5-cubes & dice
TRANSCRIPT
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7/21/2019 5-Cubes & Dice
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M@BQKJ KINJNQ^
Anh`
Anh` ns k tcr`` anm`bsndbkj fngur` wntc kjj df nts
surfkh`s bumi`r`a.
Nmpdrtkbt pdnbts kidut Anh` 5
(n) K anh` cks snx surfkh`s kba kjj df tc`m kr`
bumi`r`a frdm ; td 6.
(nn) Nf tc` surfkh`s df anh` kr` ubfdja`a kba pjkh`a
db k pjkb`, tc` fngur` df anh` sd ditknb`a wnjj
jddl jnl` db` df tc` fdjjdwnbg fngur`s.
C`r`, nb tc` fngur` ; w` fnba tckt
;
9 8 1
0
6
Bumi`r ; ns dppdsnt` td 0
Bumi`r 9 ns dppdsnt` td 1
Bumi`r 8 ns dppdsnt` td 6.
C`r` nb tc` fngur` 9 w` fnba tckt
; 9
8
1
0 6
Bumi`r ; ns dppdsnt` td 6
Bumi`r 9 ns dppdsnt` td 1
Bumi`r 8 ns dppdsnt` td 0.
C`r` nb tc` fngur` 8 w` fnba tckt
;
9
8 1
0
6
Bumi`r ; ns dppdsnt` td 8
Bumi`r 9 ns dppdsnt` td 0
Bumi`r 1 ns dppdsnt` td 6.
(nnn) K bumi`r db db` surfkh` df anh` cks db`
bumi`rs dppdsnt` td nt kba fdur anff`r`bt bumi`rs
db nts kaekh`bt surfkh`s.
[us stndb db anh` ckv` i``b hjkssnfn`a uba r tcr``
anff`r`bt hkt`gdr ns. Nb tc` fdjjdwnbg pkrkgrkpcsanff`r`bt typ`s df qu`stndbs ckv` i``b anshuss`a wntc
tc` c`jp df `xkmpj`s uba`r anff`r`bt hkt`gdrn`s.
@SKM_J@XHkt`gdry N
@x.; K anh` cks i``b tcrdwb fdur tnm`s kba prdauh`s
fdjjdwnbg r`sujts.
8 9 0 8
6 0 8 9
; 1 ; 6
(n) (nn) (nnn) (nv)
Wcnhc bumi`r wnjj kpp`kr dppdsnt` td tc`
bumi`r 8 4
(K) 1 (I) 0 (H) 6 (A) ;
Xdj. Frdm tc` fngur`s (n), (nn) kba (nv) w` fnba tckt
bumi`rs 6, ;, 0 kba 9 kpp`kr db tc` kaekh`bt
surfkh`s td tc` bumi`r 8. Qc`r`fdr`, bumi`r 1
wnjj i` dppdsnt` td bumi`r 8. C`bh` dptndb (k) ns
tc` kbsw`r.
Hkt`gdry NN
@x. 9 Qc` fngur`s gnv`b i`jdw scdw tc` twd anff`r`btpdsntndbs df k anh`. Wcnhc bumi`r wnjj kpp`kr td
bumi`r 9.
(K) 8 (I) 1 (H) 0 (A) 6
8 1
1 0
9 6
(n) (nn)
0HCK_Q@Y
HP @X ANH@
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7/21/2019 5-Cubes & Dice
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M@BQKJ KINJNQ^
Xdj. Qc` kidv` qu stndb, wc r` dbjy twd pdsntndbs
df k anh` kr` gnv`b, hkb `ksnjy i` sdjv`a wntc tc`
fdjjdwnbg m`tcda.
Xt`p N 5 Qc` anh`, wc`b ubfdja`a, wnjj kpp`kr ks scdwb
nb tc` fngur` gnv`b db tc` rngct sna`.Xt`p NN5 Wrnt` tc` hdmmdb bumi`r td idtc tc` anh` nb tc`
mnaaj` ijdhl. Xnbh` hdmmdb bumi`r ns 1, c`bh`
bumi`r 1 wnjj kpp`kr nb tc` h`btrkj ijdhl.
Xt p NNN5 Hdbsna`r tc` fngur` (n) kba wrnt` tc` fnrst bumi`r
nb tc` kbtn-hjdhlwns` anr`htndb df bumi`r 1,
(hdmmdb bumi`r) nb ijdhl N kba s`hdba bumi`r
nb ijdhl NN. Qc`r`fdr`, bumi`rs 8 kba 9 i`nbg tc`
fnrst kba s`hdba bumi`r td 1 nb kbtnhjdhlwns
anr`htndbs r`sp`htnv`jy, wnjj kpp`kr nb ijdhl N &
NN r`sp`htnv`jy.
NN
9
NNN 6 1 8 N
0 NV
;
Xt`pNV5 Hdbsna`r fng (nn) kba wrnt` fnrst kba s`hdba
bumi`r nb tc` kbtnhjdhlwns` anr`htndb td bumi`r
1, (hdmmdb bumi`r) nb ijdhl (nnn) & (nv) C`bh`
bumi`r 6 kba 0 wnjj kpp`kr nb tc` ijdhls NNN kba
NV r`sp`htnv`jy.
Xt`pV5 Wrnt` r`mknbnbg bumi`r nb tc` r`mknbnbg ijdhl.
Qc`r`fdr`, bumi`r ; wnjj hdm` nb tc` r`mknbnbg
ijdhl.
Bdw, frdm tc` ubfdja`a fngur`s w` fnba tckt bumi`r
dppdsnt` td 6 ns 8, bumi`r dppdsnt` td 9 ns 0 kba
bumi`r dppdsnt` td 1 ns ;. Qc`r`fdr`, dptndb (h) ns
dur kbsw`r.
Hkt`gdry NNN
@x.8 Frdm tc` fdjjdwnbg fngur`s df anh`. Fnba wcnhc
bumi`r wnjj hdm` nb pjkh` df 4
9 ; 6
1 0 ;
8 9 4
(n) (nn) (nnn)
(K) 1 (I) 0 (H) 9 (A) 8
Xdj. Nf tc` kidv` anh` ns ubfdja`a, nt wnjj jddl jnl` ks
tc` fng gnv`b i`jdw. (Xtua`bt scduja fdjjdw tc`
m`tcdas ks `xpjknb`a nb tc` pr`vndus `xkmpj` td
fnba tc` kpprdprnkt` pjkh` df tc` bumi`rs
kpp`krnbg db tc` anff`r`bt surfkh`s df tc` anh` nbtc` fngur`).
1
0 9 8
;
6
Bdw tc` bumi`r nb pjkh` df '4' hkb i` ditknb`a
iy mklnbg k sjngct hckbg` nb tc` fngur` ks gnv`b
c`r`.
Bdw hdmpkrnbg fng (nnn) ks kidv`, w` g`t tckt
bumi`r nb pjkh` df 4 ns 8
1
0 9
; 8
6
Hkt`gdry NV
@x.1 Wcnhc df tc` fdjjdwnbg anh`s ns na`btnhkj td tc`
ubfdja`a fngur` ks scdws c`r` 4
6 8 6 9
8 ; 9 8
1 0 1 1
(k) (i) (h) (a)
9
0 ;
1 8
6
Xdj. Frdm tc` ubfdja`a fngur` df anh , w` fnba tcktbumi`r dppdsn t` td 9 ns 1, fdr 0 nt ns 8 kba fdr
; nt ns 6. Frdm tcns r`sujt w` hkb a`fnbnt`jy sky
tckt fngur` (i), (h) kba (a) hkb bdt i` tc` kbsw`r
fngur` ks bumi`r jynbg db tc` dppdsnt` pknr df
surfkh` kr` pr`s`bt db tc` kaekh`bt surfkh`s.
C`bh` fng (k) ns dur kbsw`r.
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M@BQKJ KINJNQ^
@x.0 K an` ns tcrdwb fdur tnm s kba nts fdur anff`r`bt
pdsntndbs kr` gnv`b i`jdw. Fnba tc` bumi`r db
tc` fkh` dppdsnt` tc` fkh` scdwnbg 9.
(n) (nn) (nnn) (nv)
(k) 8 (i) 1 (h) 0 (a) 6
Xdj. C`r`, tc` bumi`r 9 kpp`krs nb tcr`` anh`, bkm`jy
(n), (nn) kba (nv). Nb tc`s` anh`, w` dvs`rv` tckt tc`
bumi`rs 9, 1, ; kba 6 kpp`kr kaekh`bt td 8. Xd,
bdb` df tc`s` bumi`rs hkb i` pr`s`bt dppdsnt`
9. Qc` dbjy bumi`r j`ft ns 0.
C`bh`, 0 ns pr`s`bt db tc` fkh` dppdsnt` 9.
@x.6 Xcdwb i`jdw kr`, fdur anff r`bt pdsntndbs df tc`
skm` anh`. Fnba tc` bumi`r db tc` fkh` dppdsnt`
tc` fkh` scdwnbg 6.
(n) (nn) (nnn) (nv)
(k) ; (i) 9 (h) 1 (a) 0
Xdj. Nb tcns hks , tc` bumi`r 6 kpp`krs nb dbjy twd
anh` frdm wcnhc w` dis`rv` tckt tc` bumi`rs ;,
8 kba 0 kpp`kr kaekh`bt td 6, sd tckt 9 dr 1 hkb
kpp`kr dppdsnt` 6.
Xd, w` i`ngb fnbanbg k bumi`r wcnhc kpp`krs kt
j`kst nb tcr`` df tc` gnv`b anh`. 8 ns suhc k
bumi`r, wcnhc kpp`krs nb (n), (nn) kba (nnn). W`
dis`rv` nb tc`s` anh` tckt, tc` bumi`rs ;, 1, 0kba 6 kpp`kr kae`h`bt td 8. Xd, tc`y hkbbdt
kpp`kr dppdsnt` 8. Qc` dbjy bumi`r tckt hkb
kpp`kr dppdsnt` 8 ns 9.
Xd, 9 hkbbdt kpp`kr dppdsnt` 6.
C`bh`, 1 kpp`krs dppdsnt` 6, sd tckt (h) ns tc`
kbsw`r.
W` kr` bdw nb k pdsntndb td sdjv` tc` fdjjdwnbg
`x`rhns`.
@SKM_J@X
@x.7. Wckt ns tc` bumi`r df strkngct jnb`s nb tc`fdjjdwnbg fngur` 4
(K) ;; (I) ;1 (H) ;6 (A) ;7
Xdj. Qc` fngur` ns jki`jj`a ks scdwb.
K I H A
L E N C G
J F
M B
_ [
YD
@
X
Hj`krjy, tc`r` kr` 8 cdrnzdbtkj jnb`s bkm`jy K@,
JF kba LG.
Qc`r` kr` 0 v`rtnhkj jnb s 5 KL, IE, HN, AC kba @ G.
Qc r kr 6 sjkbtnbg jnb s 5 JH, L@, NF, JN, KG kba HF.
Qcus, tc`r` kr` 8 + 0 + 6 = ;1 strkngct jnb`s nb
tc` fngur`.
C`bh`, tc` kbsw`r ns (i).
@x.3 Hdubt tc` bumi`r df trnkbgj`s nb tc` fdjjdwnbgfngur`.
(K) 3 (I) ;? (H) ;9 (A);1
Xdj. W` fnrst jki`j tc` `btnr` fngur` ks scdwb. Hdubt
tc` bumi`r df snmpj`st trnkbgj s. Qc`s` kr` KIC,
ICD, IHA, IDA, A@F, AFD, FGC kba FCD.
Qcus,tc`r` kr` 3 suhc trnkbgj`s. B`xt hdubt tc`bumi`r df trnkbgj`s wcnhc kr` hdmpds`a df twd
hdmpdb`bts `khc. Xuhc trnkbgj`s kr` CIA, IAF,
AFC kba FCI. Qcus, tc`r` kr` 1 suhc trnkbgj`s.
Qc` tdtkj bumi`r df trnkbgj s nb tc` gnv`b
fngur` = 3 + 1 = ;9
Qcus, (H) ns tc` kbsw`r.
@x.> Cdw mkby squkr`s ad`s tc` fngur` ckv` 4
(K) 6 (I) 7 (H) > (A) ;?
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M@BQKJ KINJNQ^
Xdj. Qc` fngur mky i` jki jj`a ks scdwb 5
Qc` squkr`s hdmpds`a df twd hdmpdb`bts `khc,kr` KILE, IHJL, HA@J, J@FG, LJGC kba ELCN.
Qcus, tc`r` kr` 6 suhc squkr`s.
Dbjy db` squkr , LH@G ns hdmpds a df fdur
hdmpdb`bts.
Qwd squkr`s bkm`jy, KHGN kba IAFC kr`
hdmpds`a df `ngct hdmpdb`bts `khc.
Qcus, tc`r` kr` 9 suhc squkr`s.
Qc`r` kr` 6 + ; + 9 = > squkr`s nb tc` gnv`b
fngur`. C`bh`, (H) ns tc` kbsw`r.
@x.;? Cdw mkby pkrkjj`jdgrkms kr` tc`r` nb tc` fngur`i`jdw 4
(K) ;1 (I) ;0 (H) ;6 (A) ;3
Xdj. W` hkb jki j tc` fngur` ks scdwb.
Qc` snmpj`st || gms kr` KIF@, IHGF, HACG, @FEN,
FGLE kba GCLJ. Qc`s` kr` 6 nb bumi`r.
Qc` || gms hdmpds`a df twd hdmpdb`bts `khc,
kr` KHG@, IACF, @GLN, FCJE, KIEN, IHLE kba
HAJL.
Qcus, tc`r` kr` 7 suhc || gms.Qc` || gms hdmpds`a df fdur hdmpdb`bts `khc,
kr` KHLN kba IAJE. n. `. 9 nb bumi`r. Qc`r` ns
dbjy db` || gm hdmpds`a df snx hdmpdb`bts,
bkm`jy, KAJN.
Qcus, tc`r` kr` 6 + 7 + 9 + ; = ;6 pkrkjj`jdgrkms
nb tc` fngur`.
C`bh`, (H) ns tc` kbsw`r.
@x.;; Wckt ns tc` bumi`r df r`htkbgj s nb tc fdjjdwnbg
fngur` 4
(K) 6 (I) 7 (H) 3 (A) >
Xdj. Qc` fngur ns jki jj`a ks scdwb 5
Xnmpj`st r`htkbgj`s kr` K@CG, @FNC, FILE, ELHJ
kba GNJA. n. `. tc`r` kr` 0 suhc r`htkbgj`s.
Qc` r`htkbgj`s hdmpds`a df twd hdpdb`bts `khc
kr` KFNG kba FIHJ.
Qcus, tc`r` kr` 9 suhc trnkbgj`s.Dbjy db` r`htkbgj`, bkm`jy KFJA ns hdmpds`a
df 8 hdmpdb`bts kba dbjy db` r`htkbgj`, bkm`jy
KIHA ns hdmpds`a df 0 hdmpdb`bts.
Qcus, tc`r` kr` 0 + 9 + ; + ; = > r`htkbgj`s nb
tc` fngur`.
C`bh`, (a) ns tc` kbsw`r.
@x.;9 A`t`rmnb` tc` bumi`r df p`btkgdbs nb tc`
fdjjdwnbg fngur`.
(K) 0 (I) 6
(H) 3 (A) ;?Xdj. Qc` fngur` mky i` jki jj`a ks scdwb nb tcns hks .
X n x _ ` b t kg d bs c k v` i ` `b fd rm `a i y t c `
hdminbktndb df tcr`` trnkbgj`s kba twd rcdm
ius`s - KAFCE, HFCEJ, @CEJI, GEJIA, NJIAF
kba LIAFC.
Fdur dtc`r p`btkgdbs kr` fdrm`a iy tc` hdmin
bktndb df tcr`` trnkbgj`s kba db` rcdmius -
JHFCM, JI@CM, ILFCM kba IJNFM.
Qcus, tc`r` kr` ;? p`btkgdbs nb tc` fngur`.
C`bh`, (A) ns tc` kbsw`r.
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7/21/2019 5-Cubes & Dice
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M@BQKJ KINJNQ^
Anr`htndbs (;-3) 5
Nb `khc df tc` fdjjdwnbg qu`stndbs fdur pdsntndbs df tc`
skm` anh` ckv` i``b scdwb. ^du ckv` td s`` tc`s` fngur`s
kba s`j`ht tc` bumi`r dppdsnt` td tc` bumi`r ks ksl`a
nb `khc qu`stndb.
[.; 0 6 8 ;
8 8 9 1
6 ; ; 6
(n) (nn) (nnn) (nv)
Wcnhc bumi`r ns db tc` dppdsnt` surfkh` df
bumi`r 8 4
(K) 1 (I) 6 (H) 0 (A) ;
[.9 1 ; 0 0
9 0 ; 6
0 9 6 1
(n) (nn) (nnn) (nv)
Wcnhc bumi`r ns dppdsnt` td bumi`r ; 4
(K) 1 (I) 6 (H) 9 (A) 8
[.8 0 6 9 0
8 9 0 6
1 ; 1 8
(n) (nn) (nnn) (nv)
Wcnhc bumi`r ns dppdsnt` td bumi`r 0 4
(K) 6 (I) 0 (H) ; (A) 8
[.1 9 9 8 ;
6 1 ; 6
1 8 9 0
(n) (nn) (nnn) (nv)
Wcnhc bumi`r ns dppdsnt` td bumi`r 9 4
(K) 1 (I) 6 (H) 0 (A) 8
[.0 ; 1 8 1
9 6 0 ;8 ; 9 9
(n) (nn) (nnn) (nv)
Wcnhc bumi`r ns dppdsnt` td bumi`r 0 4
(K) 1 (I) 6 (H) ; (A) 8
[.6 6 9 0 8
8 0 8 9
; 1 ; 6
(n) (nn) (nnn) (nv)
Wcnhc bumi`r ns db tc` dppdsnt` surfkh` df
bumi`r 8 4
(K) 9 (I) 8 (H) 1 (A) 6
[.7 K A H H
@ H A I
@ I K
(n) (nn) (nnn) (nv)
H
Wcnhc j`tt`r wnjj i` dppdsnt` td j`tt`r A 4(K) K (I) I (H) @ (A) F
[.3 =
D
=
(n) (nn) (nnn) (nv)
=
Wcnhc symidj wnjj hdm` dppdsnt` td symidj
(K) ? (I) = (H) (A)
Anr`htndb (>-98) 5
Nb `khc df tc` fdjjdwnbg qu`stndbs, s`j`ht tc` hdrr`ht dptndb
fdr tc` qu`stndb ksl a.
[.> ;
0
(n)
96
1
9
(nn)
(n) Wcnhc bumi`r wnjj hdm` dppdsnt` td bumi`r 9 4
(K) 0 (I) ; (H) 6 (A) 8
(nn) Wcnhc bumi`r wnjj hdm` dppdsnt` td bumi`r 6 4
(K) ; (I) 0 (H) 1 (A) 8
(nnn) Wcnhc bumi`r wnjj hdm` dppdsnt` td bumi`r 14
(K) 8 (I) 0 (H) ; (A) 9
[.;? 1
;
(n)
99
6
8
(nn)
Wcnhc wnjj i` bumi`r kt tc` idttdm nf 0 ns kt
tdp 4
(K) ; (I) 9 (H) 8 (A) 6
@S@YHNX@
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M@BQKJ KINJNQ^
[.;; Kjj tc` surfkh`s df tc` anh` hdbtknb anff r`bt
bumi`rs nb tc` fdrm df adts. Hdbsna`r idtc tc`
fngur`s df anh` kba t`jj cdw mkby adts kr`
hdbtknb`a db tc` fkh` dppdsnt` td tc` fkh` tckt
hdbtknbs fdur adts 4
(K) 9 (I) 8 (H) 0 (A) 6
[.;9 Wckt ns tc` bumi`r df adts db tc` fkh` dppdsnt`
td tc` fkh` tckt hdbtknbs 9 adts 4
(K) ; (I) 8 (H) 1 (A) 6
[.;8 Db tc` iksns df twd fngur`s df anh , ydu ckv td
t`jj wckt bumi`r wnjj i` db tc` dppdsnt` fkh` df
bumi`r 0 4
8
6
(n)
1;
0
8
(nn)
(K) ; (I) 8 (H) 1 (A) 0
[.;1 Wc`b bumi`r ; ns kt tc` tdp, wcnhc bumi`r wnjj
i` kt tc` idttdm 4
(K) ; (I) 9 (H) 8 (A) 6
[.;0 Wcnhc kjpcki`t wnjj hdm` db tc` dppdsnt` surfkh`
db wcnhc kjpcki`t F kpp`krs 4
K A F
H @ @
H
(n) (nn) (nnn)
I K
(K) I (I) K (H) H (A) @
[.;6 Wcnhc symidj wnjj kpp`kr db tc` dppdsnt`
surfkh` td tc` symidj 'S' 4
(n)
(K) (I) = (H) (A) D
[.;7 Qwd pdsntndb df k anh` kr` scdwb i jdw. Wc`b
tc`r` kr` twd adts kt tc` idttdb, tc` bumi`r df
adts kt tc` tdp wnjj i`
(K) 9 (I) 8 (H) 0 (A) 6
[.;3 Qcr`` pdsntndb df tc` skm` anh` kr` gnv`b i`jdw.
Dis`rv` tc` fngur`s hkr`fujjy kba t`jj wcnhc
bumi`r wnjj hdm` nb pjkh` df '4'
6 0 9
; 8 1
1
(n) (nn) (nnn)
8 4
(K) ; (I) 6 (H) 8 (A) 0
[.;> Db tc` iksns df tc` fdjjdwnbg fngur`s ydu ckv`
td t`jj wcnhc bumi`r wnjj hdm` nb pjkh` df '4'
(K) 9 (I) 8 (H) 6 (A) 1
[.9? Wcnhc j tt`r wnjj hdm` db tc` ijkbl surfkh`4]
I K K
K @ 4
F
(n) (nn) (nnn)
A I
(K) H (I) K (H) A (A) @
[.9; Wcnhc symidj wnjj kpp`kr nb pjkh` df '4'
= D
4
D
(n) (nn) (nnn)
(K) (I) = (H) (A)
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M@BQKJ KINJNQ^
[.99 Wcnhc bumi`r wnjj hdm` nb pjkh` df '4'
9 8 1
; 1 8
0
(n) (nn) (nnn)
8 4
(K) ; (I) 6 (H) 0 (A) 9
[.98 K hui` db wcds` fkh s j`tt`rs ckv` i``b wrntt`b,
ns scdwb i`jdw nb anff`r`bt pdsntndbs ks hkb i`
s``b frdm anff`r`bt anr`htndbs. Fnba tc` mnssnbg
j`tt`rs 4
M M P P
A A ^ W
W 4
(n) (nn) (nnn) (nv)
X A
(K) X (I) A (H) ^ (A) W
Anr`htndbs (91-9>) 5
Qc` fngur` (k) gnv`b i`jdw ns tc` ubfdja`a pdsntndb df k
huinhkj anh`. Nb `khc df tc` fdjjdwnbg qu`stndbs tcns
ubfdja`a fngur` ns fdjjdw`a iy fdur anff`r`bt fngur`s df
anh`. ^du ckv` td s`j`ht tc` fngur` wcnhc ns na`btnhkj td
tc` fngur` (K).
[.91
;
6 9
8
1 0
(k)
9 0 8 0
6 6 9 6
9 ;
(K) (I) (H) (A)
1 6
[.90
;
1 8 6
9
0
(k)
8 6 9 ;
9 1 8 1
8 8
(K) (I) (H) (A)
0 0
[.96K
H
A I
@
F
(k)
@ H @ F
H K H I
I K
(K) (I) (H) (A)
A K
[.97
=
+
(k)
+ =
(K) (I) (H) (A)
[.93
F
K @
I
H A
(k)
I @ I @
F F @ A
A K
(n) (nn) (nnn) (nv)
@ H
(K) (n) dbjy (I) (nn) dbjy
(H) (n) kba (nnn) dbjy (A) Kjj
-
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M@BQKJ KINJNQ^
[.9>
D
(K) (n) dbjy (I) (nn) dbjy
(H) (n) kba (nnn) dbjy (A) (nv) dbjy
Anr`htndb 5
Nb qu`stndbs 8? td 80, fdur anh`s kr` scdwb. Qc` fnrst
tcr`` anspjky kjpcki`ts nb sdm` s`qu`bh`s. Fnba tc`
s`qu`bhnkj ruj` kba fnjj tc` ijkbl spkh`s nb tc` fdurtc anh`
frdm tc` gnv`b fdur kjt`rbktnv`s.
X`qu`bh` df j`tt`rs
K I H A @ F G C N E L J M
B D _ [ Y X Q P V W S ^ U
[.8?
Q
V
S
E
M
_
X
K
@
N
(K) J, B (I) H, F(H) @, D (A) M, G
[.8;
S
K
P
J
Q F
@
C
I
_
(K) M, E (I) X, P
(H) X, V (A) J, G
[.89
_
B
L
D
U
W
Y
[
P E
(K) [, X (I) Q, @
(H) Q, F (A) B, U
[.88
N
M
@
J
@ K
^
Y
J C
(K) Q, _ (I) S, U
(H) X, V (A) A, E
[.81
H
K
F
H
C B
@
C J
M
(K) [, J (I) [, V
(H) _, X (A) C, [
[.80
Y
Q
V
I
V C
F
D
L X
(K) H, N (I) B,
(H) F, M (A) J, U
[.86 Cdw mkby strkngct jnb`s kr` tc`r` nb tc` fdjjdwnbg
fngur` 4
(K) > (I) ;; (H) ;0 (A) 13
[.87 Wckt ns tc` bumi`r df strkngct jnb`s nb tc`
fdjjdwnbg fngur` 4
(K) ;? (I) ;9 (H) ;8 (A) ;7
[.83 Wckt ns tc` bumi`r df strkngct jnb`s nb tc`
fdjjdwnbg fngur` 4
(K) ;8 (I) ;0 (H) ;7 (A) ;>
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[.8> Xtkt` tc` mnbnmum bumi`r df strkngct jnb`s
r`qunr`a td mkl` tc` fngur` gnv`b i`jdw 4
(K) ;6 (I) ;7 (H) ;3 (A) ;>
[.1? Wckt ns tc` bumi`r df strkngct jnb`s kba tc`
bumi`r df trnkbgj`s nb tc` gnv`b fngur` 4
(K) ;? strkngct jnb`s kba 81 trnkbgj`s
(I) > strkngct jnb`s kba 81 trnkbgj`s
(H) > strkngct jnb`s kba 86 trnkbgj`s
(A) ;? strkngct jnb`s kba 86 trnkbgj`s
[.1; Cdw mkby trnkbgj`s kba squkr`s kr` tc`r` nb tc`
fdjjdwnbg fngur` 4
(K) 93 trnkbgj`s, 0 squkr`s
(I) 91 trnkbgj`s, 1 sqkur`s
(H) 93 trnkbgj`s, 1 squkr`s
(A) 91 trnkbgj`s, 0 squkr`s]
[.19 Hdubt tc` bumi`r df trnkbgj`s kba squkr s nb tc`
fdjjdwnbg fngur` 4
(K) 93 trnkbgj`s, ;? squkr`s
(I) 93 trnkbgj`s, 3 squkr`s
(H) 89 trnkbgj`s, ;? squkr`s
(A) 89 trnkbgj`s, 3 squkr`s
[.18 Hdubt tc` bumi r df trnkbgj s nb tc` fdjjdwnbg
fngur` 4
(K) ;0 (I) ;6 (H) ;7 (A) ;3
[.11 Cdw mkby r`htkbgj`s ad`s tc` fdjjdwnbg fngur`
ckv` 4
(K) ;? (I) ;9 (H) ;8 (A) ;1
[.10 Cdw mkby trnkbgj s kr` tc r` nb tc fngur` i`jdw 4
(K) 0 (I) 6 (H) 3 (A) ;?
[.16 Hdubt tc` bumi`r df trnkbgj`s nb tc` fngur` gnv`b
i`jdw 4
(K) ;3 (I) 9? (H) 91 (A) 97
[.17 Hdubt tc` bumi`r df trnkbgj`s nb tc` fdjjdwnbg
fngur` 4
(K) 97 (I) 90 (H) 98 (A) 9;
[.13 Hdubt tc` bumi`r df squkr`s nb tc` fngur` gnv`b
i`jdw 4
(K) ;8 (I) ;6 (H) ;> (A) 9?
[.1> Cdw mkby r`htkbgj`s kr` tc r` nb tc` gnv`b
fngur`4
(K) 6 (I) 7 (H) 3 (A) >
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[.0? Cdw mkby trnkbgj`s ad`s tc` fdjjdwnbg fngur`
ckv 4
(K) 86 (I) 1? (H) 11 (A) 13
[.0; Cdw mkby squkr`s kr` tc`r` nb tc` fngur` gnv`b
i`jdw 4
(K) ;9 (I) ;8 (H) ;6 (A) ;7
[.09 Hdubt tc` bumi r df trnkbgj s nb tc` fdjjdwnbg
fngur` 4
(K) 3 (I) ;? (H) ;; (A) ;9
[.08 Cdw mkby trnkbgj s kr` tc`r` nb tc` fdjjdwnbg
fngur` 4
(K) 9? (I) 91 (H) 93 (A) 89
[.01 Cdw mkby trnkbgj s kr` tc`r` nb tc` fdjjdwnbg
fngur` 4
(K) ;9 (I) ;8 (H) ;1 (A) ;0
[.00 Hdubt tc` bumi r df trnkbgj s nb tc` fdjjdwnbg
fngur` 4
(K) 98 (I) 97 (H) 9> (A) 8;
[.066 6
8 9 1 6 1
0 ;
1 99
(n) (nn) (nnn) (nv)
Wcnhc bumi`r ns db tc` dppdsnt` 6 4
(K) ; (I) 9 (H) 8 (A) 1
[.07 ; 88
8 81
1
90
0
;
9
(n) (nn) (nnn) (nv)
Wckt bumi`r ns dppdsnt` 1 4
(K) ; (I) 9 (H) 0 (A) 6
[.03
(n) (nn) (nnn) (nv)
Wckt bumi`r ns dppdsnt` 8 4
(K) 9 (I) 8 (H) 1 (A) 6
[.0>
(n) (nn) (nnn) (nv)
Wcnhc bumi`r ns db tc` fkh` dppdsnt` 1 4
(K) ; (I) 9 (H) 8 (A) 0
[.6?
(n) (nn) (nnn) (nv)
Wcnhc bumi`r ns dppdsnt` 8 4
(K) ; (I) 9 (H) 1 (A) 6
[.6; Nf tc` tdtkj bumi`r df adts db dppdsnt` fkh`s df
k huinhkj ijdhl ns kjwkys 7, fnba tc` fngur` wcnhc
ns hdrr`ht 4
(K) (I) (H) (A)
Anr`htndb 5
[u`stndb 69 td 61 kr` iks`a db tc` fdjjdwnbg njjustrktndbs,
Wcnhc kr` fdur vn`ws df k hui`.
(n) (nn) (nnn) (nv)
[.69 Qc` symidj kt tc` idttdm df (nv) ns -
(K) (I) (H) (A)
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[.68 Qc` symidj dppdsnt` tc` fkh` ckvnbg tc symidj =
ns -
(K) (I) (H) (A)
[.61 Qc` symidj dppdsnt` tc` fkh` ckvnbg trnkbgj` ns-
(K) (I) (H) = (A)
[.60 Nf tc` tdtkj bumi`r df adts db dppdsnt fkh`s df
k huinhkj ijdhl ns kjwkys 7, fnba tc` fngur` wcnhc
ns hdrr`ht.
(K) (I) (H) (A)
[.66
(n) (nn) (nnn)
Wckt scduja i` tc` bumi`r dppdsnt` 8 4(K) ; (I) 6 (H) 0 (A) 1
[.67
(n) (nn) (nnn)
Cdw mkby adts jn` dppdsnt` 9 adts 4
(K) ; (I) 8 (H) 0 (A) 6
[.63
(n) (nn) (nnn)Fnba tc` bumi`r df adts db tc` fkh` dppdsnt` tc`
fkh` i`krnbg 8 adts 4
(K) 0 (I) 6
(H) 1 (A) Hkbbdt i` a`t`rmnb`a
[.6>
(n) (nn) (nnn)
Wckt bumi`r dhhur kt tc` idttdm fkh` nb tc`
tcr`` pdsntndbs df tc` skm` an` 4
(K) 6, 6, 9 (I) 0, 6, ;
(H) 0, 0, 0 (A) 6, 0, 9
[.7?
(n) (nn) (nnn)
Fnba tc` kjpcki`t dppdsnt` K.
(K) I (I) H (H) A (A) @
[.7;
(n) (nn) (nnn)
K hui` cks snx anff`r`bt symidjs arkwb dv`r nts
snx fkh`s. Qc` symidjs kr` adt, hnrhj`, trnkbgj`,squkr`, hrdss kba krrdw. Qcr`` anff`r`bt pdsntndbs
df tc` hui` kr` scdwb nb fngur`s (n), (nn) & (nnn).
(n) Wcnhc symidj ns dppdsnt` tc` adt 4
(K) Hnrhj` (I) Qrnkbgj`
(H) Krrdw (A) Hrdss
(nn) Wcnhc symidj ns dppdsnt` tc` Krrdw 4
(K) Hnrhj` (I) Qrnkbgj`
(H) Adt (A) Hrdss
(nnn) Wcnhc symidj dhhurs kt tc` idttdm df fng. (nn).
(K) Krrdw (I) Qrnkbgj`
(H) Hnrhj` (A) Adt
[.79
(S) (^) (U)
(n) Wcnhc bumi`r jn`s kt tc` idttdm fkh` df tc`
an` S 4
(K) ; (I) 9 (H) 8 (A) 1
(nn) Wcnhc bumi`r jn`s kt tc` idttdm fkh` df tc`
an` ^ 4
(K) 6 (I) 0 (H) 9 (A) ;
(nnn) Wcnhc bumi`r jn`s dppdsnt` 6 4
(K) ; (I) 9 (H) 1 (A) 0
(nv) Wcnhc bumi`rs kr` cnaa`b i`cnba tc` bumi`rs
6 kba 0 nb tc` an` U 4
(K) ; & 1 (I) ; & 8
(H) 1 & 8 (A) ; & 9
(v) Wcnhc df tc` cnaa`b bumi`rs kaekh`bt td 0 nb
an` S kr` hdmmdb td tc` cnaa`b bumi`rs kaekh`bt
td 0 nb an` U 4
(K) ; & 1 (I) 9
(H) 6 (A) Bdb`
[.78 Qwd pdsntndbs df k anh` kr` scdwb i`jdw. Nf ;ns
kt tc` idttdm, wcnhc bumi`r wnjj i` db tc`
tdp 4
(n) (nn)
(K) 9 (I) 8 (H) 1 (A) 0
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M@BQKJ KINJNQ^
KBXW@Y L@^
86.\I] Qc` fngur` ns jki`jj`a ks scdwb-
Cdrnzdbtkj jnb`s kr` A@, FC, NJ kba IH n. `. 1 nb
bumi`r. Xjkbtnbg jnb`s kr` NM, FB, AD, KH, KI,
@M, kba CB n. `. 7 nb bumi`r.
Qdtkj bumi`r df jnb`s ns 1 + 7 = ;;.
87.\I] W` hkb jki`j tc` fngur` ks scdwb-
[u`s. ; 9 8 1 0 6 7 3 > (n) > (nn) > (nnn) ;? ;; ;9 ;8 ;1 ;0 ;6 ;7 ;3 ;> 9?
Kbs. K K H H A H K K A K I I A K H A H A I K I H
[u`s. 9; 99 98 91 90 96 97 93 9> 8? 8; 89 88 81 80 86 87 83 8> 1? 1; 19Kbs. K K H H A I H I I H K H K I K I I K I H K H
[u`s. 18 11 10 16 17 13 1> 0? 0; 09 08 01 00 06 07 03 0> 6? 6; 69 68 61
Kbs. H A A H H I A A A A A A H K K H K H H A K K
[u`s. 60 66 67 63 6> 7? 7;(n) 7;(nn) 7;(nnn) 79(n) 79(nn) 79(nnn) 79(nv) 79(v) 78 71 70
Kbs. K I H I H A A I H H H K I A I A K
CNBQX & XDJPQNDB
Nb tcns fngur` 5
Qc` cdrnzdbtkj jnb`s kr` KH, CA kba G@ n. `. 8 nb
bumi`r 5tc` v`rtnhkj jnb`s kr` KG, IF kba H@ n. `. 8 nb
bumi`r5 kba tc` sjkbtnbg jnb`s kr` K@, HA, KA,
HG, A@ kba GA n. `. 6 nb bumi`r.
Qcus, tc`r` kr 8 + 8 + 6 = ;9 jnb`s nb kjj.
83.\K] Qc` fngur` mky i` jki`jj a ks fdjjdws 5
V rtnhkj jnb`s kr` KA, @C, NJ, FG, IH kba EL n. `. 6.
Cdrnzdbtkj jnb s kr` NE, KI, @F, MB, CG, AH, JL
n. `. 7.
Qdtkj bumi`r df jnb`s ns 6 + 7 = ;8
[.71
(n) (nn)
Wckt ns tc` bumi`r df adts kt tc` idttdm fkh`
df tc` j`ft ckba sna` anh` 4(K) 8 (I) 1 (H) 0 (A) 6
[.70 Qwd pdsntndbs df k anh` wntc ; td 6 adts db nts
sna`s kr` scdwb i`jdw. Nf tc` anh` ns r`stnbg db
tc` sna` wntc tcr`` adts, Wckt wnjj i` tc` bumi`r
df adts db tc` sna` kt tc` tdp 4
(n) (nn)
(K) ;dr 0 (I) 9 (H) 8 (A) 6
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M@BQKJ KINJNQ^
Qc`r` ns dbjy db` trnkbgj`s hdmpds`a df tw`jv`
hdmpdb`bts. Qcns trnkbgj` ns KIH.
C`bh`, tc`r` kr` ;? + ;? + 9 + 7 + 6 + ; = 86
trnkbgj`snb tc` fngur`.
[.1;\K] Qc` fngur` mky i` jki`jj`a ks scdwb 5
Qrnkbgj`s
Xnmpj st trnkbgj`s kr` KIN, ING, GNC, CNK, IHE,
HFE, FGE, GIE, HAL, A@L, @FL, kba FHL n. `. ;9
nb bumi`r.
Qrnkbgj`s hdmpds`a df twd hdmpdb`bts kr` KIG,
IGC, GCK, CKI, IHF, HFG, FGI, GIH, HA@,
A@F, @FH kba FHA. n.`. ;9 nb bumi`r.
Qrnkbgj`s hdmpds`a df fdur hdmpdb`bts kr` KGH,
IFA, CIF kba GH@ n. `. 1 nb bumi`r.
Qcus, tc`r` kr` ;9 + ;9 + 1 = 93 trnkbgj`s nb tc`
fngur`.
Xqukr`s
Xqukr`s hdmpds`a df twd hdmpdb`bts kr` INGE
kba HEFL n. `. 9 nb bumi`r.
Xqukr`s hdmpds`a df fdur hdmpdb`bts kr`
KIGC,IHFG kba HA@F n.`. 8 nb bumi`r.
19.\H] W` mky jki`j tc` fngur` ks scdwb 5
Qrnkbgj`s
Qc` snmpj`st trnkbgj s kr` NE[, EL[, LJ[, JM[,
MB[, BD[, D_[ kba _N[ n.`. 3 nb bumi`r.
Qc` trnkbgj`s hdmpds`a df twd hdmpdb`bts kr`
KI[, IH[, HA[, A@[, @F[, FG[, GC[, CK[,
NL[, LM[, MD[ kba DN[ n. `. ;9 nb bumi`r.
8>.\I] Qc` fngur` mky i` jki`jj a ks fdjjdws 5
Cdrnzdbtkj jnb`s kr` NE, KI, CG kba AH n. `. 1 trnkbgj`s nb tc`
fngur`.
06.(K) Frdm Fng.(n), (nn) kba (nnn), nt ns hj`kr tckt tc`
bumi`rs 8, 9, 1 kba 0 jn` kaekh`bt td tc` bumi`r
6. sd, ; jn`s dppdsnt` 6.
07.(K) Frdm fng. (n), (nn) kba (nnn), nt ns hj`kr tckt tc`
bumi`rs ;, 0, 1 kba 9 jn` kaekh`bt tds tc` bumi`r
8. Xd, 6 jn`s dppdsnt` 8. Fdrm fng. (nn) kba (nnn), nt
fdjjdws tckt 8, 0 kba 9 jn` kaekh`bt td 1. Xd, ; kba
6 hkb jn` dppdsnt` 1. Iut 6 jn`s dppdsnt` 8.
Qc`r`fdr`, ; jn`s dppdsnt` 1.
03.(H) Frdm, fng.(n), (nnn) kba (nv), nt ns hj`kr tckt bumi`rs 6,
;, 0 kba 9 jn` kaekh`bt td 8. Xd, 1 jn`s dppdsnt` 8.
0>.(K) Frdm fng. (nnn) kba (nv), nt ns hj`kr tckt tc` bumi`r 0,
6, 9 kba 8 jn` kaekh`bt td 1. Xd ; jn`s dppdsnt` 1.
6?.(H) Frdm fng. (n), (nnn) kba (nv), w` fnba tckt tc` bumi`rs
kaekh`bt td 1 kr` 0, 6, ; kba 9. Xd, 8 jn`s dppdsnt`
1 n.`. 1 jn`s dppdsnt` 8.
6;.(H) Xnbh` tc` sum df tc` bumi`r df adts db dppdsnt`
fkh`s df tc` ijdhl ns kjwkys 7, w` hkbbdt g`t ;
adt kaekh`bt 6 adts, 9 adts kaekh`bt 0 adts dr 8
adts kaekh`bt 1 adts. Xd, tc` fngur` (K), (I) kba
(A) hkbbdt i` hdrr`ht.
69.(A) Frdm fng. (n) (nnn) kba (nv), w` dis`rv tckt tc`
symidjs D D = & jn` kaekh`bt td . sd
jn`s dppdsnt` .
68.(K) Frdm fng. (nn) kba (nnn), nt ns hj`kr tckt tc` symidjs
, D kba jn` kaekh`bt td = sngb.
Xd, D jn`s dppdsnt` =
61.(K) Frdm fng.(nnn) kba (nv), nt fdjjdws tc` symidjs ,
= & D jn` kaekh`bt td . Xd db` df tc` symidjs
Ddr hkb jn` dppdsnt` . Iut jn`s
dppsnt` . Xd, D jn`s dppdsnt` .
60.(K) Xnbh` tc` sum df tc` bumi`r df adts db dppdsnt`
fkh`s df tc` ijdhl ns kjwkys 7, w` hkbbdt g`t ;
adt kaekh`bt 6 adts, 9 adts kaekh`bt 0 adts dr 8
adts kaekh`bt 1 adts. Xd, tc` fngur`s (I), (H) kba
(A) hkbbdt i` hdrr`ht.
66.(I) Frdm tc` tcr ` gnv`b fngur`s, nt ns hj`kr tckt tc`
bumi`rs ;, 9, 0 kba 1 kpp`kr kaekh`bt td 8, sd
bdb` df tc`s` hkb kpp`kr dppdsnt` 8. Qc`r`fdr`,
6 kpp`krs dppdsnt` 8.
67.(H) Frdm fngur s (nn) kba (nnn), nt ns hj`kr tckt db ,
tcr`` fdur kba snx adts hkbbdt kpp`kr dppdsnt`
fnv` adts. Qc`r`fdr`, twd adts kpp`kr dppdsnt`fnv` adts.
63.(I) Frdm fng.(nn) kba (nnn) nt ns hj`kr tckt snx, fdur, db`
kba tcr`` adts hkbbdt kpp`kr dppdsnt` twd adts.
Xd, fnv` adts kpp`kr dppdsnt` twd adts. Frdm fng.
(n) kba (nnn) nt ns hj`kr tckt fdur, twd kba db` adts
hkbbdt kpp`kr dppdsnt` tcr`` adts. Kjsd, snbh`
fnv` adts kpp`kr dppdsnt` twd adts sd tc`y
hkbbdt kpp`kr dppdsnt` tcr`` adts. Qc`r`fdr`,
snx adts kpp`kr dppdsnt` tcr`` adts.
6>.(H) Frdm tcr` fngur`ss nt ns hj`kr tckt tc` bumi`rs
9,8,; kba 6 hkbbdt kpp`kr dppdsnt` 1. Xd,0kpp`krs dppdsnt 1. Xnbh` nb `khc db` df tc`
tcr`` anh`, 1 kpp`krs db tc` tdp. Xd, 0 kpp`krs
kt tc` idttdm fkh` df `khc anh`.
7?.(A) Frdm fng. (nn) kba (nnn) nt ns hj`kr tckt H, A, I kba
F hkbbdt kpp`kr dppdsnt` @. Xd, K kpp`krs
dppdsnt` @. n.`. @ ns tc` kjpcki`t dppdsnt` K.
7;. Frdm fng. (n) kba (nnn), nt ns hj`kr tckt adt, trnkbgj`,
hrdss kba krrdw hkbbdt kpp`kr dppdsnt` tc`
hnrhj`. Xd, tc` squkr` jn`s dppdsnt` tc` hnrhj`.
Frdm fng. (nn) kba (nnn), nt ns hj`kr tckt trnkbgj`,
squkr`, krrrdw kba hnrhj` hkbbdt kpp`kr dppdsnt`tc` hrdss. Xd, tc` adt jn`s dppdsnt` tc` hrdss.
Divndusjy, tc` trnkbgj` jn`s dppdsnt` tc` krrdw.
(n) (A) Ks anshuss`a `krjn`r, tc` hrdss jn`s dppdsnt tc`
adt.
(nn) (I) Ks anshuss`a `krjn`r, tc` trnkbgj` jn`s dppdsnt`
tc` krrdw.
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M@BQKJ KINJNQ^
(nnn)(H) Xnbh` tc` squkr` jn s kt tc` tdp df fng. (nn) kba tc`
hnrhj` jn`s dppdsnt` tc` squkr`, sd tc` hnrhj` jn`s
kt tc` idttdm df fng. (nn)
79. Frdm fng. (^) & (U), nt ns hj kr tckt tc bumi`rs 1,
8, 9 kba 0 hkbbdt kpp`kr dppdsnt 6. Xd, ; jn s
dppdsnt` 6. Frdm fng. (S) & (^), nt ns hj`kr tckt tc`bumi`rs 0, ;, 6 kba 8 hkbbdt kpp`kr dppdsnt` 1.Xd,
9 jn`s dppdsnt` 1. Divndusjy, 8 jn`s dppdsnt` 0.
(n) (H) 5 Xnbh` 0 jn`s kt tc` tdp dr anh` (S) kba 8 jn`s
dppdsnt` 0, Xd 8 jn`s kt tc` idttdm fkh` df anh`
(S).
(nn) (H) 5 Xnbh` 1 jn`s kt tc` tdp df anh` (^) kba 9 jn`s
dppdsnt` 1, sd 9 jn`s kt tc` idttdm fkh` df anh`
( ).
(nnn) (K) 5 Ks anshuss`a `krjn`r, ; jn`s dppdsnt` 6.
(nv) (I) 5 Qc` bumi`rs wcnhc kr` cnaa`b i`cnba tc`bumi`rs 6 kba 0 nb anh` (U) kr` tc` bumi`rs
jynbg dppdsnt` 0 kba 6 n.`. 8 kba ;.
(v) (A) 5 Nb fng. (S), tc` cnaa`b bumi`rs kaekh`bt td 0 kr`
tcds` jynbg dppdsnt` ; kba 1. Qc`s` kr` 6 kba 9.
Nb fng. (U), tc` cnaa`b bumi`rs kaekh`bt td 0 kr`
tcds` jynbg dppdsnt` 9 kba 6. Qc`s` kr` 1 kba ;.
Hj`krjy, tc`r` ns bd bumi`r hdmmdb td {6, 9 }
kba {1, ;}.
78.(I) 5 Frdm tc` twd fngur`s nt ns hj`kr tckt fdur, fnv , twd
kba snx adts hkbbdt kpp`kr dppdsnt` tcr``
a d ts. Xd , d b ` a d t k p p` k rs d p pd snt ` t cr` `
adts.Qc`r`fdr`, nf db` adt ns kt tc` idttdm, tc`b
tcr`` adts wnjj jn` db tc` tdp.
71.(A) 5 Frdm tc` twd fngur`s nt ns hj`kr tckt tcr``, fdur,
db` kba fnv` adts hkbbdt kpp`kr dppdsnt` twd
adts. Xd, snx adts kpp`kr dppdsnt` twd adts. Nb
tc`s j`ft ckba sna` anh`, tc`r` kr` twd adts db
tc` tdp< sd, tc`r` scduja i` snx adts kt tc`
idttdm fkh`.
70.(K)5 Frdm tc` twd fngur`s nt ns hj`kr tckt twd, snx kba
fdur adts hkbbdt kpp`kr dppdsnt` tcr`` adts. Xd,
`ntc`r db` dr fnv` adts hkb kpp`kr dppdsnt` tcr``
adts. Qc`r`fdr`, nf tc` anh` ns r`stnbg db tc` sna`
wntc tcr`` adts, tc`b `ntc`r db` dr fnv` adts wnjj
kpp`kr db tc` tdp.