PROFIL_NAYa_MATEMATIKA_-_variant_3__REShENIYa_Statgrad

Формат документа: pdf
Размер документа: 0.24 Мб





Прямая ссылка будет доступна
примерно через: 45 сек.



  • Сообщить о нарушении / Abuse
    Все документы на сайте взяты из открытых источников, которые размещаются пользователями. Приносим свои глубочайшие извинения, если Ваш документ был опубликован без Вашего на то согласия.

‹KlZl=jZ^ímq]
Lj_gbjh\hqgZyjZ[hlZihF:L?F:LBD?deZkk
k_gly[ jy]h^ Z
<Zjb ZglF:
ijhnbevgucmjh\_gv
<uiheg_gZN BH
_________________________________
deZkk
______
Bgkljmdpbyihuiheg_gbxjZ[hlu
GZ \uiheg_gb_ jZ[hlu ih fZl_fZlbd_ hl\h^blky  qZkZ  fbgml
(235 fbg ml JZ[ hlZkhkl hblba^\ moqZkl_c\dexqZxsb o\k_[yaZ^ Zgbc
QZklv  kh^_j`bl  aZ^Zgbc [Zah\h]h mjh\gy keh`ghklb k djZldbf
hl\_lhf Q Zklv  kh^_ j`bl  aZ^ Zgby ih\ ur_ggh]h mjh\gy keh`ghkl b
k djZldbf hl\_lhfb  aZ^Zgbcih\ur_ggh]hb \ukhdh]h mjh\g_ckeh`ghklb
kjZazjgmlufhl_lhf Hl\_lu d aZ^Zgbyf  – aZib ku\ Zxlky \ \b^_ p_eh]h qbkeZ beb
dhg_qghc^_kylbqghc^jh[b
Ijb\uiheg_gbbaZ^Zgbc –lj_[m_lky aZibkZlvihegh_ j_r_gb_gZ
hl^_evghfebkl_[ mfZ]b Ijb \uiheg_gbb aZ^Zgbc fh`gh ihevah\Zlvky q_jgh\bdhf AZibkb
\ q_jgh\bd_g_mqblu\Zxlkyijbhp_gb\ZgbbjZ[hlu ;Zeeuihemq_ggu_<ZfbaZ\uiheg_ggu_aZ^Zgbykmffbjmxlky
IhklZjZcl_kv \uihegblv dZd fh`gh [hevr_ aZ^Zgbc b gZ[jZlv
gZb[hevr__dhebq_ kl\h[ Zeeh\
@_eZ_fmki_oZ
FZl_fZlbdZdeZkk<ZjbZglF: 2
‹KlZl=jZ^ímq]
QZklv
Hl\_lhf d dZ`^hfm ba aZ^Zgbc – yey_lky dhg_qgZy ^_kylbqgZy ^jh[v p_eh_ qbkeh beb ihke_^hZl_evghklv pbnj AZibrbl_ hl_lu  d aZ^Zgbyf\ihe_hl\_lZ\l_dkl_jZ[hlu H^gh] h jmehg Z h[ h_\ o\ Zl Z_l ^ey hde_cdb ihehku hl iheZ ^h ihlhedZ rbjbghc  f Kdhevdh jmehgh\ h[ h_\ g m`gh dmiblv ^ey hde_cdb ijyfhm]hevghcdhfgZlujZaf_jZfbfgZf"
Hl\_lBBBBBBBBBBBBBBBBBBBBBBBBBBB
GZ jbkmg d_ `bjgufb lhqdZfb ihdZaZgh kmlhqgh_ dhebq_ kl\h hkZ^dh\ 
\uiZ^Z\rbo \ Ydmlkd_ k  ih  hdly[jy  ]h^Z Ih ]hjbahglZeb
md Zau\ Zxlky  qbkeZ f_ kyp Z i h \_jlbd Zeb —dhebq_kl\h hkZ^dh\ uiZrbo
\khhl\_lkl\mxsbc^_gv\ fbeebf_ljZo>eygZ]ey^ghklb`bjgu_l hqdb g Z
jbkmgd_ kh_^bg_gu ebgb_c Hij_^_ebl_ ih jbkmgdm dZdh_ fZdkbfZevgh_
dhebq_ kl\h hkZ^dh\ \ui Z^Zeh \ i_jbh^ k  i h  hdly[jy Hl\_l ^Zcl_
fbeebf_ljZo Hl\_lBBBBBBBBBBBBBBBBBBBBBBBBBBB
GZde_lqZlhc[mfZ]_kjZaf_jhfde_ldb
11 u
bah[jZ`zg
lj_m]hevgbd
ABC
 GZc^bl_ ^ebgm _]h f_^bZgu ijh\_^zgghc
ba\_jrbgu
C.
Hl\_lBBBBBBBBB__________________.
1
2 3
A
B
C

FZl_fZlbdZdeZkk<ZjbZglF:3
‹KlZl=jZ^ímq]
:\lhf Zlbq_kd Zy ebgby ba]hl Z\eb\Z_l [ Zl Zj_cdb <_jhylghklv l h]hqlh
]hlh\Zy[ZlZj_cdZg_bkijZ\gZjZ\gZI_j_^miZdh\dhcdZ`^Z y[ Zl Zj_cdZ
ijhoh^bl kbkl_fm dhgljhey <_jhylghklv lh]h qlh kbkl_fZ aZ[jZd m_l
g_bkijZ\gmx [ZlZj_cdm jZ\gZ  <_jhylghklv lh]h qlh kbkl_f Z ih
hrb[d_ aZ[jZdm_l bkijZ\gmx [ZlZj_cdm jZ\gZ  GZc^bl_ \_jhy lghklv
lh]h qlh kemqZcgh \u[jZggZy ba]hlh\e_ggZy [ZlZj_cdZ [m^_l aZ[j Zdh\ Zg Z
kb kl_fhcd hgljhey 
Hl\_lBBBBBBBBBBBBBBBBBBBBBBBBBBB
J_rbl_mjZ\g_gb_
35 12 xx

? kebmjZ\g_gb_bf__l[he__h^gh]hdhjgy
\hl\_l_aZibrbl_f_gvrbcbadhjg_c
Hl\_lBBBBBBBBBBBBBBBBBBBBBBBBBBB
GZc^bl_ ohj^m gZ dhlhjmx hibjZ_lky m]he
30q
,
\ibkZgguc\hdjm`ghklvjZ^bmkZ
Hl\_lBBBBBBBBBBB________________.
4
5 6
FZl_fZlbdZdeZkk<ZjbZglF:
4
‹KlZl=jZ^ímq]
GZ jbkmgd_ bah[jZ`zg ]jZnbd nmgdpbb

yfxc

— ijhba\h^ghc nmgdpbb
fx
 hij_^_ezgghc gZ bgl_j\Ze_

5; 7

 GZc^bl_ lhqdm wdklj_fmfZ
nmgdpbb
fx
ijbgZ^ e_` Zsmxhlj_adm
>@1; 4

.
Hl\_lBBBBBBBBBBBBBBBBBBBBBBBBBBB
GZc^bl_ iehsZ^v ih\_joghklb fgh]h]jZggbdZ
bah[jZ`zggh]h gZ jbkmgd_ \k_ ^\m]jZggu_ m]eu
ijyfu_ 
Hl\_lBBBBBBBBBBBBBBBBBBBBBBBBBBB
7
8
4
2
2
2
3

FZl_fZlbdZdeZkk<ZjbZglF:5
‹KlZl=jZ^ímq]
QZklv
GZc^bl_agZq_gb_\ujZ`_gby
63
48
2
12
17 7
4
mm
m
_keb
2, 9
m
.
Hl\_lBBBBBBBBBBBBBBBBBBBBBBBBBBB
<h^heZaguc dhehdhe kh^_ j`Zsbc
#
fhey \ha^moZ ijb ^Z\e_gbb
1
2, 4
p
Zlfhkn_ju f_^e_ggh hi mkdZxl gZ ^gh \h^ hzfZ Ijb wl hf
ijhbkoh^bl bahl_jfbq_kdh_ k`Zlb_ \ha^moZ ^h dhg_qgh]h ^Z\e_gby
2p.
JZ[hlZ kh\_jrZ_fZy \h^hc ijb k`Zlbb \ha^moZ hij_^_ey_lky \ujZ`_gb_f
2
2
1
.# ORJ p
AT
p

 ]^_
>`
.
fhev D

˜
—ihklhyggZy
300
T
D —
l_fi_jZl mjZ \ ha^ moZ GZc^bl_ d Zdh_ ^ Z\e_gb_
2p
Zlf [m^_lbf_lvha^mo
\dhehd he_ _ kebijb k` Zlbb\ha^ moZ[ueZ kh\_jr_g Z jZ[hl Z\  200>`
Hl\_lBBBBBBBBBBBBBBBBBBBBBBBBBBB
Ba i mgdlZ : \ i mgdl < jZkklhygb_ f_`^m dhlhjufb  df h^gh\j_ f_ggh
\u_oZeb Z\lhfh[beb kl b \_ehkbi_^bkl Ba\_klgh qlh \ qZk Z\lhfh [bebkl
ijh_a`Z_lgZdf[hevr_q_f\_ehkbi_^bklHij_^_ebl_kdhjhkl v
\_ehkbi_^bklZ  _ keb b a\_klgh qlh hg ijb[ue \ i mgdl <  g Z  q ZkZ fbgml
iha`_ Z\lhfh[beb kl Z Hl\_l^Zcl_\d fq
Hl\_l ___________________________.
GZc^bl_ gZbf_gvr__ agZq_gb_ nmgdpbb
2 41
41
x
yx e


gZ hlj_ad_
>@39, 5; 47
.
Hl\_lBBBBBBBBBBBBBBBBBBBBBBBBBBB
9
10 11 12
FZl_fZlbdZdeZkk<ZjbZglF:
6
‹KlZl=jZ^ímq]
>eyaZibkbj_r_gbcbhl\_lh\gZaZ^Zgby–bkihevamcl_hl^_evguc ebkl AZibrbl_ kgZqZeZ ghf_j \uihegy_fh]h aZ^Zgby   b l ^   Z aZl_f ihegh_ h[hkgh\Zggh_ j_r_gb_ b hl\_l Hl\_lu aZibku\Zcl_ qzldhbjZa[hjqb\h Z J_rbl_mjZ\g_gb_
2
5Œ
14cos 3cos2
12
xx

.
[ GZc^bl_\k_dhjgbwlh]hmjZ\g_gbyijbgZ^e_`Zsb_hlj_adm
9 Œ Œ
2
ªº

«»
¬¼
.
<hkgh\ZgbbijZ\bevghcq_lujzom]hevghcibjZfb^u
MABCD
e_`bld\Z^jZl
ABCD
kh klhjhghc  Ijhlb\hiheh`gu_ [hdh\u_ jz[jZ ibjZfb^u ihiZjgh
i_ji_g^bdmeyjgu Q_j_a k_j_^bgu jz[_j
MA
bMB
ijh\_^_gZ iehkdhklv
.,
iZjZee_ evg Zyj_[jm
M K
.
Z >hdZ`bl_qlhk_q_gb_iehkdhklvx
.
ibjZfb^u
MABC
y\ey_l ky
iZjZee_eh]jZffhf
[ GZc^bl_iehsZ^vk_q_gbyibjZfb^u
MABC
iehkdhklvx
..
J_rbl_g_jZ\_gkl\h
2
32
30 8
51
8
xx
xx
x
 d

.
>\_hdjm`ghklbdZkZxlky\g_rgbfh[jZahf\lhqd_
K Ijyf Zy  d ZkZ_l ky
f_gvr_c hdjm`ghklb \ lhqd_
A Z [hevr_c  — lhqd_
B hlebqghc hl 
A.
IjyfZy
A K
\lhjbqgh i_j_k_dZ_l [yevrmx hdjm`g hkl v \ lhqd_
D ijyfZy
BK
\lhjbqghi_j_ k_dZ_lf_gvrmxhdjm`g hklv\lhqd_
E.
Z >hdZ`bl_qlhijyfZy
AE
iZjZee_ evg Zijyfhc
BD
.
[  Imklv
L
— hl ebqgZy hl
D
lhqdZ i_j_k_q_gby hlj_adZ
DE
k [ hev r_c
hdjm`ghklvxGZc^bl_
EL
_kebjZ^bmkuhdjm`ghkl_cjZ\gub
13
14 15 16

FZl_fZlbdZdeZkk<ZjbZglF:7
‹KlZl=jZ^ímq]
Ih [bag_k-ieZgmij_^iheZ]Z_lky\eh`blv\q_lujzoe_lgbcijh_dl p_eh_
qbkeh feg jm[e_c Ih blh]Zf dZ`^h]h ]h^Z ieZgbjm_lky ijbjhkl kj _^kl\
\deZ^qbd Z g Z    ih kjZ\g_gbx k g Zq Zehf  ]h^ Z  GZqbke_ggu_ i jhp_ glu
hklZxlky \eh`_ggufb \ ijh_dl Djhf_ wlh]h kjZam ihke_ gZqbke_gb c
ijhp_glh\ gm`gu ^hihegbl_evgu_ \eh`_gby ih  feg jm[e_c \ i_j \uc b
\lhjhc ]h^u Z lZd`_ ih  feg \ lj_lbc b q_l\zjluc ]h^u GZc^b l_
gZbf_gvrbc jZaf_j i_j\hgZqZevguo \eh`_gbc ijb dhlhjhf h[sZy km ffZ
kj_^kl\\deZ^qbdZdgZqZemlj_lv_]h]h^ZklZg_l[hevr_fegZddhgpm
ijh_dl Z —[hevr_fegjm[e_c
GZc^bl_\k_agZq_gby
aijbdhl hjuomjZ\g_gb_
2
242
22()
xxa x xa   
bf__l_^bgkl\_ggh_j_r_gb_gZhlj_ad_
>@1; 1

.
Imklv
()
Kn
h[hagZqZ_lkmffmd\Z^jZlh\\k_opbnjgZlmjZevgh]hqbkeZ n.
Z Kms_ kl\ m_lebl Zdh_ ljz oag Zqgh_qbkeh nqlh
( ) 179
Kn
?
[ Kms_kl\m_leblZdh_ljzoagZqgh_qbkeh nqlh
( ) 184
Kn
?
\ DZdh_gZbf_gvr__agZq_gb_fh`_lijbgbfZlv\ujZ`_gb_
() 2
Kn n 
,_keb
n — ljzoagZqgh_qbkeh"
17
18 19

От_l ы на тренироhqgu_\ZjbZglu1 09 - 10 112 (профильный уро_gv hl0 .09 .2018

1 2 3 4 5 6 7 8 9 10 11 12
101 09 397 3 3 10 0,017 1,25 13 2 1 0 4 1 8 90 14 0
101 10 125 27 98 0,032 3 ,5 9 3 8 7 8 30 1 2 0
101 11 1 2 0,5 4 0,0684 5 37 3 48 1, 5 9,6 1 0 0
101 12 9 4 5 0,0673 5 25 7 58 4,5 5 20 0

FZl_fZlbdZdeZkk<ZjbZglF:1
‹KlZl=jZ^ímq]
Djbl_jbbhp_gbZgbyaZ^ZgbckjZazjgmlufhl_lhf
Z J_rbl_mjZ\g_gb_
2
5 Œ
14cos 3cos2
12
xx

.
[ GZc^bl_\k_dhjgbwlh]hmjZ\g_gbyijbgZ^e_`Zsb_hlj_adm
9 Œ Œ
2
ªº

«»
¬¼
.
J_r_gb_
Z A Zibr_fbkoh^gh_ mjZ\g_gb_\\b^_
2
5 Œ
14cos 3cos2
12
xx

;
2

14sin 3cos2
12
xx

;
2

21 2sin 1 3cos2
12
xx
§·

¨¸
©¹
;


2 cos 2 1 3 cos 2
6
xx

;
31
2 cos 2 sin 2 1 3 cos 2
22 xx x
§·

¨¸
©¹
;
3cos2 sin2 1 3cos2 xx x
;
sin 2 1 x 
.
Ke_^h\Zl_evgh

22 Œ
2
xk
 
]^_
k� ]
hldm^ Z


4
xk
 
]^_
k� ]
.
[ Kihfhsvxqbkeh\hchdjm`g hklbhl[_jzfdhjgbijbgZ^ e_`Zsb_h lj_adm
9Œ
;3 Œ
2
ªº

«»
¬¼
.
IhemqbfqbkeZ
17Π4

;
13Œ
4

.
4
2
4
13
FZl_fZlbdZdeZkk<ZjbZglF:
2
‹KlZl=jZ^ímq]
Hl_lZ 


4 k

]^_
k � ]
[ 
17 Œ
4

;
13

4

.
Kh^_
j`Zgb_d
jbl_
jby ;Zeeu
H[ hkg h\Zgghi hemq_gu\_jgu_hl\_lu\ h[ hboi mgdl Zo 2 H[ hkg h\Zgghi hemq_g\_jguchl\_l\i mgdl_Zbeb\imgdl_ [.
BE B
Ihemq_gg_\_jguchl\_lba -aZ\uqbkebl_evghchrb[dbghijbwlhf
bf__lky\_jgZyihke_^h\Zl_evghklv\k_orZ]h\j_r_gby 1 J_r_gb_ g_ khhl\_lkl\m_l gb h^ghfm ba djbl_jb_\ i_j_qbke_gguo
ur_
0
FZdkbfZevguc[Zee
2
<hkgh\ZgbbijZ\bevghcq_lujzom]hevghcibjZfb^u
MABCD
e_`bld\Z^jZl
ABCD
kh klhjhghc  Ijhlb\hiheh`g u_ [hdh\ u_ jz[ jZ ibjZfb^u ihiZjgh
i_ji_g^bdmeyjgu Q_j_a k_j_^bgu jz[_j
MA
bMB
ijh\_^_gZ iehkdhklv
.,
iZjZee_ evg Zyj_[jm
M K
.
Z >hdZ`bl_qlhk_q_gb_iehkdhklvx
.
ibjZfb^u
MABC
y\ey_l ky
iZjZee_eh]jZffhf
[ GZc^bl_iehsZ^vk_q_gbyibjZfb^u
MABC
iehkdhklvx
..
J_r_gb_
Z  Imklv lhqdZ
Q
— k_ j_^bgZ j_[ jZ
MA
 Z l hqdZ
K
—k_j_^bgZ j_[jZ
MB
.
Iehkdhklv
.
i_j_k_dZ_l iehkdhklv
BMC
ih hlj_adm
KL
 iZjZee_evghfm
j_[jm
MC
 Ke_^h\Zl_evgh iehkdhklv
.
i_j_k_ dZ_l  iehkdhklv 
AMC
ih
ijyfhc i ZjZee_ evghc j_[jm
MC
 GZ wlhc ijyfhc e_`bl kj_^gyy ebgby
lj_m]hevgbdZ
AMC
 ihwlhfm iehkdhklv
.
ijhoh^bl q_j_a lhqdm
O

k_j_^bg m hl j_ad Z
AC
 LZdbf h[jZahf k_q_gb_ —q_lujzom]hevgbd
QKLO
,
\ dhlhjhf klhjhgu
KL
b
QO
iZjZee_ evg u hl j_adm
MC
b jZ\gu _] h
iheh\bg_AgZqbl
QKLO
— iZjZee_eh]jZff
[ Hlf_lbflhqdm
F
— k_ j_^bg m hl j_ad Z
QK
b jZkkfhljbf iehkdhklv
MOF
IjyfZy
QK
i_ji_g^bdmeyjg Zijyfuf
FM
bMO
ke_^h\ Zl_evghhg Z
14

FZl_fZlbdZdeZkk<ZjbZglF:3
‹KlZl=jZ^ímq]
i_ji_g^bdmeyjgZ iehkdhklb
MFO
 ihwlhfm hg Z i_ji_g^bdmeyjg Z hlj_ adm
OF
LZdbfh[ jZahfhlj_ahd
OF
kem`bl\ukhlhciZjZee_ eh] jZffZ
QKLO
.
K_q_gb_ ibjZfb^u
MABCD
iehkdhklvx
MOF
— jZ\gh[_^ j_gguc
lj_m]hevgbd
NMG
 Hlj_ahd
OF
y\ey_lky f_^bZghc ijyfhm]hevgh]h
lj_m]hevgbdZ
MOG
ijh\_^zgghcd_]h]bihl_g ma_ihwlhfm
1 2
OF MG

.
Ihmkeh\bxlj_ m] hevgbd
AMC
ijyfhm]hevgucbjZ\gh[_^j_ggucihwlhfm 11 24 4
22
AM AC

,
b lh `_  \_jgh ^ey ^ jm]b o [ hdh\ uo jz[_j Ke_^h\ Zl_evgh \k_  [hdh u_]jZgb
ibjZfb^u —jZ\ghklhjhggb_lj_m]hevgbdbLh]^Z
23
MG
b
3
OF .
IehsZ^vi ZjZee_ eh] jZffZ
QKLO
jZ\gZ
2323
OL OF
.
Hl_l
23.
Kh^_
j`Zgb_d
jbl_
jby ;Zeeu
Bf__lky \_jgh_ ^ hdZaZl_evkl\h ml\_ j`^_gby i mgdlZ Z b
h[hkgh\Zgghihemq_g\_jguchl\_l\imgdl_ [2 <_jgh^ hdZaZgi mgdl
Z
BE B
<_jghj_rzgi mgdl [ijbhlkmlkl\bbh[hkgh\Zgbc\imgdl_ Z1 J_r_gb_ g_ khhl\_lkl\m_l gb h^ghfm ba djbl_jb_\ i_j_qbk
-
e_gguo\ ur_ 0
FZdkbfZevguc[Zee
2
J_rbl_g_jZ\_gkl\h
2
32
30 8
51
8
xx
xx
x
 d

.
J_r_gb_
AZibr_fb koh^gh_g_jZ\_gkl\h\\b^_


32 2
5 8 30 8
1
8
xxx xx
x

d

;
433 2 2
5 8 40 30 8 8 0
8
xxx x xx x
x
  
d

;
43 2
310 0
8
xx x
x

d

;
2
52
0
8
xx x
x

d

,
hldm^ Z
2
x d,
0
x b58 x
d
.
Hl_l

@
;2
f 
; 0;
>

5; 8 .
15
FZl_fZlbdZdeZkk<ZjbZglF: 4
‹KlZl=jZ^ímq]
Kh^_
j`Zgb_d
jbl_
jby;Zeeu
H[ hkg h\Zgghi hemq_g\_jguchl\_l 2 J_r_gb_kh^_j`bl\uqbkebl_evgmxhrb[dm\hafh`ghijb\_^rmx
dg_\_jghfmhl\_l mghi jbwl hfbf__lky\_ jgZyihke_^h\ Zl_evghklv
\k_orZ]h\j_r_gby 1 J_r_gb_ g_ khhl\_lkl\m_l gb h^ghfm ba djbl_jb_\ i_j_qbke_gguo
ur_
0
FZdkbfZevguc[Zee
2
>\_hdjm`ghklbdZkZxlky\g_rgbfh[jZahf\lhqd_
K Ijyf Zy  d ZkZ_l ky
f_gvr_c hdjm`ghklb \ lhqd_
A Z [hevr_c  — lhqd_
B hlebqghc hl
A.
IjyfZy
A K
\lhjbqgh i_j_k_dZ_l [yevrmx hdjm`g hkl v \ lhqd_
D ijyfZy
BK
\lhjbqghi_j_ k_dZ_lf_gvrmxhdjm`g hklv\lhqd_
E.
Z >hdZ`bl_qlhijyfZy
AE
iZjZee_ evg Zijyfhc
BD
.
[  Imklv
L
— hl ebqgZy hl
D
lhqdZ i_j_k_q_gby hlj_adZ
DE
k [ hev r_c
hdjm`ghklvxGZc^bl_
EL
_kebjZ^bmkuhdjm`ghkl_cjZ\gub
J_r_gb_
Z  Imklv h[sZy dZkZl_evgZy d ^Zgguf hdjm`ghklyf ijh\_^zggZy q_ j_ a
lhqdm
C i_j_k_dZ_l h[smx dZkZl_evgmx
AB
 lhqd_
M
 L h]^Z
MA MC MB
 lh _klv f_^bZg Z
CM
lj_m]hevgbdZ
ABC
jZ\gZ iheh\bg_
klhjhg u
AB
 AgZqbl
90
ACB
q
 Lh]^Z
90
ACE
q
 ihwl hfm
AE

^bZf_lj f_gvr_c hdjm`ghklb Ke_^ h\Zl_ evgh ijyfZy
AE
i_ji_g^bdmeyjgZ
ijyfhc
AB
:gZeh]bqgh^hdZ`_fqlhijyfZy
BD
i_ji_g^bdmeyjgZ ijyfhc 
AB
 Ijyfu_
AE
b
BD
i_ji_g^bdmeyjgu h^ghc b lhc `_ ijyfhc
AB
,
agZqblhgbiZjZee_evgu
[  Imkl v jZ^b mku hdjm`ghkl_c jZ\gu
r
b
R ]^_
rR 
 Hi mklbf
i_ji_g^bdmeyj
OH
ba p_gljZ
O
f_gvr_c hdjm`ghklb gZ ^bZf_lj
BD
[hevr_cLh]^ Z
16

FZl_fZlbdZdeZkk<ZjbZglF:5
‹KlZl=jZ^ímq]
22
2
4
OH R r R r rR  
.
Hi mklbfi_ji_g^bd me yj
EF
balhqdb
EgZ
BD
Lh]^Z
2
22 2
DE EF DF OH BD BF  
2
22
422 2 294627rR R r R r Rr
  
.
Hlj_ ahd
AC
— \ukhl Z i jyfhm]hevgh] h lj_ m]hevgbdZ
ABE
 ijh\_^zggZy ba
\_jrbgu ijyfh]h m] eZ Z
EB
b
ED
— k_ dmsb_ d [ hevr_c hdjm`ghklb
ijh\_^zggu_bah^ghclhqdbihwlhfm
2
EL DE EC EB AE
.
Ke_^h\ Zl_evgh
2
16 8 7 7
27
EC EB AE
EL
ED ED˜

.
Hl_l
87 .
7
Kh^_
j`Zgb_d
jbl_
jby ;Zeeu
Bf__lky \_jgh_ ^hdZaZl_evkl\h ml\_j`^_gby i mgdl Z Z b
h[hkgh\Zgghihemq_g\_jguchl\_l\imgdl_ [3 H[ hkg h\Zgghi hemq_g\_jguchl\_l\i mgdl_
[.
BE B
Bf__lky \_ jgh_ ^ hdZaZl_evkl\h ml\_ j`^_gby i mgdl Z Z b ijb
h[hkgh\Zgghf j_r_gbb imgdlZ [ihemq_g g_\_jguc hl\_l ba -aZ
Zjbnf_lbq_kdhchrb[db 2 Bf__lky\_jgh_ ^hdZaZl_evkl\hml\_j`^_gbyi mgdl Z
Z.
BE B
Ijbh[hkgh\Zgghfj_r_gbbimgdlZ [ihemq_gg_\_jguchl\_lba -aZ
Zjbnf_lbq_kdhchrb[db
BE B
H[hkgh\Zggh ihemq_g \_jguc hl\_l \ imgdl_ [k bkihevah\Zgb_f
ml \ _ j` ^ _ g b yi mg dl Z  Zijbwl hfi mgdl Zg_\uiheg_g 1 J_r_gb_ g_ khhl\_lkl\m_l gb h^ghfm ba djbl_jb_\ i_j_qbke_gguo
ur_
0
FZdkbfZevguc[Zee
3
FZl_fZlbdZdeZkk<ZjbZglF: 6
‹KlZl=jZ^ímq]
Ih [bag_k-ieZgmij_^iheZ]Z_lky\eh`blv\q_lujzoe_lgbcijh_dl p_eh_
qbkeh feg jm[e_c Ih blh]Zf dZ`^h]h ]h^Z ieZgbjm_lky ijbjhkl kj _^kl\
\deZ^qbdZ gZ   ih kjZ\g_gbx k gZqZehf ]h^Z GZqbke_ggu_ ijhp_glu
hklZxlky \eh`_ggufb \ ijh_dl Djhf_ wlh]h kjZam ihke_ gZqbke_gbc
ijhp_glh\ gm`gu ^hihegbl_evgu_ \eh`_gby ih  feg jm[e_c \ i_j \uc b
\lhjhc ]h^u Z lZd`_ ih  feg \ lj_lbc b q_l\zjluc ]h^u GZc^b l_
gZbf_gvrbc jZaf_j i_j\hgZqZevguo \eh`_gbc ijb dhlhjhf h[sZy km ffZ
kj_^kl\\deZ^qbdZdgZqZemlj_lv_]h]h^ZklZg_l[hevr_fegZddhgpm
ijh_dl Z —[hevr_fegjm[e_c
J_r_gb_ Imklv
S
feg —i_j\hg ZqZevgu_  \eh`_gby D g Zq Zem 2-]h ]h^Z ihemqblky
1, 3 2 0 S
feg Z d gZq Zem  -]h ]h^Z —

1, 3 1, 3 20 20 1, 69 46SS 
 Ih
mkeh\bx
1, 69 46 190 S!
hldm^ Z
144
85, 2...
1, 6 9
S
!
DgZq Zem -]h]h^Zbf__f

1, 3 1, 6 9 4 6 1 5 S
Z\dhgp_ijh_dl Z—


1, 3 1, 3 1, 69 46 15 15 2, 8561 77, 74 34, 5 2, 8561 112, 24 SSS   
.
Ihmkeh\bx
2, 8561 112, 24 360 S!
hldm^ Z
247, 76
86, 7...
2,8561
S
!
:agZqblfbgbfZevgh_\hafh`gh_p_eh_agZq_gb_
87
S
.
Hl_l fegjm[ e_c
Kh^_
j`Zgb_d
jbl_
jby ;Zeeu
H[ hkg h\Zgghi hemq_g\_jguchl\_l 3 <_jgh ihkljh_gZ fZl_fZlbq_kdZy fh^_ev j_r_gb_ k\_^_gh
d bkke_^h\Zgbx wlhc fh^_eb ihemq_g g_\_jguc hl\_l ba-aZ
uqbkebl_evghchrb[db 2 <_jgh i hkljh_g Z f Zl_f Zlbq_kd Zy fh^_ ev  b j_r_gb_ k\_^_gh
dbkke_^h\Zgbxwlhcfh^_ebijbwlhfj_r_gb_g_aZ\_jr_gh
1 J_r_gb_ g_ khhl\_lkl\ m_l gb h^ghfm b a djbl_jb_\
i_j_qbke_gguo\ur_0
FZdkbfZevguc[Zee
3
17

FZl_fZlbdZdeZkk<ZjbZglF:7
‹KlZl=jZ^ímq]
GZc^bl_\k_agZq_gby
aijbdhl hjuomjZ\g_gb_
2
242
22()
xxa x xa   
bf__l_^bgkl\_ggh_j_r_gb_gZhlj_ad_
>@1; 1

.
J_r_gb_
MjZ\g_gb_

2
242
22()
xxa x xa   
jZ\ghkbevgh mjZ\g_gbyf
42 2 4 2
2( )( ) 2 2( ),
x x xa xa x xa  

2
2
0
xxa 
b
2
0.
xxa 
Iheh`bf
2
()
fx x x a 
 Ihke_^g__ mjZ\g_gb_ bf__l _^bgkl\_gguc
dhj_gv gZ hlj_ ad_
>@1; 1

lh]^Zblhevdhlh]^Zdh]^Z\uiheg_g h^bgbaljzo
kemqZ_\eb[hd\Z^jZlgucljzoqe_g
()
fx
bf__l_^bgkl\_ggucdhj_gvb wlhl
dhj_gv ijbgZ^e_`bl bgl_j\Zem
1; 1

 eb[h
()
fx
bf__l _^bgkl\_gguc
dhj_gv gZ hl j_ad_ 
>@1; 1

 jZ\guc
1
beb  eb[hd\Z^jZlgucljzoqe_g
()
fx
ijbgbfZ_lijb
1
x b
1
x g_gme_\u_agZq_gbyjZaguoagZdh\
JZkkfhljbfi_j\uc kemqZc D\Z^jZlgucljzoqe_g
()
fx
bf__l_^bgkl\_gguc
dhj_gvijbjZ\_gkl\_gmex_]h^bkdjbfbgZglZlh_klvijb
14 0 a

 beb
qlhlh`_kZfh_ ijb
0, 25
a
.
Ijb lZdhf agZq_gbb
amjZ\g_gb_
2
0
xxa  bf__l _^bgkl\_gguc dhj_gv
0, 5
x 
hgijbgZ^e_`blhlj_adm
>@1; 1

.
JZkkfhljbf \lhjhc kemqZc Bf__f (1)
fa 
b
(1) 2
fa 
 AgZqbl
(1) 0
f 
ijb
0
a  Ijb lZdhf agZq_gbb
amjZ\g_gb_
2
0
xxa  bf__l
^Z j_r_gby
0
x
b
1
x 
gZ hlj_ad_
>@1; 1

 :g Zeh]bqgh
(1) 0
f
ijb
2
a  Ijb lZdhf ag Zq_gbb
amjZ\g_gb_
2
0
xxa  bf__l _^bgkl\_ggh_
j_r_gb_
1
x gZhlj_ad_
>@1; 1

.
JZkkfhljbf lj_lbc kemqZc AgZq_gby
(1)
fa 
b
(1) 2
fa 
bf_xl jZagu_
agZdb lh]^Z b lhevdh lh]^Z dh]^Z
(2 ) 0
aa 
 beb qlh lh `_ kZfh_ ijb
20a
 
.
Ke_^h\ Zl_evghmjZ\g_gb_

2
242
22()
xxa x xa   
bf__l
_^bgkl\_ggh_j_r_gb_gZhlj_ad_
>@1; 1

lh]^Zbl hevdhl h]^ Zdh]^ Z
0, 25
a
beb
20 a
d 
.
Hl_l
>

2; 0

; 0,25.
18
FZl_fZlbdZdeZkk<ZjbZglF:
8
‹KlZl=jZ^ímq]
Kh^_
j`Zgb_d
jbl_
jby;Zeeu
H[ hkg h\Zgghi hemq_g\_jguchl\_l 4 Kihfhsvx\_jgh]hjZkkm`^_gbyihemq_gu\k_agZq_gbyaghhl\_l
kh^_j`blebrg__agZq_gb_ 3 Kihfhsvx\_jgh]hjZkkm`^_gbyihemq_gu\k_j_r_gbymjZ\g_gby
2 AZ^ZqZ\_jghk\_^_gZdbkke_^h\Zgbx\hafh`gh]hagZq_gbydhjg_c
mjZ\g_gby1 J_r_gb_g_khhl\_lkl\m_lgbh^ghfmbadjbl_jb_\i_j_qbke_gguo
ur_
0
FZdkbfZevguc[Zee
4
Imklv
()
Kn
h[hagZqZ_lkmffmd\Z^jZlh\\k_opbnjgZlmjZevgh]hqbkeZ n.
Z Kms_ kl\ m_lebl Zdh_ ljz oag Zqgh_qbkeh nqlh
( ) 179
Kn
?
[ Kms_kl\m_leblZdh_ljzoagZqgh_qbkeh nqlh
( ) 184
Kn
?
\ DZdh_gZbf_gvr__agZq_gb_fh`_lijbgbfZlv\ujZ`_gb_
() 2
Kn n 
,_keb
n — ljzoagZqgh_qbkeh"
J_r_gb_
Z LZdh_qbkehkms_kl\m_lGZijbf_j^eyqbkeZ
977
n
bf__f
2229 7 7 179 
.
[ AZf_lbfqlh^eyex[h]hp_eh]hqb keZ kqbkeh
2k
eb[h^_ebl kyg Z  _keb k
qzlgh eb[h ^Zzl ijb ^_e_gbb gZ  hklZlhd  _keb kg_qzlgh AgZqbl kmffZ
d\Z^jZlh\ \k_o pbnj ijhba\hevgh]h ljzoagZqgh]h qbkeZ nfh`_l ^_eblvky
gZlhevdh_kebd\Z^jZldZ`^hcba_]hpbnj^_eblkygZlh_ klv  d h]^Z \ k_
_]hpbnjuqzlguKe_^h\ Zl_evgh_keb
( ) 184 4 46
Kn
lh\ k_pbnjuqbkeZ
n qzlgubeb[h
222
( ) 8 8 8 192
Kn 
eb[h
22 2
( ) 8 8 6 164
Kn d
AgZqbl
bkdhfh]hqbkeZ ng_kms_kl\m_l
 Imklv
100 10
nabc 
]^_a ,b, c — pbnjuLh]^Z
222
( ) 2 ( ) 2(100 10 )
Kn nabc a bc     
222222
( 100) ( 10) ( 1) 100 10 1abc
     
.
GZbf_gvrb_ \hafh`gu_ agZq_gby \ujZ`_gbc
2
( 100) a 
,
2
( 10)b 
b
2
(1)c 
,
]^_ a, b, c — pbnju jZ\gu
291
,
21
b
20
khhl\_lkl\_gghb^hklb]Zxlkyijb
9
a ,
9
b b
1
c AgZqbl
22 2 2 22
( ) 2 91 1 0 100 10 1 1819
Kn n t    
.
19

FZl_fZlbdZdeZkk<ZjbZglF:9
‹KlZl=jZ^ímq]
Ijb
991
n
bf__f
( ) 2 163 2 991 1819
Kn n   
 Ke_^ h\Zl_ evgh
gZbf_gvr__ agZq_gb_dhlhjh_fh`_lijbgbfZlv\ujZ`_gb_
() 2
Kn n 
 _keb
n ljzoagZqgh_qbkehjZ\gh
1819

.
Hl_l Z > Z[ g_l\ 
1819

.
Kh^_ j`Zgb_d
jbl_
jby ;Zeeu
Ihemq_gu\_jgu_h[hkgh\Zggu_hl\_lu\imgdlZo Z,[ b \ 4 Ihemq_gu \_jgu_ h[hkgh\Zggu_ hl\_lu \ imgdlZo Zb [ eb[h
ihemq_gu\_jgu_h[hkgh\Zggu_hl\_lu\imgdlZo Zb \ 3 Ihemq_g \_jguc h[hkgh\Zgguc hl\_l \ imgdl_
[ imgdlu Zb \g_
j_r_gu eb[h ihemq_g \_jguc h[hkgh\Zgguc hl\_l \ imgdl_ \,
imgdlu Zb [g_j_r_gu 2 Ijb\_^zgijbf_j\imgdl_
Zi mgdlu [b \g_j_r_gu 1 J_r_gb_ g_ khhl\_lkl\m_l gb h^ghfm ba djbl_jb_\ i_j_qbke_gguo
ur_ 0
FZdkbfZevguc[Zee
4