Matematica - M1 - Subiectul I - Variante 001-100

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FTHBFCVL JF IHGHLHVYFHZ ‗ 9004 Pyoić wgykwć lh BHZFBHZKGĉ " Pyoih J

Mklkfyh zfoyfzkgć, pyomklvl yfhl, wpfgkhlkqhyfh bhzfbhzkgć"kcmoybhzkgć# Mklkfyh }oghŧkochlć, pyomklvl bklkzhy, wpfgkhlkqhyfh bhzfbhzkgć"kcmoybhzkgć# • Zohzf wvikfgzflf wvcz oilknhzoykk# Zkbpvl fmfgzk} jf lvgyv fwzf jf 7 oyf# Wf hgoyjć 80 pvcgzf jkc omkgkv# • Lh zohzf wvikfgzflf wf gfy yfqol} ćyk gobplfzf#

8# 6p 6p 6p 6p

WVIKFGZVL K $70p+ ‗ ]hykhczh 008 8# Wć wf jfzfybkcf cvbćyvl chzvyhl c jkc fnhlkzhzfh 8 % 6 % ; % ### % c 2 978 # 9# Wć wf yfqol}f kcfgvhŧkh 9 t 9 ∙ 6 t % 7 ≪ 0 # 7# Wć wf jfzfybkcf kc}fywh mvcgŧkfk ikafgzk}f m1 $0, ∓+ ↙ $8, ∓+, m$ t+ 2 t9 % 8 # :# Wf gocwkjfyć bvlŧkbfh H 2 u8,9,7,###,80r # Wć wf jfzfybkcf cvbćyvl wvibvlŧkbkloy gv zyfk flfbfczf hlf bvlŧkbkk H#

6p 6# Wć wf jfzfybkcf

jkczyf yf pvcg pvcgzf zflf lf H$9, $9, b + şk I $b, ∙9+ wć mkf :# b ∄  , hwzmfl ãcgéz jkwzhcŧh jkcz

6p =# Wć wf ghlgvlfqf gow 97π # 89





























Mklkfyh zfoyfzkgć, pyomklvl yfhl, wpfgkhlkqhyfh bhzfbhzkgć"kcmoybhzkgć# Mklkfyh }oghŧkochlć, pyomklvl bklkzhy, wpfgkhlkqhyfh bhzfbhzkgć"kcmoybhzkgć# • Zohzf wvikfgzflf wvcz oilknhzoykk# Zkbpvl fmfgzk} jf lvgyv fwzf jf 7 oyf# Wf hgoyjć 80 pvcgzf jkc omkgkv# • Lh zohzf wvikfgzflf wf gfy yfqol} ćyk gobplfzf#

9

WVIKFGZVL K $70p+ ‗ ]hykhczh 009 9:

8∙ k   #  9

6p 8# Wć wf ghlgvlfqf  6p 9# Wć wf yfqol}f ãc

7 t ∙ 8

 fgvhŧkh

t %8

2 7# t % 8 9 t ∙ 8 7# Wć wf jfzfybkcf kc}fywh mvcgŧkfk ik ikafgzk}f m 1  ↙  , m $ t+ 2 9 t ∙ 8 # %

6p 6p :# Wć wf jfzfybkcf pyoihiklkzhzfh gh, hlfnécj vc cvbćy

hi jkc bvlŧkbfh cvbfyfloy jf jovć gkmyf, wć h}fb h ≠i# 6# Wć wf ghlgvlfqf lvcnkbfh bfjkhcfk jkc H h zykvcndkvlvk HIG , vcjf H$∙9, ∙8+, I$9, 0+ 0+, G$0, =+ = +#

6p 6p =# Mkf }fgzoykk

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v 2 bk % 7 a şk } 2 $ b ∙ 9 + k ∙ a # Wć wf jfzfybkcf b 3 0 hwzmfl ãcgéz v ⊦ } #





























Mklkfyh zfoyfzkgć, pyomklvl yfhl, wpfgkhlkqhyfh bhzfbhzkgć " kcmoybhzkgć# Mklkfyh }oghŧkochlć, pyomklvl bklkzhy, wpfgkhlkqhyfh bhzfbhzkgć " kcmoybhzkgć# • Zohzf wvikfgzflf wvcz oilknhzoykk# Zkbpvl fmfgzk} jf lvgyv fwzf jf 7 oyf# Wf hgoyjć 80 pvcgzf jkc omkgkv# • Lh zohzf wvikfgzflf wf gfy yfqol} ćyk gobplfzf#

7

WVIKFGZVL K $70p+ ‗ ]hykhczh 007 6p 8# Wć wf oyjocfqf gyfwgć gyfwgćzoy cvbfyflf 9 , 7 : , : 6 # 6p 9# Wf gocwkjfyć mvcgŧkh m 1 Y ↙ Y , m $ t+ 2 : t9 ∙ 4 t % 8 # Wć wf jfzfybkcf }hlohyfh bkckbć h mvcgŧkfk m # 6p 7# Wć wf yfqol}f ãc bvlŧkbfh cvbfyfloy yfhlf fgvhŧkh ln$ t ∙ 8+ % ln$= t ∙ 6+ 2 9 # 6p :# Wć wf jfzfybkcf pyoihiklkzhzfh gh, hlfnécj vc cvbćy jkc bvlŧkbfh cvbfyfloy chzvyhlf jf jovć gkmyf, 6p

hgfwzh wh mkf pćzyhz pfymfgz# jyfpzfk fk ghyf ghyf zyfgf zyfgf pykc pykc pvcgz pvcgzvl vl H $=, $=, :+ şk fwzf pfypfcjkgvlhyć pf jyfhpzh 6# Wć wf jfzfybkcf fgvhŧkh jyfpz j 1 9t ∙ 7{ % 8 2 0 #

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6p =# œzkkcj gć wkc θ 2 , wć wf ghlgvlfqf gow9θ # 7






























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WVIKFGZVL K $70p+ ‗ ]hykhczh 00:

8 6p 8# Wć wf ghlgvlfqf 

9

8  ∙  # 8∙ k 8% k 

6p 9# Wć wf hyhzf gć }éymvl phyhiolfk 6p 7# Wć wf yfqol}f ãc 6p

%

7 t %8 %

6 t% 8 fwzf wkzvhz ãc ghjyhcvl KKK# 4

20# ; :# Wć wf jfzfybkcf pyoihiklkzhzfh gh, hlfnécj vc cvb ćy jkc bvlŧkbfh cvbfyfloy chzvyhlf jf zyfk gkmyf, hgfwzh wć hkić fthgz jovć gkmyf fnhlf#

6p 6# Wć wf jfzfybkcf 6p

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h ∄  pfczyv ghyf }fgzoykk v 2 $7 ∙ 9 h+k % $ h % 8+ a şk } 2 $9h % 8+ k % 9 a wvcz

pfypfcjkgvlhyk# =# Wć wf ghlgvlfqf lvcnkbfh lhzvykk IG h zykvcndkvlvk hwgvŧkzvcndkg HIG şzkkcj gć HI 2 = , HG 2 80 şk gć hykh zykvcndkvlvk zykvcndkvlvk HIG fwzf fnhlć gv 86 7 #






























WVIKFGZVL K $70p+ ‗ ]hykhczh 006 6p 8# Wć wf ghlgvlfqf 6p 6p 6p 6p

8 8 % 9k

%

8 8 ∙ 9k

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9# Wć wf yfqol}f ãc  kcfgvhŧkh t 9 ∙ 80 t % 89 ≪ 0 # 7# Wć wf jfzfybkcf kc}fywh mvcgŧkfk ikafgzk}f m 1 $8, ∓ + ↙ $ 0, ∓ + , m $ t+ 2 t ∙ 8 # :# Wć wf jfzfybkcf cvbćyvl mvcgŧkkloy m 1 u8, 9, 7, :r ↙ u8, 9, 7, :r gv pyopykfzhzfh gć m $8+ 2 m $:+ # 6# Wć wf jfzfybkcf gooyjochzflf }éymvlvk J hl phyhlflonyhbvlvk HIGJ jhgć H$∙9, ;+, I$>, ∙ :+, G$4, ∙ 7+ #

6p =# Zykvcndkvl HIG hyf I 2 π şk lvcnkbfh yhqfk gfygvlvk gkygvbwgykw fnhlć gv 8# Wć wf ghlgvlfqf 7

lvcnkbfh lhzvykk HG #































Mklkfyh zfoyfzkgć, pyomklvl yfhl, wpfgkhlkqhyfh bhzfbhzkgć " kcmoybhzkgć# Mklkfyh }oghŧkochlć, pyomklvl bklkzhy, wpfgkhlkqhyfh bhzfbhzkgć " kcmoybhzkgć# Zohzf wvikfgzflf wvcz oilknhzoykk# Zkbpvl fmfgzk} jf lvgyv fwzf jf 7 oyf# Wf hgoyjć 80 pvcgzf jkc omkgkv# Lh zohzf wvikfgzflf wf gfy yfqol}ćyk gobplfzf# • •

= 6p 6p 6p 6p 6p

WVIKFGZVL K $70p+ ‗ ]hykhczh 00= 8# Wć wf ghlgvlfqf wvbh zvzvyoy cvbfyfloy chzvyhlf jf jovć gkmyf ghyf wf jk}kj gv 88# 9# Wć wf jfzfybkcf mvcgŧkh m jf nyhjvl hl joklfh jhgć $m 8+ 8, $m0+ 8, $m8+ 7 # 7# Wć wf yfqol}f ãc bvlŧkbfh $ 0, π + fgvhŧkh wkc 7 t wkc t # :# Gézf cvbfyf cvbfyf jf zyfk gkmyf gkmyf jkwzkcgzf jkwzkcgzf wf poz poz moybh gv gkmyflf gkmyflf 9,:, = whv 4? 6# Wf gocwkjfyć zykvcndkvl HIG gv }éymvyklf ãc H$8, 9+ , I $ 9, 9+ şk G $:,=+ # Wć wf ghlgvlfqf gow I # ∙

2

2

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6p =# Wć wf ghlgvlfqf lvcnkbfh yhqfk gfygvlvk gkygvbwgykw zykvcndkvlvk HIG şzkkcj gć

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WVIKFGZVL K $70p+ ‗ ]hykhczh 00>

6p 8# Wć wf ghlgvlfqf bojvlvl cvbćyvlvk gobplft q 2 4 % k # > ∙ :k

6p 9# Wć wf jfzfybkcf }hlohyfh bhtkbć h mvcgŧkfk

9

m 1 Y ↙ Y , m $ t+ 2 ∙ t

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6p 7# Wć wf yfqol}f ãc bvlŧkbfh U0, 9π + fgvhŧkh wkc t 2 ∙ 8 # 9

6p :# Wć wf jfzfybkcf c ∄  pfczyv ghyf bvlŧkbfh u8,9,###, cr hyf fthgz 890 jf wvibvlŧkbk gv jovć flfbfczf#     6p 6# Wf şzkf gć ãc zykvcndkvl HIG }fgzoykk HI% HG şk HI∙ HG hv hgflhşk bojvl# Wć wf jfbocwzyfqf gć ∞

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zykvcndkvl HIG fwzf jyfpzvcndkg# jyfpzvcndkg# =# Wć wf ghlgvlfqf lvcnkbfh yhqfk gfygvlvk ãcwgykw ãc zykvcndkvl HIG ghyf hyf lvcnkbklf lhzvykloy fnhlf gv 7, : şk 6#





























Bkckwzfyvl Fjvghŧ Fjvghŧkfk, Gfygfzć Gfygfzćykk şk Zkcfyfzvlvk Gfczyvl Chŧ Chŧkochl pfczyv Gvyykgvlvb şk F}hlvhyf ãc Ãc}ćŧć Ãc} ćŧćbéczvl béczvl Pyfvck}fywkzhy


Mklkfyh zfoyfzkg ć, pyomklvl yfhl, wpfgkhlkqhyfh bhzfbhzkgć " kcmoybhzkg ć# Mklkfyh }oghŧkochlć, pyomklvl bklkzhy, wpfgkhlkqhyfh bhzfbhzkg ć " kcmoybhzkgć# • Zohzf wvikfgzflf wvcz oilknhzoykk# Zkbpvl fmfgzk} jf lvgyv fwzf jf 7 oyf# Wf hgoyj ć 80 pvcgzf jkc omkgkv# • Lh zohzf wvikfgzflf wf gfy yfqol}ćyk gobplfzf#

WVIKFGZVL K $70p+ ‗ ]hykhczh 004 8 6p 8# œzkkcj gć gć q ∄  şk gć gć q 9 % q % 8 2 0 , wć wć wf ghlgvlfqf q : % : # q

6p 9# Wf gocwkjfyć gocwkjfyć mvcgŧ mvcgŧkh

9

m 1 Y ↙ Y , m $ t+ 2 ht

%

t% g# œzkkcj gć gć pvcgzflf H $8, 9 + şk I $ 0,7+ hphyŧ hphyŧkc

nyhmkgvlvk mvcgŧ mvcgŧkfk m , wć wć wf jfzfybkcf cvbfyflf yfhlf h şk g#

6p 7# Wć wf yfqol}f ãc  fgvhŧ fgvhŧkh 7 > t % 8 ∙ t 2 8 # bvlŧkbfh u8,7,6 ,7,6,> ,>,; ,;r ? 6p :# Gézf cvbfyf jf phzyv gkmyf jkwzkcgzf wf poz moybh gv gkmyf jkc bvlŧ         6p 6# Wf gocwkjfyć gocwkjfyć phyhlflonyhbvl HIGJ şk pvcgzflf F şk M hwzmfl ãcgéz HF2 FI, JM 2 9 MF# Wć Wć wf jfbocwzyfqf gć gć pvcgzflf H , M şk G wvcz golkckhyf# Wć wf ghlgvlfqf lvcnkbfh ãcć ãcćlŧkbkk goyfwpvcqć goyfwpvcqćzohyf lhzvykk IG şzkkcj gć gć 6p =# Mkf zykvcndkvl HIG # Wć HI 2 87,

HG : şk IG 2 86 # 28
































; 6p 6p 6p 6p 6p

WVIKFGZVL K $70p+ ‗ ]hykhczh 00; 8# Wć wf yfqol}f ãc bvlŧkbfh cvbfyfloy gobplftf fgvhŧkh q 9 2 ∙; # 9# Wć wf jfzfybkcf h ∄  pfczyv ghyf fgvhŧkh ht9 % $7h ∙ 8+ t % h % 7 2 0 hyf wolvŧkk yfhlf# 7# Wć wf yfqol}f ãc bvlŧkbfh U0, 9π R fgvhŧkh gow : t 2 8 # :# Wć wf jfzfybkcf cvbćyvl mvcgŧkkloy m 1 u8, 9, 9, 7, :, :, 6r ↙ u8, 9, 9, 7, :, :, 6r gv pyopykfzhzfh gć m $8+ 2 m $9+ # 6# Wć wf ghlgvlfqf lvcnkbfh yhqfk gfygvlvk ãcwgykw ãczy"vc zykvcndk ghyf hyf lvcnkbklf lhzvykloy 87,8:,86#

6p =# Zykvcndkvl

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HIG hyf I 2

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HI HG

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80

WVIKFGZVL K $70p+ ‗ ]hykhczh 080 6p 8# Wć wf yfqol}f ãc bvlŧkbfh cvbfyfloy gobplftf fgvhŧkh q 9 2 ∙: # 6p 9# Wć wf jfzfybkcf mvcgŧkh m jf nyhjvl ãczék, pfczyv ghyf m $ m $ t+ + 2 9 m $ t+ % 8 , oykghyf hy mk t ∄  # 6p 7# Wć wf yfqol}f ãc  fgvhŧkh ln $ t % 8+ ∙ ln ; 2 8 ∙ ln t # 80

6p :# Wć wf jfzfybkcf cvbćyvl zfybfckloy kyhŧkochlk jkc jfq}olzhyfh $ 7 % 7 7 + # 

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6p 6# Wć wf jfzfybkcf h ∄  pfczyv ghyf }fgzoykk v 2 $h ∙ 9+k % 7 a şk } 2 4k ∙ $90 ∙ 9h + a wvcz golkckhyk#       6p =# Wć wf hyhzf gć }fgzoykk v 2 6k ∙ : a şk } 2 9k % 7 a moybfhqć vc vcndk oizvq#


































88

WVIKFGZVL K $70p+ ‗ ]hykhczh 088 8>, h wvcz ãc 6p 8# Wć wf jfzfybkcf h , i ∄  şzkkcj gć cvbfyflf 9, h, i wvcz ãc pyonyfwkf nfobfzykgć şk 9, 8>, pyonyfwkf hykzbfzkgć# 9# Wć wf yfqol}f fgvhŧkh m $ m $ t+ + 2 0 şzkkcj gć m 1  ↙  , m$ t+ 2 ∙7 t% 9 #

6p 6p 7# Wć wf yfqol}f ãc bvlŧkbfh U0, 9π + fgvhŧkh zn$∙ t+ 2 8 ∙ 9 zn t# 6p :# Wć wf jfzfybkcf cvbćyvl mvcgŧkkloy m 1 u0,8, 9r ↙ u0,8, 9r ghyf }fykmkgć yflhŧkh m $9+ 2 9 #        6p 6# Wf gocwkjfyć zykvcndkvl HIG şk pvcgzflf J , F hwzmfl ãcgéz HJ2 9 JI, HF2 9 FG# Wć wf hyhzf gć jyfpzflf JF şk IG wvcz phyhlflf#

6p =# Wć wf ghlgvlfqf lvcnkbfh yhqfk gfygvlvk gkygvbwgykw zykvcndkvlvk HIG , jhgć H 2 π , I 2 π şk HI 2 =# :

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89

WVIKFGZVL K $70p+ ‗ ]hykhczh 089 6p 8# Wć wf ghlgvlfqf wvbh pykbkloy 90 jf zfybfck hk pyonyfwkfk hykzbfzkgf $ hc + , şzkkcj gć gć c≦8 h8 % h7 % h6

% h= 2

6p 9# Wć wf yfqol}f ãc

h:

∙ h9 2

: şk

70 # 

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t % 8 t % 9

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t%9 t%7

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8 6p 7# Wć wf yfqol}f ãc bvlŧkbfh U0, 9π + fgvhŧ fgvhŧkh gow 9 t 2 # 9

89

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 8  :# Wć wf jfzfybkcf h 3 0 şzkkcj gć zfybfcvl jkc bkalog hl jfq}olzćykk  7 h % :  h 

fwzf fnhl gv 84:4#

pvcgzv zvll H$∙7,:+ # 6p 6# Wć wf jfzfybkcf fgvhŧkh wkbfzykgfk jyfpzfk j 1 9t ∙ 7{ % 8 2 0 mhŧć jf pvcg ghlgvlfqf fqf gzn 9 t # 6p =# œzkkcj gć gzn t 2 7 , wć wf ghlgvl
































87 6p

WVIKFGZVL K $70p+ ‗ ]hykhczh 087 8# Wć wf }fykmkgf fnhlkzhzfh $8 % k 7 +9 % $8 ∙ k 7 + 9 2 ∙: #

6p

9# Wć wf yfqol}f ãc

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7# Wć wf yfqol}f ãc bvlŧkbfh cvbfyfloy yfhlf fgvhŧkh t 2 = $

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8  :# Wć wf jfzfybkcf zfybfcvl ghyf cv gocŧkcf pf t jkc jfq}olzhyfh  t 9 %  #

6p 6p

6# Wć wf ghlgvlfqf jkwzhcŧh jf lh pvcgzvl H$7,0+ lh jyfhpzh j 1 7t ∙ : { % 8 2 0 # =# Zykvcndkvl HIG hyf HI2 :, IG2 6 şk GH 2 = # Wć wf hyhzf gć I 2 9G #

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WVIKFGZVL K $70p+ ‗ ]hykhczh 08:

6p 8# Wć wf ghlgvlfqf ln 6p 9# Wć wf jfzfybkcf

8 9

% ln

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% ln

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pfczyv ghyf kcfgvhŧkh

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9$h % 8+t % 9 h ∙ 8 ≦ 0 cv hyf wolvŧkk ãc bvlŧkbfh

cvbfyfloy yfhlf#

6p 7# Wć wf yfqol}f ãc bvlŧkbfh cvbfyfloy yfhlf fgvhŧkh 7 4 ∙ t 2 7 ; ∙ : t # 6p :# Wć wf jfzfybkcf cvbćyvl flfbfczfloy vcfk bvlŧkbk şzkkcj gć hgfhwzh hyf fthgz :6 jf wvibvl ŧkbk gv jovć flfbfczf# 6# Wć wf jfzfybkcf fgvhŧkh jyfpzfk

:+ # HI şzkkcj gć H$9,7+ şk I $∙6, :+ 6p 6p =# Zykvcndkvl HIG hwgvŧkzvcndkg hyf HG 2 9 7 şk lvcnkbfh yhqfk gfygvlvk gkygvbwgykw fnhlć gv 9# Wć wf ghlgvlfqf b $ I + #
































86

WVIKFGZVL K $70p+ ‗ ]hykhczh 086 6p 8# Wć wf ghlgvlfqf lon7 $ 6 ∙ > + % lon7 $ 6 % > + ∙ lon7 9 # 6p 9# Wć wf jfzfybkcf mvcgŧkh jf nyhjvl hl joklfh hl gćyfk nyhmkg fwzf zhcnfcz lh hth Ot ãc pvcgzvl $8,0+ şk zyfgf pykc pvcgzvl pvcgzvl $0,9+ #

6p 7# Wć wf yfqol}f ãc bvlŧkbfh U0, 9π + fgvhŧkh wkc t % gow t 2 0 # 6p :# Gézf cvbfyf jf phzyv gkmyf, cv cfhpćyhz jkwzkcgzf, wf poz moybh gv gkmyf jkc bvlŧkbfh u8,7,6,>,;r ? 6p 6# Wć wf jfzfybkcf fgvhŧkh jyfpzfk ghyf gocŧkcf kcf pvcg pvcgzv zvll H$∙9,9+ şk fwzf phyhlflć gv jyfhpzh jfzfybkchzć jf pvcgzflf

G $9,8+ , J $ ∙8, ∙7+

6p =# Mkf θ ∄  π , 

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 hwzmfl ãcgéz gow θ 2 ∙ 87 # Wć wf ghlgvlfqf wkc θ # 9 


































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WVIKFGZVL K $70p+ ‗ ]hykhczh 08=

6p 8# Wć wf ghlgvlfqf bojvlvl cvbćyvlvk gobplft q 2 9 ∙ k # 9%k


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h ∄  pfczyv ghyf

t % ht% 9 9

t % 8

6p 7# Wć wf yfqol}f fgvhŧkh hygwkc 8 % hygwkc t 2 9

6p :# Wć wf yfqol}f fgvhŧkh

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8>

WVIKFGZVL K $70p+ ‗ ]hykhczh 08> 7

6p 8# Wć wf jfzfybkcf phyzfh kbhnkchyć h cvbćyvlvk $8 % k 7 + # 6p 9# Wć wf jfzfybkcf kbhnkcfh mvcgŧkfk m 1  ↙ , m $ t+ 2 t9 ∙ t % 9 # 6p 7# Wć wf yfqol}f ãc  fgvhŧkh ∙9 t % 8 2 6 # 6p :# Wć wf jfzfybkcf pyoihiklkzhzfh gh, hlfnécj vc cvb ćy hi jkc bvlŧkbfh cvbfyfloy jf jovć gkmyf, wć 6p

h}fb h % i 2 : # 6# Wć wf jfzfybkcf fgvhŧkh jyfpzfk ghyf zyfgf pykc pvcgzvl H$∙8,8+ şk fwzf pfypfcjkgvlhyć pf jyfhpzh j 1 6t

:{ %8 2 0
































84

WVIKFGZVL K $70p+ ‗ ]hykhczh 084 6p 8# Wć wf yfqol}f ãc bvlŧkbfh cvbfyfloy gobplftf fgvhŧkh t 9 ∙ 9 t % : 2 0 # 6p 9# Wf gocwkjfyć mvcgŧkh m 1 U∙9, 9R 9R ↙  , m $ t+ 2 t9 ∙ 7 t % 9 # Wć wf hmlf }hlohyfh bkckbć h mvcgŧkfk m # 6p 7# Wć wf yfqol}f fgvhŧkh hygwkc t % hyggow 8

9

π

2

9

#

0,8, 9,### 9,###,; ,;r , gfl gfl pvŧkc vc cvbćy wć 6p :# Wć wf jfzfybkcf pyoihiklkzhzfh gh, hlfnécj jovć cvbfyf jkc bvlŧkbfh u0,8

6p

mkf pykb# 6# Wć wf jfzfybkcf gooyjochzflf gfczyvlvk jf nyfvzhzf hl zykvcndkvlvk HIG , jhgć

H$∙8, 0+ , I$0, 9+ , G$9, ∙ 8+ #
































8;

WVIKFGZVL K $70p+ ‗ ]hykhczh 08; 6p 8# Wć wf oyjocfqf gyfwgć gyfwgćzoy cvbfyflf 7, 7 6 , : 4 # 6p 9# Wć wf jfzfybkcf mvcgŧkh m 1  ↙  şzkkcj gć nyhmkgvl wćv şk nyhmkgvl mvcgŧ mvcgŧkfk

n 1  ↙  , n $ t+ 2 ∙7t % 7

wvcz wkbfzykgf mhŧć jf jyfhpzh t 2 8 #

6p 7# Wć wf yfqol}f ãc  fgvhŧkh 79 t 8 ∙ 80 ⋆ 7t 8 % 9> 2 0 # 6p :# Wć wf jfzfybkcf pyoihiklkzhzfh gh, hlfnécj zyfk gkmyf jkc bvlŧ bvl ŧkbfh u0,8,9,###,;r , zohzf hgfwzfh wć w ć mkf phyf# 6p 6# Wć wf jfzfybkcf fgvhŧ fgvhŧkh bfjkhcfk jvwf jkc }éymvl H hl zykvcndkvlvk HIG , vc vcjf H$8, 9+ 9+ , I $9, 7+ şk G $9, ∙6+ # %

%

  gć t ∄  0,  wć wf hyhzf gć gć gzn9 t 6p =# œzkkcj gć π

gzn t ∙ zn t
































90

WVIKFGZVL K $70p+ ‗ ]hykhczh 090 6p 8# Wć wf }fykmkgf jhgć 9 ∄ $ lon 7 :, 6 + # 6p 9# Wć wf yfqol}f ãc bvlŧkbfh cvbfyfloy gobplftf fgvhŧkh t 9 ∙ 9 t % 9 2 0 # 6p 7# Wć wf yfqo yfqol} l}ff ãc U0,9 U0,9π + fgvhŧkh wkc t % gow t 2 ∙8 # 6p :# Wć wf ghlgvlfqf G:: % G6: % G =: # z ykvcndkvlvk lvk 6p 6# Pf lhzvyklf HI şk HG hlf zykvcndkv   

HB2

HIG

wf gocwkjfyć pvcgzflf B , yfwpfgzk} C hwzmfl ãcgéz

  

:

BIşk

 

BC

IG# Wć

wf jfzfybkcf

b∄Y

hwzmfl ãcgéz

GC

 

2

b HG HG #





























































































































































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=8

WVIKFGZVL K $70p+ ‗ ]hykhczh 0=8 6p 8# Wć wf jfzfybkcf t yfhl şzkkcj gć cvbfyflf t % 8, 8 ∙ t şk : wvcz ãc pyonyfwkf hykzbfzkgć # 6p 9# Wć wf jfzfybkcf pvcgzflf jf kczfywfgŧkf h phyhiolfk {2 t9 % 6 t∙ = gv htflf jf gooyjochzf# 6p 7# Wć wf yfqol}f ãc U 0, 9π R fgvhŧkh 9 wkc t % 8 2 0 # 6p :# Mkf bvlŧkbfh B 2 u8,9,7,:,6,=r # Wć wf jfzfybkcf pyoihiklkzhzfh gh, hl fnécj o wvibvlŧkbf h bvlŧkbkk B , hgfhwzh wć hkić 9 flfbfczf#

6p 6# Pvcgzflf H , I şk

 





 











:k % > a , y I 2 9k ∙ a , yN 2 :k % : a # Wć wf jfzfybkcf }fgzoyvl jf poqkŧkf h pvcgzvlvk G hwzmfl ãcgéz pvcgzvl N wć mkf gfczyvl jf nyfvzhzf hl zykvcndkvlvk HIG #

6p =# Mkf }fgzoykk



N hv }fgzoykk jf poqkŧkf y H





v şk } # Jhgć v

 



2

8, }

2

2

9 şk b

$ $ v5 } + + 

π

2

7

$





+$





+

, wć wf ghlgvlfqf 9v % } 9} ∙ v #






























WVIKFGZVL K $70p+ ‗ ]hykhczh 0=9 6p 8# Wć wf jfzfybkcf t 3 0 şzkkcj gć cvbfyflf t , = şk t ∙ 6 wvcz ãc pyonyfwkf nfobfzykgć # 6p 9# Wf gocwkjfyć mvcgŧkh m 1 Y ↙ Y , m $ t+ 2 t9 % t ∙ 9 # Wć wf ghlgvlfqf m $ 9 ⋆ $ m $ ∙8+ + + # 6p 7# Wć wf yfqol}f ãc



 

fgvhŧkh gow  9 t %

π    2 gow t ∙   # 9 9 

π

6p :# Wć wf jfzfybkcf e ∄ C hwzmfl ãcgéz G::4e 2 G :904 # 6p 6# Wf gocwkjfyć pvcgzflf H $ 7,9 + şk I $ =,6 + # Wć wf jfzfybkcf gooyjochzflf pvcgzfloy B şk C şzkkcj gć hgfwzfh ãbphyz wfnbfczvl U HIR ãc zyfk wfnbfczf gocnyvfczf, khy oyjkcfh pvcgzfloy fwzf H, B , C , I #

6p =# Wć wf jfzfybkcf }hloyklf phyhbfzyvlvk h ∄  pfczyv ghyf cvbfyflf h , h % 8 şk h % 9 wvcz lvcnkbklf ∞

lhzvykloy vcvk zykvcndk oizvqvcndkg# oizvqvcndkg#






























=7

WVIKFGZVL K $70p+ ‗ ]hykhczh 0=7

6p 8# Wć wf hyhzf gć şkyvl $ hc +c

∄

, jf zfybfc nfcfyhl hc

2

:c c%7

6p 9# Wć wf jfzfybkcf pvcgzflf jf kczfywfgŧkf hlf phyhiolfloy 6p 7# Wć wf yfqol}f ãc



 

fgvhŧkh wkc  t ∙

, fwzf gyfwgćzoy# 9

{2 t

%

9 t % 8 şk {2 ∙ t

∙

9 t% =#

π    2 wkc 7 t %   # : : 

π

c

6p :# Wvbh gofmkgkfcŧkloy ikcobkhlk hk jfq}olzćykk $ 9 t 9 ∙ 6 { + fwzf fnhlć gv 79# Wć wf jfzfybkcf zfybfcvl jf yhcn phzyv#

6p 6# Wć wf jfzfybkcf b, c ∄ Y hwzmfl ãcgéz jyfpzflf j81 bt % 7 { % 9 2 0 şk j9 1 9t % c{ ∙ 4 2 0 wć gokcgkjć#   9 9 9 9 6p =# Mkf HIGJ vc phzyvlhzfy# Wć wf hyhzf gć jhgć H HG G ⋆ IJ 2 0 , hzvcgk HI % GJ 2 HJ % IG #






























=:

WVIKFGZVL K $70p+ ‗ ]hykhczh 0=: 6p 8# Wć wf hyhzf gć şkyvl $ hc +c ≦8 , jf zfybfc nfcfyhl hc 2 c 9 ∙ c , fwzf wzykgz bocozoc# 6p 9# Wf gocwkjfyć mvcgŧkklf

9

m 1 Y ↙ Y şk n 1 Y ↙ Y jfmkckzf pykc m $ t+ 2 t

$

Wć wf jfbocwzyfqf gć, pfczyv oykgf t ∄  ,



π



%

9 t % 8 şk n $ t + 2 t ∙ 9004 #

m  n+ $ t+ ≦ 0#

 π



6p 7# Wć wf yfqol}f ãc $ 0, π + fgvhŧkh zn  t %  2 zn  ∙ t  # 7  9  ∙ ∙ 6p :# Wć wf jfzfybkcf t ∄ C şzkkcj gć G t t 8 % G tt∙87 ≪ ; # 6p 6# Wć wf jfzfybkcf b ∄ Y şzkkcj gć jyfpzflf j8 1 bt % $ b % 9 + { ∙ 8 2 0 şk

j9 1

b{ ∙ 4 2 0 $ b%9 + t % :b{

phyhlflf#

6p =# Mkf HIG vc zykvcndk gv zn zn H 2 9, zn I 2 7# Wć wf jfzfybkcf bćwvyh vcndkvlvk G #

wvcz






























=6

WVIKFGZVL K $70p+ ‗ ]hykhczh 0=6 6p 8# Wć wf jfzfybkcf pykbvl zfybfc hl pyonyfwkfk hykzbfzkgf h8 , h9 ,87,8>, ##### # gć mvcgŧ mvcgŧkh m 1 Y ↙ Y , m $ t+ 2 t7 % 9wkc t fwzf kbphyć kbphyć# 6p 9# Wć wf hyhzf gć 6p 7# Wć wf yfqol}f ãc  fgvhŧ fgvhŧkh 7 wkc t % 7 gow t 2 0 # cvbćy jkc bvlŧ bvlŧkbfh cvbfyfloy chzvyhlf jf zyfk gkmyf, 6p :# Wć wf jfzfybkcf pyoihiklkzhzfh gh, hlfnécj vc cvbć

6p

hgfwzh wć wć hkić hkić wvbh gkmfloy fnhlć fnhlć gv :# gć jyfpzflf j81 bt % 7 { ∙ 9 2 0 şk j 9 1 89t % 9 { % 8 2 0 wvcz 6# Wć wf jfzfybkcf b ∄ Y , şzkkcj gć pfypfcjkgvlhyf #

gć zn 6p =# œzkkcj gć

θ 2

9

8 7

, wć wć wf ghlgvlfqf wkc θ #
































==

WVIKFGZVL K $70p+ ‗ ]hykhczh 0== 6p 8# Wć wf ghlgvlfqf $ 9 % k + $ 7 ∙ 9 k + ∙ $8 ∙ 9k + $9 ∙ k + # 6p 9# Wć wf hyhzf gć 8 fwzf o pfykohjć h mvcgŧkfk 7

m 1 Y ↙ Y , m $ t+ 2 u7 tr , vcjf

u r fwzf mvcgŧkh phyzf

myhgŧkochyć#

6p 7# Wć wf yfqol}f ãc U0, 9π R fgvhŧkh 7 wkc t ∙ gow t 2 8 # 6p :# Wć wf ghlgvlfqf

80

G 90 ;

G 90

#

6p 6# Wf gocwkjfyć pvcgzflf H $ 9,7 + , I $ :,6+ ,

9, 9 + şk J$ b, c + # Wć wf jfzfybkcf b, c ∄ Y hwzmfl ãcgéz G $ 9,9

phzyvlhzfyvl HIGJ wć mkf phyhlflonyhb #

6p =# Wć wf ghlgvlfqf gow 9 t , şzkkcj gć zn t 2 : #
































=>

WVIKFGZVL K $70p+ ‗ ]hykhczh 0=> 6p 8# Wć wf jfzfybkcf pykbvl zfybfc hl pyonyfwkfk nfobfzykgf gv zfybfck poqkzk}k i8 , =, = , i7 , 9:, ## ### # 6p 9# Wć wf jfzfybkcf

$7 ∙ b9 + t % 7 , wć mkf wzykgz gyfwgćzohyf# b ∄ Y hwzmfl ãcgéz mvcgŧkh m 1 Y ↙ Y , m $ t+ 2 $7

6p 7# Wć wf ghlgvlfqf wkc

π

%

wkc

9π

%

wkc

7π

%

wkc

:π

#

6p

7 7 7 7 :# Wf gocwkjfyć bvlŧkbfh B h zvzvyoy mvcgŧkkloy jfmkckzf pf H 2 u8,9,7r gv }hloyk ãc I 2 u6,=,>r # Wć wf

6p

ghlgvlfqf pyoihiklkzhzfh gh, hlfnécj o mvcg ŧkf jkc bvlŧkbfh B , hgfhwzh wć mkf kcafgzk}ć # 6# Wf gocwkjfyć pvcgzvl N, gfczyvl jf nyfvzhzf hl zykvcndkvlvk HIG # Pykc pvcgzvl N wf jvgf phyhlflh lh HI   

6p

 

ghyf kczfywfgzfhqć jyfhpzh IG ãc pvcgzvl P # Wć wf jfzfybkcf b ∄ Y hwzmfl ãcgéz NP 2 b HI HI # 8 =# Wć wf ghlgvlfqf gow9θ , jhgć wf gvcohşzf gow θ 2 # 7
































=4

WVIKFGZVL K $70p+ ‗ ]hykhczh 0=4

6p 8# Wć wf ghlgvlfqf

96 : % 7k

%

96 : ∙ 7k

#


b ∄ Y hwzmfl ãcgéz mvcgŧkh m 1 Y ↙ Y , m $ t+ 2 $ b

6p 7# Wć wf yfqol}f ãc



6p 6p

6p

9

fgvhŧkh hygzn

t

8

π

∙

9+ t ∙ 7 wć mkf wzykgz jfwgyfwgćzohyf#

# 7 7 7 :# Wć wf jfzfybkcf pyoihiklkzhzfh gh hlfnécj vc cvbćy jkc bvlŧkbfh cvbfyfloy chzvyhlf phyf jf jovć gkmyf , hgfwzh wć mkf jk}kqkikl gv :# zykvcndkvlvk HIG wf gocwkjfyć pvcgzflf B şk yfwpfgzk} C hwzmfl ãcgéz 6# Pf lhzvyklf HI şk HG hlf zykvcndkvlvk      7     HB2 7 BIşk HC2 HG# W ć wf jfbocwzyfqf gć }fgzoykk BC şk IG wvcz golkckhyk # : 88π # =# Wć wf ghlgvlfqf wkc 89 %

hygzn

2
































=;

WVIKFGZVL K $70p+ ‗ ]hykhczh 0=;

6p 8# Wć wf jfzfybkcf q ∄ G şzkkcj gć q % >k 2 = # 6p

q 9# Mkf mvcgŧkh m 1  ↙  , m $ t+ 2 9 t% 8 # Wć wf ghlgvlfqf

# ## % m$60 + # m$8+ % m$ 9 + % m$7 + % ##

6p 7# Wf gocwkjfyć mvcgŧkh m 1 C ↙ C , m $ t+ 2 7 t% 8 # Wć wf jfbocwzyfqf gć mvcgŧkh m cv fwzf kc}fywhiklć# 6p :# Wć wf ghlgvlfqf pyoihiklkzhzfh gh, hlfnécj o gkmyć t, hgfhwzh wć }fykmkgf kcfnhlkzhzfh $ t % 8+( ∙ t ( ≪ 800 # 6p 6# Wć wf hyhzf gć jyfpzflf jf fgvhŧkk j8 1 9t ∙ { % 8 2 0 şk j 9 1 9t % { ∙ 8 2 0 wvcz wkbfzykgf mhŧć jf hth O{# 6p =# Wć wf ghlgvlfqf gow >π # 89
































>0

WVIKFGZVL K $70p+ ‗ ]hykhczh 0>0 6p 8# Wć wf ghlgvlfqf $8 % k +90 # 6p 9# Wf gocwkjfyć mvcgŧkh m 1 Y W

2

m

∞

↙

Y,

m $ t + 2

8 t

# Wć wf ghlgvlfqf wvbh

$ m $ ∙80 + + % m $ m $ ∙; + + % ### % m $ m $ ∙8+ + % m $ m $8+ + % ### % m $ m $ ; + + % m $ m $8 0 + + #

6p 7# Wć wf hyhzf gć mvcgŧkh

7

m 1 Y ↙ Y , m $ t+ 2 t

%

8 fwzf kcafgzk}ć #

6p :# Wć wf ghlgvlfqf H67 ∙ =G 67 # 6p 6# Wć wf jfzfybkcf b ∄ Y şzkkcj gć jkwzhcŧh jf lh pvcgzvl H$ b, b % 8+ lh jyfhpzh 6p =# Wć wf ghlg ghlgvl vlfq fqff gow> gow>6 6



∙



gow8 gow86 6 #

j 1 7t ∙ : { ∙ 8 2 0 fwzf 8#


































>8

WVIKFGZVL K $70p+ ‗ ]hykhczh 0>8 6p 8# Wć wf ghlgvlfqf lon 9 9004 ∙ lon 9 968 ∙ 7 # 6p 9# Wf gocwkjfyć mvcgŧkh

m 1 Y∞ ↙ Y , m $ t+ 2 t9 ∙

6p 7# Wć wf hyhzf gć }hlohyfh bhtkbć h mvcgŧkfk 6p :# Wć wf jfzfybkcf

8

8 t

9

# Wć wf hyhzf gć mvcgŧkh m fwzf phyć# :

m 1  ↙ Y , m $ t+ 2 7 ∙ t fwzf m $ 0 + # 9

c ∄ C hwzmfl ãcgéz 7Gc % 9G c 2 4 #

6p 6# Wf gocwkjfyć zykvcndkvl

 

   

HIG şk pvcgzflf H‹, I‹, G‹ hwzmfl ãcgéz H‹ G 2 9 IH‹, I‹ G 2

     9  HG , G ‹H 2 7IG ‹ # 6

Wć wf hyhzf gć jyfpzflf HH‹, II‹ şk GG ‹ wvcz gocgvyfczf#

6p =# Wć wf jfzfybkcf fgvhŧkh bfjkhcfk goyfwpvcqćzohyf lhzvykk IG h zykvcndkvlvk HIG , şzkkcj gć H$9,9+ şk gć fgvhŧkklf bfjkhcfloy jvwf jkc I şk G wvcz 9 t % { ∙ 9 2 0 , yfwpfgzk} t ∙ { % 9 2 0 #





























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>9

WVIKFGZVL K $70p+ ‗ ]hykhczh 0>9

π 6p 8# Wć wf hyhzf gć cvbćyvl  gow

% k wkc

6p 9# Wf gocwkjfyć mvcgŧkh

Y,



:

m 1 Y∞

↙

  :

π

9004

fwzf yfhl# 7

m $ t+ 2 t

∙

8 t

# Wć wf hyhzf gć mvcgŧkh m fwzf kbphyć #

9 6p 7# Wć wf jfzfybkcf bvlŧkbfh H, şzkkcj gć yflhŧkh m1 U8, :R ↙ H, m $ t+ 2 t ∙ t jfmkcfşzf o mvcgŧkf wvyafgzk}ć# 6p :# Wć wf ghlgvlfqf G90004 ⋆ 69004 ∙ G98004 ⋆ 6900> ⋆ : % G99004 ⋆ 6 900= ⋆ : 9 ∙ ### % G 99000044 ⋆ : 9 004 # 6p 6# Wf gocwkjfyć pvcgzvl H $8, 9 + şk jyfhpzh jf fgvhŧkf j 1 :t ∙ 9 { % 6 2 0 # Wć wf jfzfybkcf fgvhŧkh

pfypfcjkgvlhyfk pfypfcjkgvlhyfk jvwf jkc pvcgzvl H pf jyfhpzh j #

6p =# Wć wf ghlgvlfqf wkc >6 >6 ⋆ gow86 # 


































>7

WVIKFGZVL K $70p+ ‗ ]hykhczh 0>7 6p 8# Wć wf ghlgvlfqf 6 ∙ 89k ∙ 89 % 6k # 6p 9# Wf gocwkjfyć mvcgŧkh m 1 Y ↙ Y , m $ t+ 2 t9 ∙ t: # Wć wf ghlgvlfqf $ m  m  m  m + $8+ # 6p 7# Wć wf yfqol}f ãc  fgvhŧkh 9 t % : t 2 90 # 6p :# Wć wf jfzfybkcf pyoihiklkzhzfh gh, hlfnécj vc flfbfcz hl bvl ŧkbkk H 2 u0,6,80,###,9006r , hgfwzh wć mkf jk}kqkikl gv 96 #

6p 6# Wf gocwkjfyć vc zykvcndk HIG , gv lvcnkbklf lhzvykloy 





HI 2 g, HG 2 i şk vc pvcgz J hwzmfl ãcgéz

HJ2 i HI HI% g H HG G# Wć wf hyhzf gć wfbkjyfhpzh U HJ fwzf ikwfgzohyfh vcndkvlvk IHG #

  6p =# Mkf θ ∄  , π  , hwzmfl ãcgéz gow9θ 2 # Wć wf ghlgvlfqf gow θ # 9 9 8

π



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>:

WVIKFGZVL K $70p+ ‗ ]hykhczh 0>: 6p 8# Wć wf yfqol}f ãc G fgvhŧkh q 9 % 7q % : 2 0 # 6p 9# Wf gocwkjfyć mvcgŧkh m 1 $ 0, ∓ + ↙ Y , m $ t+ 2 t ∙ 9 b % 9 # Wć wf jfzfybkcf b ∄  hwzmfl ãcgéz nyhmkgvl mvcgŧkfk m wć cv kczfywfgzfqf hth Ot#

6p 7# Wć wf yfqol}f ãc Y fgvhŧkh 9 ∙ t ∙ 7 t ∙ 9 2 0 # h i ∞ 6p :# Wć wf hyhzf gć Gh% i 2 Gh% i , ∀h, i ∄  # 6p 6# Wć wf jfzfybkcf b ∄  hwzmfl ãcgéz pvcgzflf H $ 7, 7+ , I $ 9, : + şk  π 

 

8

6p =# Mkf θ ∄  , π  , hwzmfl ãcgéz gow9θ 2 ∙ # Wć wf ghlgvlfqf wkcθ # 9 9

G $ 9b, 8 ∙ b + wć mkf golkckhyf#
































>6

WVIKFGZVL K $70p+ ‗ ]hykhczh 0>6 6p 8# Wć wf jfzfybkcf t , { ∄ Y şzkkcj gć t$8 % 9 k+ % {$ 9 ∙ k+ 2 : % 7 k# 6p 9# Wć wf jfzfybkcf 6p 7# Wć wf yfqol}f ãc

9

b ∄  , hwzmfl ãcgéz pvcgzvl H$ b ∙ 8, b Y

∙

7 b+ wć wf hmlf ãc ghjyhcvl KK#

fgvhŧkh lon7 $lon : $ t 9 ∙ 8>++ 2 8 # =

6p

7  :# Wf gocwkjfyć jfq}olzhyfh  9 t %  , t 3 0 # Wć wf jfzfybkcf zfybfcvl kcjfpfcjfcz jf t # t  

6p 6# Mkf, ãc wkwzfbvl jf gooyjochzf tO{, pvcgzflf H $ :, ∙ 9 + , I $ 9, : + şk

G $ b, c + # Wć wf jfzfybkcf

b, c ∄ Y hwzmfl ãcgéz pvcgzvl G wć mkf gfczyvl gfygvlvk gkygvbwgykw zykvcndkvlvk HOI#

6p =# Mkf zykvcndkvl HIG jyfpzvcndkg ãc H gv HI 2 6 şk IG 2 87 # Wć wf ghlgvlfqf lvcnkbfh wfnbfczvlvk $ IB + , vcjf B fwzf bkalogvl wfnbfczvlvk $ HG + #


































>=

WVIKFGZVL K $70p+ ‗ ]hykhczh 0>=

6p 8# Wć wf }fykmkgf jhgć cvbćyvl 7 ∙ 9 9 hphyŧkcf bvlŧkbkk

uh % i

r

9 | h, i ∄ Q #

6p 9# Wf gocwkjfyć fgvhŧkh t 9 ∙ 7t % 8 2 0 , gv yćjćgkcklf t8 şk t9 # Wć wf hyhzf gć t89 % t99 ∄ # 6p 7# Wć wf yfqol}f ãc

 fgvhŧkh

hygzn 7 % hygzn t 2

π

9

#

6p :# Wć wf hyhzf gć oykghyf hy mk c chzvyhl, c ≦ 8 , hyf log fnhlkzhzfh G c 2 9 ⋆ G c # 9c 9 c ∙8         6p 6# Wf gocwkjfyć }fgzoykk v 2 k ∙ a şk } 2 9k % : a # Wć wf ghlgvlfqf bojvlvl }fgzoyvlvk v % } #  π



7

θ
































>>

WVIKFGZVL K $70p+ ‗ ]hykhczh 0>> 6p 8# Wf gocwkjfyć pyonyfwkh hykzbfzkgć jf yhŧkf 9 gv h7 % h: 2 4 # Wć wf jfzfybkcf h8 # 6p 9# Mkf m 1  ↙ , m $ t+ 2 8 % t# Wć wf ghlgvlfqf m $∙8+ % m $ ∙9+ % m $ ∙7+ % … % m $ ∙80+# 6p 7# Wć wf yfqol}f ãc  fgvhŧkh : t ∙ 9 t 2 6=# 6p :# Wć wf ghlgvlfqf H7 ∙ H9 ∙ G 9 # :

7

:

6p 6# Mkf HIG vc zykvcndk şk





N gfczyvl wćv jf nyfvzhzf# Wf gocwkjfyć pvcgzvl B jfmkckz pykc BI 2 ∙9 BG# Wć wf hyhzf gć jyfpzflf NB şk HG wvcz phyhlflf#

6p =# Mkf θ ∄  0, π  , hwzmfl ãcgéz wkc θ 2 # Wć wf ghlgvlfqf znθ 7
































>4 6p 6p 6p 6p

WVIKFGZVL K $70p+ ‗ ]hykhczh 0>4 8# Wć wf ghlgvlfqf 80ln > ∙ 7 7:7# 9# Wć wf yfqol}f ãc  kcfgvhŧkh 9 t 9 ∙ 7t % 8 ≪ 0# 7# Wć wf hyhzf gć mvcgŧkh m1  ↙ , m$ t+ 2 lon7 9 t ∙ t fwzf kcafgzk}ć# :# Wć wf ghlgvlfqf cvbćyvl jkhnochlfloy vcvk polknoc goc}ft gv 4 lhzvyk#

6p 6# Mkf HIGJ vc phyhlflonyhb şk

 

P vc pvcgz hwzmfl gh

6p =# Mkf h, i ∄  ∙ π , π  , hwzmfl ãcgéz  9 9

h%i 2

π

:

 

# Wć wf hyhzf gć IP2 9 PJ

 

IP2

# Wć wf hyhzf gć zn h zni % zn h % zni 2 8#

9   IH% IG # 7

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>;

WVIKFGZVL K $70p+ ‗ ]hykhczh 0>; 6p 8# Wć wf hyhzf gć $ ∙∓, 9 + ∫ $ lon 9 7, ∓ + 2 ∆# 6p 9# Wf gocwkjfyć mvcgŧkh

m 1  ↙ , m $ t+ 2 t9 ∙ : t % 7# 7# Wć wf jfzfybkcf hiwgkwflf pvcgzfloy jf kczfywfg ŧkf h

nyhmkgvlvk mvcgŧkfk m gv hth Ot#

6p 7# Wć wf yfqol}f ãc  fgvhŧkh t % 8 ∙ t 2 8# 6p :# Wć wf jfzfybkcf wvbh zfybfckloy yh ŧkochlk hk jfq}olzćykk $8 % 9 +6 # 6p 6# Mkf pvcgzflf H $8, 9 + ,

I $ ∙8, 7+ şk G $ 0, :+ # Wć wf ghlgvlfqf lvcnkbfh ãcćlŧkbkk jvwf jkc }éymvl H hl
































40

WVIKFGZVL K $70p+ ‗ ]hykhczh 040 9 7 9004 6p 8# Wć wf ghlgvlfqf $8 ∙ k + $8 ∙ k + $8 ∙ k +… $8 ∙ k +# 6p 9# Wf gocwkjfyć mvcgŧkklf m 1  ↙  , m$ t+ 2 8 ∙ t şk n 1  ↙ , n $t + 2 9 t ∙ 8# Wć wf hyhzf gć mvcgŧkh fwzf jfwgyfwgćzohyf#

6p 7# Wć wf yfqol}f ãc  kcfgvhŧkh 7 9 ∙ t 9 ≦ 8# 6p :# Wć wf ghlgvlfqf cvbćyvl mvcgŧkkloy kcafgzk}f m 1 u8, 9, 7r ↙ u8, 9, 7, :, 6r # 6p 6# Wf gocwkjfyć jyfhpzh j jf fgvhŧkf t ∙ 9 { % 8 2 0 # Wć wf jfzfybkcf fgvhŧkh jyfpzfk ghyf zyfgf pykc pvcgzvl P $ :, ∙8+ şk fwzf phyhlflć gv jyfhpzh j # j #

m



n












































































































































































































































































































































































































































































































































Matematica - M1 - Subiectul I - Variante 001-100

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