J gi yf og Tow peog Fe n “oe a Neg | , <4 | Am ! = s a a amo i — ts SF B Scanned by CamScanner Mich & mH Echelon “om Scanned by CamScanner Scanned with CamScanner (ay) | ay Pye uasdoe lace - (ey a . x te -20K+64u =0 fo PIBA-UA 416 =O ACd- te) 2% (\—(c) = 0 —— lho 2 ‘. » | Ste bt ype Jtr 3 ! tue I 4. & A Ce y lal= |. u »\ = U4A-(2) —2(A-4) +3 (68), 26 »X =MNA-|2 -2A484+ 18-247 0 | | = LA= NO ET i Wy) Fox what \ahe “4 sa te allt | | en -| S, o eye: © = ~G 2 . “4 2°) Sol be Ae Scanned by CamScanner Scanner d with CamScanner = MU KH -—\ 0 5 =| 0 6 AA | + \ —| ’ 0 JA t+ O+0ra \ ! O ic L o a oe a = ee - 1\ ~& “RK <-G —; = | —6) + howu)ro Jan (o +1 (0-6) +0) = © (ae 4) + \(~¢) = 0 > Me 6 ett & = 0» ve St wut Du 3.46 = A Cot 2) —3 (s4—2) ZO Wu-3)Cu-2)20 Mo21 ust ~ Aha | > Scanned by CamScanner Scanned with CamScanner 4[On | Ay Ais “ef oe m Q inte Ry by A, Bathe Yo | | be -- — — —_—_=—|- \ = \ So Ss a oe \ 4 Sy! . b ny ny = ON Lin * “apse 9 Nes Sdiee ng, dey a sae 7 MN (reax Gn's Aim +45 | +Q Wag Te ie = oe Qe oe ley a WX +r aq a ia: Oe i Santy=Pal om or Ul a | mn Xx i (Fa © Qi Ki ce we em + ia n= by] t T ® | thig te Gy be Srrangan th “he = maltit Ti a AY=B bey Whee Ak Known at Cou tele mabix % x aug the OnKowng. ih BB w& equal % a0 than ~the Syst eq uakcon ba fol bo % HHomagencos , 5A hon — Homog enou® It ed motrin & gun b “pe fen | By ay... _. Aint by ! (oe) = A, Asn. -- - - -- Gos br ; Scanned by CamScanner Scanned with CamScanner ee a oe deddddds: CCRRRRE ER Edd dE C . Oo ACL Scanned with CamScanner @ 0° po, “. Ca) = Clapjer< nore Unknowns . eee ma we khs. yee =? Waka | ! | tt) Detesmives Y *—-Ky 427-0’ wernyrteo| | & Bxtqy-2 20 ves <) Zero Scion | |b) Non— aero Seluhion ° | | Hatt Ba Kans 6 —_f) 3x roe =O Jt malic pene w% AX= B ances eX 34k) +e CR aK) FICKKa) = 0 —B+K+2U 4 K-A=z 0 MW LAK-AWL = L+K-6 = | K-2HK+s)= 0 | = Ke 2-3 — S 2 2 : —f- r i 4 © f dddddddddededs edd COEEEL HE t Scanned with CamScanner . Ge ! it) rans Solutions 4 d= B and malo. 7 pe mae : (a) = elan)= 2< No: “| Unknawns - p rncl © For What volves “| Whe “Syslem ey) Sq ose b RSE = I \; dae YA = =X & ax + 4x02 = > carl ' | have Schuhion 2 | | Sh. ing x ytee | | ae yee IC ay re “ity tloz= y= | yy) We e" Con he a AX-6 | , |= x \o ; ! oe: Teak 6 1% Reggae’ wala ex “ \ » A [oe) = var ,ut , 4 102K Scanned by CamScanner Scanner d with CamScanner Ae == fomnn | & Fi if Go I~ | OG = ON 7” «< we ™ es A tt = A =y Kez 3k, TYt tt eoI =? Yale ok c=K, fc Ly eC) = aE =2< ho o WnKnowrd Suste Wence Wer edie will have wane - Solutions . a le Se SS es oe Geet lathen K22 th oF Loe gk (Ag) = { i ri \ O-l 2@' O 00 0% @4 eegtz = | | =) X¥2 \-akr a ~ Att = Oo 2 sho 2MKy “1a Rihtetaaaceaca A é eh 0 ‘ | Scanned by CamScanner Scanned with CamScanner fp pp » 45 ———} —— popes: = a | 5 Soluc xe yet 3 x- Kaye = 225 7 Ke oer Gy te slo. Sale Givens ne ES hese 3S — () PEt EG eh2=(6 | The v Gan be woritien a AKER | t= Kg = 4 “i te er marx ¥ 144 6) le) = ‘ 4 4 say us po > Y So ™ | G2 Bh 14 4 Y, 3 G-2¥ (ns)> 6-4 34 \6 | — rt © Uar-s Xv XK Vat & 2 a? Qty (ne) = ee ee Ss 65:8 Skee 4 SC “Us f youns | =(€ = Ye-lY “a | a ae q c | =i z2 8 CA)= €(ae)e 32 -r0 tenco ~the System will Scanned by CamScanner Scannet d with CamScanner z E i Selue tte 6! x— Yt 2t =. peed i Anol Ak dy thas F. vuees | | ! " Gtuns- ery ee. & hogy ne S | —© y at + a \l wd ae a teed —— z, oo s CORE Kx GK, Roeser, ladle a eq Rg 2k, \ Pls { ‘ rz Scanned with CamScanner (rg ted | | ‘ — | «Wimax dependine % lng indeportens. qs eter ' Ungey deperdianee s— Vectors X,, Xs, Xa- 77 be. Lenaaiy hy dapedient- la) 4 Ip\= 0+ free, the ae Walt ib Singular Asd 4 b) The Yank os K- = 4 ft et 2 me AB) = \ 4 2 | Oo 1+ y\y« 8 © -2 eo | Oo -U Vers Kak, 424 7 2, L (oxi) - V4 44 6 mI Vg ,o4y4 Ss” Ky Ry +URy °o Oo a 4 q Po ss 8. { “Ey oe es ay Se Qtze Ss =2 eo OBE A us Z=o2 “ &G) = Glas) = 3 = “no ay unos mente, ne Syelemn wo have Unique Sdutioy, °_— a Ny & sad tb OY) othe Echelon jom the matrix . \ucfoys —-GlUen, the 204 Scanned by CamScanner Scanner d with CamScanner oD) thee nt a Fw @ Salary Not alt ffv% > KX, + K2%, + Ka Xr SB Fu + KyXy = 0. ‘Nnex Indepedenses— AY Set ci \lectorvs Xi Xa Xo- --- hy wy Said b be diame Inclepedeneg a) + \al+ O bey he {WON maby & Rony eteigali. | | Cor) | Dy the Yank 4 “He wmdix tn Kelaalon 4orm 6 | equal tb ~the meg lector gitsen (ey) <4 Thee Exise = The ~punnkewus Wy Kr Ka---- Ry © 2 KIX (+ Karta t Kary t--- + Ke %y =O then, K\= Kae Kate--- =Ky= 0 Ucctorr BVHee \. State Whafliex ~The jottnoins 2 Inclepestent ! nea ly dependant” OY i) (12,3); G, 0,0) and (G0,1). | Okie let Keo Mp2 4 X52 00,0) ‘ ka = (9,0,\) : ~_ \i a2 3 - Ae 6 = #~O Oo \ i i | i | ° +- 4 i & % \lAl= ,eo ole ([o-0) a (1-6) +3(@-0) oF = 34 0. : ee eee EL Eee g — eaic we eee z, Scanned by CamScanner Scanned with CamScanner eee ws . ¥ | OE NAP APP NORD 7, ITH *) ' Sekt. z 5 Given fe \i’s > 2-2) \.. In\= A(Lo~r4) -2 (G-u)< 0 = §=4 Inj}=u FO ~ Renew, the ee sk oy que Le 4 i i | i | | Vi) (241,60) (yas) ts\4,5) Weetors ! 3 Yn : = 2+ 0 . Memee, the qiven Sel “\ ecto ae Le mM) (ya) » Cas)? G3) Sear de let Raye) Xo = 2,3) ! Xa,= (4,4) a) | 2 > re % Yi t= 2M, | Kas R-U& ‘ “(a2 i Ra &,- -58,. *( > SOR) saz Ne: 9 vectors qr ene, he er od “| que =LeD ¥) yaya) Grids rons (46,0) | ) Shi lot X= Qy2,3/) >= Cyrwn) \ Xy = (yet) Megat Aaa aaanaa a “_ ie ¢ CAKE a oO -2 -2 | Ky BRy- 2h 0 6 -2 an Kay ist { 2B —hL Ar Bo 7r34 - o ao -t a 9 -o Ky Fut, aw > Oh o 68 \ o. Ry) ah x nas Macho 9S given Menu, the atu Set gq sects Bu LD xi) 1,6) , (\, Vly v3 LY 4 \/\) & (1,0,0,\) és. Given: Wb X= Qhory: = Git) 5 Xa=Qanay Qnok Xy —7 (1,0,0,1) f- \' § © | Lt | Lit HH [I L Sq | | RK. wha- eh Kyo Bu-K, & Ky > Fr -—4t, Scanned by CamScanner Scannet d with CamScanner | IN—>x] Paynotncal Vatinowifal in A (| dagoee 1 Whe @ % Ghd the e( ayn OY dinosuy { A which & IW-xtl=0 Ghat te chamacTourlis & ) i A and the vals =the qh axe Gly Cain Values ox Characteyys cs Kalu oy Chayachit, Feats & prop Value et efent vost Te SF yl Gn \Caluus a A & Clbl the epee 4 p & KE — Thecdulur % Glleo Speitral (inex Dy & value 34 A & tn G » taacdx= 0 | | | | A a jpn ze Veclor a ~The fqn \talux *y Covreponding ‘ O- 4a the Regen Vo. luos the matnx 3d, Ayecse My OH dtinel, he, “the Corrs poncliog fe gen Vectors GU dof 2 Nol - i J) two of -more eigen volun ¥ “the nan ase ) i wl fron iz CGneponding Ecgch RC, " Oy ae tot LP Scanned I ry 2. sescssss4. ae * = we = id led, Ue LI ade a , ” ah CARR > ‘vy eo a x > » with CamScanner > Wtacleriti,, Me | ° a , sy | » i ny | soe Be ley Pr charac p lhe of 5 Speci 3 | ‘ NM \) K+ At will have Vi) Adj th kone ¢ = Kho Know habe Sem. Eiger \doliw 7 Ry ) = Trace S| A rt BA + V\ + Wy i oO LRA AAR AAA AAA AAA AAA A = : ES Fe + —2 es 8 - Ce = $e 44. 5° ~ 3 i <2 Scanned by CamScanner Scanned with CamScanner 9 & Veil That the Sum tis Vealiua qk : Mactaty A= 2 bs : e gual bb fae q A. y ot pene » § Sdh- We Know that — Sum Lge Nehurs ey 8 has a i > 1 = Tau gy & D =\4>73 > = 4, ) , The Chandaus fic ef gy # # \R-AtI=2 9 , : _o® ea | o 3 -\(\ —- \: »o0\—= 8 ad io So A | ' | | -, \|l-A o 0 | ? ls a-A e-\ © 4 >| ) o =i Ben a ae edn =y KAG-ay=18) © Fee | | | Q-N(ate NEGA) = 0 94 N2 6 r-1-4n- eG =O ) rp ) A - +hr*+ = 6 . Ne aye : Ax +S =O | »O4-2)SaA= ) ) ) Now Az \. 1 oy -\y egeaenies 3 Oo ' €-g Scanned by CamScanner Scanned with CamScanner gad Be | MOr- g)~ 2. eG) =2 Q-2) Ora) = 6 =) be oF “s we \) 27) | “ N= ede y | 84. 8 6o.-S And Spechl oad tus et Guns a(S, 4) ; Oo oo -§6 Th q QA) 2+a)sea) = 0 =Y_AZNV-245-5 - e \ 3 Arun od TS yaw : f S _B =o iD * > ” A | (\- ») (( +2) (S+A)- 0) Fel (o -0) =) e- | AAAS ASA SS - CAAAAMA GATS. Scanned by CamScanner Scanned with CamScanner = &a (omy ery =r) 4\Cgaay-0) ~ & 2 (1 -6) =. © | Q-)(oy AATALN)) +14 S2=O | Ua) ry BA+\) + AeQrO Sotelo feb Se —-\- 243 dQ= 2 NEON) cee ete meer NR harris Goshen ue | | | | Piddesdeddseasdaed CAG ig “ on - 4 Bi { a v wee uv 4 LE Scanned with CamScanner | Q-)XY+aV= © (A-1) MAI) = © ASV KeOS KES A= O1-% au the igen \Whenes f. : 2. SP Ay Ge wy. an volun OOS = 3lo,1,-24%) - (0 )\,81)+ 4O/) + BUN) - ~ (0,3, 428) ~ (ot,¢s) + (0,5) +(3,) a MOM Eze ars od MOr-6) -1(n-6) <0 | ( O-) -6) = 6 Sef ; | § | N=NSS asec the €esen aes a v | Cae (i) 3- - A=, the Corres perdig Eien = dee touts 6 fon rs (ee | | vk [ ] 5) & | ’ 2 | Syne V4 ye aah ee : 3 : Tot eeay ee \e x= K ; . = Ky < ™ | ¢ yo \!) Whhon K y= | | . Conti thon = = 6, aan the oriperding pe _ Ueto i< for by oat | | ‘§ f | G Vib ti S sy el : - * { | wl k —-X%+Qy~z~d ~ 2= \ 1 \. “ ms Sa vedlers Sa | Ss | rripen tha een eles WE Y= i \ Né repeclivey- Phy = 4 r= “4K, 2 Scanned by CamScanner Scanned with CamScanner Yo nx | § y SOY SGFCUOUVSVVYWVVVCSCCUeEvvuVvensaVuvuevysevvvY = P= Festa) +A (Stree) s GH) Vie Shae Ske O Scanned with CamScanner hy ANNE e5 ee Whar A= 2 ) the own powding ef +" : | Necloys any jer by (A- AL) R= O° Ky aks Ky re s\ Yi cet (3 _ ey ee 2 Uy rye re =O zy Bee? o Wek x Kya Z, mE 4 os X\= Ky Vin ae \"\ Whur K= J \ 6 0 ; _ ih YW = ms ss \s) When K = 1, \ \ | | Gu 8): Wha, ALB, ~The Corres ponding Elon a | ects 4 ah by (A-AT)¥,>¢ . ~2 -2) 2. ™ _ S -2 ~-g “3 l'1\: | 2-1 -§ 2 edt deeetdaaaaaaaas sansa anna ans >< (~ | ~K uw hy B, epestl , ’ ~ iy be Ss ~ 4g ae MSLTE MRE 3 Tih Scanned by CamScanner Scanned with CamScanner a st | | oe . cocina Oy Yr) X2 ape “the Eugen Ucctoxs Gs nding ve “oe \laluey 2) 2,3 respentinla Po - be Wane, XP Tn A) A trinecay A) —\A\= o C8869 wT ti NG * Pe “i > aS + bdiled (CEREETEFTEIEEEEEIIISS Scanned with CamScanner 8 Gali} a, heyy tre Comapordlig 7 Cae Gi) Wy, =1, the Covrspording Eden \ortors ote giver 7 (A-AT) X,>0 Ba) - — wes 5 Dx “4 %o = Ky (| = \e \Alu, Ky | \ \ ye e 2 (A-dT)xX2= 6 | F | A SeCsbEe AgAgA PC CCAEEEEEEECE fy x, } ¥,. ; Xy ates he eigen | jev Wu Corey po vd i, E foo, ~9 ay Pr o”lhP — Ow es z i v o \ Tdi Givens 4 — \ \ 0 eur) “AC3) 4304) [ —Gsa -—r oO \ The Chanactewitic wa B® |e atize \i-~ 2 3 3 ao \=d -r° 0 \-A| wx hau, > Time en) + AC Pormbor a8) - lal =o ( ~~ D> N— MOF) tat ae-S)— 120 | @ ——— » e = Bah -~\z 0 mg peace | _ _ —-~-= YY . 2 —q —(o) oO a oka ep A * K ‘| \) N Y B at > Ov N x Ve eo xn oO ® 2 werd \] nee a oO O > os N ha é 3S - | Sat Extivane- =. 6 - { ao © 1 ket Tk. chonactositi. 4 9 #8 loa [-A 2 3 | _ 6 . ge dis S 4dr | “ “d d ® GQ £-A 27 = We howe | N?— 2 Traw q?) +d (Sabor 4) - \wl= o| Mx (ueurcd +d (24467 4) 2420 Scanned by CamScanner Scanned with CamScanner hz) oo hed ae Sey o messes Ma ~-A Q > U \ “V6 wy Lo CA-1) 10d - Us © G-)@~¢) Ca-u))=« | A=1,4,6 aue nthe Eger, Cant ey | Ae) g— tlh, N=(, the Swap Cen Medlors cae afer be * -»TYy\=° | © a "| a | © 2 \ = a p oe s}la) le) | we sg Al lalhen k= A, Geeky lien A=Y, “te Corespers) aa Vectors Ake quer by (A- AL) ¥2> Gaye | | SEES. SIT Teese eh We reir temch ieee cmt Scanned with CamScanner oo oe cS ~ Doom (s| \nthon Ky= A oa Can Ait); i tala WZ Sy Whe ann sere \loctoms Que aioe a (Q-AT) X27 * eA Ube ~* = 4 Z = — ates ~ Mas se lator R= | | lak hue, X- NY Trace q WCAC P reanwrg 4) -\i\= YC 3ti42) + AL07 6+5)— 6 =O Scanned by CamScanner Scanned with CamScanner SE ea Se age ene * Galeg— Harton “{hagrem s— Ean Sq ease mat Saliyyies tts Frcion our’ cee tic 9” , | © 4444554 Sa 3 g £ ant _ ae \ = a 7 z a 3 gy € b. =5 ols 4 — | S sa : | Seft- Ging f= is | ‘ bi Gs = ecsiNes zo e | “Whe choyecfesuutic &) : fh & \A-ATL™ "a uy \ = ° ) 2d] =? G-HE-»)-4= © ve i > + O» \j O TSE TS ATP SS Te > See Dear POE MO EYP i i axe ET Nen ipete mMG ORIEN TRS TE e STE apa Os 1} ry Eh fp p & yw fC) Ua LCCCEEEEEEEEELE led a Scanned by CamScanner Scanned with CamScanner af PESO RI ORO BAY ¢ oer Se. Gfyonsh = B S| “Ke chataitoditin EP gq) 0% \R-ATIFO! 5-y» \~ 0 > 2dr (S-rya-a)- a= © = oS NEE RMS TIS Se a SE 6 Se Pa a | : e pp: Ba ‘). (2 * AAA YT = \"" | Vi: Re - 3 8 4 Scanned by CamScanner Scannet d with CamScanner Srey aaa a a canaries ta ~ wy, ls. ‘y Sel. Gtuens— f= \s a Tre Chasacteniintic — Se 1 A \p-xtl \" M | = 6 3% 2A Q-r) @-A) = 12 = © SB, lo=s Te veuhy CMT ut Teper Awdh R- | fe Y_>A-loy = 0 See dessins raat . : pr 6+ (AH mee O oO 2 A> “t eis | xc \irer2 atr2-( ~I1te-# 34340 b+ +o “3 to¢0 “2t3°B YH ey 2 ta tle 2 2 3 -S | 3 eS \ \S a ( oO 5 \ 29 ae Ve -18 HO FAS 3% \G =1% ne Ar GR iA- 39.5 = 23 \G -28 ¢ “05 a Scanned by CamScanner Scanned with CamScanner ee J i “the jee wat icos by using @ Fina the eutuas. GUT * ( 12 \ 2=-\y ~ty Sol! at re \i a? 2-4 “The charactentint c inv A \> 6 Y Bek ~) x = ~ GPA Gt 230 eq o A & \R-At Tes We han PRR Tree g n) +A Prune 8) To hr =* (\+ 3-44) +A(- a %-lot2) 444 zo 5 24 MH RAWY TO” oS Boe) Now replace » wilh B ee, fr- BA tN = © « oe \ % \ \ 3 ‘AERA = I! 5 ah \' oy 2-4 -Y 2 aie a. a -“ —_ —« GEIGIL AISA S SS a4 -_ ee aii = Ne ddd dbCecge < F446 \43-)d el de < Vege | eg ee BRE nawee relate 6 t4F/6 Scanned by CamScanner Scanned with CamScanner | F ao ~ a b 8) -Wal=< $44 z0 some wolbad ie clue = Lee “10 & ac fis 1 24 6 O % -% ~\@ wy { |O9 24 O\- J> la Y oO 2y rh dca Ve howe, \*% AC heg 8)+ NPralnor a ~\\\zo »- +242) +AGHHB)- 250° W-SM4FIA-%zO Nlous replace woith ‘A’ ASP FA-B= © ‘To ifr Fruesure coy Tt we Mubtuphy the above &" woth A we et, ® Scanned by CamScanner Scanned with CamScanner es A eae ae Ny) Sune h- |1 © = me 4 2k a he Chayactesuittic qf" “ § &\a-atl-s | A Oo Lb yy =| =- O } —\ WA We hae, Ctra | N) 4 ACRerninor 4%) -\qlzo \ ——_——|__ Lededdeded lS GEEEsEEgagEgEgesEgeEed PK tit) 4A eera o- 24) t w= © | ye 3r*= \EQzO be Yeplace » Lust Y ee RE fap ALB=o “Ye act Freee by GUT we taultoply i lhe above &> uth =F wees i N-an -Lagai= © | a3e'z - PgR L | a On = W-GA-L 1) (pd apn-d aL a ) ! Scanned by CamScanner Scanned with CamScanner N-N (r 4as) 4A (244820 )— WA= © VM AASA IY EO le ~seplace ) dith"B) : RP SA -lUtI= © To a Thode by ColpT wl raaaply “The abous & with A we ae Slap 4 IST - Wah= o ~\uap'= Wy BR-1SL apie pe pat lol Scanned by CamScanner Scannet d with CamScanner rea sepa baits ’ SI eres rcaramssnesaconlls , | fs BA sh AR, PLAPLA PARA PPPDADSIRADA, \\ 0 wee —nw da Chadd Scanned by CamScanner Scanned with CamScanner Q- f= A-L Mukti {he Aho r at Pr» Ve G9 Succompullly Ww AP Ae av! 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