The document provides solutions to mathematical equations and inequalities involving radicals, fractions, and variables. It contains 50 problems involving solving equations and inequalities for variables on the set of real numbers. The problems cover a range of techniques including isolating variables, combining like terms, factoring, and applying properties of radicals, fractions and inequality signs.
The document provides solutions to mathematical equations and inequalities involving radicals, fractions, and variables. It contains 50 problems involving solving equations and inequalities for variables on the set of real numbers. The problems cover a range of techniques including isolating variables, combining like terms, factoring, and applying properties of radicals, fractions and inequality signs.
This document provides 30 equations and inequalities and asks the reader to solve them on the set of real numbers. It uses variables like x, square roots, exponents, and basic arithmetic operations. The problems range from simple one-variable equations to more complex expressions with multiple variables. The goal is to calculate the value(s) of the variable(s) that satisfy each equation or inequality.
This document contains solutions to various equations and inequalities involving radicals on the set of real numbers. It is divided into 6 sections, with multiple problems provided in each section ranging from simple single-term radical equations to more complex multi-term radical equations and inequalities. The document provides the step-by-step workings for solving each problem.
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Các dạng bài tập lượng giác 11
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16. Bµi 16. Mét líp häc cã 30 häc sinh nam vµ 15 häc sinh n÷. Cã 6 häc sinh
®-îc chä ra ®Ó lËp mét tèp ca. Hái cã bao nhiªu c¸ch chän kh¸c nhau.
1. NÕu ph¶i cã Ýt nhÊt 2 n÷. 2. NÕu ph¶i chän tuú ý. Bµi 17.
Mét tæ häc sinh gåm 7 nam vµ 4 n÷. Gi¸o viªn muèn chän 3 häc sinh xÕp vµo
bµn ghÕ cña líp, trong ®ã cã Ýt nhÊt 1 nam. Hái cã bao nhiªu c¸ch
chän? Bµi 18. Chøng minh r»ng: INCLUDEPICTURE
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MERGEFORMATINET . Bµi 19. Chøng minh r»ng: INCLUDEPICTURE
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MERGEFORMATINET Bµi 20. Víi n lµ sè nguyªn d-¬ng, chøng minh hÖ thøc
sau: INCLUDEPICTURE
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MERGEFORMATINET Bµi 21. Chøng minh r»ng: INCLUDEPICTURE
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MERGEFORMATINET Bµi 22. TÝnh tæng: INCLUDEPICTURE
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MERGEFORMATINET Bµi 23. TÝnh tæng: INCLUDEPICTURE
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MERGEFORMATINET Bµi 24. Chøng minh r»ng: INCLUDEPICTURE
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MERGEFORMATINET Bµi 25. Cho n lµ mét sè nguyªn d-¬ng: a. TÝnh :
I = EMBED Equation.3 b. TÝnh tæng: INCLUDEPICTURE
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MERGEFORMATINET Bµi 26. T×m sè nguyªn d-¬ng n sao cho:
INCLUDEPICTURE
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MERGEFORMATINET Bµi 27. T×m sè nguyªn d-¬ng n sao cho:
INCLUDEPICTURE
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MERGEFORMATINET Bµi 28. T×m sè tù nhiªn n th¶o m·n ®¼ng thøc sau:
INCLUDEPICTURE
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MERGEFORMATINET Bµi 29. TÝnh tæng: INCLUDEPICTURE
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MERGEFORMATINET , biÕt r»ng, víi n lµ sè nguyªn d-¬ng:
INCLUDEPICTURE
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MERGEFORMATINET Bµi 30. T×m sè nguyªn d-¬ng n sao cho:
INCLUDEPICTURE
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MERGEFORMATINET
17. Bµi 31. T×m hÖ sè cña x8 trong khai triÓn thµnh ®a thøc cña:
INCLUDEPICTURE
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MERGEFORMATINET Bµi 32. Gäi a3n - 3 lµ hÖ sè cña x3n - 3 trong khai
triÓn thanh ®a thøc cña:(x2 + 1)n(x + 2)n. T×m n ®Ó a3n - 3 = 26n Bµi
33. T×m hÖ sè cña sè h¹ng chøa x26 trong khai triÓn nhÞ thøc Newton cña
EMBED Equation.3 BiÕt r»ng: EMBED Equation.3 Bµi 34. T×m c¸c
sè h¹ng kh«ng chøa x trong khai triÓn nhÞ thøc Newton cña:
INCLUDEPICTURE
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MERGEFORMATINET víi x 0 Bµi 35. T×m sè h¹ng thø 7
trong khai triÓn nhÞ thøc: INCLUDEPICTURE
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MERGEFORMATINET ; INCLUDEPICTURE
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MERGEFORMATINET Bµi 36. Cho : INCLUDEPICTURE
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MERGEFORMATINET Sau khi khai triªn vµ rót gän th× biÓu thøc A sÏ gåm
bao nhiªu sè h¹ng? Bµi 37. T×m hÖ sè cña sè h¹ng chøa x8 trong khai triÓn
nhÞ thøc Newton cña INCLUDEPICTURE
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MERGEFORMATINET ,
b i ¿-t
r ±-n g : I N C L U D E P I C T U R E
h t t p : / / w w w . o n t h i . c o m / i m a g e s / c t / e 4 b 7
c 6 a 6 0 2 e 0 c 0 0 f 1 0 5 8 7 8 3 b 0 c 8 2 3 5 0 a . j p g *
M E R G E F O R M A T I N E T B µ i 3 8 . k h a i
t r i Ó n b i Ó u t h ø c ( 1 - 2 x ) n t a ® - î c ® a
t h ø c c ã d ¹ n g : I N C L U D E P I C T U R E
h t t p : / / w w w . o n t h i . c o m / i m a g e s / c t / 6 c 2 5
3 4 b 0 e 6 5 e 5 7 0 9 d 4 8 6 a f 5 5 c 0 4 7 7 5 c 0 . j p g *
M E R G E F O R M A T I N E T . T ì m h Ç- s Ñ- c ç-a
I N C L U D E P I C T U R E
h t t p : / / w w w . o n t h i . c o m / i m a g e s / c t / b 1 c d
1 0 6 a f a 3 7 8 f f 9 e 8 c c 4 2 b 5 3 8 0 b 4 e 1 8 . j p g *
M E R G E F O R M A T I N E T , b i ¿-t a o + a 1 + a 2 =
7 1 B µ i 3 9 . T × m h Ö s è c ñ a x 5 t r o n g
k h a i t r i Ó n ® a t h ø c :
I N C L U D E P I C T U R E
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* MERGEFORMATINET Bµi 40. T×m sè h¹ng kh«ng chøa x trong khai triÓn
nhÞ thøc EMBED Equation.3 BiÕt r»ng: INCLUDEPICTURE
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MERGEFORMATINET Bµi 41. Gi¶i c¸c ph-¬ng tr×nh: INCLUDEPICTURE
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MERGEFORMATINET INCLUDEPICTURE
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MERGEFORMATINET INCLUDEPICTURE
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MERGEFORMATINET INCLUDEPICTURE
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MERGEFORMATINET Bµi 42. Gi¶i c¸c hÖ ph-¬ng tr×nh: INCLUDEPICTURE
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19. chuyªn ®Ò 3. Ph-¬ng ph¸p quy n¹p To¸n häc Bµi 1. Chøng minh r»ng a) 1.2 +
2.5 + 3.8 + ... + n(3n - 1) = n2(n + 1) víi n ( N* b) 3 + 9 + 27 +
... + 3n = EMBED Equation.DSMT4 (3n + 1 - 3) víi n ( N* c) 12 + 32
+ 52 + ... + (2n - 1)2 = EMBED Equation.DSMT4 víi n ( N* d)
13 + 23 + 33 + ... + n3 = EMBED Equation.DSMT4 víi n ( N* e) 12
+ 22 + 32 + ... + n2 = EMBED Equation.DSMT4 víi n ( N* f)
EMBED Equation.3 víi n ( N* g) EMBED Equation.3 víi
n ( N* h) EMBED Equation.3 víi n ( N* i) EMBED
Equation.3 víi n ( 2 k) EMBED Equation.3 víi n (
N* Bµi 2. Chøng minh r»ng víi mäi n ( N* ta cã: a) n3 + 2n chia hÕt cho
3 b) n3 + (n + 1)3 + (n + 2)3 chia hÕt cho 9 c) n3 + 11n chia hÕt cho
6 d) 2n3 - 3n2 + n chia hÕt cho 6 e) 4n + 15n - 1 chia hÕt cho 9 f) 32n +
1 + 2n + 2 chia hÕt cho 7 g) n7 - n chia hÕt cho 7 h) n3 + 3n2 + 5n chia
hÕt cho 3 Bµi 3. Chøng minh c¸c bÊt ®¼ng thøc sau a) 2n + 2 2n + 5
víi n ( N* b) 2n 2n + 1 víi n ( N*, n ( 3 c) 3n n2 + 4n
+ 5 víi n ( N*, n ( 3 d) 2n - 3 3n - 1 víi n ( 8 e) 3n - 1 n(n +
2) víi n ( 4
20. Chuyªn ®Ò 4: d·y sè D¹ng 1. X¸c ®Þnh mét sè sè h¹ng cña d·y sè. X¸c ®Þnh
sè h¹ng tæng qu¸t Bµi 1. ViÕt 5 sè h¹ng ®Çu cña d·y sè sau: a) un =
EMBED Equation.DSMT4 b) un = EMBED
Equation.DSMT4 b) EMBED Equation.DSMT4 (n 2) c)
un = EMBED Equation.DSMT4 d) EMBED Equation.DSMT4 (víi k (
1) e) u1 = 2; un + 1 = EMBED Equation.DSMT4 (un + 1) g) un =
cos EMBED Equation.DSMT4 h) nsin EMBED
Equation.DSMT4 + n2cos EMBED Equation.DSMT4 Bµi 2. T×m sè h¹ng
tæng qu¸t cña d·y sè a) (un): 1; 2; 4; 8; 16; … b) (un): EMBED
Equation.DSMT4 ; c) (un):
… EMBED Equation.DSMT4 (víi n (
1) d) (un): EMBED Equation.DSMT4 ; Bµi 3. Cho d·y sè (un): u1 =
…
EMBED Equation.DSMT4 , un+ 1 = 4un + 7 víi n ( 1 a) TÝnh u2, u3, u4,
u5, u6 b) Chøng minh r»ng: un = EMBED Equation.DSMT4 víi n (
1 Bµi 4. Cho d·y sè (un): u1 = 1; un + 1 = un + 7 víi ( 1 a) TÝnh u2, u3,
u4, u5, u6 b) Chøng minh r»ng: un = 7n Ŕ 6 Bµi 5. Cho (un): u1 = 2; un +
1 = 3un + 2n Ŕ 1 Chøng minh r»ng: un = 3n - n D¹ng 2. XÐt tÝnh ®¬n ®iÖu
cña mét d·y sè Bµi 6. XÐt tÝnh ®¬n ®iÖu cña c¸c d·y sè sau a) un =
EMBED Equation.DSMT4 ; b) un = EMBED Equation.DSMT4
c) un = EMBED Equation.DSMT4 d) un = EMBED
Equation.DSMT4 e) un = EMBED Equation.DSMT4
f) un = EMBED Equation.DSMT4 g) un = EMBED
Equation.DSMT4 h) un = EMBED Equation.DSMT4 D¹ng 4. XÐt
tÝnh bÞ chÆn cña d·y sè Bµi 7. XÐt tÝnh bÞ chÆn cña c¸c d·y sè a) un = 2n
Ŕ 1 b) un = EMBED Equation.DSMT4 c) un =
3.22n Ŕ 1 d) un = EMBED Equation.DSMT4 e) un = EMBED
Equation.DSMT4 f) un = EMBED Equation.DSMT4 bµi tËp
tù luyÖn Bài 1. tìm các
gi Û-i
h ¡-n s a u :
21. E M B E D E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4
23. E M B E D E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4
25. E M B E D E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4
26. E M B E D E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4
28. E M B E D E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4 E M B E D
E q u a t i o n . D S M T 4
29. B à i 5 . t ì m c á c g i Û-i h ¡-n s a u : 1 .
E M B E D E q u a t i o n . D S M T 4 2 .
E M B E D E q u a t i o n . D S M T 4 5 . l i m
E M B E D E q u a t i o n . D S M T 4 B à i 6 t ì m
c á c g i Û-i h ¡-n s a u : E M B E D
E q u a t i o n . D S M T 4 2 . E M B E D
E q u a t i o n . D S M T 4 3 . E M B E D
E q u a t i o n . D S M T 4 4 . E M B E D
E q u a t i o n . D S M T 4 6 . E M B E D
E q u a t i o n . DSMT4 6. EMBED
Equation.DSMT4
31. CHUY£N §Ò 6. ®¹o hµm I. TÝnh ®¹o hµm b»ng ®Þnh nghÜa Bµi 1. Dïng ®Þnh
nghÜa tÝnh ®¹o hµm cña c¸c hµm sè sau t¹i c¸c ®iÓm: 1) f(x) = 2x2 + 3x +
1 t¹i x = 1 2) f(x) = sinx t¹i x = EMBED Equation.DSMT4 3) f(x) =
EMBED Equation.DSMT4 t¹i x = 1 4) f(x) = EMBED Equation.DSMT4
t¹i x = 0 5) f(x) = EMBED Equation.DSMT4 t¹i x = 2 6) f(x) =
EMBED Equation.DSMT4 t¹i x = 0 7) f(x) = EMBED Equation.DSMT4
t¹i x = 0 8) f(x) = EMBED Equation.DSMT4 t¹i x = 0 Bµi 2.
Dïng ®Þnh nghÜa tÝnh ®¹o hµm cña c¸c hµm sè sau: 1) y = 5x Ŕ 7 2) y
= 3x2 Ŕ 4x + 9 3) y = EMBED Equation.DSMT4 4) y = EMBED
Equation.DSMT4 5) y = x3 + 3x Ŕ 5 6) y = EMBED
Equation.DSMT4 + x II. Quan hÖ gi÷a tÝnh liªn tôc vµ sù cã ®¹o
hµm Bµi 3. Cho hµm sè f(x) = EMBED Equation.DSMT4 Chøng minh
r»ng hµm sè liªn tôc trªn R nh-ng kh«ng cã ®¹o hµm t¹i x = 0. Bµi 4. Cho
hµm sè f(x) = EMBED Equation.DSMT4 1) Chøng minh r»ng hµm sè
liªn tôc trªn R 2) Hµm sè cã ®¹o hµm t¹i x = 0 kh«ng? T¹i sao?. Bµi 5.
Cho hµm sè f(x) = EMBED Equation.DSMT4 T×m a, b ®Ó hµm sè cã ®¹o
hµm t¹i x = 1 Bµi 6. Cho hµm sè f(x) = EMBED Equation.DSMT4 T×m
a, b ®Ó hµm sè cã ®¹o hµm t¹i x = 0 Bµi 7. Cho hµm sè f(x) = EMBED
Equation.DSMT4 T×m a ®Ó hµm sè kh«ng cã ®¹o hµm t¹i x = 3. III.
TÝnh ®¹o hµm b»ng c«ng thøc: Bµi 8. TÝnh ®¹o hµm cña c¸c hµm sè sau:
1) y = EMBED Equation.DSMT4 x3 Ŕ 2x2 + 3x 2) y = - x4 +
2x2 + 3 3) y = (x2 + 1)(3 Ŕ 2x2) 4) y = (x Ŕ 1)(x Ŕ 2)(x Ŕ 3)
5) y = (x2 + 3)5 6) y = x(x + 2)4 7) y = 2x3 Ŕ 9x2 + 12x Ŕ 4
8) y = (x2 + 1)(x3 + 1)2(x4 + 1)3 Bµi 9. TÝnh ®¹o hµm cña c¸c hµm
sè sau : 1) y = EMBED Equation.DSMT4 2) y = EMBED
Equation.DSMT4 3) y = EMBED Equation.DSMT4 4) y =
EMBED Equation.DSMT4 5) y = EMBED Equation.DSMT4 6) y
= EMBED Equation.DSMT4 7) y = EMBED Equation.DSMT4
8) y = EMBED Equation.DSMT4 Bµi 10. TÝnh ®¹o hµm cña c¸c hµm
sè sau: 1) y = EMBED Equation.DSMT4 2) y = EMBED
Equation.DSMT4 3) y = (x Ŕ 2) EMBED Equation.DSMT4 4) y
= EMBED Equation.DSMT4 5) y = EMBED Equation.DSMT4
6) y = x + EMBED Equation.DSMT4 7) y = EMBED
Equation.DSMT4 8) y = EMBED Equation.DSMT4 + EMBED
Equation.DSMT4 III. ViÕt ph-¬ng tr×nh tiÕp tuyÕn cña då thÞ t¹i mét
®iÓm Bµi 11. Cho hµm sè y = EMBED Equation.DSMT4 x3 Ŕ 2x2 + 3x
(C) 1) ViÕt ph-¬ng tr×nh tiÕp tuyÕn ( víi ®å thÞ (C) t¹i ®iÓm cã hoµnh ®é
lµ x = 2. 2) Chøng minh r»ng ( lµ tiÕp tuyÕn cã hÖ sè gãc nhá nhÊt Bµi
12. Cho hµm sè y = -x3 + 3x + 1 (C) 1) ViÕt ph-¬ng tr×nh tiÕp tuyÕn (
cña (C) t¹i ®iÓm cã hµnh ®é lµ x = 0 2) Chøng minh r»ng tiÕp tuyÕn ( lµ
tiÕp tuyÕn cña (C) cã hÖ sè gãc lín nhÊt. Bµi 13. 1) ViÕt ph-¬ng tr×nh
tiÕp tuyÕn víi ®å thÞ cña hs: y = x3 Ŕ 3x2 + 2 t¹i ®iÓm (-1; -2) 2) ViÕt
ph-¬ng tr×nh tiÕp tuyÕn víi ®å thÞ cña hµm sè y = EMBED Equation.DSMT4
t¹i ®iÓm cã hoµnh ®é x = 0 IV. ViÕt ph-¬ng tr×nh tiÕp tuyÕn cña ®å
thÞ (C) khi biÕt hÖ sè gãc k. Bµi 14. 1) ViÕt ph-¬ng tr×nh tiÕp tuyÕn víi
®å thÞ cña hµm sè y = EMBED Equation.DSMT4 biÕt hÖ sè gãc cña tiÕp
tuyÕn lµ EMBED Equation.DSMT4 . 2) ViÕt
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ŦŠ„Š„ꊄŠ„ţŠ„‚p^RJJ j hÒO- U mH sH
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f 5 •CJ OJ QJ mH sH # h Uf hÒO- 5 •CJ OJ QJ mH sH h Uf hê-
f CJ mH sH jªţ h¥?¶ hê-f EHúÿU aJ + j-å!H
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hÒO- CJ OJ QJ U V aJ
168. hÒO- mH sH X Y Z ` i j • ‚ ƒ „ Œ • ¤ ¥ ¦
§ ¨ ¬ ® ² ³ Ê Ë Ì Í Ð Ñ è îâ×Ïâı âÄâÄ•|âqÄÏÄ
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169. hÒO- mH sH hfE hÒO- mH sH j hÒO- U mH sH ! j§À hü Ö
hÒO- EHøÿU mH sH è é ê ë ð ñ ô ö ý þ
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sH%j¤L
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mHsH%j¤L
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171. hÒO- mH sH h[5x hÒO- mH sH h`' hÒO- 5 •mH sH h[5x hÒO-
H* mH sH
172. hÒO- mH sH
j pð hÒO- mH sH h[5x hÒO- mH sH j hÒO- U mH sH !
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hÒO- CJ OJ QJ U V aJ
# $ % ' 0 1 B C Z [ ] p q x y } ~ ¤
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h[5x hÒO- mH sH
177. „zÿd ]„zÿgdÓO- ù ú 1 2 I J K L S T k
l m n € … Ÿ • ţ Ÿ ¢ ¥ § ª « - ® öîöîâ
“϶ˆ“ˆ“«šˆ“•â•|kâ•`X`L`X hŸH² hÒO- H* mH sH
178. hÒO- mH sH hŸH² hÒO- mH sH ! jSð h)ţ hÒO- EHäÿU mH sH %
j«¤L
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‡ÿUmHsH%jŒ¤L
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hÒO- CJ OJ QJ U V aJ j hÒO- U mH sH
179. hÒO- mH sH hÒO- H* mH sH ® Á Ã Æ Ç Û Ý ß á ä
å è é î ï ó ô
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181. - - - - - - -- - !- $- %- )- - =- - ?- K- L- c-
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æÿUmHsH%j0¤L
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hÒO-
sH
185. hÒO- mH sH h)ţhÒO- mH sH j hÒO- U mH sH ! jsú h)ţ
hÒO- EHøÿU mH sH ? @ A L M d e f g m n … Ÿ ‡ ˆ-
‹ Œ £ ¤ ¥ ¦ - ® Å Æ-
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hÒO- CJ OJ QJ U V aJ
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% j1-¤L
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- EHøÿUmHsH!j…
h)ţ hÒO- EHøÿU mH sH h•W# hÒO- mH sH
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hÒO- CJ OJ QJ U V aJ h8QÎ hÒO- mH sH hÒO- H* mH sH
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%jª ¤L
hÒO- CJ OJ QJ U V aJ • ŗ ¬ - ´ µ · ¸ Ë Ì Õ
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192. hÒO- mH sH hfE hÒO- H* mH sH hfE hÒO- mH sH h Uf hÒO-
mH sH
193. hÒO- mH sH h`' hÒO- 5 •mH sH #a# b# c# d# l# m# }# ~#
ƒ# „# ›# œ# •# ţ# ª# «# º# »# Ä# Å# Ü# Ý# Þ# ß# ì# í#
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hÒO- CJ OJ QJ U V aJ h•W# hÒO- mH sH hÒO- H* mH sH
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%jž!¤L
hÒO- CJ OJ QJ U V aJ U$ V$ ^$ c$ |$ „$ …$ ‹$ Œ$ £$ ¤
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ŗ% œ% •% £% ¨% Ç% È% è% P Q y Ÿ ‡ ţ Ÿ ¡ ´
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h-Yþ CJ OJ QJ U V aJ hGL h-Yþ mH sH j h-Yþ Uj?ŗ
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