# UNIT # 01 (PART II) JEE-Physics KINEMATICS EXERCISE I. 2h g. 8. t 1 = (4 1)i ˆ (2 2) ˆj (3 3)kˆ 1. ˆv = 2 2h g. t 2 = 2 3h g

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1 J-Physics UNI # (PR II) KINMICS XRCIS I ( )i ˆ ( ) ˆj ( )kˆ i. ˆ ˆ ˆ j i ˆ ˆ j ˆ 6i ˆ 8ˆj 8. h h h C h h.. elociy m/s h D. ˆ i i cos6ˆi sin 6ˆj f ˆ i ˆj i ˆ ˆj i ˆ ˆj i ˆ ˆj f i ˆj ˆj m/s. For,, nd d d so. y d d ; d d here d d c ; y dy d b herefore (c b ) (c b ) 6. () ( + ) î +( + ( ) ) ĵ 7 î + ĵ () 9 m/s 9. Reqired rio : ( ) : ( ) h : : h ( ). Velociy fer sec is eql o + () () m/s Disnce coered in sec is eql o ()() m Now from + s. () (.)(9 ) ms. I hppens when in his ime inerl elociy becomes zero in ericl moion m/s 7. s s s () + () + + () : : : :. C ; C C C C node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6

2 J-Physics. Displcemen [+] 7 m Disnce m. S S +. [ + ]. R / M j n R R / M M R i j n R/M R i M. + sec. When he secn from P o h poin becomes he nen h poin 6. wo les of elociy ( he sme insn) is no possible. 7. d d chne in elociy w.r.. he ime For O elociy decreses so is neie For elociy consn so is zero. For C elociy consn so is zero. For CD elociy increses so is posiie. 8. Iniilly elociy increses downwrds (neie) nd fer rebond i becomes posiie nd hen speed is decresin de o ccelerion of riy (). For shores ime o cross, elociy shold be mimm owrds norh s rier elociy does no ke ny pr o cross.. Fl blows in he direcion of resln of V & W V V m/s R Vw 6m/s V W V V m/s /R V V 6j (i j) ( i j)nw W N-W direcion.. mg ( rm ) ( rg ) () () m/s rm node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6. Upwrd re of - rph ies he chne in elociy m/s for cqirin iniil elociy, i in chnes by sme mon in neie direcion. Slope of cre /. ime.. sec ol ime + 8 sec. Iniilly he speed decreses nd hen increses. -V m/s mg rg 6. he resln elociy shold be in he direcion of resln displcemen So ime 6 r mr m m 6m rm m/s

3 J-Physics 7. km. ime o rech he rond sec R R cos, 6m/s s sin R R R cos m So horizonl displcemen + 6 m cos cos 7 R R sin7 km/hr. y sin sin y / cos ; cos cos 8. For shores ime hen mimm elociy is in he direcion of displcemen. 9. ˆ ˆ ˆ ˆ QP i j i j i ˆ ˆj P (,) So from sine rle min n ( ) Q n (,) Q sin 9 min sin m sin cos cos 6 7 y. cos ˆi ( sin ) ˆj sin cos cos or y cos sin - (sin cos ). i ˆ (b c) ˆj ime o rech mimm heih (when ĵ comp. of elociy becomes zero) b c b c b ime of flih c rne horizonl elociy ime of flih b c. ime of collision of wo bo / sec. s ien in qesion i.e. he ime of flih of sone is lso eql o sec. so ericl componen of sone iniilly is m/s nd he horizonl componen w.r.. moorbo eqls o m/s. ence ˆ ˆ G i j. i ; j y y 6. ime o rech rond h h In his ime horizonl displcemen d d h d h node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6

4 J-Physics 7. Disnce m sin7 ;? cos 7 (orizonl) O R Ne ccelerion of m () 8. d. - ms - ms node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6 P y M ere y + d. So d dy y d d dy d d y d y () cos O R Componen of elociy lon srin ms be sme so M cos M 9. y + d d d dy d y dy d y Q cos d d sin O R d y \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Componen of elociy lon srin ms sme so M cos M. Ne ccelerion of lod cos sin. Ne ension on M M m sin () Now from ccelerion ension consn M () m () m () M C consn C C + + ( ) + ( ) + C C + C ms. Gien + ccelerion of C is ms pwrds d d d d rd/sec. Cenripel ccelerion R R R R R. mnide of ccelerion r 6. Gien r m m/s nlr elociy fer second reolion r finl iniil

5 J-Physics XRCIS II. ; y cos r.6 ; y dy cos d 7. consn, r,, R R ; / ins R R R y w w sin sin. d d So rio ins 8. 6cm,?, rd/s. 6 So 6 cm/s mm/s Difference cm/s mm/s 9. nd remin sme b nd is proporionl o r hs hlf he rdis, ' & ' cm/s R d n( ) d n ( +). + ( + ) ccelerion ( ) hn. Le is he disnce of poin P from O, he, from fire n h or O P h wll m d d h sec d d S d d h sec So pin les h, 8 (9 + ) we e ( )..6 m/s. nlr elociy bo cenre.. rd/sec R.8. m/s... cm/s R 6. 7m/s M 7m/s m (7) + ; m m/s (7) (7) ; ; 6. m/s C C / / C/ P / ( i) 6 (i) 6 j i i j C 7 m / s Q node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6

6 6. ime of fll of sone sec orizonl displcemen of rck in sec S +. Lenh of rck 6m 7. s ien 9 y/6 y m ere elociy of pricle Displcemen ime 6 9m/s 6 9 y J-Physics. When ccelerion is consn he insnneos elociy is eql o he ere elociy in mid of he ime inerl... < spce > < ime > ds sds ds ds d d d d s : 8. Disnce coered by : rin I (re of ) rin I m rin II (re of ) rin II 8 m So he seperion ( + 8) m. 9. r ( + 6) î + ĵ ; ( ) î + ĵ ( î + ĵ ); when hen ; s. +. Speed d d + ± sec () m., () () () 6 sin m cos m/s nd s ime o cross m is sin... o oid n cciden Displcemen + cos sin 8 + co sin 6 sin sin sin cos m.s ( + ) ( ) m/s node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6 min 6.6 m/s [ ( cos + b sin ) hs m. le b ]. ˆi + ˆj ( & y / ) () i ˆ ˆj ; () 8i ˆ 6ˆj i ˆ 9ˆj < > ( 6i ˆ.ˆj ) m/s 7 6. m/s m/s ime of scen ime of descen 8 d h h m/s m/s

7 J-Physics 7. ime ken o rech he drop o rond () ().8 m 9.8 () m 9 8. ime o fll R cos cos so i does no depend on i.e. he chord posiion. 9. () + () 6 Rio :. For mn on rolley L L o m/s m/s m wih respec o rond : + L L + L L L L. ime of flih sin cos 6...(i) (nle of projecion ) Disnce relled by Q on incline in secs is + L S L L & he rne of pricle 'P' is cos + cos ; so 9 from eqion (i) m/s 6 8 sin 9. PQ R PQ y P (,) m/s 7. ime o fll Rne 8 h Q 6 (,y) (, 6, ) orizonl elociy ime h h h h. mimm heih ericl componen of elociy becomes zero. + s sin6 For : For : 6 sin 6 h h sin 6 (/) 8h h h ;. ; y ; d d dy y sec. d y becomes zero mimm heih y m. 6. r ˆi ( ) ˆj ; dr i ˆ ( ) ˆj d d r i ˆ 6j ˆ d node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6

8 J-Physics node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p (For ) ± /,. For prllel o -is dy d dy d sec i becomes zero so (c) ˆ ˆ (,) i 6 j i ˆ ˆj 7. re of he cre ies disnce. 8. ccelerion Re of chne of elociy i.e. elociy cn be chned by chnin is direcion, speed or boh. 9.. elociy Displcmen ime F D () > nd () > so [] d/d 6 9, so ence pricle reerses is direcion only once ere cc. chne in elociy /ime. In inerl ( o 6), pricle does no reerse is elociy nd lso moes in srih line so disnce displcemen.. Moion o C s 7m/s C 7m/s Moion o 7 + s 7 7 () 89 9 () < > 7 (C) 7 (D) < C >, 7 m/s m/s 7 m/s. ( ) + ( )...(i) d d + ( ) herefore () 6, C 9 d d () [From (i)].. ence [C]. + +, y + 8 d d ence [D] + 8, y dy d +6 8; y 6; 8i ˆ 6ˆj consn y ( + ); y ( ) ( + + ) which is he eqion of srih line.. () ( )i ˆ (.) ˆj...(i) For mimm posiie coordine when becomes zero sec hen r().i ˆ.ˆj.. [] Disnce Displcemen ere speed ere elociy [] ± ± [C] [D] elociy cn chne by chnin is direcion ere elociy depends on displcemen in ime inerl e.. circlr moion fer one reolion displcemen become zero hence ere elociy b insnneos elociy neer becomes zero drin moion. In srih line moion ; here ms be reersl of he direcion of elociy o rech he desinion poin for mkin displcemen zero nd hence insnneos elociy hs o be zero les once in ime inerl. 6. ˆ ; [ speed] Velociy my chne by chnin eiher speed or direcion nd by boh. 7. ; d d 9m [ ] d d m/s m/s & i increses s increses so i neer becomes neie. 8. ere elociy Displcemen ime inerl re nder cre ime

9 J-Physics. Rne sin For & (9 ) nles, rne will be sme so for & (9 )6, projecions boh srike he sme poin. For ime of flih, ericl componens re responsible, h h sin sin sin sin 6 9. For rernin, he srin poin re of (O) re of (CD). Reqired ime s ir dr redces he ericl componen of elociy so ime o rech mimm heih will decrese nd i will decrese he downwrd ericl elociy hence ime o fll on erh increses.. orizonl componen of elociy remins consn 'sin cos (from fire) ' co sin (9 ) 'cos P cos So from y y + y ' cos sin 'cos cos sin sin cosec. s ien horizonl elociy m/s cos ; sec, heih m sin / sin Iniil ericl componen sin m/s s hoop is on sme heih of he rjecory. So by symmery will be m.. Rne sin 9 8 sin 98 (9 ) projecion nle hs sme rne. P ime of flih : 8m Q sin sin(9 ) ; ' 6. y, y ; dy d d d y (, ) y m/s ; ˆ ˆ i j ; Slope of line y is n nd lso he slope of elociy is. 7. fer sec, he speed increses wih sin 7 6 m/s Y sin7 6 m/s speed 8 6 m/s 8. New horizonl rne R + R + sin 9. h m R + ( sin m/s s so no. of blls in one min k + c [k >, c > ] d d k c n ( k + c) k k c e k. Le ccelerion of î hen ccelerion of w.r.. ˆ i ˆ j his ccelerion ms be lon he inclined plne so n 7 ) î node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6

10 J-Physics. () () ( ) b d d s, RU/FLS XRCIS III. ccelerion depends on chne in elociy no on he elociy.. Velociy nd displcemen re in sme direcion.. S rd S S () () m + ms. For : Ne ccelerion ms. +. Iniilly pcke cqires blloon elociy which is in pwrds direcion so i moes pwrds for some ime & hen in downwrd.. ecse ll bodies hin sme ccelerion in dow nwrds direcion. 6. hihes poin, ericl elociy becomes zero nd ol elociy de o horizonl componen of elociy & ccelerion de o riy which cs lwys ericlly downwrds Grees heih, nd...horizonl displcemen, 6 + ccelerion of D + C D + 6 ms downwords. lock will in comes o res if c i.e. () ½ s + R 8. Insnneos elociy is nenil o he rjecory. 9. rjecory of pricle depends on he insnneos elociy no on ccelerion.. d d speed of pricle where lwys N R cs owrds he cenre or o he insnneos elociy. node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6 6. Gien d d ; d r ds r S n r 7. n e S / r S / r S / r e e ds d r s s R d / d R / s R s o. No, becse ll msses hin sme ccelerion is in downwrd direcion.. Firsly riy decreses he speed when pricle moes pwrds nd hen in increses by sme mon in downwrd direcion.. When he ericl elociy componen becomes zero, hen he pricle is he op i.e. i hs only horizonl componen h ime which neer chnes so i is min. he op.

11 J-Physics FILL IN LNKS. X ' ' ' ol ime + ( + ) V '. y X ' rjecory eqion is y n cos cos cos 6 m/s. sin sin X cos cos 6 s ien cos sin cos cos ( ) MC COLUMN. [] X + V d d 6 d d [] V 8 d d 8 [D] For chnin he direcion 6, sec. Slope of.. cre ies ccelerion (insnneos) h poin d d., () m/s; ; (6) Chne (6) (); m/s V sin sin sin - 6 cos sin ime of flih sec sin sin ere ccelerion chre in elociy ime ere elociy Displcemen ime inerl 6 m/s ol displcemen re of 's (wih +e or e) h m m (nis). De o riy, i cqires ericl elociy nd de o horizonl force i cqires horizonl componen of force nd when elociy hin boh componens hen he ph of he pricle becomes prbolic.. sin y cos cos y ( sin) nd cos ere elociy 6 m/s () slope of line which eis I n. R rdin R / / R node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6

12 J-Physics Displcemen R sin / sin Disnce m ere elociy Displcemen ime ere ccelerion sin Chne in elociy sin 8 sin ime. Velociy & heih of he blloon fer sec: + m/s h / m Iniil elociy of drop pricle is eqls o he elociy of blloon m 8. ecse iniil ericl elociy componen is zero in boh cses. 9. Inclined plne, in downwrds jorney. he componen of riy is lon inclined sppors in displcemen b no in he oher cse.. Mimm heih depends on he ericl componen of elociy which is eql for boh.. Speed is he mnide of elociy which cn' be neie.. If he ccelerion cs opposie o he elociy hen he pricle is slowin down.. Free fll implies h he pricle moes only in presence of riy. s m/s s fer frher s s Comprehension# heih s s m from iniil posiion of blloon eih from rond + m. Dis n ce d / Displcemen d m() SSRION & RSON. For m. rne sin, he projecion nle() shold be.. m(n) (SW) m(n) node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6 So iniil elociy i + bj n b b. Wheneer pricle hin wo componens of elociy hen he ph of projecile will be prbolic, if pricle is projecs ericlly pwrds hen he ph of projecile will be srih.. ccelerion depends on chne in elociy no on elociy.. If displcemen is zero in ien ime inerl hen is ere elociy lso will be zero. e.. pricle projecs ericlly pwrds.. o mee, co-ordines ms be sme. So in frme of one pricle, second pricle shold pproch i. 6. In ir, he relie ccelerion is zero. he relie elociy becomes consn which increses disnce linerly which ime. 7. Yes, rier elociy does no ny help o cross he rier in minimm ime.., y ;, y 6 Displcemen ( ) (y y ) m Comprehension #. Posiie slopes he posiie ccelerion, neie slopes he neie cclerion.. ccelered moion hin posiie re on - rph hs conce shpe.. Mimm displcemen ol re of rph m. ere speed Dis n ce 6 8 m / s ime 7 7. ime inerl of rerdion o 7.

13 J-Physics Comprehension #. y rjecory eqion is y n n 6 &. M. heih. Rne of. ime of flih sin, cos cos m 8 sin sin sin. he op mos poin cos cos6 m m R m ; R R m Comprehension #. If he projecion nle is incresed, mimm heih will increse.. Projecion nle is & V y m/s, projecion speed is V sin V. y he y rph he ccelerion is. 8 K Iniil kineic enery / mv m/s If mss dobles, hen we cn sec from ( y ) cre hen elociy becomes hlf of preios. m / m ence [] 6. Posiion of he cble he m. heih poin. (V sin) V Comprehenison # 6. In rond frme [] i is simply projecile moion. in [] frme horizonl componen of he displcemen is zero i.e. in his frme only ericl comp. pper which is responsible for he mimm heih.. s obserer obseres h pricle moes norh-wrds. V PC N Comprehension #. R C n Pin d from ble: 8 C n.8 C n.8 8. C depends on he nle of projecion.. R C n 8 C n nd 8 R C n R m.9 n n W S obserer. Frme [D], which is ched wih pricles iself so he minimm disnce is eql o zero. V C. b m/s ; D m/s bd m/s Force cin on body N V P node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6

14 J-Physics Comprehensi on#7. In ericl direcion h sin + sin h sin...(i) In horizonl direcion cos + cos cos...(ii) From (i) nd (ii) n. mimm heih y m sin 8. mimm rne ericl displcemen sin. So rne R sin cos sin sin cos n XRCIS IV. y obserion, for eql inerl of ime he mnide of slope of line in - cre is rees in inerl.. y obserin he rph, posiion of (Q) is reer hn posiion of (P) i.e. lies frher hn nd lso he slope of - cre for & ies heir elociies >.. where & re consns d d d d For d < > d 6 6. S n + (n ) by pin he le of n7 nd 9, find he le of &, 7 m/s & m/s.. fer sec disnce coered / 9 9m elociy of lif 6 m/s p 6m/s, heih ( 9) 9 m ime o rech he rond sec node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6 6. (i) ol ime ken by objec o rech he rond sec. ime o rech on he rond by lif So inerl 6.7. sec sec..9 ~. m.

15 J-Physics..9 ; (ii) eih [ +.9] Decelerion of rin, s km/hr ime o rech plform hr ol disnce relled by he bird 6 km (i) re nder cre he chne in elociy + ;. m/s. m / s ( ) po sec :. m/s m/s m/s ( ) m/s.6.. V m/s m/s m Disnce relled o sop 6 6 Soppin disnce m 8. m 9. (i) eih pwrd re nder cre m sec rond (ii) ol ime of flih + sec. ol ime. +. s (m/s)... From ien siion : (i) (ii) re [ + 6]. km 6 km/hr 6 Cr Cr ol disnce m coers by boh cr Reminin disnce 6 89 m. Le ( ) ime 6 elociy of rin M RIN 6 M RIN sec m / s m/s m/s. Direcion of fl Resln direcion of he wind elociy nd he opposie of bo elociy 7 (i j) j W 6 i (6 ) j 6 i S 6. For : S/ 6 ( ) / hr (ere S is he ol disnce nd is ime p o which 's speed is km/hr) node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6

16 J-Physics For : S km/hr (i) ().7 hr (b) 6. hr (ii) here is no oerkin. 7. Relie elociy of w.r o, 8. d V ime cos ( cos ) 6s, drif w w 6s; w d m sec d n d 7 d w R w w w 9 w / m / sec / d V 6 d 6 m w R. sin () sin () sin sin m/s sin M. heih m. Vericl displcemen of pricle R 6 6 ime for his 6 R R R ˆ ˆ ˆ R () i j i ˆj i ˆ R ˆj. 78 sin sin 6 sin 6 m/sec 6 i.e. food pcke dropped before secs m/s h (6) 8 m. node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6 9. () cos ˆi sin ˆ j () cos ˆi ( sin ) ˆj () cos ( sin ) () () < () > cos î + ( sin ) ĵ ccordin o qesion ( cos ) ( sin ) V ( cos ) sin cos + ( sin ) cos + sin sin + sin sin sin m.6 omber + (.6) (i) (i) y solin eqion (i), we e m/s. (ii) Mimm heih : 8 + (.6 ) 8 + (.6 ) 98m (iii) horizonl disnce orizonl elociy ime of flih cos 7 6m (i) horizonl componen cos 7 8 m/s sec 7 m/s srike 8i ˆ ˆj, 8

17 J-Physics. disnce coered by free fllin body ; 8. ; d d N ( ) R () () N / sin / cos In sme ime, projecile lso rel ericl disnce, hen sin 9. ( + ) R, (.7 +.) 7 sec. sec ccelerion of. m/s R. r ; r r ;. () sec (b) + m/s sin...(i) lso cos ; cos...(ii) From eqion (i) nd (ii) n sin cos d. cos sin d m. O (,). r. m, ne m/s S O m m/s d d () Rdil ccelerion cos (b) / m/s m/s 6. ere () ( ) () 7. ˆ ˆ 6i R j kˆ rd/s ˆ 8 j R ˆ i kˆ rd/s r 8 (c) nenil ccelerion sin. ccordin o Usin 7 R R R 6 7 R R R 6 7 R R R R 6 6 nlr elociy : + 7 R 7 R R 6 R R R nlr ccelerion R 89 R m/s node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6

18 J-Physics. XRCIS IV V V 8, V V 6; V V V V + V 6 km/h 6 V 6 6 V. (i) re [ + ] km / h 9 min [ + ] Velociy of second ship ˆi ˆj / n (i) sec, minimm disnce km. m/s fer sec heih of blloon m (i) Minimm speed (ii) ; 7 m/s ( ) ; 7 m/s ( ) + m/s m/s 8 +. ( ) + m 8 sec m ( + ) node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p ; 6 8 (ii) Dis. relled / 6 8 m. I's elociy is ˆ i (, ) (, ) P O Q displcemen fer ime '' ˆ i 9 6. Le ime of ccelered moion of he helipcoper. Disnce relled by helicoper Disnce relled by sond ( ) 8 sec Finl elociy of helicoper m/s 7. V ( + ) ˆi ˆj, V ( + ) ˆ i ˆj 7 km/hr Rier ime o cross he rier ;

19 J-Physics Drif 6 m ; Drif Reminin disnce ; m ( ol ) ; 8 8 ; 6 6. P cos m/s h cos 6 O Q Y-is 6 sec sec 8. From fire () ime o cross d V Minimm ime d Rio 9. n 6 Cr V V V C C m/s Norml Cr d (i) () ( cos ) î sin ĵ From ien siion cos sec (ii) Velociy, cos + (iii) Disnce PO m/s cos 9 sin PO m h sin m (i) Mimm heih h + sin6. Rne (O) /C sin 6 8 () + Disnce PQ OQ cos OQ 6. m m/s Y-is P Q m O 8 8 h 8 n 6 cos 6 ime o srike cos h m / s PQ PO O m. For sone : h h ( sin ) h (consn) h Pole & h ( sin ).8h orizonl displcemen : cos h h h node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6

20 J-Physics ( ) h h cos cos. cos D...(i) sin...(ii) sin sin D ( sin ) D n h ( D n ) m h + + h D D n ( D n ) cos XRCIS V-. Kineic enery of projecile he hihes poin cos () where is he kineic enery of projecion, is he nle of projecion.. R hihes poin (cos ) sin R sin 6 m R 8.66 m. oh horizonl direcion speed is sme m/s m cos cos 6. When body is projeced n nles nd 9 ; he rnes for boh nles re eql nd he correspondin ime of flihs for he wo rnes re nd. sin cos sin sin 9 R node6\ : \D\\Ko\J-dnced\SMP\Phy\Solion\Uni & \ Kinemics.p6. s + ( sin ) sin sin sin For horizonl moion : cos cos sin cos 6 R. K hihes poin [K Poin of projecion ] cos K K K cos 6 K 6. K(yi ˆ j) ˆ 7. R m similrly dy d ; Ky; d d Ky K ence dy d y dy d, y by inerin y + c. ; re r R m 8. m m nd R m m 9. nd n so by y n y ( + n ) (+) y

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