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Exercise 9.2 (Solutions)
Notes of Mathematical Method
By S.M. Yusuf, A. Majeed and M. Amin
Published by Ilmi Kitab Khana, Lahore.
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Chap 09-solutions-ex-9-2-method

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  • 2. _l_ ’_ . __ , , .__...1 “HR -‘ 1--V’ "’-"f"-"-f_",_._fi,,,___;Z1;,.__31iT.:'_'..“ - ;-T'~ », " ‘ ‘ ' ;.'7*_ . ' v ‘z ifi“‘éi 1 / ‘ F, Y , / <2’ 1, - “'“§;_.‘:z’-93,.‘ K 2 ” - ti} *‘?»,'4 ‘ L ( 4 ~ ( S»~Q-J~ _---~- , V. ‘ . ._._._---- ‘1'):__i_ 7_x17 +17 2.x .2. =1 _1)$’__~_! Q 0“ 7C/+>¢3) »_: 31$--1_>¢-~~?-"'7"x_7+ix7. W mu‘ ' L . ‘-: J *1 ‘::j(-_)£l-L ,;____;~:) ' Y ?- =~_ 4: <1+~(‘f‘>5--”’.‘3 ' , T“? ’ V - Y 5 k Iv!/» 3 (in. ) +311 L . /~= ..._»<‘ 6/K 1: .:L<7c>>@~-0~‘/<'-I+*" 4.-Q ?§— 7"’) ‘<1 47> I - .4“;i~ I 1 _,», ,.-..-.-1 r ""‘ 4 gar 3 ‘QM ‘ ')¢7~“‘l ‘ -. 2.. 2" J '1 1'17‘. ‘VET ~— 3.1- A :57‘ = l4**C’*">*‘4‘ ""-“{l'7] 3 P /' ' L . 3 +¢ "-ax -é>“"'é‘¢ad 2 LA H">_> QM P___~[~2 ;;_- Q ac —--*"—""'”'""W"’—" N‘ K wé Q“’~Q_~f{I___.; Z_.;,é __ 3 7:, 5% '*"é~'¢ @ - L - -’ ‘ a,. ...}- 7 X '7: O d-AT (2 - L ~¢ evsll Y; 7': dx >__ 2.73-'3'>’-"é“'*6C'2- - 6 . . $A.->( LV‘[J___~j[ :1 6-1 L _ . >5 2.‘:-L3*‘3'*"6”( 65- ‘ __J§;A,(,,1M ‘;1~1( :1. . <2. f ’ ~ ¢ z.i.$~-w15"’- "éx 7 C Qpi. ‘L .. 7*’ 2 _<-cw "" ' -.1; A ' v—~~'"*"' I ® +_2.K.§.]»+)_;},<3or_ -+tv_L-¥—2_X/§J~*/ :2: 0 ;__ = (;¢»¢>=<' -+-Q cf ,.p- I . _,.._..__-A._._--x__.___..._.._._- :. 4' .- 7': /+qk+_,/"_{.~n-‘_7" {’Q’_{.;,./;<7(_~1~!/’} /K -§-2,>+@-+1->)0‘}'*' + »<L*+1->"*/ ' ' § -5'01 .12, v¢(_><+z)¢LI7 =‘ O +* wt"/~+> C7-+9 -'B - ::- U$X)+7[HX"> I 5 -=0 ‘ . >¢+1_ aw fj= <r+»<W+7/‘ I '>¢(‘/L“) ‘E . l ' " z-‘U1: -::0 4 .<'_<!. ¢,»y + —'-' ._-_/-"" j M‘ 4', L, ' {% A ( , 1 XL/wx/<»."»< jaxwng L I L- L ~3—-'i ~~_1+1~ - ; -2 V ’_ .+'>_§__<+' -X,‘-+2_> %1,,.=.— '.>(. A ' (1 ’ ' Kw » 2’. ' , ‘ ' . L 5»., V V L 6/ //v(7_/gm). ;~. -if @;q(1.»&‘l><)+ 5- " 7 5 >'l,x.1L K .) ___C__'J_‘ ' A >1-"“"_"'_"_'__ X W)" " A /:,Z1',T;_1 . '. Available at MathCity.org
  • 3. -.-...._.. . ....._.-..__...:- .¢.>--..- Q4‘ //t’, "‘~; ’ , » . Q 1 ~ / I‘ ’ 5}’! I7 4 @“?/'»I‘@*»~+*”‘¥T"*/’"*. J4; gléxatypo . V L_ Tb-7 7&7?‘ =jo~J~», ::)jQ’EE’4!,»; 1*-Ii“. 0 >& ~..§<_¢,x_ Y (1,-4.11.7 " pvt :2 V‘-if‘ V 4.1% < - “>7 7" *5»! -1 = I -..,, S 1 xdwtyr V ,4», - ~ W ~ /:11: W =1” i }_yCf’*”* 4* ‘“jsM70L7 Jo =8 ‘*7 "* _ . = 17-4,?- 2,12-¢2*"”’ M/Marw " *4 ~,~ ‘X. 2012. /- V -- ~ .. ---V -‘ . 'n,_,___,_____‘,_ _)‘>¢‘“vH‘,:":x~4;_‘,__‘__‘,,‘/ . V -_ . - 3 . 1 » I 2. Z *1 7_ ,_ (H>f>0|’7 ,,_0 rgj ,7. t I V 7L .. =0};/W A ’ , . 4-7 76 A ".‘ 1 S V >+#*¢“<"“> :70 +J_~3'”‘ *' R‘ ”';1".T9 Z11¢ ‘tL’_{-:> _, .____-7, + J t1__’+])M_+jE'i‘;_’;i_“/ d{_,_.= Q n-,~-F: ti." ’ — I Y %”4" . 9;?=-"JO :7 ' 9- ' fpf-I ‘ % L mqwl "L‘ rr 1 [‘)+ ‘ F‘ '/{,7 ""<’ ’/77*‘ *~ "’*‘*="* $1‘/l»<: * .’ .. .-> 4- .- -Q ‘A _(’ -1 C,-7.»_~.;L, ‘= _.>C=.»*V_.$~-»._v-1-c » :9 *]_ [(~;+1}{@_ , 7 ’ -» -C-=».»l<(c-<.‘i¢>L, »<./*1); _-_;>- --@_..i....v._._.i__-_._-_......_ --~ .. . H,“ y_- @ (J+,I)v"‘*/ = (*I+93 __lJ--‘IL _ V 4_-L7 (_(:"+|) (‘1-H) .SJ“-;L + C ~ . . I (C ._S~1;">c)éLat?-5”‘. *7 J6?‘ an '{§§€’v =‘>f<~Y<%"” ~ @ Q] <;;)»/.¢]_____.;/_»< ,_ _.-_-,_.- ("F 6”‘-a-I Z; "x+lC"2 M J ; JM - » ' ’ :2) ;'..@»_({+:) =1 F/n(¢Z‘+1>+@M¢ ;_-__-_ ' ‘A0 /' am‘: 5 in I /7? ~12 7 —-*-Y*-11+ w»+‘- * "W C. . » <' 1 . 1 i ~ ‘ =~c, “/ _1% if*'l<@~ *9 Q, . ~ '. i i I I 1 . 1 .‘~ - a ~‘ : . ; . 5 s ‘ i1 » -M,-<..4. , .» _ I ' y ,. ~ __ '- —~—--__ >...._--..'-'7"’-___A-........---_..:.;...~:... " . . » _ ~ . __ .il'_ ‘~ . I If ,_ I, » ¢;~;.~_~...._ ____='v.r.l'-mwv.;Qg£%5~ §a~.y»-r..;w.. _-<=:-_~.:..—:;.’_-:¢=;_.,_ _—;.__~__ 77 _-,~;.<.. ..,'_ .. ,_- - ~ ‘ ‘ -V ~< ~, - '- ~ _vF~:~‘*=-.~—73'4-.--vme<=‘T.. V ._-4 1 _@,-,.,._,_._.._ 7.... __. . - ~ ------~--.-., __.-.,...._.__~.. _.¢._.,_A_...-».. . . - ~ _._»_~_m ___ ~ ~ » — --_. M. , -QiIi__+ "ant: __. 7 =»-.. _,_,_____ V_,__:,, “I-:_________;____:_ . . , if
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  • 5. “ 1 O .1 ,_ s.aM%u~;m;m~@m,@~1@Z'-~ . ;@;§;5;1< % 6 2'1. _ -»a=!.14.'s _,..:.»"”‘"l _ _ ...¢.u -.-. ‘¢- .1;YIJX .“u-.;4¢m.J¢-;—l0l§vQI'An d....-1.?“__...._.-. ._. ..._-=»~.4>. .—~ V‘ ‘J 17 _X’1_+yd/VL - 7H-Y go Pu_fz_=;Hy Q . =;fxc/M - :___-7_,,*—‘!_ Q2/Q) ‘ @® 'Q7((vl+£%f> _=-‘oceay » @ anal .;_ 71‘! 7x xa.¢0"’= :7|+J»1 ‘=__xe.¢ F < > Y ,14_§_L}V ='X_€ N,t;,f,,_,L,,l>L, 1°‘? #47 Z7017 iLJe(1><.>J- ==Lf‘1e-"l7Ll‘i 5%: ‘Q /+33 at; 4-; ¢”‘.=xJ»<:<ll- ..z ‘ Je,%1xJ°'jw' =1-_= 3: I I é-'._ 4- 2- 1_ ‘I0, I237’,1. 3514-(L 1. - .. '2 "7L7'f- ,+C .:= ’2_¢___"+C )- 'X?#J‘/ L L “I4- .- = ~‘7e7 ~e.7 +¢'7(.+ ' an Q. 7 r:-, F-(7.‘:-z:+C) V 1. . V 1- * - -7 . 6 =. -11¢ (*/+!)+ -5-IL (71,-+j)_ i =-_- @v<,'7,;E:+C) , L i 7 =. Z»/LZg+C>-7C q'S"'~ K V K): ’:- kc»: XL __~! . /- " *— 2. = »-14. (‘11'.l)+c. *’~-$4 x~———-—"'——'—"—""""“"' ML-‘>4’? =- <a¢»~~»~+1>¢»~ Y<°>=~' - - , ' “ £~%'="’ % *J<1-‘W = f@>~"+‘<>+’->°”* % ~» *7 -JV-‘f Y I ‘ 3 *’ 3 . .- I ~. » Irv . I. ‘ , _ .. - - »,, =, .,._~_.v,-. M4 ,- ».,_,.,»,:,q ‘,4 H - ., ~ -- »- >-_-_ » i--,- ;_. ~»-- _—».i; ;; , __ If ~ .,. ~ .1(lI:-‘-7) = 3'>¢3i+"1:'*1°"*¢- ' f?‘.5I§_'§T,,_ -5* ,_ L __%‘j::-It xv-;—L1.~" I I L-'-:*Y> X3 -+:l'>.L-&:l')‘§YC. | ..<-1) -1(4) == *>"‘°+°*‘l “ / ; ._. ~;(¢,_> __= .. » ‘]’(o)=_--I (£@umA:A»Lb~ 7 .—..l+/’1:,’_i;-1{+z1..', ' 00' ~ -I ==l+ln1"*‘‘I-‘>1’ —' 1:: ‘*1. f)(§AL' =4 "—"-7"" ’ llml ,1.» 3 1. . (O) 7="‘4.1"+u+11*+‘r ‘L }z7_--7~7é= '>£ ~f~I1.'>+l?+'35. 1. _-)=]_;-*--———D+arCM’s _~ ‘A V‘/z;-9-‘(+11 == >¢3?+ AL*L+1-“*'°""' : "' = “P v"'t“~“ , . , ,| _ S‘-D ‘ ’ é: ‘J-'>'.t+L'[>'f~(1 é’ ,/' 7,’; I“/x3+2.xL'*l74Al(_ D . _ 7J' '- =-.t/7~1.+2.-»+?JL*'Lf " ( ’ ~'~ "“"' "J my ¢,¢b.,~,;;;.,q-g»u.,--‘¢._~¢ liw aw-r 3 I i € 1 2 3 Available at MathCity.org
  • 6. ‘,_’; . :,~ ~ ; ?—3-L7 ()/94)(x”+5‘x4-4) .2; v ',K— ‘.1; H .5-w ‘-- ' ~» ’ ~ - ,2 ‘ V ’ ‘ V‘ y . 1; ‘.,;,Qi__:_ »6(axi+8)O1_fq),1M_z,7(,,+5Ma)»l7 , Y£l_):=_2_ ._~» ..._ - -< -_~ . ’ - “”%,,»;§: - -_ ;» e=~_*!}. , . - wt. ‘ .-' >0 ¢ ‘ ; . 1 *1-.,"-'* 1 8,7 PM Pk“/W9 l 444--¢u&/*Y=@ W_—=9 3'x+$ L 1.- 1+4 -3x+x - ¢ 'X.+_3-><+l ___/-- ~ ~ I - ‘‘ - 1_,+S' 2wit <; *4 * ,.. F‘! ,= I». - ; ': (J; Li} er’, * omen k -v I L . j <:V 1,. ~ PJ?¢*‘ ’° "“”*’ lP<=’-K "<:.> * % '—’? E ‘W .<-1- v:< ."v ,3 -. Q3‘ {:3 I 7‘ 144 " an =-. Ji,‘+-§- "Q3 '% '<s¢»~>> W M»ow . Z. _) /(DA '"&.Lx+ - ’ 3 x + 8 = . ;_3 , P,_1">~¥3"Q ‘:91 Q ‘.[§::lAI ii .2 z. 1- ' - ‘- ‘ Q-,(.¢2)(_a(+3>) 1 I I =7 “J11 v‘~*4 .jg_aLv~+JJ*K ._9.J§1.»';_i'q”L7 "’_g°°l“ -_____ ~"-_ . I , -x_.’.;_, $45 "F . ‘ --> 9~¢-L*+*> +% "~<““> '1“ * LK3c_~§f5_(><~+-3)] 3&1 C‘/5+4)‘ " - v QML , "5‘u”8 f @¢+z)°*(><+s~) 2 c__________,__,..- Y, (x/Lv+A‘*)L ' 7- A Q.).Z_5_<H'.3) .. -,4 , "< %¢<= . 4*“ '>>@=»/ { ~ ¢ M /4 1 .2: ’ == 1 ' » ' . _ 14 - 1é<v<4-9LV<“""“-3) =‘ (7L‘W> gv“ - A . I ~.. . _ ,< f - _ . »'u _", ,,, ' Availabée at - — 1' __f/ V www“mathcity.orgL » I. K . I
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