Natural va butun sonlar

6 - §. Sonning butun va kasr qismi.
x haqiqiy sonning butun qismi deb, x dan oshmaydigan eng katta butun songa aytiladi.
X sonining butun qismi [x] ko’rinishida yoziladi.
Misol uchun [3,8] = 4, [-3,8] = -4
Har qanday x haqiqiy son uchun quyidagi tenglik o’rinlidir.
[x] x <[x]+1 >
X haqiqiy sonining kasr qismi deb, x va uning butun qismi ayirmasiga aytiladi.
X sonining kasr qismi {x} ko’rinishida yoziladi.
Har qanday x haqiqiy son uchun quyidagi tengliklar o’rinlidir.
x = [x] + {x}, {x} = x - [x]
Misol uchun {3,8} = 3,8-3 = 0,8 , {-3,8} = (-3,8) – (-4) =0,2
X haqiqiy sonning butun qismi doimo butun son bo’ladi: Ya’ni [x] Z
X haqiqiy sonning kasr qismi doimo quyidagi shartni qanoatlantiradi.
0 {x}<1 >

X haqiqiy son va n butun sonlar uchun quyidagi tenglik doimo qanoatlantiradi.

[x+n] = [x] + n
Misol uchun [3,8 + 4] = [3,8]+4

6-Mashq.
1.[x-1] + [x+3] = 12 tenglamani yeching
2. [x] – [x+2] +[x+4] = 18 tenglamani yeching.
3. [-2x] + [1-2x] + [3-2x] = 1 tenglamani yeching.
4. [x] = x tenglamani yeching.
5. [2x-1] = x+1 tenglamani yeching.
6. [ ] = tenglamani yeching.
7. [3x² -x] = x+1 tenglamani yeching.
8. [ x²] = x tenglamani yeching.
9. Agar a sonini m soniga bo’lganda qoldiq r bo’lsa,
[ ] = tenglik to’g’ri bo’lishini ko’rsating.
10.Agar nN shart o’rinli bo’lsa, tenglik to’g’ri bo’lishini ko’rsating.
11. Agar EKUB (a;4) = 1 shart o’rinli bo’lsa, tenglik to’g’ri bo’lishini ko’rsating.
12. [x] = 3x+1 tenglamani yeching.
13. [x]{x} = 1 tenglamani yeching.
14. {x} = 1-x tenglamani yeching.
15. x = 2[x] –{x} tenglamani yeching.

7 - §. Algebraik ifodalar
Ko’phadlarning umumiy ko’rinishi P(x) = a_nxⁿ +a_n-1x^n-1+…….+a₂x² + a₁x+a₀ shaklda bo’ladi. Agar x=1 qiymatni P(x) ko’phadga qo’ysak
P(1) = a_n1ⁿ +a_n-11^n-1+…….+a₂ 1² + a₁ 1+a₀ = a_n +a_n-1+a_n-2+……+a₂ + a₁ + a₀ (1)
ko’rinishda bo’ladi. Bundan shuni xulosa qilish mumkinki ko’phadning koeffitsientlari yig’indisini topish uchun P(1) ko’phadning qiymatini hisoblash kerak ekan.
Faraz qilaylik agar n juft son bo’lsa, x=-1 qiymatda P(-1) = a_n(-1)ⁿ +a_n-1(-1)^n-1+…….+a₂ (-1)² + a₁ (-1)+a₀ = a_n -a_n-1+a_n-2+……+a₂ - a₁ + a₀ (2)
shaklda bo’ladi.
Agar yuqoridagi (1) va (2) ifodalarni qo’shib yuborsak
P(1)+P(-1) = 2a_n + 2a_n-2 + ……+2a₂ +2a₀ (3)
shakl hosil bo’ladi. Bu yuqoridagi (3) ifodani quyidagicha shaklda yozib olamiz.
a_n + a_n-2 + ……+a₂ +a₀ = (P(1)+P(-1)) . Bu hosil bo’lgan tenglik P(x) ko’phaddagi x ning juft darajalari oldidagi koeffitsiyentlari yig’indisi deyiladi.
Agar yuqoridagi (1) va (2) ifodalarni ayirib yuborsak
P(1) – P(-1) = 2a_n-1 + 2a_n-3 +…..+2a₃ +2a₁ (4)
tenglik hosil bo’ladi. Bu yuqoridagi (4) ifodani quyidagicha shaklda yozib olish mumkin.
a_n-1 + a_n-3 +…..+a₃ +a₁ = (P(1)-P(-1)) . Bu hosil bo’lgan tenglik P(x) ko’phaddagi x ning toq darajalari oldidagi koeffitsiyentlari yig’indisi deyiladi. Biz n son juft bo’lgan holatdagi ko’phad bo’yicha ko’rib chiqdik. Yuqoridagi hosil bo’lgan natijaviy shartlar n toq son bo’lgan holatda ham o’rinli bo’ladi. Bunga o’zingiz mustaqil ravishda ishonch hosil qilib ko’ring.
Sizga algebraik ifodalarning boshlang’ich nazariyasidan qisqa ko’paytirish formulalarining oddiy shakllari ya’ni (ab)² va (ab)³ ko’phadlar yoyilmasi shaklda yozish ma’lumdir.
Bu qisqa ko’paytirish formulalarni ko’phad yoyilmasi shaklda yozishda siz darajada qaysi son turgan bo’lsa, shuncha ko’paytuvchilarga ajratib so’ngra ko’paytmani hisoblash kerak edi. Ya’ni
(ab)² = (ab) (ab) = a²2ab+b²
(ab)³ = (ab) (ab) (ab)= a³3a²b+3ab²b³
Siz bu amallarni har doim takrorlamasligiz uchun oxirida hosil bo’lgan ko’phad yoyilmasini yod olishga majbur edingiz. Lekin siz bu yuqoridagi amallarni darajasi 2,3 yoki ko’pi bilan 4 bo’lgan holatda bajara olishingiz mumkin. Siz bilan birgalikda umumiy holatda (a+b)ⁿ ko’phad yoyilmasi shaklda yozishda 2 ta buyuk matematik Paskal va Nyutonning ishlarini ko’rib chiqamiz.

1. Paskal aniqlagan usul.

   
   

1
1 1

1 2 1
1 3 3 1

1. 4 6 4 1
   

1 5 10 10 5 1
1 6 15 20 15 6 1
……………………….
Bu yuqoridagi ajoyib uchburchak matematikada Paskal uchburchagi deb nom olgan. Bu uchburchakning har bir qatoridagi son o’zidan oldingi turgan qatorning o’sha o’rindagi va bitta oldingi sonning yig’indisiga teng. Ya’ni 6 – qatorning 3 o’rnida 10 soni turibdi. Bu son 5 - qatorning 3 o’rnidagi va 2 o’rnidagi sonlar yig’indisiga tengdir.
Bu uchburchakning har bir qatoridagi sonlar (a+b)ⁿdarajani ko’phad yoyilmasi shaklda yoyishdagi koeffitsiyentlarga tengdir.
Misol: (a+b)⁵ va (a-b)⁵ darajalarni Paskal uchburchagi yordamida ochib chiqamiz.
(a+b)⁵ = a⁵+5a⁴b+10a³b²+10a²b³+5ab⁴+b⁵
(a-b)⁵ = (a+(-b))⁵ = a⁵ +5a⁴(-b)+10a³(-b)²+10a²(-b)³+5a(-b)⁴+(-b)⁵= a⁵-5a⁴b+10a³b²-10a²b³+5ab⁴-b⁵

2)Nyuton aniqlagan usul.

Bu Nyuton tomonidan aniqlangan usul Nyuton binomi deyiladi. - binomial koeffitsiyent deyiladi va tenglik orqali hisoblanadi.

Misol: (x+y)⁶ darajani Nyuton binomi usulida ochib chiqamiz.


=x⁶+6x⁵y+15x⁴y²+20x³y³+15x²y⁴+6xy⁵+y⁶

1. Ko’phadlarni ko’paytuvchilarga ajrating.

   
   

1.1) x⁴+x²+1 1.2) x¹⁶-1 1.3) x⁸+x⁴+1 1.4) 2x⁴+x³+4x²+x+2 1.5) x⁴-12x³+47x²-60x 1.6) x⁵+x⁴+1 1.7) y²z²(y²-z²) + x²z²(z²-x²)+x²y²(x²-y²) 1.8) x(y²-z²)+y(z²-x²)+z(x²-y²) 1.9) (b-c)(b+c)²+(c-a)(c+a)²+(a-b)(a+b)² 1.10) a(b-c)³+b(c-a)³+c(a-b)³ 1.11) x(y+z)(y²-z²) + y(z+x)(z²-x²)+z(y+x)(x²-y²) 1.12) x⁶+27 1.13) x⁴+3x²+4 1.14) (x+2)(x+3)(x+4)(x+5)-24 1.15) x⁴+4y⁴ 1.16) 27x³-27x²+18x-4 1.17) (x+1)(x+3)(x+5)(x+7)+15 1.18) x³+y³+z³-3xyz 1.19) (x²+x+1)² + 3x(x²+x+1)+2x² 1.20) x²y²-x²+4xy-y²+1 1.21) [a(x+y)+b(x-y)]²-[a(x-y)+b(x+y)]² 1.22) x⁶-y⁶+x⁴+x²y²+y⁴ 1.23) 9x³+7x²-7x-9 1.24) 2x⁴+7x³-2x²-13x+6 1.25) x(x+1)(x+2)(x+3)+1 1.26) a⁴+b⁴+c⁴-2a²b²-2a²c²-2b²c² 1.27) (x²+x-1)(x²+3x-1)+x² 1.28) x⁴+5x²+9 1.29) (x+1)⁴ +2(x+1)³+x(x+2) 1.30) x³+9x²+11x-21 1.31) x²-3xy+2y²-2x+4y 1.32) x³+2y³-3xy² 1.33) (x+y)⁴ +x⁴+y⁴ 1.34) x²y + xy²+x²z+xz²+y²z+yz²+2xyz 1.35) x⁵+x+1 1.36) x⁵+x⁴+1 1.37) a⁴(b-c) + b⁴(c-a)+c⁴(a-b) 1.38) (x-y)(y+z)(z+x)+(y-z)(z+x)(x+y)+(z-x)(x+y)(y+z) 1.39) x⁹ + 5x⁵+ x⁴+4x+4

2. Quyidagi darajalarni ko’phad shaklni aniqlang.

2.1) (x+2y)⁶ 2.2) (x-2y)⁶ 2.3) 2.4) 2.5) 2.6) 2.7) 2.8)

3. Berilgan binom yoyilmasida x qatnashmagan had koeffitsiyentini toping.
4. Agar binom yoyilmasida beshincha had x ga bog’liq bo’lmasa, n ning qiymatini aniqlang.
5. (1+x-3x²)²⁰¹⁸ ifodada qavslarni ochib, o’xshash hadlari ixchamlagandan so’ng hosil bo’lgan ko’phadning koeffitsiyentlari yig’indisini toping.
6. (1+x-3x²)²⁰¹⁸ ifodada qavslarni ochib, o’xshash hadlari ixchamlagandan so’ng hosil bo’lgan ko’phadning juft darajalari oldidagi koeffitsiyentlari yig’indisini aniqlang.
7. (1+x-3x²)²⁰¹⁸ ifodada qavslarni ochib, o’xshash hadlari ixchamlagandan so’ng hosil bo’lgan ko’phadning toq darajalari oldidagi koeffitsiyentlari yig’indisini aniqlang.
8. (1+x)ⁿ +(1+x)ⁿ⁺¹ yoyilmaning koeffitsiyentlari yig’indisi 1536 ga teng bo’lsa, x⁶ ning oldidagi koeffitsiyentini aniqlang.
9. (x+3)¹⁰⁰ = a₀+a₁x +……+a₁₀₀x¹⁰⁰ shaklda bo’lsa, quyidagilarning qiymatini aniqlang.
9.1) a₄₅ 9.2) a₀+a₁ +……+a₁₀₀ 9.3) a₀-a₁+a₂-a₃+……+a₁₀₀
2. Agar nol bo’lmagan a va b haqiqiy sonlar uchun ab=a-b shart o’rinli bo’lsa, ifodaning qiymatini toping.
A)-2 B) C) D)2
10. ni soddalashtiring

11. ni soddalashtiring

12. c ni soddalashtiring

13. ni soddalashtiring

14. ni soddalashtiring
(A) 1 B) 0 C) D)
15. ni soddalashtiring.
A) x+y+z B)x+y-z C)0 D)1

16. ni soddalashtiring.

A)1 B) C) D)
17. Agar a- 3 = x+4y bo’lsa, ax + 3x + 4ay + 12y + 9 ifodaning qiymatini toping.

A) x² B) y² C)a² D)2a²

18. ifodaning x = 6bo’lgandagiqiymatini toping.
A) 3 B)2 C)1,5 D)1,25

19. ifodaning x = 10 bo’lgandagiqiymatini toping.

A) 13/9 B) 7/5 C) 15/11 D) 4/3

20. ifodani x=1 bo’lgandagiqiymatini toping.

A)1 B)0,5 C)2 D)3

21. ifodanisoddalashtiring.
A)1 B)x C) x²⁰¹⁸D) x²⁰¹⁹
22. ifodanisoddalashtiring.
A)1 B)xC)x²⁰²⁰D)x²⁰¹⁹
23.Agar x² + x+1=0 bo’lsa, x²⁶ + x^–26ifodaningqiymatini toping.

A) -1 B)2 C)0 D)1

24. Agar x² + x+1=0 bo’lsa, x²⁹ + x^–29 ifodaningqiymatinitoping.

A) -1 B)2 C)0 D)1

25. Agar x² + x+1=0 bo’lsa, x³² + x^–32ifodaningqiymatini toping.
A) 1 B)2 C)0 D)– 1
26. Agar x² + x+1=0 bo’lsa, x⁹⁹ + x^–99ifodaningqiymatini toping.
A) 1 B)2 C)0 D)–1
27. Agar x² + x+1=0 bo’lsa, x³³³³ + x³³³ + x³³ + x³ +2018 ifodaningqiymatini toping.
A)2018 B)2020 C)2022 D)2014
28.(a² + b²)(x² + y²) = P²(x,y) + Q²(x, y) ayniyatbajarilsa, P(0;0) + Q(0; 0) ifodaningqiymatini toping.

A) 0 B) a+b C) a–b D) 2a

29. (a² + b²)(x² + y²) = P²(x,y) + Q²(x,y)ayniyatbajarilsa, P(1;0) + Q(1; 0) ifodaningqiymatinitoping.
A) 0B) a+b C) a– b D) 2a
30. (a² + b²)(x² + y²) = P²(x,y) + Q²(x, y) ayniyatbajarilsa, P(0; 1) + Q(0; 1) ifodaningqiymatini toping.
A) 0B) a+bC) a-b D) 2a
31. (a² + b²)(x² + y²) = P²(x,y) + Q²(x, y) ayniyatbajarilsa, P(1;1) + Q(1; 1) ifodaningqiymatini toping.
A) 0B) a+b C) a– b D) 2a
32. n ning qanday qiymatlarida 1+x²+x⁴+……+x^2n-2 ifoda 1+x+x²+x³+……+x^n-1 ifodaga qoldiqsiz bo’linadi?
A) n=1 , n=3 B) n=2, n=4 C) barcha toq sonlarda D) barcha juft sonlarda
33. 1+x²+x⁴+……+x^2n-2 ifoda 1+x+x²+x³+……+x^n-1 ifodaga qoldiqsiz bo’linadigan n ning barcha natural qiymatlari yig’indisini toping.
A) 4 B) 6 C) bunday qiymatlar mavjud emas D) cheksiz kattalikka ega
34. ayniyatdan B ni toping.
A)2 B)3 C)1 D) -1
35. Yig’indini hisoblang.

A)0 B)1 C) a+b+c D) aniqlab bo’lmaydi

36. Yig’indini hisoblang.

A)0 B)1 C) a+b+c D) aniqlab bo’lmaydi

37. Yig’indini hisoblang.



A)0 B)1 C) a+b+c D) aniqlab bo’lmaydi
38. Yig’indini hisoblang.


A)0 B)1 C) a+b+c D) aniqlab bo’lmaydi

39. Yig’indini hisoblang.

A)0 B)1 C) a+b+c D) aniqlab bo’lmaydi

40. Yig’indini hisoblang.

A)0 B)1 C) a+b+c D) aniqlab bo’lmaydi

41. Yig’indini hisoblang.

A)0 B)1 C) a+b+c D) aniqlab bo’lmaydi

42. P( ) = x²+x+1 shartniqanoatlantiruvchi P(x) ko’phadberilgan. P(3x) = 7 shartiniqanoatlantiradiganbarchax ningqiymatlariyig’indisini toping.

A)–1/9 B) 1/9 C)–2/27 D) 2/27

43. P( ) = x²+x+1 shartniqanoatlantiruvchi P(x) ko’phadberilgan. P(3x) = 7 shartiniqanoatlantiradiganbarchax ningqiymatlariko’paytmasini toping.
A)– 1/9 B)1/9 C)– 2/27 D) 2/27
44. P(x) = (1+2x–x²)⁴ko’phadning x⁷oldidagikoeffitsiyentinianiqlang.
A)– 6 B)6 C)8 D)– 8
45. Agar 2²⁰⁰⁷ – 2²⁰⁰⁶ – 2²⁰⁰⁵ + 2²⁰⁰⁴ = k∙2²⁰⁰⁴ifodaberilganbo’lsa, k ningqiymatini toping.
A) 1 B)2 C)3 D)4
46. Agar x² + y² = 1 bo’lsa, 3x⁴ + 2y⁴ +5x²y² + y²ningqiymatini toping.
A) 2 B)3 C)9 D)10
47. Agar x² + y² = 1 bo’lsa, 9x⁴ + 6y⁴ +15x²y² +3 y²ningqiymatini toping.
A) 6 B)18C)9 D)10
48. Agar x² + y² = 1 bo’lsa, 6x⁴ + 4y⁴ +10x²y² + 2y²ningqiymatini toping.
A) 4B)6 C)9 D)10
49. Agar x² + y² = 1 bo’lsa, 12x⁴ + 8y⁴ +20x²y² + 4y²ningqiymatini toping.
A) 8 B)12 C)16 D)10
50. (a+1)⁴ + 4 (a+1)³ + 4a(a+ 2) ko’phadnechtaratsionalhadliko’paytuvchilargaajraladi.
A) 3 B)2 C)4 D) ko’paytuvchilargaajralmaydi
51. (a+1)⁴ + 4 (a+1)³ + 4a(a+2)= P(x)∙Q(x) shaklidaberilganifodadan|P(x) – Q(x)|ningqiymatini toping. ( Bunda P(x) va Q(x) ratsionalhadliko’phadlar)
A) 8a+6 B) 2a² + 8a+6 C) 4 D) 8a+4
52. (a+1)⁴ + 4 (a+1)³ + 4a(a+2)= P(x)∙Q(x) shaklidaberilganifodadan P(x) +Q(x)ningqiymatini toping. ( Bunda P(x) va Q(x) ratsionalhadliko’phadlar)
A) 2a² + 8a+4 B) 2a² + 8a+6C) 4 D) 8a+4
53.a³ + b³ +c³ +3a²(b+c)+3b²(a+c)+3c²(a+b)+6abc ifodani ko’paytuvchilarga ajrating.
A) (a+b+c)(a²+b²+c²) B) (a+b+c)(a²+b²+c²-2ab-2bc-2ac) C) (a+b+c)³ D) ko’paytuvchilarga ajralmaydi
54. a³ + b³ +c³ +3a²(b+c)+3b²(a+c)+3c²(a+b)+6abc I fodani a=7, b=2, c=1 bo’lgandagi qiymatini toping.
A) 540 B) 80 C) 1000 D) 900
55. P(x)= (2x²-2x+1)¹⁹⁹⁹ (x³-2x²)²⁰⁰¹ ko’phadni barcha koeffitsientlarini yig’indisini toping.
A) 1 B)-1 C)0 D)2
56. P(x,y)=(5x²-xy-2y)¹² ko’phadni barcha koeffitsientlarini yig’indisini toping.
A)512 B)1024 C)2048 D)4096
57.Agar 3z+4(2─z)=8 bo’lsa, 1+ ifodaning qiymatini toping.
A)2,5 B)3,5 C)1,5 D)4,5
58.(a+b+c)³─a³─b³─c³ifodaquyidagilarningqaysibirigabo’linmaydi?
A) a+bB) a+b+c C) 3 D) barchasigabo’linadi
59.(a+b+c)³─a³─b³─c³ifodanechtako’paytuvchigaajraladi?
A)3 B)5 C)4 D)6
60. Agar a:b:c=2:4:7 bo’lsa, (7a+b)/(c─2) ningqiymatini toping.

A) aniqlabbo’lmaydi B)6 C)8 D)9

61. P(x) ko’phadni (x-1) ga bo’lganda qoldiq 2 chiqadi. Agar (x-2) ga bo’lganda qoldiq 1 bo’lsa, P(x) ko’phadni (x-1)(x-2) ga bo’lganda qoldiqni toping.

A) 3-x B) x+3 C)x-1 D)x-2

62. M=x³-7x²+2 ko’phad
M=(x+1)³+a(x+1)²+b(x+1)+c ko’phad ko’rinishida
keltirilsa, a ning qiymatini toping.
A)17 B)-6 C)-10 D) aniqlab bo’lmaydi
63. M=x³-7x²+2 ko’phad
M=(x+1)³+a(x+1)²+b(x+1)+c ko’phad ko’rinishida
keltirilsa, b ning qiymatini toping.

A)17 B)-6 C)-10 D) aniqlab bo’lmaydi

64. M=x³-7x²+2 ko’phad
M=(x+1)³+a(x+1)²+b(x+1)+c ko’phad ko’rinishida
keltirilsa, c ning qiymatini toping.
A)17 B)-6 C)-10 D) aniqlab bo’lmaydi
65. Soddalashtiring.



66. Soddalashtiring.



67. Soddalashtiring.


68. x, y va z haqiqiy sonlar bo’lsa, (x+y+z)³- (y+z-x)³-(z+x-y)³-(x+y-z)³ ifoda quyidagilardan qaysi biriga teng.
A)36 xyz B)24 xyz C)12 x²y²z² D) 12 xyz
69. x, y va z haqiqiy sonlar bo’lsa, (x-y)(y+z)(z+x)+(y-z)(z+x)(x+y)+(z-x)(x+y)(y+z) ifoda quyidagilardan qaysi biriga teng.
A)(x-y)(y-z)(z-x) B) -(x-y)(y-z)(z-x) C) (x+y)(y+z)(z-x) D) (x+y)(y+z)(z+x)
70.x, y va z haqiqiy sonlar bo’lsa, P(x,y,z)=x³+y³+z³-3xyz ko’phad uchun quyidagilardan qaysi biri qanoatlantiradi.
A) P(x,y,z)=P(x+y,y+z,z+x) B) P(x,y,z)=P(x-y,y-z,z+x) C) P(x,y,z)= P(x+y,y+z,z+x)
D) P(x,y,z)=P(x+2y,y+2z,z+2x)
71.x, y va z haqiqiy sonlar bo’lsa, P(x,y,z)=x³+y³+z³-3xyz ko’phad uchun quyidagilardan qaysi biri qanoatlantiradi.

A) P(x,y,z)= P(-x+y+z,x-y+z,x+y-z) B) P(x,y,z)=P(-x+y+z,x-y+z,x+y-z) C) P(x,y,z)= P(-x+y+z,x-y+z,x+y-z) D) P(x,y,z)=P(-2x+y+z,x-2y+z,x+y-2z)

72. a,b,c nol bo’lmagan haqiqiy sonlar uchun a+ shart qanoatlantirsa, ning qiymatini hisoblang.

A)2 B) 3 C)1 D) aniqlab bo’lmaydi.

73. a,b,c nol bo’lmagan haqiqiy sonlar uchun a+ shart qanoatlantirsa, ning qiymatini hisoblang.

A)2 B) 3 C)4 D) aniqlab bo’lmaydi.

74. ko’phadni ko’phadga bo’lgandagi qoldiqni toping.
A) B) x+1 C)0 D) aniqlab bo’lmaydi
75. a va b o’zaro tub sonlarning nechta juftida kasr qisqaradi.
A) 2 B) 1 C)bunday juftliklar mavjud emas D) aniqlab bo’lmaydi
76. Ifodani soddalashtiring.

A) a+b+c B) a+b-c C) a+2b-2c D) 2a-2b+2c

77. kasr P²(x;y)+3Q² (x;y) ko’rinishida yozilgan bo’lsa, P(x;y) ko’phadni aniqlang.

A) B) C) D) shart yetarli emas

78. kasr P²(x;y)+3Q² (x;y) ko’rinishida

yozilgan bo’lsa, Q(x;y) ko’phadni aniqlang.

A) B) C) D) shart yetarli emas
79. kasr P²(x;y)+3Q² (x;y) ko’rinishida

yozilgan bo’lsa, P(x;y) ko’phadni aniqlang.

A) B) C) D) shart yetarli emas
80. kasr P²(x;y)+3Q² (x;y) ko’rinishida

yozilgan bo’lsa, Q(x;y) ko’phadni aniqlang.

A) B) C) D) shart yetarli emas
81. kasr 5P²(x;y) -Q² (x;y) ko’rinishida

yozilgan bo’lsa, P(x;y) ko’phadni aniqlang.

A) B) C) D) aniqlab
bo’lmaydi
82. kasr 5P²(x;y) -Q² (x;y) ko’rinishida
yozilgan bo’lsa, Q(x;y) ko’phadni aniqlang.

A) B) C) D) aniqlab bo’lmaydi

83. kasr 5P²(x;y)-Q² (x;y) ko’rinishida yozilgan

bo’lsa, P(x;y) ko’phadni aniqlang.

A) B) C) D) aniqlab
bo’lmaydi

84. kasr 5P²(x;y)-Q² (x;y) ko’rinishida yozilgan

bo’lsa, Q(x;y) ko’phadni aniqlang.

A) B) C) D) aniqlab bo’lmaydi

85. Agar x+y+z=0 bo’lsa, x³+y³+z³ ning qiymati
quyidagilarning qaysi biriga doimo teng bo’ladi.
A) xyz B) 3xyz C) 3x²y²z² D) 0
86. (a+b+c)³-a³-b³-c³ ifoda nechta ratsional
ko’paytuvchilarga ajraladi.
A) 3 B)5 C)4 D)2
87. kasrni soddalash€tiring.
A) 3 B) C)1 D)
88. a va b sonlar uchun shart qanoatlantisa, kasrning qabul qilishi mumkin bo’lgan qiymatlarini toping.
A) 1 B)2 C)3 D) A va C
89. a va b sonlar uchun shart qanoatlantirsa, kasrning qabul qilishi mumkin bo’lgan qiymatlarini toping.
A) 1 B) C) D)2
90. Agar x² –x-1= 0 shart qanoatlantirsa, x³ - ning qiymatini toping.

A) 4 B)3 C)2 D)1

91. Agar x² –x-1= 0 shart qanoatlantirsa, x⁴ + ning qiymatini toping.

A) 7 B)5 C)6 D)4

90. a⁴ – 47 a²+1=0 shart qanoatlantirsa, ( a>0) ning qiymatini toping.
A) 2 B)3 C)4 D)1
92. Agar 16x² + 144x + (a+ b) to’lakvadrathamda b – a =– 7shartlarbajarilsa, a ningqiymatni toping.

A)165,5 B) 158,5 C)166 D)159

93. Agar 16x² + 144x + (a + b) to’lakvadrathamda b – a = – 7shartlarbajarilsa, b ningqiymatni toping.

A)165,5B) 158,5 C)166 D)159

94. ayniyatdana ningqiymatni toping.

A)1 B)2 C)0 D) aniqlabbo’lmaydi

95. ayniyatdanb ningqiymatni toping.
A)1B)2 C)0 D) aniqlabbo’lmaydi

96. ayniyatdanc ningqiymatni toping.

A)1B)2C)0 D) aniqlabbo’lmaydi

97. Soddalashtiring.

98. Agar n juft son bo’lsa, S = 1-2+3-4+…+(-1)ⁿ⁺¹n yig’indini hisoblang.
A) –n B) C) D)
99. Agar n toq son bo’lsa, S = 1-2+3-4+…+(-1)ⁿ⁺¹n yig’indini hisoblang.

A) –n B) C) D)

100. Agar n juft son bo’lsa, S = 1² -2² +3² -4² +…+(-1)ⁿ⁺¹n² yig’indini hisoblang.

A) B) C) D)

101. Agar n toq son bo’lsa, S = 1² -2² +3² -4² +…+(-1)ⁿ⁺¹n² yig’indini hisoblang.

A) B) C) D)

102. Agar a, b va c haqiqiy sonlar uchun shart o’rinli bo’lsa, ifodaning qabul qilishi mumkin bo’lgan qiymatlaridan eng kattasini toping.
A)0 B)-1 C)2 D) bir tomonlama aniqlanmaydi
103. . Agar a, b va c haqiqiy sonlar uchun shart o’rinli bo’lsa, ifodaning qabul qilishi mumkin bo’lgan qiymatlaridan eng kichigini toping.
A)0 B)-1 C)2 D) bir tomonlama aniqlanmaydi
104. Ko’phadningozodhadini toping.
f(x)=(3x²─1)³⁰(4x+1)¹⁵─x³⁰+25
A) 24 B) 25
C) 26 D) 125
105) Ko’phadningozodhadini toping.
f(x)=(4x³+1)¹⁵(3x⁴+1)⁴+(x─3)²+17
A) 24 B) 33 C) 27 D) 17
106) Ko’phadningozodhadini toping.
f(x)=(3x+1)³(4x+5)⁴(x+1)²⁰⁰+(x─1)¹⁹+19
A) 18 B) 20C) 643 D) 606