Kislik va uning

Tekislikka doir asosiy masalalar

1. 
   Berilgan M₀(x₀,y₀,z₀) nuqtadan o‘tuvchi tekisliklar dastasining tenglamasi qayerda to’g’ri ifodalangan?
   

A) A(x+x₀)+B(y+y₀)+C(z+z₀)=0 . B) Axx₀+Byy₀+Czz₀=0.

_C) x - x₀
A

• 
  y - y₀
  

B

• 
  z - z₀
  

C
= 0 . D)
x + x₀
A

• 
  y + y₀
  

B

• 
  z + z₀
  

C
= 0 .

*E) A(x–x₀)+B(y–y₀)+C(z–z₀)=0 .

2. 
   Fazoning M(1,2,-3) nuqtasidan o‘tuvchi tekisliklar dastasi tenglamasini ko‘rsating.
   

A) Ax+2By-3Cz=0. *B) A(x-1)+B(y-2)+C(z+3)=0. C) A(x+1)+B(y+2)+C(z−3)=0. D) Ax+2By-3Cz+D=0.
E) A(x-1)+B(y-2)+C(z-3)+D=0.

3. 
   M₁(3,2,-1), M₂(0,3,1) va M₃(4,5,0) nuqtalardan o‘tuvchi tekislik tenglamasini ko‘rsating.
   

A) 2x-y+z−3=0.	B)	x-2y+z+2=0.	*C) x-y+2z+1=0.
D) x-y+z=0.	E)	x-2y+2z+3=0.

4. 
   M₁(3,2,-1), M₂(0,3,1) va M₃(4,5,0) nuqtalardan o‘tuvchi tekislikda yotuvchi M₀(x,−4, 7) nuqtaning abssissasini toping.
   

A) x₀=−3 . B) x₀=5 . C) x₀=−1,5 . *D) x₀=9 . E) x₀=0 .

5. 
   Berilgan M₀(x₀,y₀,z₀) nuqtadan o‘tuvchi va n=(A,B,C) vektorga perpendikulyar tekislik tenglamasini ko‘rsating.
   

A) A(x+x₀)+B(y+y₀)+C(z+z₀)=0 . B) Axx₀+Byy₀+Czz₀=0.

_C) x - x₀
A

• 
  y - y₀
  

B

• 
  z - z₀
  

C
= 0 . D)
x + x₀
A

• 
  y + y₀
  

B

• 
  z + z₀
  

C
= 0 .

*E) A(x–x₀)+B(y–y₀)+C(z–z₀)=0 .

6. 
   M(1,2,3) nuqtadan o‘tuvchi va n=(3,2,1) normal vektorga ega tekislik tenglamasini yozing.
   

*A) 3x+2y+z-10=0 . B) x+2y+3z-14=0 . C) 2x+3y+z-11=0 .
D) x+3y+2z-13=0 . E) 3x+y+2z-11=0.

7. 
   Berilgan M₀(3,−4,0) nuqtadan o‘tuvchi va n=(1,2,−3) normal vektorga ega bo’lgan tekislikda yotuvchi N(−3,8, z) nuqtaning aplikatasini toping.
   

A) z=0 . B) z=5 . C) z= −1 . *D) z=6 . E) z= −3,5 .

8. 
   M₀(x₀,y₀,z₀) nuqtadan Ax+By+Cz+D=0 tekislikkacha bo‘lgan masofani topish formulasini ko’rsating.
   

A) d =
Ax₀

• 
  By₀
  
• 
  Cz₀
  

+ D . B) ^{d =} .

*C) d =
Ax₀

• 
  By₀
  
• 
  Cz₀
  
• 
  D
  

. D)


d = .

A² + B ²

• 
  C ²
  

E) d =
Ax₀ + By₀ + Cz₀ + D _.
A² + B² + C ²

9. 3x+4y-2 toping.
z+14=0 tekislikdan koordinata boshigacha bo‘lgan masofani

^{A) 14. B) 7. *C) 2. D) 1. E)} 2 6 ^.

10. 
    Ushbu 4x+3y-5z-8=0 va 4x+3y-5z-12=0 parallel tekisliklar orasidagi masofani toping.
    

A) 4 . B) 20 . C)
. *D) ^{2 2}
5
. E) ^{2 2} .

10. 
    x+y-z-1=0 va 2x-2y-2z+1=0 tekisliklar orasidagi burchak kosinusini toping.
    

A) 0. B) 1. C)
3 / 4 ^{. *D) 1/3. E) 3/4.}

10. 
    x+y-18=0 va y+z-72=0 tenglamalar bilan berilgan tekisliklar orasidagi burchak topilsin.
    

A) 30⁰ . B)


arccos
³ . C) 45⁰ . D)
4


arccos
³ . *E) 60⁰ .
5

10. 
    Normal vektorlari n₁=(-1, -1,0) va n₂=(0, -1, -1) bo‘lgan tekisliklar orasidagi burchak topilsin.
    

A) 45⁰ . B) 30⁰ . C) arccos ²
3
. *D) 60⁰ . E) 90⁰.

10. 
    A₁x+B₁y+C₁z+D₁=0 va A₂x+B₂y+C₂z+D₂=0 tekisliklarning parallellik sharti qayerda to‘gr’ri ko‘rsatilgan ?
    

1. 
   ^А1 А₂
   

₌ В₁ В_2
= D₁
D₂
. B)
А_{1 =} C₁ А₂ C_2
= D₁ D₂
. *C)
А_{1 =} В₁ А₂ В₂
= ^C1 .
C₂

4. 
   ^В1 В₂
   

₌ C₁
C_2
= D₁
D₂
. E)
A_{1 =} В₁ A₂ В_2
= C₁
C₂
= ^D1 .
D₂

10. 
    kx–2y+5z+10=0 va 6x– (1+k)y+10z–2=0 tekisliklar k parametrning qanday qiymatida parallel bo‘ladi ?
    

*A) k=3. B) k= – 4. C) k=2. D) k= –5. E) k=±1.

10. 
    Umumiy tenglamalari A₁x+B₁y+C₁z+D₁=0 va A₂x+B₂y+C₂z+D₂=0 bilan berilgan tekisliklarning perpendikulyarlik shartini ko‘rsating.
    

A) A₁A₂+B₁B₂+C₁C₂+D₁D₂=0. B) A₁A₂+B₁B₂+D₁D₂=0.
*C) A₁A₂+B₁B₂+C₁C₂=0. D) B₁B₂+C₁C₂+D₁D₂=0.

4. 
   ^A1 A₂
   

₌ В₁ В₂
= ^C1 .
C₂

10. 
    kx–2y−5z+10=0 va 6x–(1+k)y+10z–2=0 tekisliklar k parametrning qanday qiymatida parallel bo‘ladi ?
    

A) k=3. B) k= – 4. *C) k=6. D) k= –5. E) k=±1.

10. 
    x-y-1=0, y+z=0 va x-z-1=0 tekisliklarning kesishish nuqtasi koordinatalarining yig’indisini toping.
    

A) 0 . *B) 1 . C) −1 . D) 4 . E) −4 .

10. 
    2x-3y+4z-12=0 tenglama bilan berilgan tekislikning koordinata o‘qlari bilan kesishgan nuqtalarining koordinatalari topilsin.
    

A) ( -1,0,0), (0, -5,0), (0,0,4). B) (−6,0,0), (0, 3,0), (0,0,4).
C) (5,0,0), (0, -4,0), (0,0,−3). *D) (6,0,0), (0, -4,0), (0,0,3).
E) (1,0,0), (0, 3,0), (0,0,5).

10. 
    M(2, -1,1) nuqtadan o‘tib, 3x+2y-z+4=0 tekislikka parallel bo‘lgan tekislikning koordinata o‘qlari bilan kesishish nuqtalarining koordinatalarini toping.
    

A) ( -1,0,0), (0, -5,0), (0,0,4). B) (−6,0,0), (0, 3,0), (0,0,4).
C) (5,0,0), (0, -4,0), (0,0,−3). D) (6,0,0), (0, -4,0), (0,0,3).
*E) (1,0,0), (0, 3/2,0), (0,0,−3).

10. 
    Kesmalardagi tenglamasi
    

x ₊ y
5 - 2


+ ^z = 1 9

bo‘lgan P tekislik va koordinata tekisliklari bilan chegaralangan piramida hajmini toping.
A) 90 B) 60. C) 45. D) 30. *E) 15.

10. 
    Kesmalardagi tenglamasi
    

x ₊ y ₊
a 4


^z = 1
- 9

bo‘lgan P tekislik va koordinata tekisliklari bilan chegaralangan piramida hajmi 72 kub birlikka teng . Noma’lum a parametr qiymatini toping.
A) 2. B) −2. C) ±2. *D) ±12. E) 6.

10. 
    M(3, -2,1) nuqtadan o‘tib, a₁=(0,1,−2) va a₂=(5,0,2) vektorlarga parallel bo’lgan tekislik tenglamasini toping.
    

A) 3x+2y-z−4=0.	B) 3x−5y+2z−21=0.	*C) 2x−10y-5z−21=0.
D) 2x+5y-z+5=0.	E) 5x−12y-3z−36=0.

ADABIYOTLAR.

1. 
   SOATOV YO.U. «Oliy matеmatika», I jild, Toshkеnt, O¢qituvchi, 1992 y.
   
2. 
   PISKUNOV N.S. «Diffеrеntsial va intеgral hisob», 1-tom, Toshkеnt, O¢qituvchi, 1972 y.
   
3. 
   MADRAXIMOV X.S., G’ANIЕV A.G., MO’MINOV N.S. «Analitik gеomеtriya va chiziqli algеbra», Toshkеnt, O¢qituvchi, 1988 y.
   
4. 
   SARIMSOQOV T.А. «Haqiqiy o¢zgaruvchining funktsiyalari nazariyasi» Toshkеnt, O¢qituvchi, 1968 y.
   
5. 
   T. YOQUBOV «Matеmatik logika elеmеntlari», Toshkеnt, O¢qituvchi, 1983y.
   
6. 
   RAJABOV F., NURMЕTOV A. «Analitik gеomеtriya va chiziqli algеbra», Toshkеnt, O¢qituvchi, 1990 y.