pcm paper (09.04.2014)
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8/17/2019 PCM Paper (09.04.2014)
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QFS[ F = QETXDHX
6. Fn ixpird`int ds pirblr`ik tl lctfdn tei vfjui lb fhhijirftdln kui tl grfvdty g cy usdng f sd`pji pinkuju` lb jingte J. Dn teds ixpird`int td`i blr 611 lshdjjftdlns ds`ifsurik cy usdng f wfthe lb 6 sihlnk jifst hlunt fnk teivfjui ds 41.1 sihlnks. [ei jingte J ds ̀ ifsurik cy usdng f`itir shfji lb jifst hlunt 6 ̀ ` fnk tei vfjui ds 31.1 h`. [ei
irrlr dn tei kitir`dnftdln lb g wlujk ci=(f) 6.7% (c) 3.7%(h) ;.;% (k) 3.37%
3. [ei plsdtdln lb f prlmihtdji jfunheik brl` tei lrdgdn ft t 0 1
ds gdvin cy ( )˂ ˂;1 91 `r d m0 +r ft t 0 3s. Db tei prlmihtdji wfsjfunheik ft fn fngji q brl` tei elrdzlntfj, tein q ds (tfoi
g 0 61 ̀ s ‘3
)
(f)6 3tfn
8
-(c)
6 8tfn3
-
(h)6 7
tfn;
-(k)
6 ;tfn
9
-
8. Wftir ds bjlwdng ft f spiik lb 6.9 ̀ s ‘6 terluge f
elrdzlntfj tuci lb hrlss-sihtdlnfj frif 61 ‘3 `3 fnk ylu fritrydng tl stlp tei bjlw cy ylur pfj`. Fssu`dng teft teiwftir stlps d``ikdftijy fbtir edttdng tei pfj`, tei ̀ dnd`u`blrhi teft ylu ̀ ust ixirt selujk ci (kinsdty lb wftir 0 618
og ̀‘8)
(f) 69 N (c) 33.9 N(h) 88.7 N (k) ;9 N
;. F cjlho F lb ̀ fss ; og ds pjfhik ln fnlteir cjlho C lb ̀ fss9 og, fnk tei cjlho C rists ln f s`llte elrdzlntfj tfcji. Db tei ̀ dnd`u` blrhi teft hfn ci fppjdik ln F sl teft clte tei
cjlhos ̀ lvi tlgiteir ds 63 N, tei ̀ fxd`u` blrhi teft hfn cifppjdik tl C blr tei cjlhos tl ̀ lvi tlgiteir wdjj ci=(f) 81 N (c) 39 N(h) 37 N (k) ;5 N
9. [wl clkdis lb `fssis 6 og fnk ; og fri hlnnihtik tl f
virtdhfj sprdng, fs selwn dn tei bdguri. [ei s`fjjir `fssixihutis sd`pji efr`lndh `ltdln lb fngujfr briquinhy 39rfk/s, fnk f`pjdtuki 6.2 h` wedji tei cdggir ̀ fss ri`fdnsstftdlnfry ln tei grlunk. [ei ̀ fxd`u` blrhi ixirtik cy teisysti` ln tei bjllr ds (tfoi g 0 61 `s ‘3)
; og
6 og
(f) 31 N (c) 61 N
(h) 21 N (k) ;1 N2. F hyjdnkir lb `fss @h fnk speiri lb `fss @s fri pjfhik ft
pldnts F fnk C lb twl dnhjdnis, rispihtdvijy (Xii Bdguri). Db
teiy rljj ln tei dnhjdni wdtelut sdppdng suhe teft teidr
fhhijirftdlns fri tei sf`i, tein tei rftdlsdn
sdn
h
s
q
q ds=
F
C@
X
@H
q X
qH
HK
(f)5
7(c)
69
6;
(h)5
7(k)
69
6;
7. Dnkdf—s @fngfjyfn wfs sint tl tei @frs cy jfunhedng dt
dntl f trfnsbir lrcdt IL@ frlunk tei sun. Dt jifvis tei ifrteft I fnk `iits @frs ft @. Db tei si`d-`fmlr fxds lb Ifrte—s
lrcdt ds fi 0 6.9 × 6166 `, teft lb @frs lrcdt f` 0 3.35 × 6166`, tfoin Oipjir—s jfws gdvi tei istd`fti lb td`i blr @fngfjyfn tl rifhe @frs brl` Ifrte tl ci hjlsi tl=
MII @FDN LNJDNI QFQIS (14.1;.316;)
Lni lb tei jifkdng pucjdseirs dn Dnkdf, Kdsef Qucjdhftdln prlvdkis cllos fnk stuky ̀ ftirdfjs blr shelljs
fnk vfrdlus hl`pitdtdvi ixf`s cidng hlntdnulusjy eijk fhrlss tei hluntry. Kdsef's slji purplsi ds tl
inhlurfgi f stukint tl git tei cist lut lb pripfrftdln. Kdsef Qucjdhftdln lbbirs fn lnjdni cllostlri
tl eijp stukints cuy ixf` cllos lnjdni wdte ifsi. Wi, ft Kdsef prlvdki f wdki frrfy lb Cfno / Ingg./@ikdhfj & Lteir Hl`pitdtdvi Ixf` cllos tl eijp fjj telsi fspdrfnts wel wdse tl hrfho teidr rispihtdvi
kdbbirint jivijs lb Cfno / Ingg./ @ikdhfj & Lteir Hl`pitdtdvi ixf`s. Ft Kdsef Qucjdhftdln, wi strdvi
tl crdng lut tei cist gudkicllos teft stukints wlujk bdnk tl ci tei `lst usibuj blr fjj odnk lb
hl`pitdtdvi ixf`.
FCLY[ KDXEF QYCJDHF[DLN
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8/17/2019 PCM Paper (09.04.2014)
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62. [ei ̀ fgndtuki lb tei fvirfgi ijihtrdh bdijk nlr`fjjy prisintdn tei ft`lspeiri must fclvi tei surbfhi lb tei Ifrte dsfclut 691 N/H, kdrihtik dnwfrk tlwfrks tei hintir lb tei
Ifrte. [eds gdvis tei tltfj nit surbfhi hefrgi hfrrdik cy teiIfrte tl ci=
_Gdvin il 0 5.59 × 61 ‘63 H3/N-`3, S I 0 2.87 × 612 ̀ P(f) + 271 oH (c) ‘ 271 oH(h) ‘ 251 oH (k) + 251 oH
67. [erii hfpfhdtlrs, ifhe lb 8 `B, fri prlvdkik. [eisi hfnnlt ci hl̀ cdnik tl prlvdki tei risujtfnt hfpfhdtfnhi lb=(f) 6 `B (c) 3 `B(h) ;.9`B (k) 2 `B
65. F k.h. `fdn suppjy lb i.`.b. 331 R ds hlnnihtik fhrlss f
stlrfgi cfttiry lb i.`.b. 311 R terluge f risdstfnhi lb 6W.[ei cfttiry tir`dnfjs fri hlnnihtik tl fn ixtirnfj risdstfnhi―S—. [ei `dnd`u` vfjui lb ―S—, sl teft f hurrint pfssisterluge tei cfttiry tl hefrgi dt ds=(f) 7W (c) 4W
(h) 66W (k) ]irl64. [ei ̀ dk pldnts lb twl s`fjj ̀ fgnitdh kdpljis lb jingte k dn
ink-ln plsdtdlns, fri sipfrftik cy f kdstfnhi x, (x > > k).[ei blrhi citwiin tei` ds prlplrtdlnfj tl x ‘n weiri n ds=
NX N
x
X
(f) 6 (c) 3(h) 8 (k) ;
31. [ei ̀ fgnitdh bdijk lb ifrte ft tei iquftlr ds fpprlxd`ftijy ;× 61 ‘9 [. [ei rfkdus lb ifrte ds 2.; × 612 ̀ . [ein tei kdplji
`l`int lb tei ifrte wdjj ci nifrjy lb tei lrkir lb=(f) 6138 F ̀ 3 (c) 6131 F ̀ 3
(h) 6162 F ̀ 3 (k) 6161 F ̀ 3
36. Wein tei r`s vljtfgis RJ, RH fnk RS fri `ifsurikrispihtdvijy fhrlss tei dnkuhtlr J, tei hfpfhdtlr H fnk tei
risdstlr S dn f sirdis JHS hdrhudt hlnnihtik tl fn FH slurhi,dt ds blunk teft tei rftdl RJ = RH = RS 0 6 = 3 = 8. Db tei r`svljtfgi lb tei FH slurhis ds 611 R, tei R S ds hjlsi tl=(f) 91 R (c) 71 R
(h) 41 R (k) 611 R33. @fthe Jdst D (Wfvijingte rfngi lb ijihtrl`fgnitdh spihtru`)
wdte Jdst DD (@itelk lb prlkuhtdln lb teisi wfvis) fnk sijihttei hlrriht lptdln brl` tei lptdlns gdvin cijlw tei jdsts.
(6) 711 n` tl
6 ``
(d) Rdcrftdln lb ftl s̀
fnk `ljihujis.
(3) 6 n` t l
;11 n`
(dd) Dnnir seijj ijihtrlns
dn ftl`s `lvdng
brl` lni inirgy
jivij tl f jlwir jivij.
(8) : 61 ‘8
n` (ddd) Sfkdlfhtdvi kihfy lb
tei nu hjius.
(;) 6 `` tl
1.6 `
(dv) @fgnitrln vfjvi.
Jdst D Jdst DD
(f) (6)-(dv), (3)-(ddd), (8)-(dd), (;)-(d)(c) (6)-(ddd), (3)-(dv), (8)-(d), (;)-(dd)(h) (6)-(dd), (3)-(ddd), (8)-(dv), (;)-(d)
(k) (6)-(d), (3)-(dd), (8)-(ddd), (;)-(dv)38. F kdvir jllodng up terluge tei wftir siis tei lutsdki wlrjk
hlntfdnik dn f hdrhujfr elrdzln. [ei ribrfhtdvi dnkix lb wftir
ds;
8, fnk tei kdvir—s iyis fri 69 h` cijlw tei surbfhi lb
wftir. [ein tei rfkdus lb tei hdrhji ds=
(f) 69 8 9» » h` (c) 69 8 7» h`
(h)69 7
8
» h` (k)
69 8
7
» h`
3;. Ysdng `lnlherl`ftdh jdget lb wfvijingte j, fnixpird`intfjdst sits up tei Tlung—s klucji sjdt ixpird`int
dn terii wfys fs selwn.Db sei lcsirvis teft y 0 c—, tei wfvijingte lb jdget usik ds=
K
Hintrfj brdngi
(Brdngi wdkte )c
(Xhriin)
L
s6
s3
k
K
Hintrfj brdngi
(Brdngi
wdkte )c
(Xhriin)
L
s6
s3
[rfnspfrint
`dhf seiit lb tedhoniss
t 0 6.5 `
0 6.2
`
`
y
k
3K
Hintrfj brdngi
(Brdngi wdkte 0 )c¢
s6
s3
k
(f) 931 n` (c) 9;1 n`
(h) 921 n` (k) 951 n`39. [ei blhfj jingtes lb lcmihtdvi jins fnk iyi jins lb f Gfjdjifn
tijishlpi fri rispihtdvijy 81 h` fnk 8.1 h`. tijishlpi
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prlkuhis vdrtufj, iriht d`fgi lb fn lcmiht sdtuftik bfr fwfybrl` dt ft jifst kdstfnhi lb kdstdnht vdsdln brl` tei iyi jins.Dn teds hlnkdtdln, tei `fgndbydng plwir lb tei Gfjdjifn
tijishlpi selujk ci=(f) + 66.3 (c) ‘ 66.3
(h) ‘ 5.5 (k) + 5.532. Blr wedhe lb tei bljjlwdng pfrtdhjis wdjj dt ci ̀ lst kdbbdhujt tl
ixpird`intfjjy virdby tei ki-Crlgjdi rijftdlnsedp<(f) fn ijihtrln (c) f prltln
(h) fn f-pfrtdhji (k) f kust pfrtdhji37. [ei cdnkdng inirgy lb tei ijihtrln dn f eykrlgin ftl` ds
68.2 iR, tei inirgy riqudrik tl ri`lvi tei ijihtrln brl` teibdrst ixhdtik stfti lb Jd++ ds=
(f) 633.; iR (c) 81.2 iR(h) 68.2 iR (k) 8.; iR
35. Dkintdby tei gfti fnk ̀ fthe F, C, T dn crfhoit tl heiho.
FC
F.C
T(f) FNK (F 0 6, C 0 6, T 0 6)(c) LS (F 0 6, C 0 6, T 0 1)(h) NL[ (F 0 6, C 0 6, T 0 6)(k) ULS (F 0 1, C 0 1, T 0 1)
34. F trfns`dttdng fntinnf ft tei tlp lb f tlwir efs eidget 83 ̀fnk eidget lb tei rihidvdng fntinnf ds 91 `. Weft ds tei`fxd`u` kdstfnhi citwiin tei` blr sftdsbfhtlryhl``undhftdln dn jdni lb sdget (JLX) ̀ lki<(f) 99.; o` (c) ;9.9 o`(h) 9;.9 o` (k) ;99 o`
81. Fn n-p-n trfnsdstlr efs terii jifks F, C fnk H. HlnnihtdngC fnk H cy ̀ ldst bdngirs, F tl tei plsdtdvi jifk lb fn f``itir,
fnk H tl tei nigftdvi jifk lb tei f``itir, lni bdnks jfrgikibjihtdln. [ein, F, C fnk H ribir rispihtdvdjy tl=(f) I`dttir, cfsi fnk hljjihtlr (c) Cfsi, i`dttir fnk hljjihtlr
(h) Cfsi, hljjihtlr fnk i`dttir (k) Hljjihtlr, ì dttir fnk cfsi.
QFS[ C = HEI@DX[ST
86. Dn f bfhi hintirik hucdh jfttdhi ftl`s F fri ft tei hlrnir pldnts fnk ftl`s C ft tei bfhi hintirik pldnts. Db ftl` C ds`dssdng brl` lni lb tei bfhi hintirik pldnts, tei blr`ujf lb tei dlndh hl`plunk ds=
(f) FC3 (c) F9C3(h) F3C8 (k) F3C9
83. Rfn kir Wffj—s iquftdln blr f gfs ds stftik fs,
3nS[ n
p fR nc R
â ò0 - ç ÷- å õ.
[eds iquftdln rikuhis tl tei pirbiht gfs iquftdln,nS[
pR
0
wein,(f) ti`pirfturi ds subbdhdint edge fnk prissuri ds jlw.
(c) ti`pirfturi ds subbdhdint jlw fnk prissuri ds edge.(h) clte ti`pirfturi fnk prissuri fri viry edge.(k) clte ti`pirfturi fnk prissuri fri viry jlw.
88. [ei stfnkfrk ijihtrlki pltintdfjs1
@ /@I + lb blur `itfjs
F, C, H fnk K fri ‘ 6.3 R, 1.2 R, 1.59 R fnk ‘ 1.72 R,rispihtdvijy. [ei siquinhi lb kiplsdtdln lb `itfjs lnfppjydng pltintdfj ds=
(f) F, H, C, K (c) C, K, H, F(h) H, C, K, F (k) K, F, C, H
8;. Ft f hirtfdn ti`pirfturi, lnjy 91% ED ds kdsslhdftik dntl E3fnk D3 ft iqudjdcrdu`. [ei iqudjdcrdu` hlnstfnt ds=(f) 6.1 (c) 8.1(h) 1.9 (k) 1.39
89. Kdssljvdng 631 g lb f hl`plunk lb (`lj. wt. 21) dn 6111 g lb wftir gfvi f sljutdln lb kinsdty 6.63 g/`J. [ei ̀ ljfrdty lb tei sljutdln ds=(f) 6.11 @ (c) 3.11 @
(h) 3.91 @ (k) ;.11 @82. [ei efjb-jdbi pirdlk lb f bdrst lrkir rifhtdln ds 69 `dnutis.
[ei f`lunt lb sucstfnhi jibt fbtir lni elur wdjj ci=
(f)6
;
lb tei lrdgdnfj f`lunt
(c)6
5 lb tei lrdgdnfj f`lunt
(h)6
62 lb tei lrdgdnfj f`lunt
(k)6
83 lb tei lrdgdnfj f`lunt
87. F hurrint lb 61.1 F bjlws blr 3.11 e terluge fn ijihtrljytdhhijj hlntfdndng f `ljtin sfjt lb `itfj U. [eds risujts dn teikihl`plsdtdln lb 1.391 ̀ lj lb ̀ itfj U ft tei hftelki. [eilxdkftdln stfti lb U dn tei ̀ ljtin sfjt ds= (B 0 42,911 H)
(f) 6 + (c) 3 +(h) 8 + (k) ; +
85. [ei inirgy lb fn ijihtrln dn bdrst Cler lrcdt lb E-ftl` ds ‘ 68.2 iR. [ei inirgy vfjui lb ijihtrln dn tei ixhdtik stfti lb
Jd3+ ds=(f) ‘ 37.3 iR (c) 81.2 iR(h) ‘ 81.2 iR (k) 37.3 iR
84. [ei ti`pirfturi ft wedhe lxygin ̀ ljihujis efvi tei sf`irllt ̀ ifn squfri spiik fs eijdu` ftl`s efvi ft 811 O ds=
Ftl`dh ̀ fssis= Ei 0 ; u, L 0 62 u)(f) 811 O (c) 211 O (h) 6311 O (k) 3;11 O
;1. [ei stfnkfrk intefjpy lb blr`ftdln lb NE8 ds ‘ ;2.1 oM/`lj.
Db tei intefjpy lb blr`ftdln lb E3 brl` dts ftl`s ds ‘ ;82 oM/`lj fnk teft lb N3 ds ‘ 763 oM/`lj, tei fvirfgi clnk intefjpylb N - E clnk dn NE8 ds=(f) ‘ 6613 oM/`lj (c) ‘ 42; oM/`lj(h) + 893 oM/`lj (k) + 6192 oM/`lj
;6. [ei f`lunt lb lxygin dn 8.2 `ljis lb wftir ds=
(f) 669.3 g (c) 97.2 g(h) 35.5 g (k) 65.; g
;3. [ei gfs ivljvik ln eiftdng HfB3 fnk XdL3 wdte hlnhintrftikE3XL;, ln eykrljysds gdvis f wedti gijftdnlus prihdpdtfti.[ei prihdpdtfti ds=(f) eykrlbjulsdjdhdh fhdk(c) sdjdhf gij
(h) sdjdhdh fhdk(k) hfjhdu`bjulrlsdjdhfti
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;8. Hejlrl hl`plunk lb Rfnfkdu` efs lnjy spdn `fgnitdh`l`int lb 6.78 C@. [eds Rfnfkdu` hejlrdki efs tei blr`ujf=(f) RHj3 (c) RHj;(h) RHj8 (k) RHj9
;;. Fn lhtfeikrfj hl`pjix lb Hl8+ ds kdf`fgnitdh. [ei
eycrdkdsftdln dnvljvik dn tei blr`ftdln lb tei hl`pjix ds=(f) sp8k 3 (c) ksp3
(h) k 3 sp8 (k) sp8k ;9. Wedhe lb tei bljjlwdng ds nlt blr`ik wein E3X rifhts wdte
fhdkdh O 3Hr 3L7 sljutdln<
(f) HrXL; (c) Hr 3(XL;)8(h) O 3XL; (k) X
;2. Wedhe lb tei bljjlwdng efs unpfdrik ijihtrln(s)<
(f) N3 (c) 3L-
(h) 33 N + (k) 3
3L -
;7. Dn tei bljjlwdng sits lb rifhtfnts wedhe twl sits cist ixedcdt
tei f`peltirdh hefrfhtirs lb Fj3L8. xE3L<Xit 6= Fj3L8.xE3L (s) fnk LE
‘ (fq)Xit 3= Fj3L8.xE3L (s) fnk E3L (j )Xit 8= Fj3L8.xE3L (s) fnk E
+ (fq)
Xit ;= Fj3L8.xE3L (s) fnk NE8 (fq)(f) 6 fnk 3 (c) 6 fnk 8(h) 3 fnk ; (k) 8 fnk ;
;5. [ei nu`cir fnk typi lb clnks dn 33H - dln dn HfH3 fri=
(f) Lni s clnk fnk lni p-clnk(c) Lni s clnk fnk twl p-clnk(h) [wl s clnk fnk twl p-clnk(k) [wl s clnk fnk lni p-clnk
;4. [ei blr` lb drln lctfdnik brl` cjfst burnfhi ds=(f) Xtiij (c) Hfst Drln
(h) Qdg Drln (k) Wrluget Drln91. [ei hlrriht stfti`int fclut tei `fgnitdh prlpirtdis lb
_Bi(HN)2P8‘ fnk _BiB2P
8‘ ds(] 0 32) =(f) clte fri pfrf`fgnitdh.
(c) clte fri kdf`fgnitdh.(h) _Bi(HN)2P
8‘ ds kdf`fgnitdh, _BiB2P8‘ ds pfrf`fgnitdh.
(k) _Bi(HN)2P8‘ ds pfrf`fgnitdh, _BiB2P
8‘ ds kdf`fgnitdh.
96. Wedhe lni lb tei bljjlwdng rifhtdlns wdjj nlt risujt dn teiblr`ftdln lb hfrcln-hfrcln clnk<
(f) Sid`ir-[dì fn rifhtdln
(c) Brdikij Hrfbt—s fhyjftdln(h) Wurtz rifhtdln(k) Hfnndzzfrl rifhtdln
93. Dn tei eykrlhfrclrftdln - lxdkftdln rifhtdln lb prlpini wdtekdclrfni, E3L3 fnk NfLE, tei lrgfndh hl`plunk blr`ikds=(f) HE8HE3LE (c) HE8HELEHE8(h) HE8HE3HE3LE (k) (HE8)8HLE
98. [ei `fmlr prlkuht lb tei rifhtdln
LE NE3
3 3 ; NfNL /E XL ¾¾¾¾¾¾¾®
(f)
LEE
(c)
L
(h)
(k) NE
9;. Blr tei hl`plunksHE8Hj, HE8Cr, HE8D fnk HE8B,tei hlrriht lrkir lb dnhrifsdng H-efjlgin clnk jingte ds=(f) HE8B : HE8Hj : HE8Cr : HE8D
(c) HE8B : HE8Cr : HE8Hj : HE8D(h) HE8B : HE8D : HE8Cr : HE8Hj(k) HE8Hj : HE8Cr : HE8B : HE8D
99. Fjjyj peinyj iteir hfn ci pripfrik cy eiftdng=(f) H2E9Cr + HE3 0 HE ‘ HE3 ‘ LNf(c) HE3 0 HE ‘ HE3 ‘ Cr ‘ H2E9LNf
(h) H2E9 ‘ HE 0 HE ‘ Cr ‘ HE8 ‘ LNf(k) HE3 0 HE ‘ Cr ‘ H2E9 ‘ HE3 ‘ LNf
92. Dn f nuhjilpedjdh sucstdtutdln rifhtdln=
K@BS ‘ Cr Hj S ‘ Hj Cr - -+ ¾¾¾® + ,wedhe lni lb tei bljjlwdng unkirglis hl`pjiti dnvirsdlnlb hlnbdgurftdln<
(f) H2E9HEH2E9Cr (c) H2E9HE3Cr (h) H2E9HEHE8Cr (k) H2E9HHE8H2E9Cr
97. Dn wedhe lb tei bljjlwdng pfdrs F ds `lri stfcji tefn C<
F C
(f) ,
(c) ,
(h) ,
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(k) ( )8 8 8Qe H , HE H· ·
95. Xtruhturi lb sl̀ i d`plrtfnt pljỳ irs fri gdvin. Wedhe lniriprisints Cunf-X<
(f) (‘ H ‘ H 0 HE ‘ H ‘)3 3E E
HE8
n
(c) (‘ H ‘ HE 0 HE ‘ H ‘ HE ‘ HE ‘)3 3 3E E
H E2 9
n
(h) (‘ H ‘ HE 0 HE ‘ H ‘ HE ‘ HE ‘)3 3 3E E
HN
(k) (‘ H ‘ H 0 HE ‘ H ‘)3 3E E
Hj
n
94. Wedhe ds ̀ fmlr prlkuht blr`ik wein fhitlni ds eiftik wdtedlkdni fnk pltfssdu` eykrlxdki<
(f) Dlklfhitlni (c) Fhitdh fhdk(h) Dlklblr` (k) Fhitlpeinlni
21. Wedhe lni lb tei bljjlwdng hjfss lb hl`plunks ds lctfdnik cy pljy`irdzftdln lb fhityjini<
(f) Qljy-yni (c) Qljy-ini(h) Qljy-istir (k) Qljy-f`dni
QFS[ H = @F[EI@F[DHX
26. Jit Q ci tei rijftdln kibdnik ln tei sit lb fjj rifj nu`cirs
suhe teftQ 0 {(f, c) = sih3f ‘ tfn3 c 0 6}. [ein Q ds=(f) ribjixdvi fnk sy``itrdh cut nlt trfnsdtdvi.(c) ribjixdvi fnk trfnsdtdvi cut nlt sỳ `itrdh.(h) sy``itrdh fnk trfnsdtdvi cut nlt ribjixdvi.
(k) fn iqudvfjinhi rijftdln.23. Jit w (D` w ´ 1) ci f hl`pjix nu`cir. [ein tei sit lb fjj
hl`pjix nu`cir z sftdsbydng tei iquftdln
( )w wz o 6 z- 0 - , blr sl`i rifj nu`cir o, ds
(f) { }z = z 60 (c) { }z = z z0
(h) { }z = z 6´ (k) { }z = z 6, z 60 ´
28. Db iquftdlns fx3 + cx + h 0 1 ( )f,c,h S,f 1Î ´ fnk 3x3 + 8x
+ ; 0 1 efvi f hl``ln rllt, teinf = c = h iqufjs=(f) 6 = 3 = 8 (c) 3 = 8 = ;(h) ; = 8 = 3 (k) 8 = 3 = 6
2;. Db6
f fnk
6
c fri tei rllts lb tei iquftdln, fx3 + cx + 6 0
1 (f ´ 1, f, c, Î S), tein tei iquftdln,
( ) ( )8 8x x c f 8fcx 1+ + - 0 efs rllts=
(f)8
3f fnk8
3c
(c) 63fc fnk 63f c
(h) fc fnk fc
(k)8
3-
f fnk8
3-
c
29. Db f, c, h fri nln-zirl rifj nu`cirs fnk db tei systì lb iquftdlns
(f ‘ 6)x 0 y + z,(c ‘ 6)y 0 z + x,(h ‘ 6)z 0 x + y,
efs f nln-trdvdfj sljutdln, tein fc + ch + hf iqufjs=(f) f + c + h (c) fch
(h) 6 (k) ‘ 622. Db C ds f 8 × 8 `ftr dx suhe teft C3 0 1, tein kit.
_(D + C)91 ‘ 91CP ds iqufj tl=(f) 6 (c) 3(h) 8 (k) 91
27. [ei nu`cir lb tir`s dn tei ixpfnsdln lb (6 + x)616 (6 + x3 ‘ x)611 dn plwirs lb x ds=(f) 813 (c) 816
(h) 313 (k) 61625. [ei su` lb tei kdgdts dn tei undt—s pjfhi lb fjj tei ;-kdgdtnu`cirs blr`ik cy usdng tei nu`cirs 8, ;, 9 fnk 2, wdtelutripitdtdln, ds=(f) ;83 (c) 615(h) 82 (k) 65
24. Gdvin fn F.Q. welsi tir`s fri fjj plsdtdvi dntigirs. [ei su`
lb dts bdrst ndni tir`s ds griftir tefn 311 fnk jiss tefn 331. Db tei sihlnk tir` dn dt ds 63, tein dts ;te tir` ds=(f) 5 (c) 62(h) 31 (k) 3;
71. Db tei su`
3 3 3 3 3 3
8 9 7
......6 6 3 6 3 8+ + + ++ + + up tl 31 tir`s ds iqufj
tlo
36, tein o ds iqufj tl=
(f) 631 (c) 651(h) 3;1 (k) 21
76. Db b(x) ds hlntdnulus fnk4 3
b 3 4
â ò0ç ÷
å õ, tein 3x 1
6 hls 8xjd` b
x®
-â òç ÷å õ
ds iqufj tl=
(f)4
3(c)
3
4
(h) 1 (k)5
4
73. Db y 0 inx, tein
3 3
3 3
k y k x
kx ky
â òâ òç ÷ç ÷ç ÷ç ÷å õå õ
ds iqufj tl=
(f) ninx (c) ni ‘nx
(h) 6 (k) ‘ ni ‘nx
78. Db tei Sljji—s teilri` eljks blr tei bunhtdln
b(x) 0 3x8 + fx3 + cx dn tei dntirvfj _‘ 6, 6P blr tei pldnt6
h3
0 ,
tein tei vfjui lb 3f + c ds=(f) 6 (c) ‘ 6(h) 3 (k) ‘ 3
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7;. Db ( )x x
8 ;b x 6
9 9
â ò â ò0 + -ç ÷ ç ÷å õ å õ
, x S Î , tein tei iquftdln b(x) 0
1 efs =(f) nl sljutdln
(c) lni sljutdln(h) twl sljutdlns(k) `lri tefn twl sljutdlns
79.
( )
5 5
3 3
sdn x hls xkx
6 3sdn x hls x
-
-ó ds iqufj tl=
(f)6
sdn 3x h3
+
(c)6
sdn 3x h3
- +
(h) 6 sdn x h3
- +
(k) 3sdn x h- +
72. [ei dntigrfj( )
6
3
31
Dn 6 3xkx
6 ;x
+
+ó, iqufjs=
(f) Dn3;
p(c) Dn3
5
p
(h) Dn362
p(k) Dn3
83
p
77. Jit F 0 {(x, y)= y3 £ ;x, y ‘ 3x ¸ ‘ ;}. [ei frif (dn squfriundts) lb tei rigdln F ds=(f) 5 (c) 4(h) 61 (k) 66
75. Db tei kdbbirintdfj iquftdln riprisintdng tei bf`djy lb fjj hdrhjis
tluhedng x-fxds ft tei lrdgdn ds ( ) ( )3 3 kyx y g x ykx
- 0 , tein
g(x) iqufjs=
(f)6
x3
(c) 3x3
(h) 3x (k) 36 x3
74. Jit f fnk c ci fny twl nu`cirs sftdsbydng3 3
6 6 6
;f c+ 0 .
[ein, tei bllt lb pirpinkdhujfr brl̀ tei lrdgdn ln tei vfrdfcji
jdni,x y
6f c
+ 0 , jdis ln=
(f) f eypircljf wdte ifhe si`d-fxds 0 3
(c) f eypircljf wdte ifhe si`d-fxds 0 3(h) f hdrhji lb rfkdus 0 3
(k) f hdrhji lb rfkdus 0351. Gdvin terii pldnts Q, V, S wdte Q(9, 8) fnk S jdis ln tei x-
fxds. Db iquftdln lb SV ds x ‘ 3y 0 3 fnk QV ds pfrfjjij tl tei
x-fxds, tein tei hintrldk lb KQVS jdis ln tei jdni=(f) 3x + y ‘ 4 0 1 (c) x ‘ 3y + 6 0 1(h) 9x ‘ 3y 0 1 (k) 3x ‘ 9y 0 1
56. Db tei pldnt (6, ;) jdis dnsdki tei hdrhji x3 + y3 ‘ 2x ‘ 61y + p 01 fnk tei hdrhji klis nlt tluhe lr dntirsiht tei hllrkdnfti
fxis, tein tei sit lb fjj plssdcji vfjuis lb p ds tei dntirvfj=(f) (1, 39) (c) (39, 84)(h) (4, 39) (k) (39, 34)
53. Db LC ds tei si`d-`dnlr fxds lb fn ijjdpsi, B6 fnk B3 fri dts
blhd fnk tei fngji citwiin B6C fnk B3C ds f rdget fngji,tein tei squfri lb tei ihhintrdhdty lb tei ijjdpsi ds=
(f)6
3(c)
6
3
(h)6
3 3(k)
6
;
58. Iquftdln lb tei pjfni wedhe pfssis terluge tei pldnt lb
dntirsihtdln lb jdnisx 6 y 3 z 8
8 6 3
- - -0 0 fnk
x 8 y 6 z 3
6 3 8
- - -0 0
fnk efs tei jfrgist kdstfnhi brl` tei lrdgdn ds=(f) 7x + 3y + ;z 0 9;(c) 8x + ;y + 9z 0 ;4
(h) ;x + 8y + 9z 0 91(k) 9x + ;y + 8z 0 97
5;. F jdni dn tei 8-kd`insdlnfj spfhi `fois fn fngji q
13pâ ò: q £ç ÷å õ wdte clte tei x fnk y fxis. [ein tei sit lb fjj
vfjuis lb q ds tei dntirvfj=
(f) 1,;
pâ ôç ýå ÿ
(c) ,2 8
p pä ôí ýï ÿ
(h) ,; 3
p pä ôí ýï ÿ
(k) ,8 3
p pâ ôç ýå ÿ
59. Db f 3, c 8® ®
0 0 fnk 3 f c 9® ®
- 0 , tein 3 f c® ®
+ iqufjs=
(f) 67 (c) 7
(h) 9 (k) 6
52. Dn f sit lb 3n kdstdnht lcsirvftdlns, ifhe lb tei lcsirvftdlns
cijlw tei `ikdfn lb fjj tei lcsirvftdlns ds dnhrifsik cy 9
fnk ifhe lb tei ri`fdndng lcsirvftdlns ds kihrifsik cy 8.
[ein tei `ifn lb tei niw sit lb lcsirvftdlns=
(f) dnhrifsis cy 6 (c) kihrifsis cy 6(h) kihrifsis cy 3 (k) dnhrifsis cy 3
57. Db F fnk C fri twl ivints suhe teft ( ) ( )Q F C Q F CÅ 0 Ç ,tein tei dnhlrriht stfti`int f`lngst tei bljjlwdngstfti`ints ds=(f) F fnk C fri iqufjjy jdoijy
(c) ( )Q F C 1¢Ç 0
(h) ( )Q F C 1¢ Ç 0(k) Q(F) + Q(C) 0 6
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EDN[X & XLJY[DLN
Qeysdhs
6. (c) Fhhlrkdng tl tei quistdln.
t 0 (41 ¶ 6) lr,6
41
t
t
K0
j 0 (31 ¶ 1.6) lr,1.6
31
j
j
K0
% < g
g
K0
Fs wi onlw,
3 j t g
0 p
Þ3
; j g
t
3p0
lr, 3K K Kâ ò
0 ¶ +ç ÷å õ g j t
g j t
01.6 6
331 41
â ò+ »ç ÷
å õ 0 1.137
Z % 3.7% g
g
K0
3. (h) Brl̀ quistdln,
Elrdzlntfj vijlhdty (dndtdfj),
;131`/s
30 0 xu
Rirtdhfj vijlhdty (dndtdfj), 91 0 u y t +6
3 gt 3
Þ u y × 3 +6
3
(‘61) ×;
lr, 91 0 3u y ‘ 31
lr, u y 071
89`/s3
0
Z89 7
tfn31 ;
q 0 0 0 y
x
u
u
Þ Fngji q 0 tfn ‘6 7
;8. (f) Blr 6 ̀ jingte lb elrdzlntfj tuci
@fss lb wftir @ 0 kinsdty × vlju`i
0 618 × frif × jingte
0 618 × 61 ‘3 × 6 0 61 og
[eiriblri `dnd`u` blrhi 0t
KK
(rfti lb hefngi lb
`l`intu`)
0 61 × 6.9 0 69 N ;. (h) @dnd`u` blrhi ln F
0 brdhtdlnfj blrhi citwiin tei surbfhis
0 63 N
F
C
; og
9 og
X`llte tfcji
[eiriblri ̀ fxd`u` fhhijirftdln
f`fx 0363N 8`/s
;og0
Einhi `fxd̀ u` blrhi,
B`fx 0 tltfj `fss × f`fx 0 4 × 8 0 37 N
9. (h) @fss lb cdggir clky @ 0 ; og
@fss lb s`fjjir clky ̀ 0 6 og
X`fjjir `fss (` 0 6 og) ixihutis X.E.@ lb fngujfr
briquinhy w 0 39 rfk/sF`pjdtuki x 0 6.2 h` 0 6.2 × 61 ‘3
Fs wi onlw,
`[ 3
O 0 p
lr,3 `
3O
p0 p
w
lr, _ P6 6
` 6og? 39 rfk / s39 O
0 0 w 0V
lr, O 0 239 N ̀‘6.
[ei ̀ fxd`u` blrhi ixirtik cy tei systi` ln tei bjllr 0 @g + Ox + ̀ g
0 ; × 61 + 239 × 6.2 × 61 ‘3 + 6 × 61
0 ;1 + 61 + 61
0 21 N
2. (k) Fs wi onlw,
Fhhijirftdln,
3
`g sdnf
D`
r
q0
+
Blr hyjdnkir,h h
h 3
hh 3
@ .g.sdnf
@ S 6@3 S
q0
+
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61
0 h h3
hh 3
@ .g.sdn
@ S @
3S
q
+
lr, fh 0 h3 g sdn8
q
Blr speiri,
s ss
ss 3
@ g sdnf
D@
r
q0
+
0s s
3
s 3
@ g sdn
3 @S @
9 S
q
+
lr, s s9f g sdn7
0 q
gdvin, fh 0 fs
d.i.,h s
3 9g sdn g sdn
8 7q 0 q
Z h
s
9g
sdn 6973sdn 6;
g8
q0 0
q
7. (h)
5. (h) Qldssln—s rftdl,( )
( )
jftirfj strfdn
jlngdtukdnfj strfdn
cs 0
f
Blr ̀ ftirdfj jdoi hlppir, s 0 1.88Fnk, y 0 8o (6 ‘ 3 s)
Fjsl,4 6 8
y o n0 +
y 0 3n (6+ s)Einhi, n : y : o
4. (k) Fs wi onlw,
x 0 x1 i ‘ct/3`
Brl` quistdln,;1c
3`5 61i-
0 ....(d)
Xd̀ djfrjy,
t
3`9 61i-
0 ....(dd)
Xljvdng iqns (d) fnk (dd) wi git
t 6;3 sA61. (f) Fhhijirftdln kui tl grfvdty hefngis wdte tei kipte,
kg g 6
S
â ò¢ 0 -ç ÷å õ
Qrissuri, Q 0 rge
Einhi rftdl,x k
ds 6y S
â ò-ç ÷å õ
66. (f) Brl̀ quistdln,
Dn 6 sih eift gfdnik cy wftir
0 6 OW ‘ 621 M/s
0 6111 M/s ‘ 621 M/s
0 5;1 M/s
[ltfj eift riqudrik tl rfdsi tei ti`pirfturi lb wftir (vlju`i 3J) brl` 37±h tl 77±h
0 `wftir ×sp. et × Kq
0 3 × 618 × ;.3 × 91 _V `fss 0 kinsdty × vlju`iP
Fnk, 5;1 × t 0 3 × 618 × ;.3 × 91
lr,
83 61 ;.3 91t
5;1
» » »0
0 911 s 0 5 ̀ dn 31s
63. (f)
68. (c)
6;. (c) [wl pirpinkdhujfr X.E.@s fri
x 0 f6 hls wt ....(6)fnk y 0 f3 3 hls wt ....(3)Brl` iqn (6)
6
xhls wt
f0
fnk brl` iqn (3)
3
y3hls t
f0 w
Z y 0 3
6
f3 x
f
69. (f) Gdvin, f̀ pjdtuki f 0 61 h`
wfvi vijlhdty 0 3 × ̀ fxd`u` pfrtdhji vijlhdty
d.i, f33
wj w0
p plr, j 0 ;f 0 ; × 61 0 ;1 h`
62. (h) Gdvin,
Ijihtrdh bdijk I 0 691 N/H
[ltfj surbfhi hefrgi hfrrdik cy ifrte q 0 <
Fhhlrkdng tl Gfuss—s jfw.
1
qIFb 0 0
Î
lr, q 0 Î1 I F0 Î1 I p r
3.
0 5.59 × 61 ‘63 × 691 × (2.87 × 612)3.
251 Oh
Fs ijihtrdh bdijk kdrihtik dnwfrk einhi
q 0 ‘ 251 Oh
67. (k) Qlssdcji hl`cdnftdln lb hfpfhdtlrs
(d) [erii hfpfhdtlrs dn sirdis hl`cdnftdln
8¹B 8 B¹ 8 B¹
iq
6 6 6 6H 8 8 8
0 + +
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66
Ziq
66 B
H0 `
(dd) [erii hfpfhdtlrs dn pfrfjjij hl`cdnftdln
8¹B
8 B¹
8 B¹
Hiq 0 8 + 8 + 8 0 4 ¹B
(ddd) [wl hfpfhdtlrs dn pfrfjjij fnk lni ds dn sirdis
8¹B
8¹B
8¹B
Hiq 0 3¹B
(dv) [wl hfpfhdtlrs dn sirdis fnk lni ds dn pfrfjjij
Hiq 0 ;.9 ¹B
65. (h) Gdvin, i`b lb hijj I 0 311 RDntirnfj risdstfnhi lb hijjs 0 6 WK. H. `fdn suppjy vljtfgi R 0 331 R
Ixtirnfj risdstfnhi S 0 <
I Rr S
R
-â ò0 ç ÷
å õ
316 S
331
â ò0 »ç ÷
å õZ S 0 66W.
64. (k) Dn ̀ fgnitdh kdplji
Blrhi ;6r ¹
Dn tei gdvin quistdln,
Blrhi ¹ x ‘ n
Einhi, n 0 ;
31. (f) Gdvin, C 0 ; × 61 ‘9 [
S I 0 2.; × 612 ̀
Kdplji ̀ l`int lb tei ifrte @ 0 <
C 01
8
@
; k
`p
( )
7
9 82
; 61 @; 61
; 2.; 61
-
- p» »
» 0p» »
Z @ A 6138 F`3
36. (h) Gdvin, RJ = RH = RS 0 6 = 3 = 8
R 0 611 R
RS 0 <
Fs wi onlw,
( )33
S J HR R R R0 + -
Xljvdng wi git, S R 41R
33. (k) Rdcrftdln lb ftl`s fnk ̀ ljihujis 711 n` tl 6 ̀ `
Sfkdlfhtdvi kihfy lb tei nuhjius : 61 ‘8 n`
@fgnitrln vfjvi 6 ̀ ` tl 1.6 ̀
38. (k) Gdvin, ¹ 0;
8
e 0 69 h`
S 0 <
e
H
L
S
sdn 41
sdnH
±0 `
Þ3 3
6 S 8sdnH
;S e0 0 0
` +
Þ 62 S 3 0 4 S 3 + 4 e3
lr, 7S 3 0 4 e3
lr,8 8
S e 69h`7 7
0 0 »
3;. (c) Gdvin t 0 6.5 × 61 ‘2`
¹ 0 6.2
n 0 3 (brl` bdguri)
Fppjydng blr`ujf (¹ ‘ 6) t 0 nj(6.2 ‘ 6) × 6.5 × 61 ‘2 0 3j
lr,2
6.5 61 1.2
3
-» »j 0
0 9;1 n`
39. (k) Gdvin, Blhfj jingte lb lcmihtdvi, b 1 0 81 h`
blhfj jingte lb iyi jins, b i 0 8.1 h`
@fgndbydng plwir, @ 0 <
@fgndbydng plwir lb tei Gfjdjifn tijishlpi,
1 iK
i
b b @ 6
b K
â ò0 -ç ÷å õ
081 8
68 39
â ò-ç ÷å õ
_V K 0 39 h`P
033
6139
» 0 5.5 h`
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32. (k) F`lng tei gdvin pfrtdhjis `lst kdbb dhujt tl
ixpird`intfjjy virdby tei ki-crlgjdi rijftdlnsedp ds blr
f kust pfrtdhji.
37. (c) Blr bdrst ixhdtik stfti, n 0 3 fnk blr Jd + + ] 0 8
3n 368.2I ]n0 » 0 68.2 4; » 0 81.2 iR
35. (f)F. C
T
FC
T FC.FC FC FC FC FC FC0 0 + 0 + 0
Dn teds hfsi lutput T ds iqudvfjint tl FNK gfti.
34. (c) Gdvin = eS 0 83 ̀
e[ 0 91 `
@fxd`u` kdstfnhi, k@ 0 <
Fppjydng, k@ 0 [ S 3Se 3Se+
0 2 23 2.; 61 91 3 2.; 61 83» » » + » » » 0 ;9.9 o`
81. (h) Dn tei gdvin quistdln, F, C fnk H ribir cfsi, hljjihtlr
fnk i`dttir rispihtdvijy.
HEI@DX[ST
86. (k) Nu`cir lb ftl`s6
5 65
F 0 » 0
Nu`cir lb ftl`s6 9
93 3
C 0 » 0
(Xdnhi ftl` C ds ̀ dssdng brl` lni lb tei bfhi hintirik pldnt)
Z Blr`ujf lb dlndh hl`plunk 0 FC9/3 0 F3C9
83. (k) Gdvin3
nS[ n Q f
v nc v
â ò0 - ç ÷å õ-
Wedhe hfn fjsl ci wrdttin fs
( )3
3
n f Q R nc nS[
R
ä ô+ - 0í ý
í ýï ÿ
Ft jlw prissuri fnk edge ti`pirfturi tei ibbiht lb
3f
R fnk c ds nigjdgdcji einhi QR 0 nS[ [ .
88. (h) Fs tei vfjui lb rikuhtdln pltintdfj kihrifsis tei
rikuhdng plwir dnhrifsis d.i.
H : C : K : F
(1.59). (1.2). (‘1.72) (‘ 6.3)
8;. (f) 3 36 6
ED E D3 3
+
ft t 0 1 1 1
ft iq. (6 ‘ 1.9) (1.9) (1.9)91
1.9611
â òf 0 0ç ÷å õ
V
Nlw( ) ( )6/3 6/31.9 1.9
1.9H O 0 0 6
89. (c) Gdvin
f̀ss lb sljuti (w) 0 631 g`fss lb sljvint (w) 0 6111 g
@lj. `fss lb sljuti 0 21 g
kinsdty lb sljutdln 0 6.63 g/ ̀ j
Brl` tei gdvin kftf,@fss lb sljutdln 0 6111 + 631 0 6631 g
V@lj. `fss
k R
0 lr@lj. `fss
R k
0
Rlju`i lb sljutdln6631
6111 `j6.63
R 0 0 lr 0 6 jdtri
Nlw ̀ ljfrdty (@) 0( )
W
@lj. ̀ fss jdtR » 0
6313@
21 60
»
82. (h) Gdvin 6 3 69 `dnutist 0[ltfj td`i ([ ) 0 6 er 0 21 ̀ dn
Brl` [ 0 n × 6 3t
n 021
;69
0
Nlw brl` tei blr`ujf1
N 6
N 3
nâ ò0 ç ÷å õ
;6 6
3 62
â ò0 0ç ÷å õWeiri N1 0 dndtdfj f`lut
N 0 f`lunt jibt fbtir td`i t
einhi tei f`lunt lb sucstfnhi jibt fbtir 6 elur wdjj ci
662
87. (h) Fhhlrkdng tl Bfrfkfy—s bdrst jfw lb ijihtrljysds
42911
I d t W
» »0
Weiri I 0 iqudvfjint widget
0`lj. ̀ fss lb `itfj(@)
lxdkftdln stfti lb `itfj ( ) x
Xucstdtutdng tei vfjui dn tei blr`ujf
42911
@ d t W
x
»0 »
lr 42911
@ d t xW
»0 » 0 61 3 21 2142911 1.391
» » »»
0 8
Gdvin = nl. lb ̀ ljis 0 1.391 @
W
ä ô0í ýï ÿEinhi lxdkftdln stfti lb ̀ itfj ds (+8)
85. (h) Blr Jd3+ dln
I 0 ‘ 68.2 ×3
3
]
n
0( )
( )
3
3
868.2
3
- »
-
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68
068.2 4
81.2 iR;
- »0 -
84. (k) r`s8 S[
R @
0
R r`s(L3) 0 R r`s(Ei)
3
3
L Ei
L Ei
8 8 S[ S[
@ @ 0 lr 3
3
L Ei
L Ei
[ [
@ @ 0
Z3L
811 83[ 3;11O
;
»0 0
;1. (c) Gdvin 3 3 86 8 N E NE ?
3 3+
b E ;2.1 oM / `ljK 0 -
3 b E E E ? E ;82 oM / `lj+ K 0 -
3 b N N N ? E 763 oM / `lj+ K 0 -
( )b 8 N N E E N B6 8
E NE E E E3 3
- - -K 0 K + K - K
( ) ( ) N B6 8
;2 763 ;82 E3 3
-- 0 - + - - K
Ln hfjhujftdln
N BE 42; oM / `lj-K 0 -;6. (c) 6 `lji lb wftir hlntfdns 0 62 g` lb L3
Z 8.2 ̀ lji lb wftir hlntfdns0 62 × 8.2 0 97.2 g
;3. (k) 3 3 3 ; ; 3 ;3HfB XdL E XL XdB E L HfXL+ + ¾¾® + +eykrljysds
HfXdB2 ¾¾¾¾¾®
;8. (f) ( )3n n` 0 +
( )6.78 3n n0 +
Ln hfjhujftdng tei vfjui lb n wi bdnk ` 0 6
Nl. lb unpfdrik ijihtrlns 0 6
einhi dts hlnbdgurftdln wdjj ci
R(38) 0 _FrP 8k 8; s3
R3+ 0 _FrP8k 6; s3
Z Dts hejlrdki efs tei blr`ujf RHj3;;. (h) _Hl(NE8)2P8+
(k 3 sp8)
Lhtfeikrfj fnk Kdf`fgnitdh
;9. (f) O 3Hr 3L7 + ;E3XL; + 8E3X ¾¾® O 3XL; + Hr 3 (XL;)8+ 7E3L + 8X
;2. (c) 3 * 3 33L (67) ( 3 ) ( 3 )( 3 ) x OO s s p- 0 s s s
3 3 * 3 * 6( 3 ) ( 3 ) ( 3 ) ( 3 ) z y z p p p pp p p p
lni unpfdrik ijihtrln - Qfrf`fgnitdh.
;7. (c) Fju`dndu` lxdki ds f`peltirdh lxdki cihfusi dt selws
tei prlpirtdis lb tei clte fhdkdh fnk cfsdh lxdkis. Dt
rifhts wdte clte fhdks fnk cfsis tl blr` sfjt fnk wftir.
Fj3L8. xE3L + 3NfLE ¾¾® NfFjL3 + E3L
Fj3L8. xE3L + EHj ¾¾® FjHj8 + E3L
;5. (k) [ei struhturi lb HfH3 ds Hf3+ 3= =_ H H P½d.i, lni p fnk twl s clnks
;4. (c) [ei drln lctfdnik brl` cjfst burnfhi ds dn tei blr` lb
hfst drln.
91. (f) Clte fri pfrf`fgnitdh, tei lnjy kdbbirinhi ds teft HN ‘
ds f strlng bdijk jdgfnk weirifs B ‘ ds f wifo bdijk jdgfnk.
96. (k) Hfndzzfrl—s rifhtdln ds f kdsprlplrtdlnftdln rifhtdln
lb fjkieyki dn wedhe lni ̀ ljihuji lb fjkieyki rikuhis
tl fjhlelj weirifs lteir rikuhis tl sfjt lb hfrclxyjdh
fhdk
H2E9HEL + OLE ¾¾® H2E9HE3LE + H2E9HLLO
93. (h) 3 23 3
C E
8 3 8 3 3E L ,NfLEHE HE HE HE HE HE LE0 ¾¾¾¾¾¾®
98. (c)
9;. (f) [ei hlrriht lrkir lb dnhrifsdng clnk jingte ds
HE8B : HE8Hj : HE8Cr : HE8D
99. (f) H2E9Cr + NfL ‘ HE3 ‘ HE 0 HE3 ¾¾®H2E9 ‘ L ‘ HE3 ‘ HE 0 HE3 + NfCr
Fjjyj peinyj iteir
92. (h) H2E9HEHE8Cr cidng fn lptdhfjjy fhtdvi sihlnkfry
fjoyj crl`dki unkirglis X N3 nuhjilpedjdh sucstdtutdln
rifhtdln. Einhi dt unkirglis hl`pjiti dnvirsdln lb
hlnbdgurftdln.
H Cr + Hj ‛ ‘
E H8
H E2 9
E
K@B ¾¾¾® Hj ‛ H
HE8
H E2 9
E
97. (k) 8Qe H ds `lri stfcji tefn (HE8)8H
cihfusi rislnfnhi
stfcdjdsftdln ibbiht dn 8Qe H ds `lri prlnlunhik fs
hl`pfrik tl eypirhlnmugftdln stfcdjdsftdln ibbiht dn
(HE8)8H lvirfjj stfcdjdty lrkir f`lng brii rfkdhfj ds =
[rdpeinyj`iteyj > cinzyj > fjjyj > tirtdfry fjoyj
> sihlnkfry > prd`fry > ̀ iteyj > vdnyj
95. (c) 3 36, 8-Cutfkdini
HE HE HE HEn n0 - 0 +
HE 0 HE3
Nf, Eift
Qljy`irdsftdln
Xtyrini
(‘ HE ‘ HE 0 HE ‘ HE ‘ HE ‘ HE )3 3 3 n
Cutfkdini -Xtyrini hlpljy`ir
(XCS lr CYNF - X)
‘
94. (h) HE8HLHE8 + 8D3 + ;OLE ¾¾® HED8 + HE8HLLO
+ 8ED + 8E3L21. (c) Qljy-ini. i.g
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6;
2 2fhityjini Cinzini
8 HE HE H E½ ¾¾®
8 HE ‘ H HE8 ½Qrlpyni
HE8
EH
H
HE
6, 8, 9-[rd`iteyj cinzini
HH
HE
HE8
HE8
@ftei`ftdhs26. (k) Q 0 3 3{( , ) = sih tfn 6}f c f c- 0
Blr ribjixdvi =
3 3sih tfn 6f f- 0 (trui " f)Blr sy``itrdh =
sih3 c ‘ tfn3 f 0 6
J.E.X
3 36 tfn (sih 6)c f+ - - 0 3 36 tfn sih 6c f+ - +
0 ‘ (sih3 f ‘ tfn3c) + 3
0 ‘ 6 +3 0 6
Xl, Sijftdln ds sy``itrdh
Blr trfnsdtdvi =db sih3 f ‘ tfn3 c 0 6 fnk sih3 c ‘ tfn3 h 0 6
sih3 f ‘ tfn3 h 0 (6 + tfn3 c) ‘ (sih3 c ‘ 6)
0 ‘sih3c + tfn3c + 3
0 ‘ 6 + 3 0 6
Xl, Sijftdln ds trfnsdtdvi.
Einhi, Sijftdln Q ds fn iqudvfjinhi rijftdln
23. (k) Hlnsdkir tei iquftdln
w wz - 0 (6 ),o z o S- Î
Hjifrjy 6 z ´ fnk6
w wz --
ds purijy rifj
Z 6
w wz --
06
w wz --
Þ 6
w wz
z
--
06
w wz --
Þ w wz wz wzz - - + 0 w wz wz wzz - - +
Þ 3| |w w z + 0 3| |w w z +
Þ 3( )(| | )w w z - 0 w w-
Þ | z |3 0 6 ( D` 1)w ´V
Þ | z | 0 6 fnk 6 z ´Z [ei riqudrik sit ds { z = | z | 0 6, 6 z ´ }
28. (c) Jit f, c ci tei hl``ln rllts lb clte tei iquftdlns.
Blr bdrst iquftdln 3 1fx cx h+ + 0 ,wi efvi
f + c 0c
f
-...(6)
f.c 0h
f...(3)
Blr sihlnk iquftdln 33 8 ; 1 x x+ + 0 ,wi efvi
f + c 08
3
-...(8)
f . c 03
6...(;)
Nlw, brl` (6) & (8) & brl` (3) & (;)
cf- 0 8
3- h
f 0 3
6
c
f 0
8/ 3
6
[eiriblri ln hl`pfrdng wi git f 0 6, c 08
3 &
h 0 3
puttdng teisi vfjuis dn bdrst iquftdln, wi git
3 83 1
3 x x+ + 0 lr 33 8 ; x x+ + 0 1
brl` teds, wi git f 0 3, c 0 8? h 0 ;lr f = c = h 0 3 = 8 = ;
2;. (f) Jit6
f fnk
6
c ci tei rllts lb 3 6 1fx cx+ + 0
6
f +
6
c 0c
f
â òf + c0 -ç ÷
fcå õ
6
f c 06
ff
Þ 0 fc
c 0 ( )- f + c8 8( ) ( 8 ) x x c f fcx+ + - 0 1
Þ 3 8 8( 8 ) x c fc x f+ - + 0 1
Quttdng vfjuis lb f fnk c, wi git
( ) ( ) ( )8
3 8/ 38 ( ) x ä ô+ - f + c + fc f + c + fcí ýï ÿ 0 1
Þ ( )3 8/3 8/3 8/38 8 ( ) ( ) x xä ô- f +c + fc f + c - fc f + c + fcï ÿ 0 1
Þ 3 8/3 8/3 8/3 8/3( ) x x- f + c + f c 0 1
Sllts lb teds iquftdln fri 8/3 8/3,f c29. (c) Gdvin systi` lb iquftdlns hfn ci wrdttin fs
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( 6)f x y z - - - 0 1
( 6) x c y z - + - - 0 1
( 6) 1 x y h z - - + - 0
Blr nln-trdvdfj sljutdln, wi efvi6 6 6
6 6 6
6 6 6
f
c
h
- - -
- - -
- - - 0 1
S 3 ® S 3 ‘ S 8
6 6 6
1
6 6 6
f
c h
h
- - -
-
- - - 0 1
H3 ® H3 ‘ H8
6 1 6
1
6 6
f
c h h
h h
- -+ -
- - - 0 1
Fppjy S 8 ® S 8 ‘ S 6
6 1 6
1
f
c h h
f h h
- -
+ -
- - 0 1
Þ 3 3( 6)_ P 6_ ( )P- + - - +f ch h h f c h 0 1
Þ ( 6)_ Pf ch fc fh- - - 0 1Þ fch ch fc fh- - - 0 1
Þ fc ch hf+ + 0 fch22. (f) kit _(D + C)91 ‘ 91CP
0 kit _91H1 D +91H6 C +
91H3 C3 + 91H8 C
8 + ...
+ 91H91 C91 C91 ‘ 91CP
{Fjj tir`s efvdng Cn, 3 91n£ £wdjj ci zirl cihfusi gdvin teft C3 0 1}
0 kit _D + 91C ‘ 91CP 0 kit _DP 0 6
27. (h) Gdvin ixpfnsdln ds
(6 + x)616 (6 ‘ x + x3)611
0 (6 + x) (6 + x)
611(6 ‘ x + x
3)611
0 (6 + x) _(6 + x) (6 ‘ x + x3)P611
0 (6 + x) _(6 ‘ x8)611P
Ixpfnsdln (6 ‘ x8)611 wdjj efvi 611 + 6 0 616 tir`s
Xl, (6 + x) (6 ‘ x8)611 wdjj efvi 3 × 616 0 313 tir`s
25. (c) Wdte 8 ft undt pjfhi,
tltfj plssdcji blur kdgdt nu`cir (wdtelut ripitdtdln)
wdjj ci 8 ! 0 2
Wdte ; ft undt pjfhi,
tltfj plssdcji blur kdgdt nu`cirs wdjj ci 8! 0 2
Wdte 9 ft undt pjfhi,
tltfj plssdcji blur kdgdt nu`cirs wdjj ci 8! 0 2
Wdte 2 ft undt pjfhi,
tltfj plssdcji blur kdgdt nu`cirs wdjj ci 8! 0 2
Xu` lb undt kdgdts lb fjj plssdcji nu`cirs
0 2 × 8 + 2 × ; + 2 × 9 + 2 × 2
0 2 _8 + ; + 9 + 2P
0 2 _65P
0 615
24. (h) Jit f ci tei bdrst tir` fnk k ci tei hl``lnkdbbirinhi lb gdvin F.Q.
Xihlnk tir`, f+ k 0 63 ...(6)
Xu` lb bdrst ndni tir`s,
X4 04
(3 5 )3
f k + 0 4( ; )f k +
Gdvin teft X4 ds `lri tefn 311 fnk jiss tefn 331
Þ 311 : X4 : 331Þ 311 : 4 (f + ;k ) : 331Þ 311 : 4 (f + k + 8k ) : 331Quttdng vfjui lb (f + k ) brl` iquftdln (6)
311 : 4 (63 + 8k ) : 331
Þ 311 : 615 + 37k : 331Þ 311 ‘ 615 : 615 + 37k ‘ 615 : 331 ‘ 615Þ 43 : 37k : 663Qlssdcji vfjui lb k ds ;
37 × ; 0 615
[eus, 43 : 615 : 663
Quttdng vfjui lb k dn iquftdln (6)
f + k 0 63
f 0 63 ‘ ; 0 5
;te tir` 0 f + 8k 0 5 + 8 × ; 0 31
71. (f) nte tir` lb gdvin sirdis ds3 6
( 6)(3 6)2
n
n n n
+
+ +
02
( 6)n n +
Jit nte tir`, fn 06 6
26n n
ä ô-í ý+ï ÿ
Xu` lb 31 tir`s, X31 0 f6 + f3 + f8 + .... + f31
X31 06 6 6 6 6 6
2 2 2 ...6 3 3 8 8 ;
â ò â ò â ò- + - + - +ç ÷ ç ÷ ç ÷å õ å õ å õ
6 6 6 6 6 62 2 265 64 64 31 31 36
â ò â ò â ò+ - + - + -ç ÷ ç ÷ ç ÷å õ å õ å õ
X31 06
63
-6
3
â ò+ç ÷å õ
6
8-
6
8
â ò+ç ÷å õ
6
;- ...
ä â ò+í ç ÷å õíï
6
65+
6
64-
6
64
â ò+ç ÷å õ
6
31-
6
31
â ò+ç ÷å õ
6
36
ôâ ò- ýç ÷å õ ýÿ
X31 06 631
2 636 36
â ò- 0ç ÷å õ
.....(6)
Gdvin teft X31 0 36
o .....(3)
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Ln hl`pfrdng (6) fnk (3), wi git
o 0 631
76. (c) Gdvin teft4
3 b
â òç ÷å õ 0
3
4
31
6 hls8jd`
x
x b
x®
-â òç ÷å õ
0
3
1jd`
6 hls8 x
x
x®
â òç ÷-å õ
0
3
1 3
jd`8
3sdn3
x
x
x®
â òç ÷ç ÷ç ÷å õ
0
3
1 3
4 ;. .
6 ; 4jd`83
sdn 3
x
x
x®
â òç ÷ç ÷
ç ÷å õ
01 3
3
; 6jd`
84 3sdn
3
8
3
x x
x
®
â òç ÷»ç ÷ç ÷
â òç ÷ç ÷ç ÷å õå õ
0 1
3
31
jd`63
84sdn
3jd`8
3
x
x
x
x
®
®
ä ôí ýí ý
í ýí ýâ òí ýç ÷å õí ýï ÿ
1
sdnjd` 6 x
x
x®
ê ú0à ðî þ
03 6 3
.4 6 4
ä ô0í ýï ÿ
73. (k) Gdvin teft, y 0 inx
Kdbbirindftdng clte sdkis wdte rispiht tl x
ky
kx 0 ninx
Fgfdn kdbbirintdftdng w.r.t ' x',
3
3
k y
kx0 n.ninx 0 n3inx .......(6)
y 0 inx
nx 0 jlgi y
x 06
n jlg y
Kdbbirintdftdng x wdte rispiht tl y
kx
ky 0
6 6.
n y
Fgfdn kdbbirintdftdng wdte rispiht tl y
3
3
k x
ky0
3
6 6.
n y
â ò-ç ÷å õ
0 3 36 6
nxny ni
-0 - ...... (3)
@ujtdpjydng iquftdln (6) fnk (3)
3 3
3 3
k y k x
kx ky
â ò â òç ÷ ç ÷å õ å õ
03 36( )nx nx nxn i i ni
n
--â ò 0 -ç ÷å õ
78. (c) b ( x) 0 3 x8 + fx3 + cx
jit, f 0 ‘ 6, c 0 6
Gdvin teft b ( x) sftdsby Sljj teilri` dn dntirvfj _‘6, 6P
b ( x) `ust sftdsby twl hlnkdtdlns.
(6) b (f) 0 b (c)
(3) b ¢ (h) 0 1 (h selujk ci citwiin f fnk c) b (f) 0 b (6) 0 3 (6)8 + f (6)3 + c (6) 0 3 + f + c
b (c) 0 b (‘6) 0 3 (‘6)8 + f (‘6)3 + c (‘6)
0 ‘3 + f ‘ c
b (f) 0 b (c)
3 + f + c 0 ‘ 3 + f ‘ c
3c 0 ‘ ;
c 0 ‘ 3
(gdvin teft6
3h 0 )
b ¢ ( x) 0 2 x3 + 3fx + c
ft x 06
3, b ¢ ( x) 0 1
1 0
36 6
2 33 3
f câ ò â ò
+ +ç ÷ ç ÷å õ å õ
8
3f c+ + 0 1
83
3f+ -
0 1
f 0 3 ‘8
3 0
6
3
3f + c 0 3 ×6
3 ‘ 3 0 6 ‘ 3 0 ‘ 6
7;. (c) b ( x) 08 ;
69 9
x xâ ò â ò
+ -ç ÷ ç ÷å õ å õ
Qut b ( x) 0 1
Þ 1 08 ;
69 9
x xâ ò â ò+ -ç ÷ ç ÷å õ å õ
Þ8 ;
9 9
x xâ ò â ò
+ç ÷ ç ÷å õ å õ 0 6
Þ 8 x + ; x 0 9 x ...6)Blr x 0 6
86 + ;6 > 96
Blr x 0 8
88 + ;8 0 46 : 98
Lnjy blr x 0 3, iquftdln (6) Xftdsby
Xl, lnjy lni sljutdln ( x 0 3)
79. (c) Jit D 0
5 5
3 3
sdn hls
6 3sdn hls
x x
kx x x
-
-ó
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67
0
; 3 ; 3
3 3
(sdn ) (hls )
6 3sdn hls
x xkx
x x
-
-ó
0
; ; ; ;
3 3
(sdn hls )(sdn hls )
6 3sdn hls
x x x x
kx x x
+ -
-ó
0
3 3 3 3 3
3 3 3 3
3 3
_(sdn hls ) 3sdn hls P
_(sdn hls P_sdn hls P
6 3sdn hls
+ -+ -
-ó
x x x x
x x x xkx
x x
0 hls3 x kx-ó 0sdn3
3
xh
-+ 0
6sdn3
3 x h- +
72. (h) Jit D 0
6/ 3
31
jn(6 3 )
6 ;
xkx
x
+
+ó lr
6/ 3
31
jn(6 3 )
6 (3 )
xkx
x
+
+ó
Qut 3 x 0 tfnq
Z 33
sihkx
k 0 q
q lr kx 0
3sih
3
k q q
fjsl wein x 0 1 Þ q 0 1
fnk wein x 06
;9 lr 3 ;
pÞ q 0 ±
Z D 0; 3
31
jn(6 tfn ) sih
36 tfn
k
p
+ q q q»
+ qó
D 0;
3
31
6 jn(6 tfn )sih
3 6 tfnk
p
+ q» q q
+ qó
(V 6 + tfn3 q 0 sih3 q)
D 0;
1
6jn(6 tfn )
3k
p
+ q qó ...(6)
D 0;
1
6 jn 6 tfn3 ;
k
p
ä p ôâ ò+ - q qç ÷í ýå õï ÿó
(Ysdng tei prlpirty lb kibdndti dntigrfj)
D 0;
1
tfn tfn6 ;jn 63
6 tfn tfn;
k
p pä ô- qí ý+ qí ýpí ý+ » q
ï ÿ
ó
D 0;
1
6 6 tfnjn 6
3 6 tfnk
p
- qä ô+ q
í ý+ qï ÿó
D 0;
1
6 6 tfnjn
3
p
+ qó
6 tfn+ - q6 tfn
k ä ô
qí ý+ qï ÿ
D 0;
1
6 3jn
3 6 tfnk
p
ä ô qí ý+ qï ÿó
D 0;
1
6_jn 3 jn(6 tfn )P
3k
p
- + q qó
D 0; ;
1 1
6 6jn 3. jn(6 tfn )
3 3
k k
p p
q - + q q
ó ó
D 0;6
jn 33 1
D p
q - (brl` iq. (6))
D + D 06
jn 3 13 ;
pâ ò-ç ÷å õ
3D 06
jn 33 ;
p» »
3D 0 jn 3
5
p lr jn 3
62
D p0
77. (c) Frif lb sefkik plrtdln
y x3 0 ;
(;, ;)
UU¢6
‘6
‘3
3 8 ;
3
8;
T¢
T
6
3; ;
3 3
;
3 ;
y yky ky
-
+â ò0 -ç ÷å õó ó
; ;3 8
3 3
6 6;
3 3 ; 8
y y y
- -
ä ô ä ô0 + -í ý í ý
í ý í ýï ÿ ï ÿ
_ P
6 6 2; 5{5 62} {3 5}
3 ; 8 8
ê ú0 + - - - +à ðî þ
0 69 ‘ 2 0 4 sq undts
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65
75. (h) Xdnhi bf`djy lb fjj hdrhjis tluhedng x-fxds ft tei lrdgdn
(1, 1)
(1, f)
Z Iqn ds ( x)3 + ( y ‘ f)3 0 f3
weiri (1, f) ds tei hintri lb hdrhji.
Þ 3 3 3 3 x y f fy+ + - 0 f3
Þ 3 3 3 x y fy+ - 0 1 ...(6)
Kdbbirintdfti clte sdki w.r.t ' x', wi git
3 3 3ky ky
x y fkx kx
+ - 0 1
Þ ky ky
x y fkx kx
+ - 0 1 Þ
ky x y
kxky
kx
+ 0 f
Qut vfjui lb 'f' dn iqn (6), wi git
3 3
3
ky y x
kx x y y ky
kx
ä ô+í ý
+ - í ýí ýï ÿ
0 1
Þ 3 3 3( ) 3 3ky ky
x y y xykx kx
+ - - 0 1
Þ 3 3 3( 3 ) ky
x y ykx
+ - 0 3 xy
Þ 3 3( ) ky
x ykx
- 0 3 xy ( ) g x y½
Einhi, g ( x) 0 3 x
74. (h) Jit tei bllt lb tei pirpinkdhujfr brl` (1, 1) ln tei
vfrdfcji jdni 6 66 ds ( y ) x y
xf c
+ 0 >
Einhi, pirpinkdhujfr kdstfnhi lb tei vfrdfcji jdni
6 x y
f c+ 0 brl` tei pldnt L (1, 1) 0 LF
F( , , 1) x y6 6
L (1, 1)
6 x xf c+ 0
Þ 3 36 6
3 3
6
6 6 x y
f c
-0 +
+
Þ 3 36 63 3
6
6 6 x y
f c
0 ++
Þ 3 36 6 3 36 6 6
;;
x yf cä ô0 + + 0í ýï ÿ
V ,
wedhe ds iquftdln lb f hdrhji wdte rfkdus 3.
Einhi ( x6, y6) d.i., tei bllt lb tei pirpinkdhujfr brl`
tei pldnt (1, 1) tl tei vfrdfcji jdni 6 x x
f c+ 0 ds jdis
ln f hdrhji wdte rfkdus 0 3
51. (k)
Q (9, 8)
V (5, 8) x y ‘ 3 0 3
S (3, 1)
Iquftdln lb SV ds x ‘ 3 y 0 3 ...(6)
ft y 0 1, x 0 3 _S (3, 1)P
fs QV ds pfrfjjij tl x, y-hllrkdnftis lb V ds fjsl 8
Quttdng vfjui lb y dn iquftdln (6), wi gitV (5, 8)
Hintrldk lb KQVS 05 9 3 8 8
,8 8
+ + +â òç ÷å õ 0 (9, 3)
Lnjy (3 x ‘ 9 y 0 1) sftdsby tei gdvin hl-lrkdnftis.
56. (k) T
U
r
r
H (8, 9)
[ei iquftdln lb hdrhji ds
3 3 2 61 Q x y x y+ - - + 0 1 ...(d)
3 3( 8) ( 9) x y- + - 0 3( 8; Q)-
Hintri (8, 9) fnk rfkdus 'r' 8; Q0 -Db hdrhji klis nlt tluhe lr dntirsiht tei x-fxds tein
rfkdus x : y - hllrkdfnti lb hintri Hlr 8; Q 9- :
-
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64
Þ 8; ‘ Q : 39Þ Q > 4 ...(dd)Fjsl db tei hdrhji klis nlt tluhe lr dntirsiht x-fxds
tei rfkdus r : x-hllrkdnfti lb hintri H.
lr8; Q 8 8; Q 4 Q 39- : Þ - : Þ >
... (ddd)
Db tei pldnt (6, ;) ds dnsdki tei hdrhji, tein dts kdstfnhi
brl` hintri H : r.
lr 3 3_(8 6) (9 ;) P 8; Q- + - : -
Þ 9 : 8; ‘ O Þ Q : 34 ... (dv) Nlw fjj tei hlnkdtdlns (dd), (ddd) fnk (dv) fri sftdsbdik
db 39 : Q : 34 wedhe ds riqudrik vfjui lb Q.
53. (f) Jit3 3
3 36
x y
f c+ 0 ci tei iquftdln lb ijjdpsi.
Gdvin teft B6C fnk B
3C fri pirpinkdhujfr tl ifhe
lteir.
Xjlpi lb B6C × sjlpi lb B3C 0 ‘ 6
1 1
1 1
c c
fi fi
- -â ò â ò»ç ÷ ç ÷å õ å õ- - - 0 ‘ 6
B6
(‘ , 1)f (‘ , 1)fi ( , 1)fiB3 ( , 1)f
(1, ‘ )c
C (1, )c
c c
fi fi
-â ò â ò»ç ÷ ç ÷å õ å õ 0 ‘ 6
c3 0 f3i3
i3 0
3
3
c
f
33
36
ci
f
ê úì ì0 -à ð
ì ìî þV
3
36 c
f- 0 3
3c
f
6 0
3
33
c
f Þ
3
3
6
3
c
f0
i3 0
3
3
6 66 6
3 3
c
f- 0 - 0
i3 06
3
58. (h) Gdvin iquftdln lb jdnis fri
6
8
x - 0
3 8
6 3
y z - -0 ...(6)
fnk8
6
x - 0
6 3
3 8
y z - -0 ...(3)
Fny pldnt ln jdni (6) ds Q (8 6, 3, 3 8)j + j + j + fnkln jdni (3) ds V ( ` + 8, 3` + 6, 8` + 3).Ln sljvdng 8j + 6 0 ` + 8 fnk j + 3 0 3` + 6wi git j 0 6, ` 0 6Z Qldnt lb dntirsihtdln lb twl jdnis ds S (;, 8, 9)Xl, iquftdln lb pjfni ^ tl LS weiri L ds (1, 1, 1)fnk pfssdng terluge S ds
; 8 9 x y z + + 0 915;. (h) Dt ̀ fois q wdte x fnk y-fxis.
j 0 hlsq, ` 0 hlsq, n 0 hls (p ‘ 3q)wi efvi j 3 + `3 + n3 0 6
Þ hls3q + hls3q + hls3 (p ‘ 3q) 0 6Þ 3 hls3q + (‘hls3q)3 0 6Þ 3 hls3q ‘ 6 + hls33q 0 1Þ hls3q ‘ _6 + hls3qP 0 1Þ hls 3q 0 1 lr hls3q 0 ‘ 6Þ 3q 0 p/3 lr 3q 0 p
Þ q 0 p/; lr3
pq 0
Þ q 0 ,; 3
p pä ôí ýï ÿ
59. (h) Gdvin | 3 |f c-rr
0 9
3 3(3 | |) | | 3 | 3 || | hlsf c f c+ - » qr rr r
0 9
Quttdng vfjuis lb | |fr
fnk | |cr
, wi git
3 3(3 3) (8) 3;hls» + - q 0 39
Þ hlsq 0 1
Þ q 03
p
| 3 |f c+rr
0 62 4 3;hls+ + q 0 39 0 952. (f) [eiri fri 3n lcsirvftdlns x6, x3, ..., x3n
Xl, `ifn 0
3
6 3
n
d
x
n0
é
Jit teisi lcsirvftdlns ci kdvdkik dntl twl pfrts x6,
x3, ..., xn fnk xn+6, ..., x3nIfhe dn 6st pfrt 9 ds fkkik, sl tltfj lb bdrst pfrt ds
6
9n
d x n
0
+é .
Dn sihlnk pfrt 8 ds suctrfhtik brl` ifhe
Xl, tltfj lb sihlnk pfrt ds3
6
8n
dn
x n
0 +
-é
[ltfj lb 3n tir`s fri
3 3
6 6 6
9 8 3n n n
d d d
d d n d
x n x n x n
0 0 + 0
+ + - 0 +é é é
-
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@ifn 0
3
6
3
3
nd
d
x n
n0
+é 0
3
6
63
nd
d
x
n0
+é
Xl, dt dnhrifsi cy 6.
57. (k) Jit F fnk C ci twl ivints suhe teft
Q(F C)Å 0 Q(F C)Ç
fnk Q(F C)Å 0 Q (F) + Q (C) ‘ Q(F C)Ç
lptdln (f) = sdnhi Q(F C) Q(F C)Å 0 Ç (gdvin)teiriblri F fnk C fri iqufjjy jdoijy
Xupplsi lptdln (c) fnk lptdln (h) fri hlrriht.
Z Q(F C )Ç ¢ 0 1 fnk Q(F C)¢ Ç 0 1
Þ Q(F) Q(F C)- Ç 0 1 fnk Q(C) Q(F C)- Ç 0 1
Þ Q (F) 0 Q(F C)Ç fnk Q(C) 0 Q(F C)Ç
[eus Q(F) 0 Q (C) 0 Q(F C)Ç 0 Q(F C)Å
_V Gdvin Q(F C)Ç 0 Q(F C)Å PFjsl, wi onlw
Q(F C)Å 0 Q(F) + Q (C) ‘ Q(F C)Ç
0 Q(F C)Ç + Q(F C)Ç ‘ Q(F C)Ç
0 Q(F C)Çwedhe ds trui brl` gdvin hlnkdtdln
Einhi, lptdln (f), (c) fnk (h) fri hlrriht.
55. (h) 3 sdn8f ‘ 7sdn3f + 7 sdnf ‘ 3 0 1Þ 3 sdn3f (sdnf ‘ 6) ‘ 9 sdnf (sdnf ‘ 6)
+ 3 (sdnf ‘ 6) 0 1
Þ (sdn f ‘ 6) (3 sdn3 f ‘ 9 sdnf + 3) 0 1Þ sdn f ‘ 6 0 1 lr 3 sdn3 f ‘ 9 sdnf + 3 0 1
sdn f 0 6 lr sdn f 09 39 62 9 8
; ;
¶ - ¶0
3
pf 0
lr sdn f 06
, 33
Nlw, sdn 3f ´
blr, sdn f 06
3
f 03
,8 8
p p
[eiri fri terii vfjuis lb f citwiin _1, 3pP
54. (c) hlsihq 0q
q
+-
, sdn q
q
-q 0
+
hls q 03
3 36 sdn 6( )
q p q
p q p q
â ò-¶ - q 0 - 0ç ÷å + õ +
hlt; 3
p qâ ò+ç ÷å õ 0hlt hlt 6
; 3
hlt hlt
; 3
p q-
p q+
hlt 63
hlt 6
3
q-
0q
+
0
hls sdn3 3
hls sdn3 3
q q-
q q+
Ln rftdlnfjdzdng kinl`dnftlr, wi git
hls sdn hls sdn3 3 3 3
hls sdn hls sdn3 3 3 3
q q q qâ ò â ò- +ç ÷ ç ÷ç ÷ ç ÷q q q q
+ +ç ÷ ç ÷å õ å õ
03 3
hls
sdn hls 3sdn hls3 3 3 3
qq q q q
+ +
0hls
6 sdn
q+ q
03 / ( )
( )6
pq p q
p q
p q
+-
++
0 pq
p0
q
p
41. (c) jit p 0 Db dt klis nlt rfdn
q 0 D gl tl shellj
Fhhlrkdng tl jfw lb hlntrfplsdtdvi
p Þ q ½
~q Þ ~ pd.i. ~q 0 D kl nlt gl tl shellj
~ p 0 Dt rfdns
~q Þ ~ p ds Db D kl nlt gl tl shellj, dt rfdns.
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