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QFS[ F = QETXDHX

6. Fn ixpirdìnt ds pirblrìk tl lctfdn tei vfjui lb fhhijirftdln kui tl grfvdty g cy usdng f sd`pji pinkuju` lb jingte J. Dn teds ixpirdìnt tdì blr 611 lshdjjftdlns dsìfsurik cy usdng f wfthe lb 6 sihlnk jifst hlunt fnk teivfjui ds 41.1 sihlnks. [ei jingte J ds ̀ ifsurik cy usdng fìtir 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`ìkdftijy fbtir edttdng tei pfj`, tei ̀ dndù`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 ̀ dndù` blrhi teft hfn ci fppjdik ln F sl teft clte tei

cjlhos ̀ lvi tlgiteir ds 63 N, tei ̀ fxdù` 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 ̀ fxdù` 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 sfì, 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 ìits @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 tdì 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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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 `dndù` 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

`lìnt 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 ìfsurikrispihtdvijy 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, fnixpirdìntfjdst sits up tei Tlung—s klucji sjdt ixpirdìnt

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

ixpirdìntfjjy 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`fxdù` kdstfnhi citwiin tei` blr sftdsbfhtlryhl`ùndhftdln 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`ìtir,

fnk H tl tei nigftdvi jifk lb tei f`ìtir, 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 blrùjf 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 ìtfjs

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 ìtfjs 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


(k)6


87. F hurrint lb 61.1 F bjlws blr 3.11 e terluge fn ijihtrljytdhhijj hlntfdndng f `ljtin sfjt lb ìtfj 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 sfìrllt ̀ 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`lìnt lb 6.78 C@. [eds Rfnfkdu` hejlrdki efs tei blrùjf=(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 blrìk 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 stftiìnt 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) Sidìr-[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 blrìkds=(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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2

(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 blrìk 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 pljyìrdzftdln 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`ìtrdh cut nlt trfnsdtdvi.(c) ribjixdvi fnk trfnsdtdvi cut nlt sỳ ìtrdh.(h) sy`ìtrdh 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 slì 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 blrìk 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

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-kdìnsdlnfj 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 ìkdfn lb fjj tei lcsirvftdlns ds dnhrifsik cy 9

fnk ifhe lb tei ri`fdndng lcsirvftdlns ds kihrifsik cy 8.

[ein tei ìfn 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 stftiìnt f`lngst tei bljjlwdngstftiìnts 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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8/17/2019 PCM Paper (09.04.2014)

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4

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 × vljuì

0 618 × frif × jingte

0 618 × 61 ‘3 × 6 0 61 og

[eiriblri `dndù` blrhi 0t

KK

(rfti lb hefngi lb

`lìntu`)

0 61 × 6.9 0 69 N ;. (h) @dndù` blrhi ln F

0 brdhtdlnfj blrhi citwiin tei surbfhis

0 63 N

F

C

; og

9 og

X`llte tfcji

[eiriblri ̀ fxdù` 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 ̀ fxdù` 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

+

8/17/2019 PCM Paper (09.04.2014)

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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 (vljuì 3J) brl` 37±h tl 77±h

0 `wftir ×sp. et × Kq

0 3 × 618 × ;.3 × 91 _V `fss 0 kinsdty × vljuìP

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 × ̀ fxdù` 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 + +

8/17/2019 PCM Paper (09.04.2014)

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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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63

32. (k) F`lng tei gdvin pfrtdhjis `lst kdbb dhujt tl

ixpirdìntfjjy 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 `

@fxdù` 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 Blrùjf 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

Rljuì 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 tdì ([ ) 0 6 er 0 21 ̀ dn

Brl` [ 0 n × 6 3t

n 021

;69

0

Nlw brl` tei blrùjf1

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 tdì 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 ìtfj(@)

lxdkftdln stfti lb ìtfj ( ) x

Xucstdtutdng tei vfjui dn tei blrùjf

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 blrùjf 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 =

[rdpeinyjìteyj > 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

Qljyìrdsftdln

Xtyrini

(‘ HE ‘ HE 0 HE ‘ HE ‘ HE ‘ HE )3 3 3 n

Cutfkdini -Xtyrini hlpljyìr

(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-[rdìteyj 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`ìtrdh =

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`ìtrdh

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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69

( 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,


Wdte 2 ft undt pjfhi,


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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62

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- :

8/17/2019 PCM Paper (09.04.2014)

19/20

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

8/17/2019 PCM Paper (09.04.2014)

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31

@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.

pcm paper (09.04.2014)

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