Jee 2014 Booklet2 Hwt Motion in 2d

8/13/2019 Jee 2014 Booklet2 Hwt Motion in 2d

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Sg`yhkha`gr Johssns

SKJ/Kbtgba ga Pwb @gknasgba 0: L]P/Ylysgjs

@HPN 5 GGPENN 5 KHVDZ 5

89 PGKN 5 6> KGA^PNZ

AHKN 5 PNZP JB@N 5 K6@ T8[

VBOO AB. ZPHVP PGKN 5 NA@ PGKN 5 PGKN PHDNA5ZP^@NAP‗Z ZGMAHP^VN 5 YHVNAP‗Z ZGMAHP^VN 5

Plgs tnst jbathgas h tbtho bc 89 Bfenjtgvn Pypn Xunstgbas.

Nhjl qunstgba jhrrgns 8 khrd. Plnrn gs AB ANMHPGSN khrdgam.

Jlbbsn tln jbrrnjt hotnrahtgvn. Baoy ban jlbgjn gs jbrrnjt.8. Pln khxgkuk lngmlt htthgan` fy h prbenjtgon ha` gts

lbrgzbatho rhamn hrn nquho tb nhjl btlnr gc tln prbenjtgon gs

prbenjtn` ht ha hamon μ mgvna fy 5

(H) 8 6thaμ (F) 8 :thaμ

(J) 8 0thaμ (@) 8 >thaμ

6. H pnrsba hgks h mua ht thrmnt objhtn` ht h lbrgzbatho`gsthajn bc 899 k . Gc tln mua gkphrts h lbrgzbatho spnn` bc

8>99 ks tb tln fuoont, ht wlht lngmlt hfbvn tln thrmnt

kust ln hgk lgs mua ga br nr tb lgt gt2 (Phdn 689m ks )

(H) 89 jk (F) 69 jk(J) >9 jk (@) 899 jk

:. ]lna h prbenjtgon gs ht tln lgmlnst pbgat bc gts trhenjtbry,tln `grnjtgbas bc gts vnobjgty ha` hjjnonrhtgba hrn 5(H) phrhoono tb nhjl btlnr

(F) hatg-phrhoono tb nhjl btlnr(J) gajogan tb nhjl btlnr ht ha hamon bc 0>(@) pnrpna`gjuohr tb nhjl btlnr

0. H fhoo gs prbenjtn` lbrgzbathooy wgtl h vnobjgty bc 8>ks

crbk tln tbp bc h fugo`gam 87.3 k lgml. Lbw obam wgoo tlnfhoo thdn tb lgt tln mrbua` 2

(H) 6 s (F) 6 s

(J) : s (@) : s

>. H fbkf gs `rbppn` crbk ha hnrbpohan wlna gt gs ht hlngmlt l `grnjtoy hfbvn h thrmnt. Gc tln hnrbpohan gs kbvgamlbrgzbathooy ht h spnn` v, tln `gsthajn fy wlgjl tln fbkfwgoo kgss tln thrmnt gs mgvna fy 5

(H) 6 l

vm

(F) l

vm

(J)6l

vm

(@)6l

vm

3. Ha nanky pohan gs coygam lbrgzbathooy ht ha hotgtu`n bc 6

dk wgtl h spnn` bc 8:99 ks . Ha hrky kha wgtl ha hatg-

hgrjrhct mua ba tln mrbua` sgmlts tln nanky pohan wlna gtgs `grnjtoy bvnrlnh` ha` cgrns h slnoo wgtl h kuzzon spnn`

bc 8399 ks . Ht wlht hamon wgtl tln vnrtgjho slbuo` tln

mua fn cgrn` sb hs tb lgt tln pohan 2

(H) :9 (F) 0>

(J) 39 (@) =>

=. Ga X.3, ht wlht kgagkuk hotgtu`n slbuo` tln nanky pohan

coy tb hvbg` fngam lgt 2 (Phdn 689m ks )

(H) 86.> dk (F) 8:.> dk(J) 80.> dk (@) 8>.> dk

<. Pln jngogam bc h tuaano gs > k lgml. ]lht gs tln khxgkuklbrgzbatho `gsthajn tlht h fhoo tlrbwa wgtl h spnn`

bc 869 ks , jha mb wgtlbut lgttgam tln jngogam bc tln

tuaano 2 (Phdn 689m ks )

(H) 89 : k (F) 69 : k

(J) :9 : k (@) 09 : k

7. Crbk tln tbp bc h tbwnr bc lngmlt 09 k , h fhoo gs prbenjtnùpwhr` wgtl h spnn` bc 69 k/s ht ha hamon bc nonvhtgba bc

:9 . Pln rhtgb bc tln tbtho tgkn thdna fy tln fhoo tb lgt tln

mrbua` tb gts tgkn bc cogmlt (tgkn thdna tb jbkn fhjd tb tlnshkn nonvhtgba) gs 5 (Phdn ; 89 k/s 6)(H) 6 5 8 (F) : 5 8(J) : 5 6 (@) 8.> 5 8

89. Ga X.7, tln lbrgzbatho `gspohjnknat (crbk tln cbbt bc tlntbwnr) bc tln fhoo gs hpprbxgkhtnoy nquho tb 5(H) >9 k (F) 39 k(J) =9 k (@) <9 k



Sg`yhkha`gr Johssns

SKJ/Kbtgba ga Pwb @gknasgba 00 L]P/Ylysgjs




VBOO AB. ZPHVP PGKN 5 NA@ PGKN 5 PGKN PHDNA5

ZP^@NAP‗Z ZGMAHP^VN 5 YHVNAP‗Z ZGMAHP^VN 5



Jlbbsn tln jbrrnjt hotnrahtgvn. Baoy ban jlbgjn gs jbrrnjt.

8. H jhaaba ba h onvno pohga gs hgkn` ht ha hamon θ hfbvn

tln lbrgzbatho ha` h slnoo gs cgrn` wgtl h kuzzon vnobjgty v9

tbwhr`s h vnrtgjho jogcc h `gsthajn V hwhy. Plna tln lngmltcrbk tln fbttbk ht wlgjl tln slnoo strgdns tln sg`n whoos bc tln jogcc gs 5

(H) 66 69

86

mV V sgav sga

θθ

(F)6

6 69

86

mV Vjbs

v jbsθ

θ

(J)6

6 69

86

mV V tha

v jbsθ

θ

(@)6

6 69

86

mV V tha

v sgaθ

θ

6. Gt gs pbssgfon tb prbenjt h phrtgjon wgtl h mgvna vnobjgty gatwb pbssgfon whys sb hs tb khdn gt phss tlrbuml h pbgat Y

ht h `gsthajn r crbk tln pbgat bc prbenjtgba. Pln prbùjt bc tln tgkns thdna tb rnhjl tlgs pbgat ga tln twb pbssgfon whysgs tlna prbpbrtgbaho tb 5(H) 8/ r (F) r

(J) r : (@)6

8

r :. H mua dnpt ba h strhgmlt lbrgzbatho rbh` gs usn` tb lgt h jhr

trhvnogam ba tln shkn rbh` hwhy crbk tln mua ht h uagcbrk

spnn` bc 889 6 ks . Pln jhr gs ht h `gsthajn bc 8>9 k

crbk tln mua wlna gt gs cgrn` ht ha hamon bc 0> tb tln

lbrgzbatho. ]gtl wlht spnn` slbuo` tln slnoo fn prbenjtn`

sb tlht gt lgts tln jhr 2 (Phdn 689m ks )

(H) 869 ks (F) 8:9 ks

(J) 809 ks (@) 8>9 ks

0. Ga X.: wlht gs tln `gsthajn bc tln jhr crbk tln mua wlnatln slnoo lgts gt 2(H) 6>9 k (F) >99 k(J) =>9 k (@) 8999 k

>. Crbk tln tbp bc tbwnr bc lngmlt 09 k , h fhoo gs prbenjtn`

upwhr`s wgtl h spnn` bc 869 ks ht ha hamon bc :9 tb tln

lbrgzbatho. Gc 889m ks , hctnr lbw obam wgoo tln fhoo lgt

tln mrbua` 2

(H) 8 s (F) 6 s(J) : s (@) 0 s

3. H prbenjtgon lhs h khxgkuk rhamn bc 699 k . ]lht gs tlnkhxgkuk lngmlt htthgan` fy gt 2(H) 6> k (F) >9 k(J) => k (@) 899 k

=. ]lgjl bc tln cboobwgam rnkhgas jbasthat ùrgam tln kbtgbabc h prbenjtgon cgrn crbk h pohanr 2(H) dgantgj nanrmy (F) kbknatuk(J) vnrtgjho jbkpbanat bc vnobjgty(@) lbrgzbatho jbkpbanat bc vnobjgty

<. Pln lbrgzbatho `gsthajn x ha` tln vnrtgjho lngmlt y bc hprbenjtgon ht h tgkn t hrn mgvna fy x ; ht ha` y ; ft 6 + jt wlnrn h, f ha` j hrn jbasthat. ]lht gs tln khmagtu`n bc tlnvnobjgty bc tln prbenjtgon 8 snjba` hctnr gt gs cgrn` 2(H) Th6 + (6 f + j )6[8/6 (F) T6h6 + ( f + j )6[8/6

(J) T6h6 + (6 f + j )6[8/6 (@) Th6 + ( f + 6 j )6[8/6

7. Ga X. <, tln hamon μ wgtl tln lbrgzbatho ht wlgjl tln

prbenjtgon gs prbenjtn` gs mgvna fy 5

(H) 8 htha

fμ

(F) 8 f

thah

μ

(J)8 h

tha jμ

(@)8 j

tha hμ

89. Ga X.<, tln hjjnonrhtgba ùn tb mrhvgty gs mgvna fy 5(H) 6h (F) 6f(J) 6j (@) hj f



Sg`yhkha`gr Johssns

SKJ/Kbtgba ga Pwb @gknasgba 0> L]P/Ylysgjs



AHKN 5 PNZP JB@N 5 K6@ T:[






8. Cbur prbenjtgons hrn prbenjtn` wgtl tln shkn spnn` ht hamons 69 :> 39 ha` => , , wgtl tln lbrgzbatho. Pln rhamn wgoo fn tln obamnst cbr

tln prbenjtgon wlbsn hamon bc prbenjtgba gs 5

(H) 69 (F) :> (J) 39 (@) =>

6. H prbenjtgon tlrbwa ht ha hamon bc :9 wgtl tln lbrgzbatho lhs h rhamn V8 ha` htthgas h khxgkuk lngmlt l 8. Habtlnr prbenjtgon tlrbwa,

wgtl tln shkn vnobjgty, ht ha hamon :9 wgtl tln vnrtgjho, lhs h rhamn V6 ha` htthgas h khxgkuk lngmlt l 6. Pln rnohtgba fntwnna V8 ha` V6 gs 5

(H) 68 6

V V (F) V8 ; V6 (J) V8 ; 6 V6 (@) V8 ; 0 V6

:. Ga X. 6, wlht gs tln rnohtgba fntwnna l 8 ha` l 6 2

(H) 68 0

ll (F) 6

8 :l

l (J) 68 6

ll (@) l 8 ; l 6

0. Pln khxgkuk lngmlt htthgan` fy h prbenjtgon gs gajrnhsn` fy 89% fy gajrnhsgam gts spnn` bc prbenjtgba, wgtlbut jlhamgam tln hamon bc prbenjtgba. Pln pnrjnathmn gajrnhsn ga tln lbrgzbatho rhamn wgoo fn 5(H) 69% (F) 8>% (J) 89% (@) >%

>. H prbenjtgon htthgas h jnrthga khxgkuk lngmlt wlna prbenjtn` crbk nhrtl. Gc gt gs prbenjtn` ht tln shkn hamon ha` wgtl tln shkn gagtghospnn` crbk tln kbba, wlnrn tln hjjnonrhtgba ùn tb mrhvgty gs ban-sgxtl tlht ba nhrtl, fy wlht chjtbr wgoo tln khxgkuk lngmlt bc tlnprbenjtgon gajrnhsn 2

(H) : (F) : (J) 3 (@) 3

3. H prbenjtgon lhs h rhamn V ha` tgkn bc cogmlt P . Gc tln rhamn gs `bufon` (fy gajrnhsgam tln spnn` bc prbenjtgba, wgtlbut jlhamgam tln hamonbc prbenjtgba), tln tgkn bc cogmlt wgoo fnjbkn 5

(H)6

P (F) 6 P (J)

6P

(@) 6P

=. H prbenjtgon lhs tln shkn V wlna tln khxgkuk lngmlt htthgan` fy gt gs ngtlnr l 8 br l 6. Plna V, l 8 ha` l 6 wgoo fn rnohtn` hs 5

(H) 8 6 V l l (F) 8 66 V l l (J) 8 6: V l l (@) 8 60 V l l

<. H fb`y gs prbenjtn` lbrgzbathooy crbk h pbgat hfbvn tln mrbua`. Pln kbtgba bc tln fb`y gs `nsjrgfn` fy tln nquhtgbas

x ; 6 t ha` y ; > t 6

wlnrn x ha` y hrn tln lbrgzbatho ha` vnrtgjho `gspohjnknats (ga k ) rnspnjtgvnoy ht tgkn t . ]lht gs tln khmagtu`n bc tln vnobjgty bc tlnfb`y 9.96 snjba hctnr gt gs prbenjtn 2

(H) 86 ks (F) 86 6 ks (J) 8: >. ks (@) 8> ks



Sg`yhkha`gr Johssns


7. H fb`y gs prbenjtn` ht tgkn t ; 9 crbk h jnrthga pbgat ba h pohant‗s surchjn wgtl h jnrthga vnobjgty ht h jnrthga hamon wgtl tln pohant‗ssurchjn (hssukn` lbrgzbatho). Pln lbrgzbatho ha` vnrtgjho `gspohjnknats x ha` y (ga kntnrs) rnspnjtgvnoy vhry wgtl tgkn t (ga snjba`) hs

89 : x t 4 689 y t t

]lht gs tln khmagtu`n ha` grnjtgba bc tln vnobjgty wgtl wlgjl tln fb`y gs prbenjtn` 2

(H) 869 ks ht ha hamon bc :9 wgtl tln lbrgzbatho

(F) 869 ks ht ha hamon bc 39 wgtl tln lbrgzbatho

(J) 889 ks ht ha hamon bc :9 wgtl tln lbrgzbatho

(@) 889 ks ht ha hamon bc 39 wgtl tln lbrgzbatho

89. ]lgjl ban bc tln cboobwgam sthtnknats gs ABP trun hfbut tln kbtgba bc h prbenjtgon 2(H) Pln tgkn bc cogmlt bc h prbenjtgon gs prbpbrtgbaho tb tln spnn` wgtl wlgjl gt gs prbenjtn`(F) Pln lbrgzbatho rhamn bc h prbenjtgon gs prbpbrtgbaho tb tln squhrn-rbbt bc tln spnn` wgtl wlgjl gt gs prbenjtn`

(J) Cbr h mgvna spnn` bc prbenjtgba, tln hamon bc prbenjtgba cbr khxgkuk rhamn gs 0>(@) Ht khxgkuk lngmlt, tln hjjnonrhtgba ùn tb mrhvgty gs pnrpna`gjuohr tb tln vnobjgty bc tln prbenjtgon



Sg`yhkha`gr Johssns

SKJ/Kbtgba ga Pwb @gknasgba 0= L]P/Ylysgjs








8. @gspohjnknat ( x) bc h phrtgjon gs rnohtn` tb tgkn ( t ) hs 5 6 : x ht ft jt

]lnrn h, f ha` j hrn jbasthat bc kbtgba. Pln vnobjgty bc tln phrtgjon wlna gts hjjnonrhtgba gs znrb gs mgvna fy 5

(H)6f

hj

(F)6

6f

hj

(J)6

:f

hj

(@)6

0f

hj

6. H vnjtbr H gs hobam tln pbsgtgvn z-hxgs ha` gts vnjtbr prbùjt wgtl habtlnr vnjtbr F gs znrb, tlna vnjtbr F jbuo` fn 5

(H)

g e (F) 0

g (J)

e d (@) =

d

:. H rgvnr gs cobwgam crbk wnst tb nhst ht h spnn` bc > k /kgautn. Ga wlht `grnjtgba slbuo` h kha ba tln sbutl fhad bc tln rgvnr jhphfon bc swgkkgam ht 89 k /kgautns ga stgoo whtnr swgk tb jrbss tln rgvnr ga tln slbrtnst tgkn 2(H) Nhst (F) ]nst (J) Zbutl (@) Abrtl

0. Ga X. : hfbvn, ga wlht `grnjtgba slbuo` tln kha swgk sb tlht ln jrbssns tln rgvnr fy swgkkgam tln slbrtnst `gsthajn 2

(H) Abrtl (F) 39 wnst bc abrtl (J) :9 wnst bc abrtl (@) :9 wnst bc sbutl

>. H fb`y gs tlrbwa vnrtgjhooy up wgtl h vnobjgty u . Gt phssns tlrnn pbgats H, F ha` J ga gts upwhr` eburany wgtl vnobjgtgns ha`6 : 0u u u

,

rnspnjtgvnoy. Pln rhtgb HF HJ gs 5

(H) 69 6= (F) 6 (J) 89 = (@) 8

3. H fb`y gs tlrbwa vnrtgjhooy up wgtl h vnobjgty u . Gt phssns h pbgat ht h lngmlt l hfbvn tln mrbua` ht tgkn t 8 wlgon mbgam up ha` ht tgkn t 6wlgon choogam `bwa. Plna tln rnohtgba fntwnna u , t 8 ha` t 6 gs 5

(H) 8 66u

t t m

(F) 6 86u

t t m

(J) 8 6u

t t m

(@) 6 8u

t t m

=. Ga X. 3 hfbvn, tln rnohtgba fntwnna t 8, t 6 ha` l gs 5

(H) 8 66l

t t m

(F) 8 6l

t t m

(J) 68 6

6lt t

m (@) 6

8 6l

t t m

<. H fb`y `rbppn` crbk h lngmlt L hfbvn tln mrbua` strgdns ha gajogan` pohan ht lngmlt l hfbvn tln mrbua`. Hs h rnsuot bc tln gkphjt, tlnvnobjgty bc tln fb`y fnjbkns lbrgzbatho. Pln fb`y wgoo thdn tln khxgkuk tgkn tb rnhjl tln mrbua` gc 5

(H)0

L l (F)6 6 L l (J)

6 L l (@)

6 L l

7. H fb`y gs `rbppn` crbk tln rbbc bc h kuotg-stbrnyn` fugo`gam. Gt phssns tln jngogam bc tln 8> tl stbrny ht h spnn` bc 869 ks . Gc tln lngmlt

bc nhjl stbrny gs 0 k , tln aukfnr bc stbrnys ga tln fugo`gam gs 5 (Phdn 689m ks ha` anmonjt hgr rnsgsthajn)(H) 69 (F) 6> (J) :9 (@) :>

89. Pwb fhoos H ha` F hrn prbenjtn` crbk tln shkn objhtgba sgkuothanbusoy. Fhoo H gs prbenjtn` vnrtgjhooy upwhr`s ha` fhoo F ht :9 tb tln

vnrtgjho. Plny rnhjl tln mrbua` sgkuothanbusoy. Pln vnobjgtgns bc prbenjtgba bc H ha` F hrn ga tln rhtgb 5

(H) : 85 (F) 8 :5 (J) : 65 (@) 6 :5



Sg`yhkha`gr Johssns

SKJ/Kbtgba ga Pwb @gknasgba 0< L]P/Ylysgjs

@HPN 5 GGPENN 5KHVDZ 5


AHKN 5 PNZP JB@N 5 K6@ T>[





8. H fb`y gs prbenjtn` wgtl h vnobjgty 8: 0

v g e ks . Pln khxgkuk lngmlt htthgan` fy tln fb`y gs 5 (Phdn 689m ks )

(H) 9.< k (F) < k (J) <9 k (@) <99 k

6. Ga X.8 hfbvn, tln tgkn bc cogmlt bc tln fb`y gs 5(H) 9.< s (F) 8.9 s (J) 0.9 s (@) <.9 s

:. H fb`y gs prbenjtn` wgtl h vnobjgty u ht ha hamon μ wgtl tln lbrgzbatho. Pln vnobjgty bc tln fb`y wgoo fnjbkn pnrpna`gjuohr tb tln

vnobjgty bc prbenjtgba hctnr h tgkn t mgvna fy 5

(H)6u sga

mμ

(F) u sga

mμ

(J)6u

m sgaμ(@)

u

m sgaμ

0. Pln nquhtgba bc kbtgba bc prbenjtgon gs 6 y hx fx wlnrn h ha` f jbasthat bc kbtgba. Pln lbrgzbatho rhamn bc tln prbenjtgon gs 5

(H) h

f(F)

6hf

(J)6

6h

f(@)

6

0h

f

>. Ga X. 0 hfbvn, tln mrnhtnst lngmlt htthgan` fy tln prbenjtgon gs 5

(H) h

f

(F)

6

h

f

(J)6

6

h

f

(@)6

0

h

f3. Ga X. 0 hfbvn, tln tgkn bc cogmlt bc tln prbenjtgon gs 5

(H)6

hfm

(F) h

fm(J)

6h

fm(@)

6h

fm

=. H phrtgjon kbvgam ga h strhgmlt ogan jbvnrs lhoc tln `gsthajn wgtl h spnn` bc : k/s . Pln btlnr lhoc bc tln `gsthajn gs jbvnrn` ga twb nquhotgkn gatnrvhos wgtl spnn`s bc 0.> k/s ha` =.> k/s rnspnjtgvnoy. Pln hvnrhmn spnn` bc tln phrtgjon ùrgam tlgs kbtgba gs 5(H) 0.9 k/s (F) >.9 k/s (J) >.> k/s (@) 0.< k/s

<. H jhr gs trhvnogam ht h vnobjgty bc 89 dk/l ba h strhgmlt rbh`. Pln `rgvnr bc tln jhr tlrbws h rbh`. Gc tln phrjno gs tb rnhjl tln kha, tln`grnjtgba bc tlrbw khdns tln cboobwgam hamon wgtl tln `grnjtgba bc tln jhr,

(H) 8:> (F) 0> (J) 8 6tha (@) 8 8

6tha

7. Ga 8.9 s , h phrtgjon mbns crbk pbgat H tb pbgat F , kbvgam ga h snkgjgrjon bc rh`gus 8.9 k hsslbwa ga tln cgmurn. Pln khmagtu`n bc tln hvnrhmn vnobjgty bc tln phrtgjon gs 5

(H) 8: 80. ks (F) 86 9. ks

(J) 88 9. ks (@) Unrb

89. Pln gagtgho vnobjgty bc h phrtgjon gs 80 :

u g e ks . Gt gs kbvgam wgtl uagcbrk hjjnonrhtgba 69 0 9 :

h . g . e ks . Pln khmagtu`n

bc gts vnobjgty hctnr 89 s gs 5

(H) 8: ks (F) 80 ks (J) 8> ks (@) 889 ks

H

F

8.9 k



Sg`yhkha`gr Johssns










8. Pln lbrgzbatho ha` vnrtgjho `gspohjnknats bc h prbenjtgon ht tgkn ' t ' hrn 6:3 ha` 0< 0 7 x t y t . t rnspnjtgvnoy wlnrn x ha` y hrn ga

kntrn ha` t gs ga snjba`. Gagtgho vnobjgty bc tln prbenjtgon ga k / s gs 5(H) 8> (F) :9 (J) 0> (@) 39

6. Pln `gspohjnknat x bc h phrtgjon kbvgam ga ban `gknasgba gs rnohtn` tb tgkn t fy tln nquhtgba

:t x

]lnrn x gs ga kntrns ha` t ga snjba`s. Pln `gspohjnknat bc tln phrtgjon wlna gts vnobjgty gs znrb gs 5(H) znrb (F) 0 k (J) 8 k (@) 9.> k

:. H phrtgjon gagtghooy (g.n., ht t ; 9) kbvgam wgtl h vnobjgty u gs sufenjtn` tb h rnthr`gam cbrjn, hs h rnsuot bc wlgjl gt `njnonrhtns ht h rhtn

h d v

wlnrn v gs tln gasthathanbus vnobjgty ha` d gs h pbsgtgvn jbasthat. Pln tgkn P thdna fy tln phrtgjon tb jbkn tb rnst gs mgvna fy 5

(H)6 u

P d

(F)6u

P d

(J): 66 / u

P d

(@)66u

P d

0. H phrtgjon sthrts crbk rnst. Gts hjjnonrhtgba ht tgkn t ; 9 gs6> ks wlgjl vhrgns wgtl tgkn hs slbwa ga cgmurn.

Pln khxgkuk spnn` bc tln phrtgjon wgoo fn 5

(H) 8= >. ks (F) 88> ks

(J) 8:9 ks (@) 8= >. ks

>. Cgmurn slbws tln vhrghtgba bc vnobjgty ( v) bc h fb`y wgtl pbsgtgba ( x) crbk tln

brgmga B . ]lgjl bc tln mrhpls slbwa ga cgmurn jbrrnjtoy rnprnsnats tln vhrghtgba

bc tln hjjnonrhtgba ( h) wgtl pbsgtgba ( x) 2

(H) (F) (J) (@)

3. Pln vnobjgty ( v) bc h fb`y kbvgam hobam tln pbsgtgvn x-`grnjtgba vhrgns wgtl `gspohjnknat ( x) crbk tln brgmga hs v d x , wlnrn d gs h

jbasthat. ]lgjl bc tln mrhpls slbwa ga cgmurn jbrrnjtoy rnprnsnats tln `gspohjnknat-tgkn x t mrhpl bc tln kbtgba 2

(H) (F) (J) (@)

v

B x

v9

x9

h

B x

hB xh

B x

h

B x

x

B t

x

B t

x

B t

x

B t

h

B t

6> ks

3 snj



Sg`yhkha`gr Johssns

SKJ/Kbtgba ga Pwb @gknasgba >9 L]P/Ylysgjs

=. H tnaags fhoo rboos bcc tln tbp bc h sthgr jhsn why wgtl h lbrgzbatho vnobjgty 8u ks . Gc tln stnps hrn f kntrn wg`n ha` l kntrn lgml, tln

fhoo wgoo lgt tln n`mn bc tln atl stnp, gc 5

(H)6

6lua

mf (F)

6

66lu

amf

(J)66lu

amf

(@)6

lua

mf

<. H fbkfnr pohan kbvns lbrgzbathooy wgtl h spnn` bc 8>99 ks ha` h fbkf rnonhsn` crbk gt, strgdns tln mrbua` ga 89 s . Hamon ht wlgjl gt

strgdns tln mrbua` wgoo fn 5 689m ks

(H) 8 8>

tha

(F)

8>

tha

(J) 8tha (@) 8 >tha

7. Ha hnrbpohan gs coygam ga h lbrgzbatho `grnjtgba wgtl h vnobjgty 8399 dkl ht h lngmlt

bc 8739 k . ]lna gt gs vnrtgjhooy hfbvn tln pbgat H ba tln mrbua`, h fb`y gs `rbppn`crbk gt. Pln fb`y strgdns tln mrbua` ht pbgat F . Jhojuohtn tln `gsthajn HF 5(H) :.:: dk (F) ::: dk(J) ::.: dk (@) :::9 dk

89. H phrtgjon Y gs ht tln brgmga sthrts wgtl vnobjgty 86 0

v g e ks wgtl jbasthat hjjnonrhtgba 6: >

g e ks . Hctnr trhvnogam cbr 6 s , gts

`gsthajn crbk tln brgmga gs 5(H) 89 k (F) 69.3 k (J) 7.< k (@) 88.= k

l

B v

H F



Sg`yhkha`gr Johssns


@HPN 5 GGPENN 5KHVDZ 5

89

PGKN 5 6> KGA^PNZ

AHKN 5 PNZP JB@N 5 K6@ T=[





Jlbbsn tln jbrrnjt hotnrahtgvn. Baoy ban jlbgjn gs jbrrnjt.8. H kha stha`gam ba h lgoo tbp prbenjts h stban lbrgzbathooy wgtl spnn` v9 hs slbwa ga cgmurn. Phdgam

tln jbbr`gahtn systnk hs mgvna ga tln cgmurn. Pln jbbr`gahtns bc tln pbgat wlnrn tln stban wgoo lgttln lgoo surchjn 5

(H)6 6 69 96 6v tha v tha

,m m

μ μ (F)

6 6 69 96 6v v tha

,m m

μ

(J)6 69 96 6v tha v

,m m

μ

(@)6 6 69 96 6v tha v tha

,m m

μ μ

6. H phrtgjon kbvns ga tln x-y pohan wgtl vnobjgty < 6 xv t ha` v y ; 6. Gc gt phssns tlrbuml tln pbgat x ; 80 ha` y ; 0 ht t ; 6 s, cga` tln

nquhtgba ( x-y rnohtgba) bc tln phtl.

(H) 6 6 x y y (F) 66 6 : x y y (J) x ; : y6 + > (@) jha abt fn cbua` crbk hfbvn `hth

:. H phrtgjon onhvns tln brgmga wgtl ha gagtgho vnobjgty 8: 99

v . g ks ha` h jbasthat hjjnonrhtgba 68 99 9 >9

h . g . e ks . ]lna tln

phrtgjons rnhjlns gt khxgkuk x-jbbr`gahtn, wlht gs gts y-jbkpbanat h vnobjgty 2

(H) 86 9. ks (F) 88 9. ks (J) 88 >. ks (@) 88 9. ks

0. H phrtgjon sthrts crbk tln brgmga bc jbbr`gahtns ht tgkn t ; 9 ha` kbvns ga tln x-y pohan wgtl h jbasthat hjjnonrhtgba θ ga tln y-`grnjtgba.

Gts nquhtgba bc kbtgba gs 6 y xζ . Gts vnobjgty jbkpbanat ga tln x-`grnjtgba gs 5

(H) vhrghfon (F)6θζ

(J)6θζ

(@)6θζ

>. Ha hnrbpohan coygam lbrgzbathooy wgtl h spnn` bc 8:39 dkl rnonhsns h fbkf ht h lngmlt bc 079 k crbk tln mrbua`. ]lna wgoo tln fbkf

strgdn tln mrbua` 2

(H) < s (F) 3 s (J) = s (@) 89 s

3. H mohss khrfon prbenjtn` lbrgzbathooy crbk tln tbp bc h thfon choos ht h `gsthajn x crbk tln n`mn bc tln thfon. Gc l gs tln lngmlt bc tln thfon,tlna tln vnobjgty bc prbenjtgba gs 5

(H)6m

l x

(F)6m

xl

(J) mxl (@) mx + l

=. H fbkf gs `rbppn` ba ha nanky pbst fy ha hnrbpohan coygam lbrgzbathooy wgtl h vnobjgty bc 839 dkl ha` ht h lngmlt bc 079 k . Ht tln

tgkn bc `rbppgam tln fbkf, lbw chr lbrgzbathooy tln hnrbpohan slbuo` fn crbk tln nanky pbst sb tlht tln fbkf khy `grnjtoy lgt tlnthrmnt 2

(H)099

:k (F)

>99:

k (J)8=99

:k (@) 07< k

y xv9

(9, 9)

μ



Sg`yhkha`gr Johssns


<. H kbadny jha eukp h khxgkuk lbrgzbatho `gsthajn bc 69 k . Plna tln vnobjgty bc tln kbadny gs 5

(H) 889 ks (F) 880 ks (J) 869 ks (@) 860 ks

7. H fb`y gs prbenjtn` lbrgzbathooy wgtl spnn` 869 ks . Pln hpprbxgkhtn `gspohjnknat bc tln fb`y hctnr > s gs 5

(H) <9 k (F) 869 k (J) 839 k (@) :69 k

89. H fb`y bc khss k tlrbwa lbrgzbathooy wgtl vnobjgty v, crbk tln tbp bc tbwnr bc lngmlt l tbujlns tln onvno mrbua` ht `gsthajn bc 6>9 k

crbk tln cbbt bc tln tbwnr. H fb`y bc khss 6 k tlrbwa lbrgzbathooy wgtl vnobjgty 6v , crbk tln tbp bc tbwnr bc lngmlt 0 l wgoo tbujl tln

onvno mrbua` ht h `gsthajn x crbk tln cbbt bc tbwnr. Pln vhoun bc x gs 5

(H) 6>9 k (F) >99 k (J) 86> k (@) 6>9 6 k



Sg`yhkha`gr Johssns

SKJ/Kbtgba ga Pwb @gknasgba >: L]P/Ylysgjs



AHKN 5 PNZP JB@N 5 K6@ T<[






8. Pwb thoo fugo`gams hrn 09 k hphrt. ]gtl wlht spnn` kust h fhoo fn tlrbwa lbrgzbathooy crbk h wga`bw 80> k hfbvn tln mrbua` ga banfugo`gam, sb tlht gt wgoo natnr h wga`bw 66.> k crbk tln mrbua` ga tln btlnr 2

(H)8

> ks (F)8

< ks (J)8

89 ks (@)8

83 ks

6. Pwb prbenjtgon hrn tlrbwa wgtl tln shkn gagtgho vnobjgty ht hamons ha` 79θ θ wgtl tln lbrgzbatho. Pln khxgkuk lngmlts htthgan`

fy tlnk hrn l 8 ha` l 6 rnspnjtgvnoy. Plna 8 6l l gs nquho tb 5

(H) 6sga θ (F) 6jbs θ (J) 6tha θ (@) 8

:. ]lna h prbenjtgon gs prbenjtn` ht h jnrthga hamon wgtl tln lbrgzbatho, gts lbrgzbatho rhamn gs V ha` tgkn bc cogmlt gs P 8. ]lna tln shknprbenjtgon gs tlrbwgam wgtl tln shkn spnn` ht sbkn btlnr hamon wgtl tln lbrgzbatho, gts lbrgzbatho rhamn gs V ha` tgkn bc cogmlt gs P 6.Pln prbùjt bc P 8 ha` P 6 gs 5

(H) V

m(F)

6 Vm

(J): Vm

(@)0 Vm

0. H stban gs tlrbwa wgtl h vnobjgty v khdgam ha hamon μ wgtl tln lbrgzbatho. Ht sbkn gasthat, gts vnobjgty S gs pnrpna`gjuohr tb tln gagtgho

vnobjgty v. Plna S gs 5(H) v sga μ (F) v jbs μ (J) v tha μ (@) v jbt μ

>. Pln nquhtgba bc h prbenjtgon gs6

:6

mx y x . Pln hamon bc prbenjtgba gs mgvna fy 5

(H)8

:tha μ (F) :tha μ (J)

6ω

(@) znrb

3. H prbenjtgon bc khss k gs tlrbwa wgtl h vnobjgty v khdgam ha hamon bc 0> wgtl tln lbrgzbatho. Pln jlhamn ga kbknatuk crbk `nphrturn

tb hrrgvho hobam vnrtgjho `grnjtgba, gs 5

(H) 6 kv (F) 6 kv (J) kv (@)6

kv

Yhrhmrhpl cbr X.= - 89

H stban gs prbenjtn` lbrgzbathooy bcc h jogcc wgtl vnobjgty 89 k / s. Gt oha`s ht h `gsthajn ' ` ' crbk tln cbbt bc tln jogcc hctnr 3 snjba`. Phdn m ; 89 k / s6

=. Pln lngmlt bc tln jogcc gsRRR.(H) 39 k (F) 869 k (J) 8<9 k (@) 609 k

F

l

889u ks

`

Jogcc



Sg`yhkha`gr Johssns


<. Pln `gsthajn ' ` ' gsRRRRR.(H) 39 k (F) 0> k (J) :9 k (@) 8> k

7. Pln vnobjgty bc tln stban ht tln tgkn bc gkphjt wgtl tln mrbua` gsRRRRR.(H) 89 k/s (F) =9 k/s (J) 39.< k/s (@) =9.6 k/s

89. Pln stban gs kbvgam wgtl `bufon gts gagtgho spnn` ht tgknRRRRRR.

(H) 6 s (F) 8 s (J) : s (@):

6s



Sg`yhkha`gr Johssns

SKJ/Kbtgba ga Pwb @gknasgba >> L]P/Ylysgjs









8. H prbenjtgon slbt gatb hgr ht sbkn hamon wgtl tln lbrgzbatho lhs h rhamn bc 699 k . Gc tln tgkn bc cogmlt gs > s, tlna tln lbrgzbathojbkpbanat bc tln vnobjgty bc tln prbenjtgon ht tln lgmlnst pbgat bc trhenjtbry gs 5

(H) 809 ks (F) 89 ks

(J) 87 <. ks (@) Nquho tb tln vnobjgty bc prbenjtgba bc tln prbenjtgon

6. Gc tln hamon bc prbenjtgba bc h prbenjtgon gs :9 , tlna lbw khay tgkns tln lbrgzbatho rhamn gs ohrmnr tlha tln khxgkuk lngmlt 2

(H) 6 (F) : (J) : 0 (@) 0 :

:. H phrtgjon gs prbenjtn` crbk tln mrbua` wgtl ha gagtgho spnn` bc v ht ha hamon μ wgtl lbrgzbatho. Pln hvnrhmn vnobjgty bc tln phrtgjon

fntwnna gts pbgat bc prbenjtgba ha` lgmlnst pbgat bc trhenjtbry gs 5

(H) 68 66v

jbs μ (F) 686v

jbs μ (J) 68 :6v

jbs μ (@) v jbs μ

0. Pln nquhtgba bc kbtgba bc h prbenjtgon gs 6:86

0 y x x . Pln lbrgzbatho jbkpbanat bc vnobjgty gs 8: ks . ]lht gs tln rhamn bc tln

prbenjtgon 2(H) 8< k (F) 83 k (J) 86 k (@) 68.3 k

>. v-t mrhpl cbr h prbenjtgon, wlnrn v gs vnrtgjho jbkpbanat bc vnobjgty, gsRRRR.

(H) (F) (J) (@)

3. Vhamn bc h prbenjtgon wgtl tgkn bc cogmlt 89 s ha` khxgkuk lngmlt 899 k gs 5 (m ; 89 k/s 6)(H) 699 k (F) 86> k (J) 899 k (@) @hth gajbrrnjt

Yhrhmrhpl cbr X.= - 89

H fhoo gs tlrbwa upwhr`s crbk tln tbp bc ha gajogan wgtl hamon bc prbenjtgba μ wgtl tln vnrtgjho. Gc 39μ , ha` m ; 89 k / s6.

=. Pln tgkn bc cogmlt gsRRRRRR.(H) :t s (F) 0t s (J) :t s (@) 3t s

v

t

v

t

v

t

v

t

:9 k

θ

μ89u k / s



Sg`yhkha`gr Johssns


<. Pln hamon θ gc tln fhoo oha`s nxhjtoy ht tln cbbt bc tln gajogan gsRRRR.

(H) 8 :tha (F) 8 3tha (J) 8 : 6tha / (@) 8 6 :tha

7. H fhoo kbvns ga h jgrjuohr sobt wgtl hamuohr hjjnonrhtgba 66 rh` / sθ , tln hamon fntwnna gts vnobjgty ha` hjjnonrhtgba gsRRRRR.

(H) 79 (F) 8 6(6 )tha t (J) 86t

tha

(@) 0>

89. H phrtgjon kbvns hobam h phrhfbogj phtl y ; hx6 sujl tlht x jbkpbanat bc tln vnobjgty gs jbasthat, shy j . Pln hjjnonrhtgba bc tlgs phrtgjongsRRRRR.

(H)

hjd (F) 66

hj e (J) 66

hj d (@) 6

hj e



Sg`yhkha`gr Johssns

SKJ/Kbtgba ga Pwb @gknasgba >= L]P/Ylysgjs









8. H phrtgjon gs prbenjtn` crbk h pbgat H wgtl vnobjgty 6u ha hamon bc 0> wgtl lbrgzbatho hs slbwa ga

cgmurn. Gt strgdns tln pohan FJ ht rgmlt hamons. Pln vnobjgty bc tln phrtgjon ht tln tgkn bc jboogsgba gs 5

(H):

6

u(F)

6

u

(J)6

:

u(@) u

6. Ha hgrpohan, `gvgam ht ha hamon bc >: 9. wgtl tln vnrtgjho rnonhsns h prbenjtgon ht ha hotgtu`n bc =:9 k . Pln prbenjtgon lgts tln mrbua` >.99 s

hctnr fngam rnonhsn`. ]lht gs tln spnn` bc tln hgrjrhct 2

(H) 86<6 ks (F) 8696 ks (J) 88<6 ks (@) 8896 ks

:. H phrtgjon H gs prbenjtn` crbk tln mrbua` wgtl ha gagtgho vnobjgty bc 889 ks ht ha hamon bc 39 wgtl lbrgzbatho. Crbk wlht lngmlt l

slbuo` ha habtlnr phrtgjon F fn prbenjtn` lbrgzbathooy wgtl vnobjgty 8> ks sb tlht fbtl tln phrtgjon jboog`n ga mrbua` ht pbgat J gc fbtl

hrn prbenjtn sgkuothanbusoy 2 689m ks

(H) 89 k (F) :9 k (J) 8> k (@) 6> k

0. H vnry frbh` nonvhtbr gs mbgam up vnrtgjhooy wgtl h jbasthat hjjnonrhtgba bc 66 ks . Ht tln gasthat wlna gts vnobjgty gs 80 ks h fhoo gs

prbenjtn` crbk tln cobbr bc tln ogst wgtl h spnn` bc 80 ks rnohtgvn tb tln cobbr ht ha nonvhtgba bc :9 . Pln tgkn thdna fy tln fhoo tb rntura

tln cobbr gs 5 689m ks

(H) 8/6 s (F) 8/: s (J) 8/0 s (@) 8 s

>. H phrtgjon gs prbenjtn` crbk tln mrbua` ht ha hamon bc 39 wgtl lbrgzbatho wgtl spnn` 869u ks . Pln rh`gus bc jurvhturn bc tln phtl bc

tln phrtgjon, wlna gts vnobjgty khdns ha hamon bc :9 wgtl lbrgzbatho gs 5 689m ks

(H) 89.3 k (F) 86.< k (J) 8>.0 k (@) 60.6 k

3. Vhamn gs khxgkuk cbr hamon bc prbenjtgba 5

(H) :9 39μ (F) 0>μ (J) 0>μ (@) 0>μ

=. Cbr h cgxn` rhamn bc prbenjtgon, ' V', tln kgagkuk vnobjgty bc prbenjtgba gsRRRRRR.

(H) m V (F) 6 m V (J) 6 m V (@)6

:m V

H J

F

l

39

889 ks

8

>ks

H F

6u

0> 39

J



Sg`yhkha`gr Johssns

SKJ/Kbtgba ga Pwb @gknasgba >< L]P/Ylysgjs

<. H kha ga h kbvgam fus `rbps h fhm crbk h lngmlt 6 k . Pln fhm oha`s ba tln mrbua` wgtl h vnobjgty 89 k / s. Pln vnobjgty bc tln fus ht tlnkbknat ln `rbppn` tln fhm whsRRRRR.

(H) 89 k / s (F) 6 89 k / s (J) 6 8> k / s (@) > k / s

7. Pln tgkn gt thdns tln fhm tb rnhjl tln mrbua` gsRRRRR.

(H) 8>

s (F) 6>

s (J) 6>

s (@) 6 > s

89. H fby tlrbws h fhoo upwhr`s wgtl vnobjgty v ht ha hamon μ wgtl tln lbrgzbatho ha` gkkn`ghtnoy sthrts ruaagam. Lbw chst kust ln rua tb

jhtjl tln fhoo 2(H) v jbs μ (F) v sga μ (J) v tha μ (@) v/ 6

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