Natural va butun sonlar

(1-x₁)²+(x₁ -x₂)²+(x₂ -x₃)²+……+(x_n-1 –x_n)²+x_n² = tenglamani yagona yechimi borligini ko’rsating. Hamda bu yechimlarni o’zini toping.
45. Shunday nN sonini topingki tenglamalar sistemasi qanoatlantiruvchi x₁, x₂ …….x_n musbat sonlar mavjud bo’lsin. Topilgan n natural soni bo’yicha sistemani yeching.
46. Quyidagi tengsizliklarni doimo to’g’riligini ko’rsating.
1) (a>0)
2) )
3.

4.

5. (2n)! < (n(n+1))ⁿ
6. (
7. Shlyomilx masalasi. n>2 uchun quyidagi tengsizlikni doimo o’rinli ekanligiligi ko’rsating.



8.
9.
10.
11. ab²c³d⁴
12. x²+y²+z²xy+yz+xz
13. x²+y²+1 xy+x+y
14. x⁴+y⁴+88xy
15. a²b²+b²c²+a²c²abc(a+b+c)
16. 2(a³+b³+c³) ab(a+b)+ac(a+c)+bc(b+c)
17. x² -5xy+7y² 0
18. 4xy –3x² -8y²0

19. x² +2xy+3y²+2x+6y+4>0

20. (n>1)
21.
22. Agar a,b,c musbat sonlar va abc =1 shartlar o’rinli bo’lsa, a+b+c+ab+bc+ac6 tengsizlik doimo to’g’riligini ko’rsating.
23. Agar a>b va ab=2 shartlar o’rinli bo’lsa, tengsizlik doimo to’g’riligini ko’rsating.
24. Agar a,b,c nomanfiy sonlarning yig’indisi 1 ga teng bo’lsa, (1+a)(1+b)(1+c)8(1-a)(1-b)(1-c)
25.Agar a₁, a₂,…..a_n musbat sonlar uchun tengsizlik doimo to’g’riligini ko’rsating.
26. x² + 3
27. x+ 3 (x>0)

28.

29. nⁿ>1 (

30.

47. tengsizliklar sistemasini yeching.
A) [0;100] B) [0;100) C) [0;101] D) [0; )
48. a, b, c va d butun sonlarda P(x) = ax³+bx²+cx+d (a ko’phad uchun P(15)=3 va P(21)=12 shartlarni qanoatlantirsa, a+b+c+d ning qiymatini toping.
A) 10 B)25 C)36 D)

48. Agar x>0, y>0, z>0 va x+y+z =12 shartlar o’rinli bo’lsa, U = x²yz³ ifodaning eng katta qiymatini toping.

A)6912 B)6000 C) 2000 D)aniqlab bo’lmaydi

12. 
    
    - §. Modulli ifodalar
    

x haqiqiy sonning absolyut qiymati (moduli) deb nomanfiy bo’lgan ko’rinishdagi songa aytiladi. Uning umumiy ko’rinishi quyidagicha bo’ladi.

ning geometrik ma’nosi koordinata boshidan x gacha bo’lgan masofani anglatadi.
Absolyut qiymatlar bo’yicha quyidagi munosabatlar o’rinlidir.

1. 
   
   
   
2. 
   
   
   

3. -

   
   
4. 
   
   (b ) tengsizlik -b tengsizlikka ekvivalent hisoblanadi.
   
5. 
   
   
   
6. 
   
   
   
7. 
   
   
   

8. Max(a;b) =

   
   

9. min(a;b) =

   
   

1. ifodaningqiymatinimodulbelgisisizyozing.

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

2. ifodaningqiymatinimodulbelgisisizyozing.
A)1B)2 C)3 D)4
3.
ifodaningqiymatinimodulbelgisisizyozing.
A)1B)2C)3 D)4
4.
ifodaningqiymatinimodulbelgisisizyozing.
A)1B)2 C)3 D)4
5.

Ifodaning qiymatini modul belgisisiz yozing.

A) B) C) D)
6. ifodanisoddalashtiring. ( xϵ(– ∞; 1) )
A) x² +x+1 B) – (x²+x+1) C)x² +x D)x²– x+1
40. ifodanisoddalashtiring.
( xϵ(1; 3) )
A) x² +x+1 B) –(x² +x+1) C)x² +x D)x²– x+1
41. ifodanisoddalashtiring.( xϵ(3; ∞) )

A) x² +x+1B) – (x² +x+1) C)x² +x D)x²– x+1

42. ifodanisoddalashtiring. ( aϵ(– ∞; – 2) )

A)– a/2 B) (a – 1)/2 C) a(a – 1)/2 D) a(a + 1)/2

43. ifodanisoddalashtiring.( aϵ(– 2; ∞) )
A) – a/2 B) (a – 1)/2C) a(a – 1)/2 D) a(a + 1)/2
44. ifodanisoddalashtiring. ( mϵ(– ∞; – 2) )

A) 1/(m + 2) B) – 1/(m + 2) C) 1/(m – 2) D)– 1/(m – 2)

45. ifodanisoddalashtiring. ( mϵ(– 2; 0) )

A)1/(m + 2) B) – 1/(m + 2) C) 1/(m – 2) D) – 1/(m – 2)

46. ifodanisoddalashtiring. ( mϵ(3; ∞) )

A) 1/(m + 2) B) – 1/(m + 2) C) 1/(m – 2) D) – 1/(m – 2)

47. ifodanisoddalashtiring. ( mϵ(0; 3) )
A) 1/(m + 2) B) – 1/(m + 2) C) 1/(m – 2) D) – 1/(m – 2)
48. ifodanisoddalashtiring. ( xϵ(– ∞; – 1) )

A) x– 2 B)(x² + 4)/(x – 2) C) –(x+2) D) x+2

49. ifodanisoddalashtiring.( xϵ(– 1; 1) )
A) x– 2B)(x² + 4)/(x – 2) C) –(x+2) D) x+2
50. ifodanisoddalashtiring. ( xϵ(1; 2) )
A) x– 2 B)(x² + 4)/(x – 2)C) –(x+2) D) x+2
51. ifodanisoddalashtiring. ( xϵ(2; ∞) )
A) x– 2 B)(x² + 4)/(x – 2) C) –(x+2) D) x+2
52. ifodanisoddalashtiring. ( xϵ(– ∞; – 1) )

A)x/(x – 1) B) x/(1 – x) C) – x/(x + 1) D) x/(x + 1)

53. ifodanisoddalashtiring. ( xϵ(– 1; 0) )
A) x/(x – 1) B) x/(1 – x) C) – x/(x + 1) D) x/(x + 1)
54. ifodanisoddalashtiring. ( xϵ[0; 1) )
A) x/(x – 1) B) x/(1 – x) C) – x/(x + 1) D) x/(x + 1)
55. ifodanisoddalashtiring. ( xϵ(1; ∞) )
A) x/(x – 1) B) x/(1 – x) C) – x/(x + 1)D) x/(x + 1)
56. ifodanisoddalashtiring. ( xϵ(– ∞; 2) )

A) x² – 4x–12 B) (x+2)² C) x² – 4x+12 D)x² + 4x+12

57. ifodanisoddalashtiring. ( xϵ(2; ∞) )
A) x² – 4x–12B) (x+2)² C) x² – 4x+12 D)x² + 4x+12
58. ifodanisoddalashtiring. ( xϵ(–∞; 0) )

A)– 1/x B) 1/x C)x D)–x

59. ifodanisoddalashtiring.( xϵ(0; 1) )

A) – 1/x B) 1/x C)x D)– x

60. ifodanisoddalashtiring.( xϵ(1; 2) )

A)– 1/x B) 1/x C)x D)– x

61. ifodanisoddalashtiring. ( xϵ(2; 3) )
A)– 1/xB) 1/x C)x D)– x
62. ifodanisoddalashtiring.( xϵ(3; ∞) )
A)– 1/xB) 1/x C)x D)– x
63. ifodanisoddalashtiring.( xϵ( – ∞; – 3) )

A) 1/(a + 1) B) 1/(a + 3) C) 1/(a – 1) D) 1/(a – 3)

64. ifodanisoddalashtiring. ( xϵ(– 3; – 1) )

A) 1/(a + 1) B) 1/(a + 3) C) 1/(a – 1) D) 1/(a – 3)

65. ifodanisoddalashtiring. ( xϵ(–1; 2) )

A) 1/(a + 1) B) 1/(a + 3) C) 1/(a – 1) D) 1/(a – 3)

66. ifodanisoddalashtiring.( xϵ(2; ∞) )
A) 1/(a + 1) B) 1/(a + 3) C) 1/(a – 1) D) 1/(a – 3)
67. ifodanisoddalashtiring. ( xϵ(– ∞; 0) )

A) B) C) D)

68. ifodanisoddalashtiring.( xϵ(0; 1) )

A) B) C) D)

69. ifodanisoddalashtiring. ( xϵ[ 1; ∞) )

A) B) C) D)

70. ifodanisoddalashtiring.( xϵ(– ∞; 0) )

A) B) C) D)

71. ifodanisoddalashtiring.( xϵ[ 0; 1/3) )

A) B) C) D)

72. ifodanisoddalashtiring.( xϵ(1/3; 1) )

A) B) C) D)

73. ifodanisoddalashtiring. ( xϵ(1; ∞) )

A) B) C) D)

74. ifodanisoddalashtiring.( xϵ( – ∞; – 3/2) )

A) B) C) D)

75. ifodanisoddalashtiring.( xϵ(– 3/2; 0) )

A) B) C) D)

76. ifodanisoddalashtiring.( xϵ(0; 3) )

A) B) C) D)

77. ifodanisoddalashtiring.( xϵ(3; ∞) )

A) B) C) D)

78. |−abc| = −abc, |a−b| = −b + a va |−b| = b bo’lsa, quyidagilardan qaysi biri har doim o’rinli.

A) b<0 B) 0 C) c<0 D) b

79. funksiyaning eng katta qiymatini toping
A) 9 B) 12 C) 15 D) 24

12.