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8/19/2019 Class 9 Imo 5 Years EBook
1/69
;
8/19/2019 Class 9 Imo 5 Years EBook
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Nlpyrhift © :2;= ̂ nhm`nm Lcyephkd Glu`dkthl`. Rrh`tmd whtf tfm pmrehsshl` lg ̂ nhm`nm Lcyephkd Glu`dkthl`. @l pkrt
lg tfhs pujchnkthl` eky jm rmprldunmd, trk`sehttmd, h` k`y glre lr jy k`y emk`s, mcmntrl`hn, emnfk`hnkc, pfltlnlpyh`i,rmnlrdh`i lr ltfmrwhsm, whtflut tfm prhlr pmrehsshl` lg tfm nlpyrhift flcdmr. Lw`mrsfhp lg k` mjlla dlms `lt ihvm tfm
plssmsslr tfm mjlla nlpyrhift.
Kcc dhsputms sujomnt tl Dmcfh ourhsdhnthl` l`cy.
Dhsnckhemr ? Tfm h`glrekthl` h` tfhs jlla hs tl ihvm ylu tfm pktf tl sunnmss jut ht dlms ̀ lt iukrk`tmm ;22% sunnmss ks
tfm strktmiy hs nlepcmtmcy dmpm`dm`t l` hts mxmnuthl`. K`d ht hs jksmd l` ckst ymkrs pkpmrs lg HEL mxke.
Rujchsfmd jy ? ETI Cmkr`h`i Emdhk (R) Ctd.
Nlrplrktm Lgnm ? Rclt 88, :`d Gcllr, ^mntlr 44 H`sthtuthl`kc Krmk, Iurikl`, Fkryk`k.
Rfl`m ? 2;:4 - 48=;:22
Ymj? eti.h` Mekhc? h`glBeti.h`
Zmid. Lgnm ? 427, Tko Kpt., Zh`i Zlkd, @mkr ^kgdkrou`i Flsphtkc, @mw Dmcfh-;;22:8
Whsht www.eti.h ̀glr juyh`i jllas l`ch`m.
8/19/2019 Class 9 Imo 5 Years EBook
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4ª tf HEL :2;2
=ª tf HEL :2;;
7ª tf HEL :2;: - ^MT K
7ª tf HEL :2;: - ^MT J
9ª tf HEL :2;> - ^MT K
9ª tf HEL :2;> - ^MT J
6ª tf HEL :2;4 - ^MT K
6ª tf HEL :2;4 - ^MT J
Nl`tm`ts
NCK^^ ;:
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H`tmr`kthl`kcEktfmekthns
Lcyephkd
H`tmr`kthl`kc
Ektfmekthns
Lcyephkd
H`tmr`kthl`kc
Ektfmekthns
Lcyephkd
@kthl`kc^nhm`nm
Lcyephkd
@kthl`kc
^nhm`nm
Lcyephkd
@kthl`kc
^nhm`nm
Lcyephkd
@kthl`kc
Nyjmr
Lcyephkd
@kthl`kc
Nyjmr
Lcyephkd
H`tmr`kthl`kcM`ichsf
Lcyephkd
H`tmr`kthl`kc
M`ichsf
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H`tmr`kthl`kc
M`ichsf
Lcyephkd
ETIN^^
^mrhms
ETI
N^^
^mrhms
ETI
N^^
^mrhms
ETI YLZAJLLA^ / LCXERHKD JLLA^
NCK^^ ;
NCK^^ :
NCK^^ >
8/19/2019 Class 9 Imo 5 Years EBook
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H`tmr`kthl`kc
Ektfmekthns
Lcyephkd
H`tmr`kthl`kc
Ektfmekthns
Lcyephkd
H`tmr`kthl`kc
Ektfmekthns
Lcyephkd
@kthl`kc
^nhm`nm
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^nhm`nm
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^nhm`nm
Lcyephkd
@kthl`kc
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Lcyephkd
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Nyjmr
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M`ichsf
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H`tmr`kthl`kc
M`ichsf
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M`ichsf
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ETI
N^^
^mrhms
ETI
N^^
^mrhms
ETI
N^^
^mrhms
NCK^^ 4
NCK^^ =
NCK^^ 7
ETI YLZAJLLA^ / LCXERHKD JLLA^
8/19/2019 Class 9 Imo 5 Years EBook
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H`tmr`kthl`kc
Ektfmekthns
Lcyephkd
Ektf H[
Nfkccm`im
Ektf H[
Nfkccm`im
H`tmr`kthl`kc
Ektfmekthns
Lcyephkd
@kthl`kc
^nhm`nm
Lcyephkd
^nhm`nm H[
Nfkccm`im
^nhm`nm H[
Nfkccm`im
@kthl`kc
^nhm`nm
Lcyephkd
@kthl`kc
Nyjmr
Lcyephkd
Rsynflcliy lg
^unnmss
@kthl`kc
Nyjmr
Lcyephkd
H`tmr`kthl`kc
M`ichsf
Lcyephkd
Tfm ^mnrmts lg
^]NNM^^
H`tmr`kthl`kc
M`ichsf
Lcyephkd
ETI
N^^
^mrhms
Ekstmr Em`tkc
Kjhchty h` >2 Dkys
Ekstmr Em`tkc
Kjhchty h` >2 Dkys
ETI
N^^
^mrhms
ETI YLZAJLLA^ / LCXERHKD JLLA^
NCK^^ 9
NCK^^ 6
8/19/2019 Class 9 Imo 5 Years EBook
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H`tmr`kthl`kc
Ektfmekthns
Lcyephkd
@kthl`kc
^nhm`nm
Lcyephkd
@kthl`kc
Nyjmr
Lcyephkd
H`tmr`kthl`kc
M`ichsf
Lcyephkd
ETI YLZAJLLA^ / LCXERHKD JLLA^
NCK^^ ;2
H`tmr`kthl`kcEktfmekthns
Lcyephkd
@kthl`kc^nhm`nm
Lcyephkd
@kthl`kcNyjmr
Lcyephkd
H`tmr`kthl`kcM`ichsf
Lcyephkd
NCK^^ 8
Ektf H[
Nfkccm`im
^nhm`nm H[
Nfkccm`im
Rsynflcliy lg
^unnmss
Tfm ^mnrmts lg
^]NNM^^
Ekstmr Em`tkc
Kjhchty h` >2 Dkys
Ektf H[
Nfkccm`im
^nhm`nm H[
Nfkccm`im
Rsynflcliy lg
^unnmss
Tfm ^mnrmts lg
^]NNM^^
Ekstmr Em`tkc
Kjhchty h` >2 Dkys
8/19/2019 Class 9 Imo 5 Years EBook
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@kthl`kc
Nyjmr
Lcyephkd
H`tmr`kthl`kc
Ektfmekthns
Lcyephkd
@kthl`kc
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ETI YLZAJLLA^ / LCXERHKD JLLA^
NCK^^ ;:
@kthl`kcNyjmr
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Tfm ^mnrmts lg
^]NNM^^
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Kjhchty h` >2 Dkys
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Nfkccm`im
^nhm`nm H[
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^unnmss
Tfm ^mnrmts lg
^]NNM^^
Ekstmr Em`tkc
Kjhchty h` >2 Dkys
8/19/2019 Class 9 Imo 5 Years EBook
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Nckss
; tl ;2
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GL]@DKTHL@ NL]Z^MGlr Nckssms 9, 6, 8 & ;2
OMM (Ekh` & Kdvk`nmd) | KHRET | JLKZD^ | LCXERHKD | @T^M
Nckss 9
Nckss 8
Nckss ;2
Nckss 6
8/19/2019 Class 9 Imo 5 Years EBook
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4tf
Xmkr :2;2
Nckss 8
Nlpyrhift©:2;=. ̂ nhm`nm Lcyephkd Glu`dkthl` (^LG)
8/19/2019 Class 9 Imo 5 Years EBook
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:4tf HEL | Cmvmc-H | Nckss 8
^MNTHL@ H ? CLIHNKC ZMK^L@H@I
;. K rkt ru`s :2 e tlwkrds Mkst k`d tur`s tl rhift, ru`s ;2 e k`d tur`s tl rhift, ru`s 8 e k`d kikh`
tur`s tl cmgt, ru`s = e k`d tur`s tl cmgt, ru`s ;: e k`d gh`kccy tur`s tl cmgt k`d ru`s 7 e. @lw wfhnf
dhrmnthl` hs tfm rkt gknh`i<
(K) Mkst (J) Ymst (N) @lrtf (D) ^lutf
:. Nfllsm tfm ldd `uemrkc pkhr/irlup h` tfm ihvm` qumsthl` ?
(K) : „ 6 (J) > „ :9 (N) 4 „ >: (D) = „ ;:=
>. Tfm ihvm` qumsthl` nl`tkh`s glur prljcme ghiurms ekramd ;, :, > k`d 4. ^mcmnt k ghiurm grle kel`ist
tfm lpthl`s wfhnf whcc nl`th`um tfm skem smrhms ks ihvm` h` tfm Rrljcme ghiurms.
; : > 4
<
Rrljcme Ghiurms
(K) (J) (N) (D)
4. Yfkt nlems `mxt h` tfm ihvm` pkttmr` <
;2, ;8, 42, 99, ;=6, \\\;; (J) >29 (N) >2; (D) :88
=. Nlu`t tfm `uejmr lg nujms h` tfm ihvm` ghiurm.
(K) :>
(J) :4
(N) :=
(D) @l`m lg tfmsm
7. Twl plshthl`s lg k dhnm krm ihvm`. Yfm` : hs kt tfm tlp, wfhnf lg tfm glcclwh`i
`uejmrs nlucd jm kt tfm jlttle<
(K) ; (J) :
(h) (hh)(N) > (D) 4
9. Hg 9 ; 5 74Ͱ > 8 5 ;44. Yfkt hs tfm vkcum lg = 7 <
(K) :: (J) == (N) 77 (D) ;:;
6. :2 grhm`ds emmt kt k ekrrhkim rmnmpthl` lg tfmhr nleel` grhm`d. Mvmryl`m lg :2 grhm`ds sflla fk`ds
whtf mknf ltfmr l`cy l`nm. Gh`d tfm `uejmr lg fk`dsfkams.
(K) 422 (J) >62 (N) 42 (D) ;82
8. Yfhnf lg tfm ihvm` lpthl`s whcc nlem `mxt h` tfm ihvm` smrhms <
_]K, VLN, WHM, TNI, \\\\
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> 4tf HEL | Cmvmc-H | Nckss 8
;2. Tfm ihvm` qumsthl` nl`shsts lg twl smts lg ghiurms. Ghiurms ;, :, > k`d 4 nl`sthtutm tfm Rrljcme
smt. Tfmrm hs k dmgh`htm rmckthl`sfhp jmtwmm` ghiurms ; k`d :. Mstkjchsf k shehckr rmckthl`sfhp jmtwmm`
ghiurms > k`d 4 jy nfllsh`i ghiurm 4 grle tfm kctmr`kthvms.
; : > 4
<
Rrljcme ^mt
(K) (J) (N) (D)
;;. L`m mvm`h`i jmglrm su`smt twl grhm`ds ^ueht k`d Elfht wmrm tkcah`i tl mknf ltfmr gknm tl gknm. Hg
Elfht's sfkdlw wks mxkntcy tl fhs rhift shdm, wfhnf dhrmnthl` wks ^ueht gknh`i<
(K) @lrtf (J) ^lutf (N) Ymst (D) @l`m lg tfmsm
;:. ^krtfka rmemejmrs tfkt fhs jrltfmr' s jhrtfdky hs kgtmr ;=tf jut jmglrm ;6 tf lg Gmjrukry wfmrmks fhs
shstmr Aksfhsf rmemejmrs tfkt fmr jrltfmr's jhrtfdky hs kgtmr ;7 tf jut jmglrm ;8 tf lg Gmjrukry. L`
wfhnf dky h` Gmjrukry hs ^krtfka's jrltfmr's jhrtfdky<
(K) ;7tf (J) ;9tf (N) ;6tf (D) ;8tf
;>. Lut lg tfm glur ghiurms (K), (J), (N) k`d (D), tfrmm krm shehckr h` k nmrtkh` wky. Flwmvmr, l`m ghiurm
hs `lt cham tfm ltfmr tfrmm. Nfllsm tfm ghiurm wfhnf hs dhggmrm`t grle tfm rmst.
(K) (J) (N) (D)
;4. Gh`d tfm ehssh`i nfkrkntmr h` tfm ihvm` qumsthl`. 9 8 :; :9
4 : >7 ;6
8 4 =4 <(K) ;6 (J) :4
(N) >7 (D) 46
;=. H` tfm ihvm` `uejmr smqum`nm, flw ek`y sunf mvm` `uejmrs krm tfmrm wfhnf krm mxkntcy dhvhshjcm
jy hts heemdhktm prmnmdh`i `uejmr jut `lt mxkntcy dhvhshjcm jy hts heemdhktm glcclwh`i `uejmr<
> 6 4 ; = 9 : 6 > 4 6 8 > 8 4 : ; = 6 :
(K) L`m (J) Twl (N) Tfrmm (D) Glur
;7. Hg 'k' emk`s 'pcus', 'j' emk`s 'eh`us', 'n' emk`s 'eucthpchmd jy' k`d 'd' emk`s 'dhvhdmd jy' tfm`;6 n ;4 k 7 j ;7 d 4 5 <
(K) 7> (J) :=4 (N) :66 (D) ;:26
;9. Tfm `uejmrs h` tfm ihvm` ghiurms krm krrk`imd knnlrdh`i tl k pkrthnuckr rucm. ^mcmnt tfm nlrrmnt
k`swmr wfhnf hs tl jm wrhttm` kt tfm pcknm lg shi` lg h`tmrrlikthl` glr tfhrd ghiurm.
(K) >; 8
:= ;6
>7
;7
4
6; :7
48
74
:=
74 <
>7
;44(J) :>
(N) :4
(D) :=
4tf HEL - :2;2
Nlpyrhift©:2;=. ̂ nhm`nm Lcyephkd Glu`dkthl` (^LG)
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44tf HEL | Cmvmc-H | Nckss 8
;6. Flw ek`y squkrms dlms tfm glcclwh`i ghiurm nl`tkh`<
(K) :9 (J) ;8 (N) >2 (D) @l`m lg tfmsm
;8. Flw ek`y pkhrs lg cmttmrs krm tfmrm h` tfm wlrd 'CK@I]H^F' wfhnf fkvm ks ek`y cmttmrs jmtwmm`
tfme h` tfm wlrd ks h` tfm kcpfkjmt<
(K) @hc (J) L`m (N) Tfrmm (D) Glur
:2. K jus stkrts grle nhty V. Tfm `uejmr lg wlem` h` tfm j us hs fkcg lg tfm `uejmr lg em`. H`
nhty X, ;2 em` cmkvm tfm jus k`d ghvm wlem` m`tmr. @lw, `uejmr lg em` k`d wlem` hs mqukc. Flw
ek`y pkssm`imrs m`tmrmd tfm jus h` tfm jmih``h`i<
(K) ;= (J) >2 (N) >7 (D) 4=
^MNTHL@ HH ? EKTFMEKTHNKC ZMK^L@H@I
:;. Tfm `uejmr -(> >)(> + >) hs k/k` \\\\ .
(K) Zkthl`kc `uejmr (J) @kturkc `uejmr (N) Zmkc `uejmr (D) Kcc lg tfmsm
::. Yfhnf lg tfm glcclwh`i plcy`lehkcs fks ( x + ;) ks k gkntlr<
H. x > + x : + x + ; HH. x 4 + x > + x : + x + ;
HHH. x 4 + > x > + > x „: + x + ; HW. x > + x : „ (: + : )x + :
(K) L`cy H (J) L`cy HHH (N) HH k`d HW (D) H k`d HH
:>. Twl dhsth` nt ch` ms \\\\\ .
(K) Kcwkys h`tmrsmnt (J) Kcwkys mhtfmr h`tmrsmnt lr krm pkrkccmc
(N) Kcwkys fkvm twl nleel` plh`ts (D) @l`m lg tfmsm
:4. H` tfm ihvm` ghiurm, vkcum lg x hs \\\\ .
K D
N
J
>4² 4=²
>2² x
(K) 4=² (J) >4² (N) 74² (D) ;28²
:=. Yfhnf lg tfm glcclwh`i stktmem`ts hs nlrrmnt <
(K) K trhk`icm fks twl rhift k ìcms.
(J) Kcc tfm k`icms lg k trhk`icm krm elrm tfk` 72².
(N) K` mxtmrhlr k`icm lg k trhk`icm hs kcwkys irmktmr tfk` tfm lpplshtm h`tmrhlr k`icms.
(D) Kcc tfm k`icms lg k trhk`icm krm cmss tfk` 72².
:7. K plh`t hs dmgh`md ks \\\\\ .
(K) Tfkt wfhnf fks `l cm`itf `l jrmkdtf `l fmhift. (J) ^ekcc pkrt lg k ch`m whtf `l cm`itf.
(N) Dhem`shl`cmss, jut rmprmsm`tmd jy dlt l`cy. (D) ]`dmgh`md tmre k`d rmprmsm`tmd jy k dlt.
4tf HEL - :2;2
Nlpyrhift©:2;=. ̂ nhm`nm Lcyephkd Glu`dkthl` (^LG)
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= 4tf HEL | Cmvmc-H | Nckss 8
:9. L` pcltth`i tfm plh`ts L (2, 2), K (>, 2), J(>, 4), N (2, 4) k`d olh`h`i LK, KJ, JN k`d NL , wfhnf lg
tfm glcclwh`i ghiurm hs ljtkh`md <
(K) ^qukrm (J) Zmntk`icm (N) Trkpmzhue (D) Zflejus
:6. K studm`t hs ihvm` tfrmm sthnas lg cm`it fs ;: ne, 7 ne k`d 4 ne rmspmnthvmcy. Fm hs ksamd tl ekam
k trhk`icm k`d tfm` gh`d tfm krmk lg tfm trhk`icm glremd jy tfm > sthnas <(K) >7 ne: (J) ;: ne:
(N) :4 ne: (D) Rrlvhdh`i krmk `lt plsshjcm
:8. Hg ch`ms c || e k`d Òk ? Òj 5 > ? :, Ò[ 5 82² k`d Òj + Òn 5 ;22², tfm` gh`d ÒR grle ihvm` ghiurm.
c
e
K
R
[
J
k
jn
(K) :7² (J) >:² (N) ;:² (D) 7:²
>2. Yfhnf lg tfm glcclwh`i hs k gkntlr lg ( x + y )> „ ( x > + y >) <
(K) x : + y : + : xy (J) x : + y : „ xy (N) xy : (D) > xy
>;. Tfrluif flw ek`y ekxheue `uejmr lg qukdrk`ts nk` k strkhift ch`m pkss <
(K) : (J) > (N) ; (D) 4
>:. Yfkt eust jm sujtrkntmd grle 4 x 4 „ : x > „ 7 x : + x „ =, sl tfkt tfm rmsuct hs mxkntcy dhvhshjcm jy
: x : + x „ ; <
(K) = (J) ;: (N) „ 4 (D) „ 7
>>. H` tfm ihvm` ghiurm, KJ || DM , tfm` emksurm lg Ò KND hs \\\\\ .
c K
N
D
J
M
42²
4>²
(K) 4>² (J) 42² (N) 6>² (D) 89²
>4. Hg -
-
-
9 ; 9 + ;5 + 9,
9 + ; 9 ; k j tfm` gh`d tfm vkcums lg k k`d j.
(K) k 5 2, j 5>
: (J)
>
: k
-
5 , j 5 ; (N) :
2,>
k j -
5 5 (D) k 5 4, j 5 :
>=. Hg k + j + n 5 8 k`d k : + j : + n : 5 >=, tfm` gh`d tfm vkcum lg k > + j > + n > „ >kjn.
(K) =: (J) ;26 (N) :;7 (D) ;6:
>7. Hg lrdh`ktm lg k plh`t hs zmrl, tfm` tfhs plh`t kcwkys chms \\\\ .
(K) L` tfm x-kxhs (J) L` tfm y-kxhs (N) H` tfm HHHrd qukdrk`t (D) H` tfm HWtf qukdrk`t
4tf HEL - :2;2
Nlpyrhift©:2;=. ̂ nhm`nm Lcyephkd Glu`dkthl` (^LG)
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74tf HEL | Cmvmc-H | Nckss 8
>9. KJND hs k ihvm` ghmcd wflsm shdms krm h`dhnktmd. Hg ÒDKJ 5 82², tfm` krmk lg tfm ghmcd hs \\\.
D N
KJ
:6 e
;= e
42 e
8 e
(K) ;62 e: (J) ;:7 e: (N) >27 e: (D) >22 e:
>6. Rlh`ts h` wfhnf kjsnhssk k`d lrdh`ktm fkvm dhggmrm`t shi`s whcc chm h` \\\\ .
(K) Hst k`d HHHrd qukdrk`ts (J) HHHrd k`d HWtf qukdrk`ts (N) HH`d k`d HHHrd qukdrk`ts (D) HH`d k`d HWtf qukdrk`ts
>8. KJN hs k` hslsnmcms trhk`icm whtf KJ 5 KN . Kcthtudms krm drkw` tl tfm shdms KJ k`d KN grle vmrthnms
J k`d N . L`m kcthtudm NG hs glu`d tl jm 4 ne k`d JN 5 = ne. Gh`d MN , wfmrm JM hs kcthtudm tl
shdm KN .
(K) 4 ne (J) = ne (N) > ne (D) @l`m lg tfmsm
42. H` ihvm` ghiurm, hg KJ || ND || MG , R[ || Z^ , ÒZ[D 5 :=² k`d ÒN[R 5 72², tfm` Ò[Z^ hs mqukctl \\\\ .
K
N
M
J
D
G
^
Z
[
R
72²
:=²
(K) 6=² (J) ;>=² (N) ;4=² (D) ;;2²
^MNTHL@ HHH ? MWMZXDKX EKTFMEKTHN^
4;. Tfm E.R. lg k nkemrk hs>
:lg tfm N.R. k`d ^.R. hs
8
;2lg E.R. Gh`d tfm pmrnm`tkim lg prlght lr clss.
(K) :=% prlght (J) >=% prlght (N) >>.>>% clss (D) @l`m lg tfmsm
4:. R nk` dl k olj h` ;2 dkys, [ nk` dl tfm skem olj h` ;: dkys k`d Z nk` dl tfm skem olj h` ;= dkys.
H` flw ek`y dkys tfmy whcc gh`hsf tfm wlra tlimtfmr<
(K) : (J) 4 (N) 7 (D) ;:
4>. Krekk` ilms ;2 ae dhstk`nm whtf kvmrkim spmmd lg 7 ae/fr wfh cm tfm rmst :2 ae fm trkvmcs whtf
k` kvmrkim spmmd lg ;= ae/fr. Yfkt hs tfm kvmrkim spmmd lg Krekk` durh`i tfm wflcm olur`my<
(K) ;2 ae/fr (J) ;: ae/fr (N) ;> ae/fr (D) ;4.= ae/fr
44. K gktfmr hs tfrmm thems ks lcd ks fhs sl`. Kgtmr twmcvm ymkrs, fhs kim whc c jm twhnm tfkt lg fhs sl`.
Gh`d tfmhr prmsm`t kims.
(K) ;2, >2 (J) ;:, >7 (N) 7, >7 (D) ;:4, :4
4=. :=2 em` tlla k dhp h` k wktmr tk`a kt k them, wfhnf hs 62 e Ù =2 e. Yfkt hs tfm rhsm h` tfm wktmr
cmvmc hg tfm kvmrkim dhspcknmem`t lg ; ek` hs 4 e> <
(K) :: ne (J) := ne (N) ;6 ne (D) >2 ne
4tf HEL - :2;2
Nlpyrhift©:2;=. ̂ nhm`nm Lcyephkd Glu`dkthl` (^LG)
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9 4tf HEL | Cmvmc-H | Nckss 8
47. Zkfuc wm`t tl purnfksm k @lahk eljhcm fk`dsmt. Tfm sflpammpmr tlcd fhe tl pky :2% tkx hg fm ksamd
glr tfm jhcc. Zkfuc ek`kims tl imt tfm dhsnlu`t lg =% l` tfm kntukc skcm prhnm lg tfm eljhcm k`d fm
pkhd tfm sflpammpmr >>:= whtflut tkx. Jmshdms fm ek`kims tl kvlhd tl pky :2% tkx l` tfm kcrmkdy
dhsnlu`tmd prhnm. Yfkt hs tfm kelu`t lg dhsnlu`t tfkt fm fks ilt <
(K) b 9=2 (J) b >9= (N) b 69= (D) b =:=
49. KJN hs k trhk`icm. D, M k`d G krm tfm ehd-plh`ts lg tfm shdms JN , NK k`d KJ rmspmnthvmcy. Tfm rkthl
lg tfm krmk lg D KJN k`d DDMG hs \\\\\
(K) 4 ? ; (J) = ? ; (N) > ? ; (D) Nk`'t jm dmtmreh`md
46. R k`d [ nk` dl k phmnm lg wlra h` ;: dkys, [ k`d Z h` ;= dkys, Z k`d R h` :2 dkys. H` flw ek`y
dkys nk` Z kcl`m dl ht<
(K) 72 (J) =2 (N) := (D) :4
48. K tm`t hs h` tfm glre lg rhift nhrnuckr nl`m ;2.= e fhif, tfm dhkemtmr lg tfm jksm jmh`i ;> e. Hg
6 em` krm h` tfm tm`t, gh`d tfm kvmrkim `uejmr lg nujhn emtrms lg khr spknm pmr ek`.
(K) >
>::6
e> (J) =8.9= e> (N) 8
>7;>
e> (D) >
=6>:
e>
=2. Hg tfm k``ukc h`nrmksm h` tfm plpuckthl` jm :2% k`d tfm prmsm`t plpuckthl` jm ;2,222. Yfkt whcc
jm tfm plpuckthl` kgtmr > ymkrs <
(K) ;7,222 (J) ;9,:62 (N) ;4,422 (D) ;,9:6
^RKNM GLZ ZL]IF YLZA
4tf HEL - :2;2
Nlpyrhift©:2;=. ̂ nhm`nm Lcyephkd Glu`dkthl` (^LG)
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=tf
Xmkr :2;;
Nckss 8
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: =tf HEL - :2;;
^MNTHL@ H ? CLIHNKC ZMK^L@H@I
;. ^tudy tfm glcclwh`i h`glrekthl` k`d k`swmr tfm qumsthl` ihvm` jmclw ht.
K jknasehtf fks m hrl` krthnms K, J, N, D k`d M, mknf fkh`i k dhggmrm`t wmhift.
(h) K wmhifs twhnm ks eunf ks J. (hh) J wmhifs glur k`d k fkg thems ks eunf ks N.
(hhh) N wmhifs fkg ks eunf ks D. (h) D wmhifs fkg ks eunf ks M.
(v) M wmhifs cmss tfk` K jut elrm tfk` N.
Yfhnf lg tfm glcclwh`i hs tfm chiftmst<
(K) K (J) J (N) N (D) D
:. Nfllsm tfm ldd `uemrkc pkhr/irlup.
(K) 8 „ > (J) ;
:
;
6√ (N)
;
>
;
;:√ (D) :4 „ 7
>. ^mmnt k iurm grle kel`ist tfm lpthl`s wfhnf wh nl`th`um tfm skem smrhms ks mstkjhsfmd jy
tfm m Rrljme Ghiurms.
(K) (J) (N) (D)
4. Lut lg glur iurms ekramd (K), (J), (N) k`d (D) tfrmm krm shehkr h` k nmrtkh` ek``mr. Flwmmr l`m
iurm hs `lt ham tfm ltfmr tfrmm. Nfllsm tfm iurm wfhnf hs dhggmrm`t grle tfm rmst.
(K) (J) (N) (D)
=. H` tfm ektrhx tfmrm krm mhift dmshi`s k`d l`m spknm hs cmgt jck`a ks sflw` jy k qumsthl` ekra.Yfhnf k`swmr iurms wh rmpknm tfm qumsthl` ekra<
<
(K) (J) (N) (D)
7. Tfm jk`a jkk`nm lg Rurk` hs elrm tfk` tfkt lg ^usfek jut mss tfk` tfm jk`a jkk`nm lg ^h`if.
Hg tfm jk`a jkk`nm lg Rurk`, ^usfek k`d ^h`if jm x, y k`d z rmspmnthvmcy tfm` wfhnf lg tfm
glcclwh`i hs nlrrmnt<
(K) x 3 y 3 z (J) y 3 x 3 z (N) z 3 x 3 y (D) x 3 z 3 y
9. Gh`d tfm `uejmr lg trhk`ims h` tfm ihm` iurm.
(K) ;6
(J) :2
(N) :4
(D) @l`m lg tfmsm
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>=tf HEL - :2;;
6. Yfhnf `uejmr whcc rmpcknm tfm qumsthl` ekra<
6 6 4
7 7 7;4 ;4 <
4 6 =
;2 ;4 ;;;6 :: ;=
(K) 6 (J) 4 (N) ;2 (D) 7
8. Tfmrm krm tfrmm phc ckrs V, X k`d _ lg dhggmrm`t fmhifts. Tfrmm sphdmrs K, J k`d N stkrt tl nchej l`
tfmsm phcckrs sheuctk`mluscy. H` l`m nfk`nm K nchejs l` V = ne jut schps dlw` ; ne. J nchejs
l` X 7 ne jut schps dlw` > ne. N nchejs l` _ 9 ne jut schps dlw` : ne. Hg mknf rmquhrms =2
nfk`nms tl rmknf tfm tlp lg tfm phkr, wfkt hs tfm fmhift lg tfm sflrtmst phkr<
(K) ;44 ne (J) ;=: ne (N) ;=> ne (D) ;4; ne
;2. Irlup tfm ihm` iurms h`tl tfrmm nkssms ush`i mknf iurm l ̀y l`nm.
(K) >, 4, 81 =, 9, 61 ;, :, 7 ; : >
4 = 7
9 6 8
(J) ;, =, 71 :, 4, 61 >, 9, 8
(N) 4, 7, 61 >, =, 91 ;, :, 8
(D) ;, :, 91 >, =, 81 4, 7, 6
;;. Hg >:4 Ù ;=2 5 =4, :=; Ù 42: 5 46 k`d =:> Ù >4= 5 ;:2, tfm` 7=; Ù >4= 5 <
(K) ;:2 (J) 6= (N) ;44 (D) 72
;:. Gh`d tfm iurm wfhnf nl`tkh`s tfm Ghi. (V) ks hts mejmddmd pkrt.
Ghi. (V)
(K) (J) (N) (D)
;>. H` tfm `uejmr =8247:96;> rst k`d shxtf dhihts krm h`tmrnfk`imd1 k`d shehkry tfm plshthl`s lg
smnl`d k`d smvm`tf dhihts k`d sl l` krm h`tmrnfk`imd. Yfhnf lg tfm glcclwh`i whcc jm tfm 4 tf dhiht
grle tfm rhift kgtmr tfm `mw krrk`imem`t<
(K) ; (J) 8 (N) 4 (D) 2
;4. H` tfm ihvm` ektrhx tfm vkcums lg K, J k`d N rmspmnthvmcy krm \\\\. (K) K 5 ;>, J 5 ;;, N 5 8 (J) K 5 ;>, J 5 8, N 5 ;;
8
J
N
K
;2
6
;:
9
;;
(N) K 5 8, J 5 ;;, N 5 ;> (D) K 5 8, J 5 ;>, N 5 ;;
;=. Twl nkrs stkrt grle lpplshtm pknms l` k ekh` rlkd, ;=2 ae kpkrt. Ghrst nkr ru`s glr := ae k`d
tkams k rhift tur` k`d tfm` ru`s ;= ae. Ht tfm` tur`s cmgt k`d ru`s glr k`ltfmr := ae k`d tfm`
tkams tfm dhrmnthl` tl rmknf tfm ekh` rlkd. H` tfm emk` them, dum tl k eh`lr jrmka dlw` tfm ltfmr
nkr fks ru` l`cy >= ae kcl`i tfm ekh` rlkd. Yfkt wlucd jm tfm dhstk`nm jmtwmm` tfm twl nkrs kt
tfhs plh`t<
(K) 7= ae (J) 9= ae (N) 62 ae (D) 6= ae
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4 =tf HEL - :2;;
;7. @mmke, wfl hs Zlfht's dkuiftmr, skys tl H`du, "Xlur eltfmr Zmmtk hs tfm ylu`imr shstmr lg ey
gktfmr, wfl hs tfm tfhrd nfhd lg ^lfk`oh." Flw hs ^lfk`oh rmktmd tl H`du<
(K) Ektmr`kc u`ncm (J) Gktfmr (N) Irk`dgktfmr (D) Gktfmr-h`-ckw
;9. Hg Ell` hs nkmd ^tkr, ^tkr hs ^u`, ^u` hs Mkrtf, Mkrtf hs Ouphtmr, Ouphtmr hs Ekrs k`d Ekrs hs
Emrnury, wfmrm dlms ek` hm<
(K) Ekrs (J) Ouphtmr (N) Mkrtf (D) ^u`
;6. H` tfm ihm` iurm, squkrm rmprmsm`ts dlntlrs, trhk`im rmprmsm`ts kdhms k`d nhrnm rmprmsm`ts
suriml`s. Yfhnf cmttmr rmprmsm`ts tfm ckdhms wfl krm jltf dlntlrs k`d suriml`s<
R
^ T
Z
]
[
(K) ] (J) T (N) ^ (D) R
;8. H` tfm ihm` qumsthl`, wfhnf `uejmr wh rmpknm tfm qumsthl` ekra<
2 ; <
4 6 2
= 9 ;;7 7 :
9 4 :
(K) 2 (J) : (N) ;; (D) ;:
:2. Hg tfm plshthl`s lg tfm tfhrd k`d tm`tf mttmrs lg tfm wlrd DLN]EM@TKTHL@ krm h`tmrnfk`imd, k`d
hamwhsm tfm plshthl`s lg tfm glurtf k`d smm`tf mttmrs, tfm smnl`d k`d shxtf mttmrs, krm ksl
h`tmrnfk`imd, tfm` wfhnf lg tfm gllwh`i wh jm mmm`tf mttmr grle tfm rhift m`d<
(K) N (J) H (N) T (D) ]
^MNTHL@ HH ? EKTFMEKTHNKc ZMK^L@H@I
:;. Hg x 3 „;, tfm` wfhnf lg tfm gllwh`i mxprmsshl`s flds<
(K) x 3 x: 3 x> (J) x 3 x> 3 x: (N) x: 3 x> 3 x (D) x> 3 x 3 x:
::. Hg tfm pmrhemtmr lg k rmntk`iukr ikrdm` plt hs 62 emtrms, wfhnf lg tfm gllwh`i nlud jm tfm
cm`itf lg l`m lg hts shdms<
H. >2 emtrms HH. 42 emtrms HHH. =2 emtrms
(K) H l`cy (J) HH l`cy (N) HHH l`cy (D) H k`d HH
:>. H` tfm kdolh`h`i iurm, nhrnms whtf nm`trms kt C, E k`d @ mknf fkvm k rkdhus lg : u`hts k`d krmpcknmd ks sflw` fmrm. Hg R[Z^ hs tfm sekmst rmntk`im tfkt wh m`nlsm tfm > nhrnms, wfkt hs
tfm krmk lg R[Z^<
C E @
R [
Z ^
(K) 6 sq.u`hts (J) ;: sq.u`hts (N) ;7 sq.u`hts (D) >: sq.u`hts
:4. Hg >x + :y 5 ;9 k`d x k`d y krm plshthm h`tmimrs, tfm` y nlucd jm wfhnf lg tfm glcclwh`i<
(K) : (J) > (N) 4 (D) =
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==tf HEL - :2;;
:=. H` tfm ihm` iurm c;
|| c:. Yfhnf lg tfm glcclwh`i eust jm trum<
(K) g 5 j
(J) j 5 n
(N) n 5 m
(D) d 5 g
:7. H` tfm kdolh`h`i iurm, plh`t R tfm h`tmrsmnthl` lg ch`ms c; k`d c
: fks nllrdh`ktms (k, j) ks sflw`
fmrm. Yfhnf lg tfm glcclwh`i nlucd jm tfm lrdmrmd pkhr (k, j)<
X
c :
c ;
( )k,j
V
R
L
(K) = =
:,
√
(J) √
; =
6, (N) (2, 4) (D)
:
>
;
>,
:9. H` tfm ihm` iurm, tfrmm mquhktmrk trhk`ims fkm k nleel` mrtmx tfm` x + y + z 5
(K) 82² (J) ;:2² (N) ;=2² (D) ;62²
:6. Hg k rhift t rhk`icm fks shdms lg cm`itf ; k`d :, wfhnf lg tfm glcclwh`i nlucd jm tfm cm`itf lg tfm
rmekh`h`i shdm<
H. ; HH. : HHH. >
(K) H l`cy (J) HH l`cy (N) HHH l`cy (D) H k`d HHH l`cy
:8. Tfm mxprmsshl` (k + j + n): + (k + j „ n): + (k „ j + n): + (j + n „ k): mqukcs \\\\.
(K) 4(kj + kn + jn) (J) :(k: + j: + n: + kj + jn + kn)
(N) 4(k: + j: + n:) (D) (k: + j: + n:)
>2. K dhm hs rlmd ;2,=22 thems. Tfm grmqum`ny lg mknf lutnlem hs sflw` h` tfm tkjm. Yfkt hs tfm
mephrhnkc prljkjhchty lg tfm lutnlems lg eucthpcms lg ><
Lutnlem
Grmqum`ny ;79= ;9:= ;74: ;976 ;69> ;6;9
(K) ;;=>
>=22 (J)
;;=>
>4=8 (N)
;;=>
;2=22 (D)
>4=8
;2=222
>;. H` D's KJN k`d R[Z, KJ 5 R[, JN 5 [Z krm mxtm`dmd tl V k`d X rmspmnthvmcy k`d ∢ KJV 5 ∢R[X.
]`dmr wfhnf nl`irum`ny nrhtmrhl`, tfm trhk`ims krm nl`irum`t<
(K) ^^^ (J) ^KK (N) ^^K (D) ^K^
>:. Gh`d tfm kum lg ;;9748
;99;=7;
; 7ç
æééé
÷
öùùù
/
.
(K) >/;; (J) ;>/9 (N) 9/;; (D) 9/;>
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7 =tf HEL - :2;;
>>. KJND hs k trkpmzhue, h` wfhnf KJ || DN, KD P KJ k`d M & G krm plh`ts l` KD & JN rmspmnthvmcy.
JD hs olh`md. Hg ∢MGN 5 ;;2², ∢ KJD 5 72², ∢JDN 5 x ² k`d ∢DNG 5 y ², tfm` kums lg x k`d y krm
\\\\.
(K) 72², 92² (J) =2², 62² (N) 82², 42² (D) >2², ;22²
>4. Gh h` tfm jk`a ?
Hg k strkhift ch`m emmts twl ltfmr strkhift ch`ms sl ks tl ekam tfm sue lg tfm twl h`tmrhlr k`icms
l` l`m shdm lg tfm trk`smrsk mss tfk` twl rhift k`ims, tfm` tfm ltfmr strkhift h`ms mxtm`dmd
h``htmy, wh emmt \\\\ tfm trk`smrsk.
(K) L` tfkt shdm lg (J) L` k`ltfmr shdm lg (N) L` (D) Nk`'t jm dmtmreh`md
>=. Yfhnf lg tfm glcclwh`i lpth l`s h` Nlcue` H whtf rmksl`s h` Nlcue` HH hs nlrrmnt whtf rmspmnt tl
tfm glcclwh`i qumsthl` ?
Flw ek`y h`mkr mqukthl`s h` x k`d y nk` jm skthsmd jy x 5 ; k`d y 5 ><
Nlcue` H Nlcue` HH
(K) L`cy l`m (h) ks l`cy x 5 ;, y 5 > hs ihvm`.
(J) Mxkntcy twl (hh) l`m glr x 5 ;, smnl`d glr y 5 >.(N) H``htmcy ek`y (hhh) l`m glr x 5 ;, smnl`d glr y 5 > k`d tfhrd glr x + y 5 >.
(D) Mxkntcy tfrmm (hv) ks plh`t (;, >) chms l` irkpf k`d tfrluif l`m plh`t h``htm ch`ms nk` pkss, sl
wm imt h``htm irkpfs lr h``htm mqukthl`s.
>7. Hg tfm prldunt lg twl surds hs k rkthl`k `uejmr, tfm` mknf l`m lg tfme hs nkmd tfm \\\\\ lg tfm
ltfmr \\\\\.
(K) @mikthvm (J) Zkthl`kchsh`i gkntlr (N) Nl`ouiktm surd (D) Jh`lehkc surd
>9. Hg l`m cmi lg k` hslsnmcms rhift-k`icmd trhk`icm hs h`n rmksmd jy 7 ne k`d tfm ltfmr cmi hs h`n rmksmd
jy 4 ne, tfm` nfk`im h` krmk lg tfm trhk`ims hs :4 sq ne. Gh`d tfm m`itf lg tfm mi lg tfm lrhih`k
trhk`icm.
(K) =
;:ne (J)
;:
=ne (N)
;
=ne (D)
;
;:ne
>6. Zmekh`dmr wfm` k: + j: + :(kj + jn + nk) hs dhvhdmd jy (k + j + :n) hs \\\\\\\.
(K) k + j (J) kj (N) k „ j (D) 2
>8. Tfm vkcum lg x
x
x
x
x
x
c
e
c e
e
`
e `
`
c
` c
:
:
:
:
:
:
+ + +⋍ ⋍ 5 \\\\\\.
(K) x (J) „; (N) ; (D) 2
42. Kcc tfm tfrmm qukdrhcktmrkcs KDMN, KJHF k`d JNIG krm squkrms k`d ∢ KJ N 5 82². Hg tfm krmk lg
KDMN 5 x: k`d krmk lg KFHJ 5 y:(x: 0 y:), tfm` `d tfm krmk lg JNIG<
K
J N
D
M
G I
F
H
(K) x: + y: (J) :y: (N) :x: (D) x: „ y:
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9=tf HEL - :2;;
^MNTHL@ HHH ? MvMZXDKX EKTFMEKTHN^
4;. K nkss stkrts kt ;2 k.e. k`d ksts th ;.:9 p.e. Glur pmrhlds krm fmd durh`i tfhs h`tmrk. Kgtmr mmry
pmrhld, = eh`utms krm ihm` grmm tl tfm studm`ts. Tfm mxknt durkthl` lg mknf pmrhld hs \\\\.
(K) 4: eh`utms (J) 46 eh`utms (N) =; eh`utms (D) => eh`utms
4:. H` Kru`'s lph`hl`, fhs wmhift hs irmktmr tfk` 7= ai jut mss tfk` 9: ai. Fhs jrltfmr dlms `lt kirmmwhtf Kru` k`d fm tfh`as tfkt Kru`'s wmhift hs irmktmr tfk` 72 ai jut mss tfk` 92 ai. Fhs eltfmr's
hmw hs tfkt fhs wmhift nk``lt jm irmktmr tfk` 76 ai. Hg k lg tfme krm nlrrmnt h` tfmhr msthekthl`s,
wfkt hs tfm kvmrkim lg dhggmrm`t prljkjcm wmhifts lg Kru`<
(K) 79 ai (J) 76 ai (N) 78 ai (D) @l`m lg tfmsm
4>. K ek` wkamd dhkil`ky knrlss k squkrm plt. Kpprlxhektmy, wfkt wks tfm pmrnm`t skmd jy `lt
wkcah`i kcl`i tfm mdims<
(K) :2 (J) :4 (N) >2 (D) >>
44. K ek` purnfksms twl nclnas K k`d J kt k tltkc nlst lgb 7=2. Fm sms K kt :2% prlt k`d J kt k
clss lg :=% k`d imts tfm skem smcch`i prhnm glr jltf tfm nclnas. Yfkt krm tfm purnfksh`i prhnms
lg K k`d J rmspmnthvmcy<
(K) b ::=, b 4:= (J) b :=2, b 422 (N) b :9=, b >9= (D) b >22, b >=2
4=. Mhift pmlpcm krm pck``h`i tl sfkrm mqukccy tfm nlst lg k rm`tkc nkr. Hg l`m pmrsl` whtfdrkws grle
tfm krrk`imem`t k`d tfm ltfmrs sfkrm mquky tfm m`thrm rm`tk lg tfm nkr, tfm` tfm sfkrm lg mknf
lg tfm rmekh`h`i pmrsl`s hs h`nrmksmd jy \\\\ lg tfm lrhih`kc sfkrm.
(K) ;/8 (J) ;/6 (N) ;/9 (D) 9/6
47. K studm`t wks ksamd tl dhhdm k `uejmr jy 7 k`d kdd ;: tl tfm qulthm`t. Fm, flwmmr, rst kddmd
;: tl tfm `uejmr k`d tfm` dhhdmd ht jy 7, imtth`i ;;: ks tfm k`swmr. Tfm nlrrmnt k`swmr sflud
fkvm jmm` \\\\.
(K) ;:: (J) ;;6 (N) ;;4 (D) ;:4
49. Zmmek jluift x pm`s kt b :.72 mknf k`d y irmmth`i nkrds kt 62 pkhsm mknf. Hg tfm pm`s nlst
b ;: elrm tfk` tfm nkrds, tfm mqukthl` h`lh`i x k`d y hs \\\\\.
(K) ;>x „ 4y 5 7 (J) ;>x „ 4y 5 72 (N) :72x „ 6y 5 ;22 (D) :72x „ 6y 5 ;:
46. Tl ekam k ekrrhkim tm`t, plms krm pk`tmd kl`i tfm pmrhemtmr lg k squkrm md kt k dhstk`nm lg
= emtrms grle mknf ltfmr k`d tfm tltkc `uejmr lg plcms usmd hs :2. Yfkt hs tfm krmk (h` sq. emtrms)
lg tfm squkrm md<
(K) =22 (J) 422 (N) 822 (D) @l`m lg tfmsm
48. Fusskh`'s wmhift hs ; ;
4thems tfm wmhift lg Zkfhe. Tfmhr tltk wmhift hs 82 ai. Yfkt hs Fusskh`'s
wmhift<
(K) 82 ai (J) =2 ai (N) 42 ai (D) :2 ai
=2. H` k nkss, tfm `uejmr lg jlys hs elrm tfk` tfm `uejmr lg ihrs jy ;:% lg tfm tltk strm`itf. Tfm
rkthl lg jlys tl ihrcs hs \\\\\.
(K) ;= ? ;; (J) ;; ? ;4 (N) ;4 ? ;; (J) 6 ? ;;
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Xmkr :2;:
^MT K
Nckss 8
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^MNTHL@ H ? CLIHNKC ZMK^L@H@I
;. H` tfm glcc lwh`i qumsthl`, tfm syejl cs B, © , $ k`d % krm usmd whtf tfm glcclwh`i emk`h`is ks
hccustrktmd jmclw ?
'K $ J' emk`s 'K hs `lt sekccmr tfk` J'1
'K B J' emk`s 'K hs `mhtfmr sekccmr tfk` `lr mqukc tl J'1
'K J' emk`s 'K hs `mhtfmr sekccmr tfk` `lr irmktmr tfk` J'1
'K % J' emk`s 'K hs `mhtfmr irmktmr tfk` `lr mqukc tl J'.
@lw h` tfm glcclwh`i qumsthl`, kssueh`i tfm ihvm` stktmem`ts tl jm trum, `d wfhnf lg tfm
tfrmm nl`ncushl`s, H, HH k`d HHH ihvm` jmclw tfme hs/krm dm`htmcy trum k`d ihvm ylur k`swmr
knnlrdh`icy.
^tktmem`ts ? A © R, R B [, [ $ Z
Nl`ncushl`s ? H A B Z HH. Z % R HHH. [ % A
(K) L`cy H k`d HH krm trum (J) L`cy HH k`d HHH krm trum
(N) L`cy HHH hs trum (D) Kcc H, HH k`d HHH krm trum
:. ^mcmnt k iurm grle kel`ist tfm lpthl`s wfhnf whcc nl`th`um tfm skem smrhms ks mstkjchsfmd jy
tfm vm Rrljcme Ghiurms R, [, Z, ^ k`d T.
(K) +
N ^ (J)
+
N
^
(N)
^
N+
(D)
+
N^ +
>. ^tudy tfm glcclwh`i h`glrekthl` nkrmguccy k`d k`swmr tfm qumsthl` ihvm` jmclw.
R, [, Z, ^, T, W k`d Y krm smvm` pkssm`imrs k`d H, HH, HHH krm tfrmm dhggmrm`t vmfhncms. Tfmrm krm
ktcmkst twl pkssm`imrs h` mknf vmfhncm H, HH k`d HHH k`d l`m lg tfme hs k ckdy. Tfmrm krm twl
m`ih`mmrs, twl dlntlrs k`d tfrmm tmknfmrs kel`i tfme. Z hs ckdy dlntlr k`d sfm dlms `lt trkvmcwhtf tfm pkhr lg shstmrs R k`d W. [, k ekcm m`ih`mmr trkvmcs whtf l`cy Y, k tmknfmr h` vmfhncm H. ^
hs k ekcm dlntlr. Twl pmrsl`s jmcl`i tl skem prlgmsshl` dl `lt trkvmc h` tfm skem vmfhncm. R hs
`lt k` m`ih`mmr k`d trkvmcs h` vmfhncm HH.
Flw ek`y ckdy emejmrs krm tfmrm kel`ist tfme<
(K) Tfrmm (J) Glur (N) Tfrmm lr Glur (D) Dktk h`kdmquktm
4. H`spmntlr Okth` trkvmccmd grle fhs plchnm stkthl` glr 422 emtrms. Fm tfm` tur`md cmgt k`d trkvmccmd
=22 emtrms strkhift kgtmr wfhnf fm tur`md cmgt kikh` k`d trkvmccmd glr 422 emtrms strkhift. Fm tfm`
tur`md rhift k`d wkcamd glr k`ltfmr 722 emtrms strkhift. Flw gkr hs fm grle tfm Rlchnm stkthl`<
(K) ;.2 ae (J) ;.; ae (N) ;.4 ae (D) ;.6 ae
=. Zmkd tfm glcclwh`i h`glrekthl` nkrmguccy k`d k`swmr tfm qumsthl` ihvm` jmclw ht.
(h) Tfmrm hs k irlup lg shx pmrsl`s K, J, N, D, M k`d G h` k gkehcy. Tfmy krm Rsynflclihst, Ek`kimr,
Ckwymr, Omwmccmr, Dlntlr k`d M`ih`mmr.
(hh) Tfm dlntlr hs tfm irk`dgktfmr lg G wfl hs k Rsynflclihst.
(hhh) Tfm Ek`kimr D hs ekrrhmd tl K.
(hv) N, tfm Omwmccmr, hs ekrrhmd tl tfm Ckwymr.
(v) J hs tfm eltfmr lg G k`d M.
(vh) Tfmrm krm twl ekrrhmd nlupcms h` tfm gkehcy.
Flw hs K rmcktmd tl M<
(K) Jrltfmr (J) ]`ncm (N) Gktfmr (D) Irk`dgktfmr
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7. Whoky hs gtmm`tf grle tfm grl`t h` k nlcue` lg jlys. Tfmrm wmrm tfrhnm ks ek`y jmfh`d fhe ks
tfmrm wmrm h` grl`t. Flw ek`y jlys krm tfmrm jmtwmm` Whoky k`d tfm smvm`tf jly grle tfm m`d lg
tfm nlcue`<
(K) >> (J) >4 (N) >= (D) Dktk h`kdmquktm
9. Yfhnf lg tfm lpthl`s whcc nlem `mxt h` tfm smrhms<
OK_, CMV, @HW, RLT, \\ , flw whcc ylu nldm ‖JMKT‘<
(K) :=-:>-:4-9 (J) :=-:;-:7-9 (N) :=-::-:=-9 (D) :=-::-:7-9
;2. Hg ;:% pmlpcm h` k vhcckim krm suggmrh`i grle nk`nmr k`d ;>% grle jclld prmssurm, wfhnf lg tfm
glcclwh`i dhkirke jmst rmprmsm`ts tfm shna plpuckthl` lg tfm vhcckim<
(K) (J) (N)
(D)
;;. Clla kt tfm syejlc-cmttmr-`uejmr smqum`nm ihvm` jmclw.
: R O B 6 $ C J ; W # [ 7 d I Y 8 A N D > • £ = G Z 9 K X 4
Tfrm m lg tfm glcclwh`i glur krm shehckr h` rmckthl` tl tfmhr plshthl`s h` tfm kjlvm smqum`nm k`d
fm`nm glre k irlup. Yfhnf l`m dlms `lt jmcl`i tl tfkt irlup<
(K) [ A = (J) C 7 D (N) R C d (D) ; I
;:. K nfhcd wm`t 82 e h` tfm Mkst tl clla glr fhs gktfmr, tfm` fm tur`md rhift k`d wm`t :2 e. Kgtmr tfhs
fm tur`md rhift k`d kgtmr ilh`i >2 e, fm rmknfmd tl fhs u`ncm's flusm. Fhs gktfmr wks `lt tfmrm.Grle tfmrm fm wm`t ;22 e tl fhs @lrtf k`d emt fhs gktfmr h` tfm sflp. Flw gkr dhd fm emmt fhs
gktfmr grle tfm stkrth`i plh`t<
(K) 62 e (J) ;22 e (N) ;42 e (D) :72 e
;>. Ektnf tfm ihvm` `mt whtf tfm nlrrmnt phnturm.
(K)
(J)
(N)
(D)
;4. Gh`d lut wfhnf lg tfm lpthl`s nlepcmtms tfm iurm ektrhx<
(K) (J)
(N)
(D)
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;=. K ilcdsehtf fks vm ilcd rh`is, mknf fkvh`i k dhggmrm`t wmhift ?
^tktmem`t ; ? Zh`i D hs wmhifh`i twhnm ks eunf ks rh`i M.
^tktmem`t : ? Zh`i M hs wmhifh`i glur thems ks eunf ks rh`i G.
^tktmem`t > ? Zh`i G hs wmhifh`i fkcg ks eunf ks rh`i I.
^tktmem`t 4 ? Zh`i I hs wmhifh`i fkcg ks eunf ks rh`i F.
^tktmem`t = ? Zh`i F hs wmhifh`i cmss tfk` rh`i D jut elrm tfk` rh`i G.
Yfhnf lg tfm glcclwh`i rmprmsm`ts tfm dmsnm`dh`i lrdmr lg wmhifts lg tfm rh`is <
(K) D, M, I, F, G (J) M, I, F, D, G
(N) D, M, G, I, F (D) Dktk h`kdmquktm
;7. H` tfm glcclwh`i qumsthl` grle kel`ist tfm lpthl`s, smcmnt tfm l`m wfhnf skthsms tfm skem
nl`dhthl`s lg pcknmem`t lg tfm dlt ks h` Ghi. (V).
(K)
(J) (N)
(D)
;9. H` tfm ihvm` qumsthl`, wfhnf `uejmr whcc rmpcknm tfm qumsthl` ekra<
48:
7
6
8
49
=
=9:
=
4
8
67
9
>
=
=
:9
4
<
(K) ;;= (J) ;>2 (N) ;>= (D) ;42
;6. ^mcmnt k iurm grle kel`ist tfm lpthl`s, wfhnf wfm` pcknmd h` tfm jck`a spknm lg i. (V) wlucd
nlepcmtm tfm pkttmr`.
(K)
(J)
(N) (D)
;8. Gh`d tfm `uejmr lg qukdrhcktmrkcs h` tfm ihvm` iurm.
(K) ;2
(J) 7
(N) 8
(D) @l`m lg tfmsm
:2. Zhtk drhvms tl tfm `lrtf lg fmr pcknm lg stky kt K k`d kgtmr trkvmcch`i := ae, `ds tfkt sfm fks
drhvm` h` tfm wrl`i dhrmnthl`. ^fm tfm` tur`s tl fmr rhift k`d trkvmcs : ae, k`d tfm` sfm kikh`
tur`s tl tfm rhift k`d drhvms strkhift k`ltfmr := ae. Flw eunf dhstk`nm fks sfm `lw tl nlvmr tl
il jkna tl tfm stkrth`i plh`t <
(K) := ae (J) : ae (N) 4 ae (D) 42 ae
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^MNTHL@ HH ? EKTFMEKTHNKC ZMK^L@H@I
:;. Tfm mxprmsshl` :x> + kx : + jx + >, wfmrm k k`d j krm nl`stk`ts, fks k gkntlr lg x „ ; k`d cmkvms
k rmekh`dmr lg ;= wfm` dhvhdmd jy x + :. Gh`d tfm vkcum lg k k`d j rmspmnthvmcy.
(K) „>, 6 (J) >, „6 (N) „>, „6 (D) >, 6
::. Hg 'c', '
j' k`d '
f' lg k nujlhd krm h`nrmksmd, dmnrmksmd k`d h`nrmksmd jy ;%, >% k`d :% rmspmnthvmcy,
tfm` tfm vlcuem lg tfm nujlhd \\\\\\.
(K) H`nrmksms
(J) Dmnrmksms
(N) H`nrmksms lr dmnrmksms dmpm`dh`i l` lrhih`kc dhem`shl`s
(D) Nk`'t jm nkcnucktmd whtf ihvm` dktk
:>. K jlx lg nflnlcktms nl`tkh` = nflnlcktms whtf fkrd nm`trms k`d 4 whtf slgt nm`trms.
Keht tkams k nflnlcktm, smcmntmd kt rk`dle, grle tfm jlx k`d mkts ht.
Koky tfm` tkams k nflnlcktm, smcmntmd kt rk`dle, grle tfm jlx.
Gh`d tfm prljkjhchty tfkt Keht k`d Koky jltf nfllsm k nflnlcktm whtf k fkrd nm`trm.
(K)
=8
(J)
48
(N)
>
;6 (D)
=
;6
:4. H` ihvm` iurm, KJND k`d KJMG krm twl nynchn qukdrhcktmrkcs. Hg ∢JND 5 ;;2², tfm` ∢JMG 5 <
(K) ==²
G
D
N J
K
M ; ; 2 ²
(J) 92²
(N) 82²
(D) ;;2²
:=. Tfm vkcum lg >
: : > : : > : : >
> > >
( ) ( ) ( )
( ) ( ) ( ) <
k j j n n k
k j j n n k
√ + √ + √
√ + √ + √
5
(K) >(k + j)(j + n)(n + k) (J) >(k „ j)(j „ n)(n „ k)
(N) (k „ j)(j „ n)(n „ k) (D) ;
:7. Flw ek`y pck`ms nk` jm ekdm tl pkss tfrluif tfrmm dhsth`nt plh`ts<
(K) L`m hg tfmy krm nlcch`mkr (J) H``htm hg tfmy krm nlcch`mkr
(N) L`cy l`m hg tfmy krm `l` nlcch`mkr (D) Jltf (J) k`d (N)
:9. Tfm ch`m JM hs k dhkemtmr lg tfm ihvm` nhrncm. Hg ∢JKN 5 >>² k`d
∢MJN 5 =9². Tfm` ∢NKM 5
(K) =9²
(J) >>²
(N) 46²
(D) 82²
:6. Tfm tmre nl`tkh`h`i tfm fhifmst plwmr lg x h` tfm plcy`lehkc g (x) hs :x4. Twl lg tfm rllts lg
tfm mqukthl` g (x ) 5 2 krm „ ; k`d :. Ihvm` tfkt x: „ >x + ; hs k qukdrkthn gkntlr lg g (x ), `d tfm
rmekh`dmr wfm` g (x) hs dhvhdmd jy :x „ ;.
(K) ;;
6 (J) : (N) 2 (D) „;/>
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:8. Tfm sue lg k `uejmr k`d hts rmnhprlnkc hs tfrhnm tfm dhggmrm`nm lg tfm `uejmr k`d hts rmnhprlnkc.
Tfm `uejmr hs \\\\\\\\\.
(K) » : (J)
» ;
: (N) » > (D) Jltf (K) k`d (J)
>2. ^lcvm glr x ?:
6
>:
4
>
; :
x
x x
√
√ 5
( / ).
(K) :;
= (J)
;
;
= (N) >
;
= (D) ;
>
=
>;. : 7 2 6:. . \\\\\\\\\ .√ 5
(K)
;6:
88 (J)
;62
888 (N)
;9=
88 (D)
;64
888
>:. Tfm jksm lg k` hslsnmcms trhk`icm hs 4 ne k`d hts krmk hs ;7 ne:. Hg l`m lg tfm twl mqukc shdms lg
tfm trhk`icm hs a ne, tfm kpprlxhektm vkcum lg a „ ;.:4 hs \\\\\\\\\\\.
(K) 8 ne (J) 9 ne (N) ;2 ne (D) 6 ne
>>. KJND hs k rmntk`icm. Hg KJR k`d JN[ krm mquhcktmrkc trhk`icms, ∢RJ[ 5 \\\\.
(K) 7=² K
D
J
N
R
[(J) 9=²
(N) 72²
(D) 82²
>4. K nmrtkh̀ dhstk`nm hs nlvmrmd kt k nmrtkh̀ spmmd (s;). Hg fkcg lg tfhs dhstk`nm hs nlvmrmd h` dlujcm
tfm them kt spmmd (s:). Gh`d tfm rkthl lg tfm twl spmmds.
Kcsl, hg s; hs 72 ae/fr, `d s
:.
(K) ; ? ;7, >.9= ae/fr (J) 4 ? ;, ;= ae/fr (N) : ? ;, >2 ae/fr (D) : ? 6, :42 ae/fr
>=. Hg prldunt lg kjsnhssk k`d lrdh`ktm lg k plh`t hs plshthvm, tfm` tfm plh`t chms h`
(K) H qukdrk`t (J) HHH qukdrk`t (N) HW qukdrk`t (D) Jltf (K) k`d (J)
>7. Tfm twl nhrncms fkvm rkdhh x k`d >x. K plh`t hs nflsm`, kt rk`dle, h`shdm
tfm ckrimr nhrncm. Gh`d, h` hts shepcmst grknthl`kc glre, tfm prljkjhchty
tfkt tfhs plh`t hs h` tfm sfkdmd krmk.
(K) 9/6 (J) 6/8>x
x
(N) ;/8 (D) ;/6
>9. Gh`d tfm ehssh`i vkcum. ( )( ) <
;> 9;> 9
:2
> >
: :++ √5
(K) 7 (J) :2 (N) 8; (D) ;8
>6. Tfm studm`ts h` k nlccmim wmrm ksamd tl vltm glr tfmhr gkvlurhtm
sujomnt. Tfm phm nfkrt rmprmsm`ts tfm `uejmr lg vltms glr mknf sujomnt.
Ektfmekthns ljtkh`md := elrm vltms tfk` Nfmehstry. Nkcnucktm tfm
`uejmr lg studm`ts wfl tlla pkrt h` tfm survmy.
(K) :=2 (J) =22
E k t f m e
k t h n
s
Rfyshns
I m l i r k p f y
Nfmehstry
;:2²
;2=²
4=²
(N) 4=2 (D) 722
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>8. K nujlhdkc emtkc jclna lg dhem`shl`s :2 ne Ù ;7 ne Ù ;: ne wmhifs 7 ai. Gh`d tfm wmhift lg
jclna lg tfm skem emtkc lg shzm ;2 ne Ù 6 ne Ù 6 ne.
(K) > ai (J) 6 ai (N) 9 ai (D) ; ai
42. Yfkt pmrnm`tkim lg k dky hs shx flurs k`d 4= eh`utms<
(K) 9.:;6% (J) 6.>% (N) :6.;:=% (D) @l`m lg tfmsm
^MNTHL@ HHH ? MWMZXDKX EKTFMEKTHN^
4;. Tfm plpuckthl` lg k tlw` wks ;72222 tfrmm ymkrs kil. Hg ht fkd h`nrmksmd jy >%, :.=% k`d =%
h` tfm ckst tfrmm ymkrs, `d hts prmsm`t plpuckthl`.
(K) ;79>77 (J) ;99477 (N) ;99>77 (D) ;99>7=
4:. Tfm tkxh nfkrims h` k nhty nleprhsm lg k xmd nfkrim, tlimtfmr whtf tfm nfkrim lg tfm dhstk`nm
nlvmrmd. Glr k olur`my lg ;7 ae, tfm nfkrims pkhd krm Zs. ;=7 k`d glr k olur`my lg :4 ae, tfm
nfkrim pkhd krm Zs. :24. Yfkt whcc k pmrsl` fkvm tl pky glr trkvmcch`i k dhstk`nm lg >2 ae<
(K) Zs. :>7 (J) Zs. :42 (N) Zs. :46 (D) Zs. :=:
4>. ^kvhtk fks Zs. :9 h` tfm glre lg gty pkhsm k`d twm`ty-vm pkhsm nlh`s. ^fm fks twhnm ks ek`y
twm`ty-vm pkhsm nlh`s ks sfm fks =2 pkhsm nlh`s. Flw ek`y nlh`s lg mknf ah`d dlms sfm
fkvm<
(K) :9, =4 (J) >2, 72 (N) :=, =2 (D) 42, 62
44. K trkdmr purnfksms 92 ai lg tmk kt Zs. ;= pmr ai k`d >2 ai lg tmk kt Zs. ;6.=2 pmr ai. Hg tfm
pknah`i nfkrims krm : pmrnm`t, tfm` kt wfkt prhnm fm eust smcc tfm ehxturm lg twl tl ikh`
;=%<
(K) Zs ;6.6: pmr ai (J) Zs. ;6 pmr ai (N) Zs. ;6.=2 pmr ai (D) Zs.;9.62 pmr ai
4=. Tfm kvmrkim kim lg k gkehcy lg 7 emejmrs 4 ymkrs kil wks := ymkrs. Emk`wfhcm k nfhcd wksjlr` h` tfhs gkehcy k`d sthcc tfm kvmrkim kim lg tfm wflcm gkehcy hs skem tldky. Tfm prmsm`t kim
lg nfhcd hs \\\\\\\.
(K) : ymkrs (J) ; ymkrs;
: (N) ; ymkr (D) Dktk h`sugnhm`t
47. : em` k`d > wlem` `hsf :=% lg tfm wlra h` ; dky, wfhcm 7 em` k`d ;4 wlem` nk` `hsf
tfm wflcm wlra h` = dkys. H` flw ek`y dkys whcc :2 wlem` `hsf ht<
(K) :2 (J) := (N) :4 (D) @l`m lg tfmsm
49. Twl phpms V k`d X nk` cc k nhstmr` h` :4 eh`. k`d >: eh`. rmspmnthvmcy. Hg jltf tfm phpms krm
lpm`md tlimtfmr, tfm` kgtmr flw eunf them X sflucd jm nclsmd sl tfkt tfm tk`a hs gucc h` ;6
eh`utms<
(K) 7 eh`s (J) 6 eh`s (N) ;2 eh`s (D) @l`m lg tfmsm
46. ^urmsf trkvmccmd ;:22 ae jy khr wfhnf glremd (:/=) lg fhs trhp. L`m tfhrd lg tfm wflcm trhp, fm
trkvmccmd jy nkr k`d tfm rmst lg tfm olur`my fm pmrglremd jy trkh`. Gh`d tfm dhstk`nm trkvmccmd jy
trkh`. Kcsl, `d tfm spmmd lg tfm trkh` hg tfm them tkam` glr tfm trkh` tl trkvmc tfm wflcm dhstk`nm hs
6 frs.
(K) ;722 ae, >=2 ae/fr (J) 622 ae, >9= ae/fr
(N) ;622 ae, >22 ae/fr (D) 462 ae, >62 ae/fr
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48. Tfrmm chiftsfhps ksf sheuctk`mluscy kt 7?22 k.e. Tfm rst chiftsfhp ksfms mvmry ;: smnl`ds,
tfm smnl`d chiftsfhp mvmry >2 smnl`ds k`d tfm tfhrd chiftsfhp mvmry 77 smnl`ds. Kt wfkt them
whcc tfm tfrmm chiftsfhps `mxt ksf tlimtfmr<
(K) 7?28 k.e. (J) 7?;2 k.e. (N) 7?;; k.e. (D) 7?;: k.e.
=2. K gruht smccmr fks :4 ai lg kppcms. Fm smccs k pkrt lg tfmsm kt k ikh` lg :2% k`d tfm jkck`nm kt k
clss lg =%. Hg l` tfm wflcm fm mkr`s k prlt lg ;2%, tfm kelu`t lg kppcms slcd kt k clss hs \\\. (K) 4.7 ai (J) 7 ai (N) 8.7 ai (D) ;;.4 ai
^RKNM GLZ ZL]IF YLZA
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Xmkr :2;:
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Nckss 8
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^MNTHL@ H ? CLIHNKC ZMK^L@H@I
;. Ljsmrvm tfm ihvm` dhkirkes glremd jy u`ht squkrms. @lt h`ncudh`i tfm ehddcm squkrm flcm, tfm `uejmr
lg u`ht squkrms `mmdmd tl juhcd tfm tm`tf pkttmr` hs \\\\\.
(K) 82
(J) 8;
(N) 8:
(D) 8>
:. Ilh`i =2 e tl tfm slutf lg ey flusm, H tur` cmgt k`d il k`ltfmr :2 e. Tfm` tur`h`i tl tfm `lrtf ,
H il >2 e k`d tfm` stkrt wkcah`i tlwkrds ey flusm. H` wfhnf dhrmnthl` ke H wkcah`i `lw <
(K) @lrtf (J) Mkst (N) @lrtf-Ymst (D) ^lutf-Mkst
>. Hg LCXERHN^ hs nldmd ks RE_@LFJZ, k`d O]ER hs nldmd ks AWCL, tfm` NLER]TMZ wh cc jm nldmd
ks \\\\\\\.
(K) DR@[W]G^ (J) DR@[T^G[ (N) DR@[T^D[ (D) J@CLT^D[
4. Gh`d lut tfm nlrrmnt lpth l` sunf tfkt Ghi. (V ) hs mejmddmd h` l`m lg tfme.
Ghi. (V)
(K)
(J)
(N)
(D)
=. Tfm ihvm` qumsthl` nl`shsts lg twl smts lg iurms. Ghiurms ;, :, >
k`d 4 nl`sthtutm tfm prljcme smt wfhcm lpthl` iurms K, J, N k`d D nl`sthtutmtfm k`swmr smt. Tfmrm hs k dm`htm rmckthl`sfhp jmtwmm` iurms ; k`d :.
Mstkjchsf k shehckr rmckthl`sfhp jmtwmm` iurms > k`d 4 jy smcmnth`i k
suhtkjcm iurm grle tfm k`swmr smt tfkt wlucd rmpcknm tfm qumsthl` ekra (,7,8
; : >
4 = 7
9 6 8
(J) ;,4,61 :,=,81 >,7,9
(N) ;,>,41 :,=,61 7,9,8
(D) ;,:,>1 4,=,71 9,6,8
9. Lut lg tfm ihvm` vm iurms, glur krm shehckr h` k nmrtkh` wky. Gh`d tfm ldd l`m lut.
(K)
(J)
(N)
(D)
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6. H` tfm kdolh`h`i Wm`` dhkirke, tfm squkrm rmprmsm`ts wlem`, tfm
trhk`icm rmprmsm`ts pmrsl`s wfl krm h` ilvmr`em`t smrvhnms, tfm
nhrncm rmprmsm`ts mdunktmd pmrsl`s k`d tfm rmntk`icm rmprmsm`ts
pmrsl`s wlrah`i h` prhvktm smntlr. Mknf smnthl` lg tfm dhkirke hs
`uejmrmd tfm` wfhnf `uejmr rmprmsm`ts tfm u`mdunktmd wlem`,
wfl fkvm ilvmr`em`t smrvhnm ks wmcc ks oljs h` prhvktm smntlrs <
(K) 7 (J) 4 (N) ;: (D) 8
8. Yfhnf lg tfm glcc lwh`i cmttmrs/`uejmrs h` tfm ihvm` cmttmr-`uejmr smqum`nm jmclw hs mxkntcy h`
tfm ehddcm lg tfm 8tf cmttmr/`uejmr grle tfm cmgt m`d k`d 9 tf cmttmr/`uejmr grle tfm rhift m`d <
J > A I = R D 9 8 M Z O H E T 7 _ F [ > Y : K
(K) H (J) Z (N) O (D) E
;2. Tfm ihvm` smt lg iurms nkrry nmrtkh` nfkrkntmrs. Kssueh`i tfkt tfm nfkrkntmrs h` smt glcclw k
shehckr pkttmr`, `d tfm ehssh`i nfkrkntmr.
(K) : (J) 8 (N) ;9 (D) ;6
;;. Tfrmm plshthl`s lg tfm skem dhnm fks jmm` sflw` h` tfm iurm. Yfkt dhiht whcc jm l` tfm tlp hg
dhiht : hs l` tfm jlttle<
(K) ; (J) 4 (N) = (D) 7
;:. Hg ht hs plsshjcm tl ekam l`cy l`m emk`h`iguc wlrd whtf tfm rst, tfm smnl`d, tfm shxtf k`d tfm
tm`tf cmttmrs lg tfm wlrd DH^NCKHEMZ wfhnf lg tfm glcclwh`i whcc jm tfm tfhrd cmttmr grle tfm cmgtm`d< Hg `l sunf wlrd nk` jm ekdm, ihvm V ks ylur k`swmr k`d hg elrm tfk` l`m sunf wlrd nk`
jm ekdm, ihvm X ks tfm k`swmr.
(K) H (J) Z (N) X (D) V
;>. Zlfht, Au`kc, Ksfhsf k`d Olf` krm studm`ts lg k snfllc . Tfrmm lg tfme stky gkr grle tfm snfllc
k`d l`m `mkr ht. Twl study h` nckss HW, l`m h` nckss W k`d l`m h` nckss WH. Tfmy study Fh`dh,
Ektfmekthns, ^lnhkc ^nhm`nm k`d ^nhm`nm. L`m hs illd kt kcc tfm glur sujomnts wfhcm k`ltfmr
hs wmka h` kcc lg tfmsm. Zlfht stkys gkr grle tfm snfllc k`d hs illd kt Ektfmekthns l`cy wfhcm
Au`kc hs wmka h` Ektfmekthns l`cy k`d stkys nclsm tl tfm snfllc. @mhtfmr Au`kc, Zlfht `lr Ksfhsf
studhms h` nckss WH. L`m wfl hs illd kt kcc tfm sujomnts studhms h` nckss W. @kem tfm jly wfl
hs illd kt kcc tfm sujomnts.
(K) Zlfht (J) Au`kc (N) Ksfhsf (D) Olf`
;4. Krrk`im tfm ihvm` wlrds h` tfm smqum`nm h` wfhnf tfmy lnnu r h` tfm dhnt hl`kry k`d tfm` nfllsm
tfm nlrrmnt smqum`nm.
;. Yrh`a cm :. Yrhii cm >. Yrhtfm 4. Yrmtnf =. Yrktf
(K) 4, =, ;, :, > (J) =, 4, :, ;, > (N) 4, :, =, ;, > (D) =, :, ;, >, 4
;=. Hg K * J emk`s ' K euct hp c hmd jy J ' 1 K B J emk`s ' K eh`us J ' 1 K $ J emk`s ' K pcus J ' 1
K # J emk`s ' K dhvhdmd jy J ' , tfm` wfkt whc c jm tfm tltkc krmk lg :; nhrncms, mknf whtf tfm
skem rkdhus lg r ne<
(K) :: * r * 9 * r # :; (J) :: # :; * r * r (N) 77 * r : # 9 (D) 77 * r * r
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;7. Tfm ihvm` qumsthl` nl`shsts lg vm iurms ekramd R, [, Z, ^ k`d T nkccmd tfm Rrljcme Ghiurms
glcclwmd jy glur lpthl`s. ^mcmnt k iurm grle kel`ist lpthl`s wfhnf whcc nl`th`um tfm skem smrhms
ks mstkjchsfmd jy tfm vm Rrljcme Ghiurms.
(K)
(J)
(N)
(D)
;9. Gh`d tfm ehrrlr hekim lg tfm ihvm` wlrd, hg tfm ehrrl r hs pcknmd vmrthnkccy tl tfm cmgt.
KDEH@H^TZKTHL@
(K) K E H @ H ^ T Z K T H L @ D
(J) K E H @ H ^ T Z K T H L @ D (N) K E H @ H ^ T Z K TH L @ D (D) K E H @ H ^T Z K T H L @ D
;6. Gh`d tfm `uejmr lg trhk`icms h` tfm ihvm` iurm.
(K) :7
(J) :9
(N) :6
(D) @l`m lg tfmsm
;8. Rlh`th`i tlwkrds k pfltlirkpf, Elfht skhd, 'Fhs shstmr hs tfm eltfmr lg Wh`ky, tfm sl` lg ey jrlt fmr'.
Flw Elfht hs rmcktmd tl Wh`ky<
(K) Jrltfmr (J) ]`ncm (N) @mpfmw (D) Jrltfmr-h`-ckw
:2. Hg ; hs sujt rkntmd grle tfm ckst dhiht lg mknf lg tfm ihvm` jmclw tfrmm dhiht `uejmrs, tfm sue lg
tfm dhihts lg flw ek`y lg tfme krm prhem `uejmrs<
89: =:7 469 >=8 :=;
(K) @l`m (J) Twl (N) L`m (D) Tfrmm
^MNTHL@ HH ? EKTFMEKTHNKC ZMK^L@H@I
:;. H` tfm kdoknm`t iurm KM 5 KG, tfm` wfhnf lg tfm glcclwh`i hs trum<
(K) k + j + n 5 ;62²
kx
n
y
K
J N
M
Gx
jD
(J) k + n 5 j
(N) :n + k 5 j
(D) :k + n 5 j
::. Tfm vkcum lg ;+:226 ;+:228 ;+:2;2 ;+:2;; :2;> hs⋍ \\\\\.
(K) :228 (J) :2;2 (N) :2;; (D) :2;>
:>. ^hepch gy ?7
: > 7+
7
> + :
4 >
7 :√ √
√
(K) 2 (J) ; (N) : (D) >
:4. Hg (x „ :y )7 5 K2x7 „ K
;x= . y; + K
:x4 . y: „ K
>x> . y> + K
4x: . y4 „ K
=x . y= + K
7y7, `d tfm vkcum lg
K2 „ K
; + K
: „ K
> + K
4 „ K
= + K
7.
(K) 2 (J) : (N) ; (D) „>
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:=. Kjsn hssk lg kcc tfm plh`ts l` x-kxhs hs \\\\\.
(K) 2 (J) ; (N) : (D) K`y `uejmr
:7. ;=22 gkehchms wht f : nfhc drm` wmrm smcmntmd rk`dlecy, k`d tfm glcclwh`i dktk wks rmnlrdmd?
@uejmr lg ihrcs h` k gkehcy : ; 2
@uejmr lg gkehchms 49= 6;4 :;;
Tfm prljkjhc hty lg k gkehcy, nflsm` kt rk`dle, fkvh`i ; ihrc hs \\\\.
(K)6;4
767 (J)
6;4
49= (N)6;4
:;; (D)
6;4
;=22
:9. H` tfm ihvm` iurm (`lt drkw` tl snkcm), T] || ^Z k`d TZ || ^W, tfm` `d ∢k k`d ∢j.
(K) k 5 ;;=², j 5 42²
R [
Z
^ T
]
W
k
j=2²
82²
:=²
(J) k 5 ;:2², j 5 7=²
(N) k 5 ;4=², j 5 ==²
(D) k 5 ;2=², j 5 >=²
DHZMNTHL@ (:6 & :8) ? Tfm ch`m irkpf sflws tfm smntlrwhsm jrmka-up lg vm`turm nkphtkc gu`dh`i h` H`dhk
glr tfm ymkr :2VV.
:6. Gu`dh`i h` tfm nl`suemr-rmcktmd smntlr mxnmmds tfkt h` tfm m`mriy-rmcktmd smntlr kpprlx
jy \\\\\.
(K) 422% (J) 722% (N) =22% (D) 4=2%
:8. Tfm rkthl lg gu`dh `i h` tfm nleputmr slg twkrm smrvhn m k`d tfm jhl-tmnf`lcliy smntlrs (tkam`
tlimtfmr) tl tfkt h` tfm h`dustrhkc prldunts k`d eknfh`mry smntlr hs \\\\\\\.
(K) ;2 ? ; (J) ; ? ; (N) : ? > (D) > ? :
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>2. H` tfm ihvm` iurm, ekra tfm rmckthl` jmtwmm` KJ k`d KD.
(K) KJ 5 KD
K
JN D
; :
(J) KJ 3 KD
(N) KJ 0 KD
(D) Dktk h`kdmquktm
>;. Ihvm` k ch`m c k`d k plh`t R l` ht. Flw ek`y ch`ms nk` jm drkw` pkssh`i tfrluif tfm plh`t R < (K) H`g`htm (J) L`m (N) Twl (D) Tfrmm
>:. Tfm plsht hvm slcuthl`s lg tfm mqukthl` kx + jy + n 5 2 kcwkys chm h` tfm \\\\\\.
(K) ;st qukdrk`t (J) :`d qukdrk`t (N) >rd qukdrk`t (D) 4tf qukdrk`t
>>. Gh`d tfm vkcum lg> 7 86
= ; / := (;=) >
> :
: > 4/> ;/>
√
√
.
(K) ;2 (J) :6 : (N) >
: (D)
;:=
:9
>4. K fk`d gk` hs ekdm jy sth tnfh `i ;2 mqukc shzm trhk`iuc kr strhps l g twl dhg gmrm`t typms lg pkpmr ks
sflw` h` iurm. Tfm dhem`shl`s lg mqukc strhps krm := ne, := ne k`d ;4 ne. Gh`d tfm krmk lg
mknf typm lg pkpmr `mmdmd tl ekam tfm fk`d gk`.
(K) :62 ne:
(J) 4:2 ne:
(N) 642 ne:
(D) >64 ne:
>=. Yfhnf lg tfm glcc lwh`i stktmem`ts hs trum <
(K) K ch`m smiem`t fks `l m`d plh`t.
(J) K rky fks dmg`htm cm`itf.(N) K rky R[
hs tfm skem ks rky [R
.
(D) Tfrmm lr elrm ch`ms krm nl`nurrm`t hg tfmy fkvm k nleel` plh`t.
>7. H` tfm ihvm` iurm, ∢x k`d ∢y krm \\\\\.
(K) x 5 92², y 5 >9²
(J) x 5 >9², y 5 92²
(N) x 5 =2², y 5 =9²
(D) x 5 =9², y 5 =2²
>9. Mxprmss ;= 9;:. h` tfm glre lgp
q, q ≢ 2.
(K) =;6=
>22 (J)
=;6=
;22 (N)
=;66
> (D)
=;6=
>>2
>6. Yfkt eust jm kddmd tl x4 + :x> „ :x: + x „ ; sl tfkt rmsuct hs mxkntcy dhvhshjcm jy x: + :x „ ><
(K) „ x + : (J) > (N) x „ : (D) 4x „ >
>8. Tfm rkdhus lg tfm nych`dmr hs fkcg hts fmhift k`d krmk lg tfm h``mr pkrt hs 7;7 sq. nes. Flw ek`y
chtrms lg ehca kpprlxhektmcy, nk` ht nl`tkh`<
(K) ;.4 C (J) ;.= C (N) ;.9 C (D) ;.8 C
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42. H` tfm ihvm` iurm, T hs k plh`t l` shdm [Z lg R[Z k`d ^ hs tfm plh`t sunf tfkt ZT 5 ^T. Tfm`
R[ + RZ 0 \\\\\.
(K) [Z
(J) [T
(N) [^
(D) ^T
^MNTHL@ HHH ? MWMZXDKX EKTFMEKTHN^
4;. Tfm prljkjhc hthms lg k studm`t imtth`i K, J, N k`d D irkdm krm 2.>=, 2.:=, 2.>= k`d 2.2=, tfm` tfm
prljkjhchty tfkt k studm`t imts kt elst irkdm N hs \\\\\\.
(K) 2.>= (J) 2.42 (N) 2.8= (D) 2.7=
4:. K pmrsl` cm`t lut k nmrtkh` sue l` shep cm h`tmrmst k`d tfm skem sue l` nlepl u`d h`tmrmst kt
nmrtkh` rktm lg h`tmrmst pmr k``ue. Fm `lthnmd tfkt tfm rkthl jmtwmm` tfm dhggmrm`nm lg nleplu`d
h`tmrmst k`d shepcm h`tmrmst lg > ymkrs k`d tfkt lg : ymkrs hs := ? 6. Tfm rktm lg h`tmrmst pmrk``ue hs \\\\\\.
(K) ;2% (J) ;;% (N) ;:% (D) ;: ;
:%
4>. K nclsmd emtkcchn nyc h`dr hnkc jlx hs ;.:= e fhif k`d hts jksm rkdhus hs >= ne. Hg tfm emtkc sfmmt
nlsts b 62 pmr e:, tfm nlst lg tfm ektmrhkc usmd h` tfm jlx hs \\\\\\.
(K) b :6;.72 (J) b :82 (N) b >42.=2 (D) b =22
44. Ckthak jluift k rkdhl smt k`d spm`t b ;;2 l` hts rmpkhrs. ^fm tfm` slcd ht tl Ikrhek kt :2% prlt,
Ikrhek slcd ht tl Emifk kt k clss lg ;2% k`d Emifk slcd ht glr b ;;66 kt k prlt lg ;2%. Yfkt hs
tfm kelu`t glr wfhnf Ckthak jluift tfm rkdhl smt < (K) b 6=2 (J) b 682 (N) b 8>2 (D) b 8=2
4=. H` tfm el`tf lg Ok`ukry, tfm Zkhcwky Rlch nm nkuift 4222 thnamtcmss trkvmccmrs. H` Gmjrukry, tfm
`uejmr rlsm jy =%. Flwmvmr, dum tl nl`stk`t vhihc jy tfm Rlchnm k`d tfm Zkhcwky stkgg, tfm `uejmr
rmdunmd jy =% k`d h` Kprhc ht gurtfmr rmdunmd jy ;2%. Tfm tltkc `uejmr lg thnamtcmss trkvmccmrs
nkuift h` tfm el`tf lg Kprhc wks \\\\\\.
(K) >;:= (J) >:== (N) >=9= (D) >=8;
47. Rrmsm`t kims lg ^kemmr k`d K`k`d krm h` tfm rkt hl lg = ? 4 rmspmnthvmcy. Tfrmm ymkrs
fm`nm, tfm rkthl lg tfmhr kims whcc jmnlem ;; ? 8 rmspmnthvmcy. Yfkt hs K`k`d's prmsm`t kim h`ymkrs <
(K) :4 ymkrs (J) :9 ymkrs (N) 42 ymkrs (D) >2 ymkrs
49. Glur mqukc shzmd ekxheue nhrnuckr pcktms krm nut lgg grle k squkrm pkpmr sfmmt lg krmk 964 ne:.
Tfm nhrnuegmrm`nm lg mknf pcktm hs \\\\\\.
(K) :: ne (J) 44 ne (N) 77 ne (D) 66 ne
46. Rrhyk pck`ts >>;:4 ek`il tr mms h` fmr ikrdm` k`d krrk`ims tfme sl tfkt tfmrm krm ks ek`y rlws
ks tfmrm krm ek`il trmms h` mknf rlw. Tfm `uejmr lg rlws krm \\\\\\.
(K) ;:6 (J) ;6: (N) :6; (D) :;6
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48. H` k` mxkeh`kthl` , k studm`t snlrms 4 ekras glr mvmry nlr rmnt k`swmr k`d clsms ; ekra glr mvmry
wrl`i k`swmr. Hg sfm kttmepts h` kcc =2 qumsthl`s k`d snlrms ;:2 ekras, tfm `uejmr lg qumsthl`s
sfm kttmepts nlrrmnt hs \\\\\.
(K) 7: (J) 44 (N) 4: (D) >4
=2. K nrhnamtmr wflsm jlwch`i kvmrkim hs ;:.4 ru`s pmr whnamt tkams = whnamts glr :7 ru`s k`d tfmrmjy
dmnrmksms fhs kvmrkim jy 2.4. Tfm `uejmr lg whnamts tkam` jy fhe thcc tfm ckst ektnf wks \\\\\.(K) 74 (J) 9: (N) 62 (D) 6=
^RKNM GLZ ZL]IF YLZA
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Xmkr :2;>
^MT K
Nckss 8
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: 9tf HEL - ^mt K
;. Nfllsm tfm nlrrmnt lpthl` sl tfkt k smrhms nk` jm
mstkjchsfmd.
;9, ;8, :>, :8, (.....), >9
K. >;
J. >> N. >=
D. >7
:. H` tfm ihvm` qumsthl`, twl rlws lg `uejmrs krm ihvm`.
Tfm rmsuctk`t `uejmr h` mknf rlw hs tl jm wlramdlut smpkrktmcy jksmd l` tfm glcclwh`i rucms k`d tfm
qumsthl` jmclw tfm rlws hs tl jm k`swmrmd. Tfm
lpmrkthl`s lg `uejmrs prlirmss grle cmgt tl rhift.
Zucms ?
(h) Hg k` mvm` `uejmr hs glcclwmd jy k`ltfmr mvm``uejmr tfmy krm tl jm kddmd.
(hh) Hg k` mvm` `uejmr hs glcclwmd jy k prhem `uejmr
tfmy krm tl jm eucthpchmd.
(hhh) Hg k` ldd `uejmr hs glcclwmd jy k` mvm` `uejmr,tfm mvm` `uejmr hs tl jm sujtrkntmd grle tfm ldd
`uejmr.
(hv) Hg k` ldd `uejmr hs glcclwmd jy k`ltfmr ldd
`uejmr, tfm rst `uejmr hs tl jm kddmd tl tfm
squkrm lg tfm smnl`d `uejmr.
(v) Hg k` mvm` ̀ uejmr hs glcclwmd jy k nleplshtm ldd`uejmr, tfm mvm` `uejmr hs tl jm dhvhdmd jy tfm
ldd `uejmr.
7= ;; ;:
;= > ;; Yfkt hs tfm sue lg tfm rmsuctk`ts lg tfm rlws<
K. >77
J. 77 N. :74
D. 47:
>. ^mcmnt tfm nlrrmnt ehrrlr hekim lg tfm ihvm` Ghi.(V),
hg tfm ehrrlr hs pcknmd kcl`i E@.
K.
J.
N. D.
4. Zhtk stk`ds tfhrd h` k rlw lg twm`ty studm`ts, krrk`imd
h` ksnm`dh`i lrdmr lg fmhift. Ghvm `mw studm`ts olh`
tfm irlup, kcc tkccmr tfk` Zhtk. Yfkt whcc jm Zhtk's
plshthl` hg tfm studm`ts krm `lw krrk`imd h` dmsnm`dh`ilrdmr lg tfmhr fmhifts<
K. ::`d
J. :>rd
N. ;6tf
D. Dktk h`kdmquktm
=. Hg 'Jlla' hs nkccmd 'Rm`', 'Rm`' hs nkccmd 'Rkpmr', 'Rkpmr'hs nkccmd 'Mrksmr', 'Mrksmr' hs nkccmd '^nkcm' k`d '^nkcm'
hs nkccmd 'Nkcnucktlr', tfm` whtf wfhnf dlms k studm`t
wrhtm<
K. Jlla
J. Rkpmr N. Rm`
D. Nkcnucktlr
7. ^mcmnt k `uejmr wfhnf whcc rmpcknm tfm qumsthl` ekra
h` tfm `uejmr pkttmr` ihvm` jmclw.
K. ;7
J. ;= N. :6
D. :;
9. ^mcmnt k iurm grle tfm lpthl`s wfhnf skthsms tfm
nl`dhthl` lg pcknmem`t lg dlts ks h` Ghi. (V).
K.
J.
N.
D.
CLIHNKC ZMK^L@H@I
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>9tf HEL - ^mt K
6. ^mcmnt k iurm grle tfm lpthl`s wfhnf whcc nlepcmtm
tfm pkttmr` h` Ghi. (V).
K.
J.
N.
D.
8. Flw ek`y trhk`icms krm tfmrm h`
tfm ihvm` iurm< K. 6
J. ;2 N. ;:
D. ;4
;2. Rkhr lg rst twl iurms l` cmgt shdm lg '? ?' fkvm k
nmrtkh` rmckthl`sfhp jmtwmm` tfme. Mstkjchsf tfm skem
rmckthl`sfhp jmtwmm` tfm pkhr lg iurms l` rhift shdm
lg '? ?', wfhnf whcc rmpcknm tfm "
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4 9tf HEL - ^mt K
;9. ^mcmnt grle tfm lpthl`s wfhnf glres tfm nlrrmnt wktmrhekim lg tfm ihvm` wlrd.
NKZH@I
K.J.
N.
D.
;6. ^mcmnt k Wm`` dhkirke grle tfm lpthl`s wfhnf jmst
rmprmsm`ts tfm rmckthl`sfhp kel`ist"Dli, Rkrrlt, Rmt k`hekcs"<
K.
J.
N.
D.
;8. ^mcmnt k iurm grle tfm lpthl`s h` wfhnf Ghi. (V) hs
mejmddmd ks l`m lg hts pkrt.
K.
J.
N.
D.
:2. Zmkd tfm glcclwh`i h`glrekthl` k`d k`swmr tfm qumsthl`
tfkt glcclw.
(h) Glur pmrsl`s K, J, N k`d D mkt ek`il, jk`k`k,
lrk`im lr pmknf grle El`dky tl Tfursdky. @ltwl pmrsl`s mkt tfm skem gruht l` skem dky. Mknf
lg tfme mkts l`cy l`m gruht l` k dky k`d dlms `lt
rmpmkt ht l` k`y ltfmr dky.
(hh) @mhtf mr N `lr D mkts lrk`i m lr pmknf l`Tumsdky.
(hhh) J mkts jk`k`k l` Ymd`msdky.
(hv) K mkts pmknf l` El`dky.
(v) N dlms `lt mkt ek`il l` Tfursdky.
(vh) D mkts jk`k`k l` El`dky.
Yfl mkts lrk`im l` Tumsdky<
K. K
J. J
N. N
D. Mhtfmr K lr J
EKTFMEKTHNKC ZMK^L@H@I
:;. H` tfm ihvm` ghiurm, KLJ hs k strkhift ch`m k`d
∢ KLV > 5 =9², ∢ V ;LV 4 5 89², ∢ V >LJ 5 ;:>² k`d∢ V 4LJ 5 76². Gh`d ∢ KLV ;.
K. 42²
J. >2²
N. :=² D. ;=²
::. Tfm krmk lg tfm trhk`icm glremd jy : x + > y 5 7 k`d
tfm nllrdh`ktm kxms hs \\\\.
K. > sq. u`hts
J. : sq. u`hts
N. 7 sq. u`hts D. = sq. u`hts
:>. Gh`d tfm rmekh`dmr wfm` : x> „ 8 x: + x + ;: hs dhvhdmd
jy : + > x.
K. 2
J. ;;7/8 N. „:
D. ;6:/:9
:4. R[Z^ hs k rflejus k`d ^ZT hs k` mquhcktmrkc trhk`icm.Hg ∢[Z^ 5 62², tfm` ∢[TZ 5
R
T
Z
^
62²[
K. 42²
J. :2² N. >2²
D. =2²
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=9tf HEL - ^mt K
:=. H` wfhnf qukdrk`t dlms tfm plh`t R( x, y) chm hg
xy 3 2<
K. H lr HH
J. HH lr HW
N. H lr HHH
D. HHH lr HW
:7. Tfm dmirmm lg tfm plcy`lehkc
> ;: > ;: ;: 4: : x x x + √ +( ) + + hsK. ;
J. :
N. 4
D. 2
:9. H` tfm ihvm` ghiurm, ∢ J 3 ∢ K k`d ∢N 3 ∢ D.
Tfm` wfhnf lg tfm glcclwh`i stktmem`ts hs trum
rmikrdh`i tfm rmckthl`sfhp jmtwmm` KD k`d JN <
K. KD 5 JN
J. KD 3 JN
N. KD 0 JN
D. @ltfh`i nk` jm skhd
:6. KJND hs k ghmcd h` tfm glre lg k qukdrhcktmrkc wflsm
shdms krm KJ 5 42 e, JN 5 ;= e, ND 5 :6 e k`d
KD 5 8 e. Hg ∢ DKJ 5 82², tfm krmk lg tfm ghmcd hs
K. >27 e:
J. >;: e:
N. ;:7 e:
D. :;6 e:
:8. R[Z^ hs k pkrkccmclirke. Hg V k`d X krm ehd-plh`ts
lg R[ k`d ^Z rmspmnthvmcy k`d dhkil`kc ^[ hs olh`md.
Tfm rkthl kr (||ie V[ZX ) ? kr (D[^Z) 5
R
Z
V
X ^
[
K. ; ? 4
J. : ? ;
N. ; ? :
D. ; ? ;
>2. Tfm prljkjhchty lg smcmnth`i k jly h` k nckss hs 2.7
k`d tfmrm krm 4= studm`ts h` k nckss. Gh`d tfm `uejmr
lg ihrcs h` tfm nckss.
K. ;6
J. ;8
N. :2 D. ;=
>;. ^hepchgy ?;
9 7
=
; 79
+ √
√ √
K. 2
J.;
:
N. ;4
D. ;
>:. Yfkt hs tfm krmk lg tfm sfkdmd pkrt h` tfm ihvm` ghiurm
ς 5
::
9<
;6 ne
K. 9: ne:
J. ;26 ne:
N. >:4 ne:
D. 746 ne:
>>. Tfm ihvm` iurm hs ̀ lt drkw` tl snkcm. Gh`d tfm vkcums
lg ∢[R^ k`d ∢TZ[ rmspmnthvmcy.
R T
Z ^
]
:>²
7=²
;>=²
[
K. 96², ;42²
J. 66², ;;2²
N. 7:², ;=2²
D. 6:², ;;2²
>4. Tfm phm nfkrt sflws tfm irkdms kttkh`md jy k irlup
lg studm`ts glr k nleel` tmst.
;2%D
:2%K
J
:=%N
=%M
Hg :42 studm`ts skt glr tfm nleel` tmst k`d tflsm
wfl ljtkh`md Irkdm D k`d M gkhcmd tfm tmst, flw
ek`y studm`ts pkssmd tfm tmst<
K. 4:
J. ;6:
N. :;= D. :24
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7 9tf HEL - ^mt K
>=. Tfrmm nujms lg emtkc wflsm mdims krm h` tfm rkthl
> ? 4 ? = krm emctmd tl glre k nujm wflsm dhkil`kc hs
;: > ne. Gh`d tfm mdims (h` ne) lg tfm tfrmm nujms.
K. 7, 6, ;2J. 8, ;:, ;=
N. 6, 8, ;;
D. >, 4, =
MWMZXDKX EKTFMEKTHN^
>7. Tfm shdms lg k qukdrk`iuckr ghmcd, tkam` h` lrdmr krm:7 e, :9 e, 9 e k`d :4 e rmspmnthvmcy. Tfm k`icm
nl`tkh`md jy tfm ckst twl shdms hs k rhift k`icm. Gh`d
hts krmk.
K. >:4 e:
J. :>6.=8 e:
N. >9=.64 e:
D. >64.78 e:
>9. Tfm tkxh gkrm h` k nhty hs b := glr ghrst ahclemtrm k`d
b ;2.= glr `mxt mknf sujsmqum`t ahclemtrm. K trkvmccmr
hs nfkrimd b ;28 ks tfm gkrm. Flw ek`y ahclemtrms
dhd fm trkvmc<
K. 6 ae
J. 8 ae
N. ;2 ae
D. @l`m lg tfmsm
>6. Glur ru``mrs stkrtmd ru``h`i tfm rknm h` tfm skem
dhrmnthl` krlu`d k nhrnuckr pktf lg 9 ae. Tfmhr spmmds krm
4 ae/fr, > ae/fr, 8 ae/fr k`d >.= ae/fr h`dhvhdukccy.
Hg tfmy fkvm stkrtmd tfmhr rknm kt 7 l' nclna h` tfm
elr`h`i, tfm` kt wfkt them tfmy whcc jm kt tfm stkrth`i
plh`t flurs
N. ;2 flurs
D. ;= flurs
>8. K jllasmccmr prlnurms 42 jllas glr b >:22 k`d smccs
tfme kt k prlght mqukc tl tfm smcch`i prhnm lg 6 jllas.
Yfkt hs tfm smcch`i prhnm lg l`m dlzm` jllas, hg tfm
prhnm lg mknf jlla hs skem<
K. b 9:2
J. b 872
N. b ;:22
D. b ;442
42. Tfm plpuckthl` lg k nhty fkd h`nrmksmd sunnmsshvmcy
kt tfm rktm lg 7%, 4% k`d :% pmr k``ue durh`i ckst
tfrmm ymkrs. Hg hts prmsm`t plpuckthl` hs ;;:4446, tfm`
wfkt wks hts plpuckthl` tfrmm ymkrs kil<
K. ;222222
J. ;2=2222
N. ;262222
D. ;;22222
4;. Tfm cm`itf lg cl`imst plcm tfkt nk` jm pcknmd l` tfmgcllr lg k rlle hs ;: e k`d tfm cm`itf lg cl`imst plcm
tfkt nk` jm pcknmd h` tfm rlle hs ;= e. Tfm fmhift
lg tfm rlle hs
K. > e
J. 7 e
N. 8 e
D. 4 e
4:. K k`d J nk` dl k phmnm lg wlra h` ;2 dkys1 J k`d
N h` ;: dkys1 K, J k`d N h` 6 dkys. H` flw ek`y
dkys K k`d N tlimtfmr nk` dl tfm skem wlra<
K. 6
J. ;=
N. :2
D. Nk`'t jm dmtmreh`md
4>. Tfmrm krm glur jmccs. Tfmy rh`i kgtmr mvmry l`m
eh`utm, twl k`d fkcg eh`utms, =2 smnl`ds k`d =
smnl`ds rmspmnthvmcy. Hg kcc tfm glur jmccs rk`i ckst
them tlimtfmr kt 6 ? :2 p.e., tfm` kt wfkt them whcc
tfmy kcc `mxt rh`i sheuctk`mluscy<
K. 6 ? :> p.e.
J. 6 ? :4 p.e. N. 6 ? := p.e.
D. 6 ? :7 p.e.
44. Tfm dhem`shl`s lg k pfltlirkpf krm 4 ne k`d
;.6 nes. Hg tfm jrmkdtf lg tfm m`ckrimd pfltl hs
4.= ne k`d ht wks m`ckrimd prlplrthl`kccy tfm` wfkt
hs tfm `mw cm`itf lg `mw pfltlirkpf<
K. 7 ne
J. =.4 ne
N. ;2 ne
D. 8 ne
4=. Tfm prhnm lg k nkr dmprmnhktms h` tfm ghrst ymkr jy
:=%, h` tfm smnl`d ymkr jy :2%, h` tfm tfhrd ymkr
jy ;=% k`d sl l`. Tfm gh`kc prhnm lg tfm nkr kgtmr
> ymkrs, hg tfm prmsm`t nlst lg tfm nkr hs b ;2,22,222
hs
K. b 9,62,222
J. b =,;2,222
N. b 7,82,222
D. b ;,92,222
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99tf HEL - ^mt K
47. Nl`shdmr tfm iurm jmclw.
H
HH
HHH
Tfm rkthl lg tfm krmk lg H tl HH hs 6 ? ;. Tfm krmk lgHHH hs ;
:
>ks eunf ks tfm krmk lg HH. Hg tfm dhggmrm`nm
jmtwmm` tfm krmk lg H k`d HH hs :=: ne:, tfm`
(R) Krmk lg H
([) Krmk lg HH + HHH R [
K. 87 ne: :66 ne:
J. :66 ne: 87 ne:
N. ;66 ne: 67 ne:
D. 67 ne: ;66 ne:
49. ^mcmnt tfm nlrrmnt lpthl`. ^tktmem`t ; ? ( x + y)> 5 x> + y> + > xy ( x + y) k`d
( x: + y:) 5 ( x + y): + : xy
^tktmem`t : ? Hg tfm` tfm vkcum lg x x
::
;9+ 5 ,
x
x
>>
;;8+ 5
K. ^tktmem`t ; hs trum jut stktmem`t : hs gkcsm.
J. ^tktmem`t ; hs gkcsm jut stktmem`t : hs trum.
N. Jltf tfm stktmem`ts krm trum.
D. Jltf tfm stktmem`ts krm gkcsm.
46. Yfhnf lg tfm glcclwh`i stktmem`ts hs nlrrmnt<
K. Kcc tfm k`icms lg k trhk`icm nk` jm cmss tfk` 72².
J. K trhk`icm nk` fkvm elrm tfk` l`m ljtusm k`icm.
N. K` mxtmrhlr k`icm lg k trhk`icm hs cmss tfk` mhtfmr
lg hts h`tmrhlr lpplshtm k`icms.
D. K` mxtmrhlr k`icm lg k trhk`icm hs mqukc tl tfm suelg tfm twl h`tmrhlr lpplshtm k`icms.
48. ^mcmnt tfm nlrrmnt lpthl`.
^tktmem`t ; ? Hg tfm kcthtudm grle l`m vmrtmx lg k
trhk`icm jhsmnts tfm lpplshtm shdm, tfm` tfm trhk`icm hs
`lt hslsnmcms. ^tktmem`t : ? Hg tfm jhsmntlr lg tfm vmrthnkc k`icm lg
k trhk`icm jhsmnts tfm jksm, tfm` tfm trhk`icm hs `lt
hslsnmcms.
K. ^tktmem`t ; hs trum jut stktmem`t : hs gkcsm. J. ^tktmem`t ; hs gkcsm jut stktmem`t : hs trum.
N. Jltf tfm stktmem`ts krm trum. D. Jltf tfm stktmem`ts krm gkcsm.
=2. k, j, n krm tfrmm smts lg vkcums lg x ?
k ? :, >, 9, ;, >, :, >
j ? 9, =, 8, ;:, =, >, 6
n ? 4, 4, ;;, 9, :, >, 4 Yfhnf l`m lg tfm glcclwh`i stktmem`ts hs nlrrmnt<
K. Emk` lg k 5 Eldm lg n
J. Emk` lg n 5 Emdhk` lg n N. Emdhk` lg j 5 Eldm lg k
D. Emk`, Emdhk` k`d Eldm lg k krm mqukc
KNFHMWMZ^ ^MNTHL@
^RKNM GLZ ZL]IF YLZA
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9tf
Xmkr :2;>
^MT J
Nckss 8
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: 9tf HEL - ^mt J
;. Gh`d lut flw whcc tfm amy Ghi. (V) clla cham kgtmr
rltkthl`<
K.
J.
N.
D.
:. Tfmrm hs k smt lg iurms nkrryh`i nmrtkh` `uejmrs.
Kssueh`i tfkt tfm ̀ uejmrs h` mknf smt glcclw k shehckr pkttmr`, `d tfm ehssh`i `uejmr.
K. :
J. 7
N. 6
D. 74
>. Yfm` tfm ihvm` iurm hs glcdmd
tl glre k nujm, flw ek`ydlts wlucd chm lpplshtm tfm
gknm jmkrh`i vm dlts<
K. ;
J. :
N. >
D. 4
4. ^mcmnt tfm pkhr grle lpthl`s tfkt fks k rmckthl`sfhp
shehckr tl tfm iurm (V).
K.
Ghiurm (V)
J.
N.
D.
=. Hg mknf lg tfm ldd dhihts h` tfm `uejmr =47>6 hs
dmnrmksmd jy ; k`d mknf lg tfm mvm` dhihts hs h`nrmksmd
jy ;, tfm` wfhnf lg tfm glcclwh`i whcc jm tfm sue lg
tfm dhihts lg tfm `mw `uejmr<
K. := J. :7
N. :6
D. :9
7. Tfm ihvm` qumsthl` hs jksmd l` tfm dhkirke ihvm`
jmclw sflwh`i glur pmrsl`s kt tfm glur nlr`mrs lg k
squkrm rmihl` ks sflw`.
Grle tfm plshthl`s h` ihvm` iurm, N k`d K elvm
dhkil`kccy tl lpplshtm nlr`mrs k`d tfm` l`m shdm mknf
nclnawhsm k`d k`th-nclnawhsm rmspmnthvmcy. J k`d D
elvm twl shdms mknf nclnawhsm k`d k`th-nclnawhsm
rmspmnthvmcy. Yfmrm hs K `lw<
K. Kt tfm `lrtf-wmst nlr`mr
J. Kt tfm `lrtf-mkst nlr`mr
N. Kt tfm slutf-mkst nlr`mr
D. Kt tfm slutf-wmst nlr`mr
9. ^lem lg tfm nrhnamt pckymrs krm tm``hs pckymrs, slem
tm``hs pckymrs krm flnamy pckymrs, `l nrhnamt pckymr
hs k flnamy pckymr. Yfhnf lg tfm glcclwh`i dhkirkes
nlrrmntcy rmprmsm`ts tfm kjlvm stktmem`ts<
K.
J.
N.
D.
CLIHNKC ZMK^L@H@I
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>9tf HEL - ^mt J
6. Tfm glcclwh`i qumsthl` nl`shsts lg ghvm ghiurms ekramd
;, :, >, 4 k`d = nkccmd tfm prljcme ghiurms. ^mcmnt k
ghiurm grle kel`ist tfm lpthl`s wfhnf whcc nl`th`um
tfm skem smrhms ks mstkjchsfmd jy tfm ghvm prljcme
ghiurms.
Rrljcme Ghiurms
K. J.
N. D.
8. H` k nldh`i ck`iukim, tfm cmttmrs lg tfm M`ichsf kcpfkjmt
krm krrk`imd h` sunf k ek``mr tfkt kcc tfm vlwmcs krm
put h` tfm m`d k`d tfm rmekh`h`i cmttmrs krm krrk`imd
grle tfm rst cmttmr l`wkrds. Tfm rmkrrk`imd kcpfkjmts
krm usmd tl dm`ltm tfm plshthl` lnnuphmd jy cmttmrs h`
tfm lrhih`kc kcpfkjmts. Yfkt hs tfm nldm lg EMTK<
K. TMKE
J. RYCW
N. CYRW
D. [IXJ
;2. Tfmrm krm dmmr k`d pmknlnas h` k zll. Jy nlu`th`i
fmkds tfmy krm 62. Tfm `uejmr lg tfmhr cmis hs :22.
Flw ek`y pmknlnas krm tfmrm<
K. :2
J. >2
N. =2
D. 72
;;. Tfmrm hs k nmrtkh` rmckthl`sfhp jmtwmm` pkhr lg `uejmrs
ihvm` l` mhtfmr shdm lg ? ?. Hdm`thgy tfm rmckthl`sfhp
jmtwmm` tfm pkhr lg `uejmrs k`d smcmnt tfm lpthl`
wfhnf whcc rmpcknm tfm qumsthl` ekra<
2.;7 ? 2.22;7 ? ? ;.2: ? <
K. 2.2;2:
J. 2.;2:
N. ;.2:2
D. ;2.:2
;:. H` tfm glcclwh`i qumsthl`, Ghiurm (V) hs glcclwmd jy
glur kctmr`kthvm ghiurms (K), (J), (N) k`d (D) sunf
tfkt Ghiurm (V) hs mejmddmd h` l`m lg tfme. Trknm
lut tfm kctmr`kthvm ghiurm wfhnf nl`tkh`s Ghiurm (V)ks hts pkrt.
Ghiurm (V)
K.
J.
N.
D.
;>. ^tudy tfm glcclwh`i h`glrekthl` nkrmguccy k`d k`swmr
tfm qumsthl` ihvm` jmclw ht?
Tfmrm krm shx pmrsl`s K, J, N, D, M k`d G. N hs tfm
shstmr lg G. J hs tfm jrltfmr lg M‘s fusjk`d. D hs tfm
gktfmr lg K k`d irk`dgktfmr lg G. Tfmrm krm twl gktfmrs,
tfrmm jrltfmrs k`d k eltfmr h` tfm irlup.
Yfl hs tfm eltfmr< K. K
J. J
N. D
D. M
;4. Nlepcmtm tfm pkttmr` ihvm` h` ghiurm (V).
K.
J.
N.
D.
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4 9tf HEL - ^mt J
;=. H` tfm ihvm` qumsthl ,̀ tfmrm hs smt lg ghvm ghiurms
nkccmd tfm prljcme smt. ^mcmnt k suhtkjcm iurm wfhnf
whcc rmpcknm tfm qumsthl` ekra h` iurm (>) sl tfkt k
smrhms hs glremd jy tfm iurms ;, :, >, 4 k`d = tkam`
h` lrdmr.
Rrljcme ^mt
K.
J.
N.
D.
;7. ^mcmnt tfm nlrrmnt ehrrlr hekim lg Ghiurm (V).
Ghiurm (V)
K.
J.
N.
D.
;9. Flw ek`y pkrkccmclirkes krm
tfmrm h` tfm glcclwh`i iurm<
K. ;7
J. ;4
N. ;2
D. @l`m lg tfmsm
;6. Hg ù emk`s + , „ emk`s ù, Ù emk`s „ k`d + emk`s Ù,
tfm`( )>7 4 6 4
4 6 : ;7 ;
´ - ´
+ ´ + ¶ 5 <
K. 2
J. 6
N. ;:
D. ;7
;8. H` tfm ihvm` qumsthl`, tfrmm iurms V, X k`d _ krmsflwh`i k smqum`nm lg glcdh`i k nhrnuckr sfmmt lg
pkpmr. Tfm tfhrd iurm _ dmphnts tfm nuts ekdm h` tfm
glcdmd pkpmr. ̂ mcmnt k iurm wfhnf wlucd elst nclsmcy
rmsmejcm tfm pkpmr h` i. (_) wfm` u`glcdmd.
K.
J.
N.
D.
:2. Tfm ihvm` qumsthl` nl`shst lg twl smts lg iurms.
Ghiurms ;, :, > k`d 4 nl`sthtutm tfm prljcme smt.
Tfmrm hs k dmgh`htm rmckthl`sfhp jmtwmm` ghiurms
; k`d :. Mstkjchsf k shehckr rmckthl`sfhp jmtwmm`
iurms > k`d 4 jy smcmnth`i k suhtkjcm iurm grle tfm
lpthl`s tfkt wlucd rmpcknm tfm qumsthl` ekra (
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7 9tf HEL - ^mt J
>:. Flw ek`y jrhnas (`uejmr `mkr tl `mxt fu`drmd)
whcc jm rmquhrmd tl juhcd k wkcc >2 e cl`i, >2 ne
tfhna k`d = e fhif whtf k prlvhshl` lg : dllrs, mknf
4 e Ù :.= e k`d mknf jrhna jmh`i :2 ne Ù ;7 ne Ù 6 ne
wfm` l`m-`h`tf lg tfm wkcc hs ccmd whtf chem<
K. ;>=22
J. ;>722
N. :22=2
D. ;6=22
>>. H` ihvm` iurm, R k`d [ krm nm`trms lg twl nhrncms
h`tmrsmnth`i kt J k`d N . KND hs k strkhift ch`m. Tfm`,
rkthl lg ∢ J[D k`d ∢ KNJ hs
K. ; ? ;
J. : ? ;
N. ; ? :
D. ; :?
>4. H` ihvm` ghiurm, KJND hs k rmntk`icm h` wfhnf
ND 5 7 ne, KD 5 6 ne. Gh`d tfm krmk lg pkrkccmclirke
NDMG .
K. >7 ne:
J. :4 ne:
N. 46 ne:
D. 72 ne:
>=. 42% lg < + 87: 5 >=% lg 862 + ;=2% lg 6=2
K. >9:
J. 64:
N. ;742
D. ;2=:
MWMZXDKX EKTFMEKTHN^
>7. Mhift pmlpcm krm pck``h`i tl sfkrm mqukccy tfm nlst
lg k rm`tkc nkr. Hg l`m pmrsl` whtfdrkws grle tfm
krrk`imem`t k`d tfm ltfmrs sfkrm mqukccy tfm m`thrm
nlst lg tfm nkr, tfm` tfm sfkrm lg tfm rmekh`h`i pmrsl`s
h`nrmksmd jy
K.;
9
J.;
6
N.;
8
D.9
6
>9. K nmrtkh` gkntlry mepclymd 722 em` k`d 422 wlem`
k`d tfm kvmrkim wkim wks b :=.=2 pmr dky. Hg k wlek`
ilt b = cmss tfk` k ek`, tfm` wfkt krm tfmhr dkhcy
wkims<
K. Ek` ? b :=1 Ylek` ? b :2
J. Ek` ? b :9.=21 Ylek` ? b ::.=2
N. Ek` ? b >21 Ylek` ? b :=
D. Ek` ? b >:.=21 Ylek` ? b :9.=2
>6. K nych`dmr whtf fmhift k`d rkdhus : ? ; hs ccmd whtf
slgt drh`a k`d tfm` ht hs thctmd sl ks tl kcclw slem
slgt drh`a tl lw lgg tl k` mxtm`t wfmrm tfm cmvmc lg
slgt drh`a oust tlunfms tfm clwmst plh`t lg tfm uppmr
elutf. Hg :.; C lg slgt drh`a hs rmtkh`md h` tfm nych`dmr,
wfkt hs tfm nkpknhty lg tfm nych`dmr<
K. >.7 C
J. 4 C
N. 4.: C
D. @l`m lg tfmsm
>8. H` k clttmry lg ;22 thnamts `uejmrmd ; tl ;22, twl
thnamts krm drkw` sheuctk`mluscy. Gh`d tfm prljkjhchty
tfkt jltf tfm thnamts drkw` fkvm prhem `uejmrs.
K.:
>>
J.9
=2
N.9
:2
D.=
77
42. Tfm prhnm lg k Ekruth nkr rhsms jy >2%, wfhcm tfm skcms
lg tfm nkr ilms dlw` jy :2%. Yfkt hs tfm pmrnm`tkimnfk`im h` tfm tltkc rmvm`um<
K. = %
J. 4 %
N. 7 %
D. 6 %
4;. Zkemsf jluift x kppcms kt b 2.=2 mknf k`d y ahwh
gruht kt b 2.62 mknf. Fm slcd tfm kppcms kt b 2.6=
mknf k`d tfm ahwh gruht kt b ;.22 mknf. Yfhnf lg tfm
glcclwh`i kcimjrkhn mxprmsshl`s nlrrmntcy sflws tfm
prlt mkr`md jy Zkemsf<
Nlpyrhift©:2;=. ̂ nhm`nm Lcyephkd Glu`dkthl` (^LG)
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99tf HEL - ^mt J
K. >= x + :2 y
J. >= xy
N. :2 xy
D. >2 x + ;= y
4:. K rmntk`iuckr nlurtykrd >.96 emtrms cl`i k`d =.:= emtrms
whdm hs tl jm pkvmd mxkntcy whtf squkrm thcms, kcc lg tfm
skem shzm. Tfm` tfm ckrimst shzm lg tfm thcm wfhnf nlucd
jm usmd glr tfm purplsm hs (` Ù >)ne. Gh`d `.
K. ;4
J. :;
N. 4: