Free Printable Funksional Analiz

1.

Stereografik proeksiya natijasida tekislikdagi har qanday aylananing aksi sferaga nima bo`lib tushadi?

a)

Sfera

b)

Aylana

c)

Yarim sfera

d)

Yarim aylana

2.

Kompleks tekislikdagi har bir nuqtaga (kompleks songa) S\{P} sferada nechta nuqtaga mos keladi?

a)

1

b)

2

c)

3

d)

ixtiyoriy

3.

Stereografik proyeksiya kompleks tekislikdagi barcha nuqtalar to`plami C bilan S\{P} sferaning nuqtalar to`plami o`rtasida qanday moslik o`rnatadi?

a)

O`zaro bir qiymatli

b)

Turli qiymatli

c)

Moslik o`rnatmaydi

d)

Ikki qiymatli

4.

Sfera qutb nuqtasi qanday bo`ladi?

a)

P(0,0,0)

b)

P(1,0,0)

c)

P(0,0,1)

d)

P(1,1,1)

5.

C to`plamdan olingan har bir kompleks songa R² tekislikda, bu sonni geometrik tasvirlovchi nechta nuqta mos keladi?

a)

Ixtiyoriy

b)

Faqat 1 ta

c)

2 ta

d)

3 ta

6.

z=x+iy kompleks sonda x>0 bo`lsa uning argumenti qaysi formula yordamida topiladi?

a)

argz = arctg $\left(\frac{y}{x}\right)$

b)

argz = arctg $\left(\frac{x}{y}\right)$

c)

argz = $\pi$ - arctg $\left(\frac{y}{x}\right)$

d)

argz = $\pi$ - arctg $\left(\frac{x}{y}\right)$

7.

z=x+iy kompleks sonda x=0, y>0 bo`lsa uning argumenti qaysi formula yordamida topiladi?

a)

argz= π /4

b)

argz=2 π

c)

argz=3 π /2

d)

argz= π /2

8.

Agar z(a) = z(b) bo`lsa, ya`ni egri chiziqning boshlang’ich va oxirgi nuqtalari ustma-ust tushsa, bunday

a)

Ochiq

b)

Yopiq

c)

Jordan chizig'i

d)

Soha

9.

Ushbu $E\cap U\left(z_0,\epsilon\right)=z_0$ bajariladigan $\epsilon$ > 0 son mavjud bo`lsa, u holda nuqta $z_0$ E to`plamning qanday nuqtasi deyiladi?

a)

Chegara

b)

Chegaralanmagan

c)

Yakkalangan

d)

Limit

10.

Agar $E\subset C$ shunday to`plam bo`lsaki, uning chegarsiga tegishli ixtiyoriy $z_1$ va $z_2$ nuqtalarni birlashtiruvchi chiziq shu chegaraga tegishli bo`lsa, u qanday to`plam bo`ladi?

a)

Ochiq to'plam

b)

Yopiq to'plam

c)

Bo'sh to'plam

d)

Bog'lamli to'plam

11.

Ixtiyoriy yopiq Jordan chizig’i kompleks tekislikni nechta bir bog’lamli sohalarga ajratadi.

a)

2 ta

b)

3 ta

c)

8 ta

d)

Cheksiz ko'p

12.

Hisoblang: $\left(\frac{1+i}{1-i}\right)^{20}$

a)

0

b)

1

c)

i

d)

-i

13.

z = z(t) , $\left(0\le t\le b\right)$ kompleks tekislikda nimani ifodalaydi?

a)

To’g’ri chiziqni

b)

Egri chiziqni

c)

Jordan chiziqni

d)

Aylanani

14.

Kompleks tekislikda ushbu z - 1 < 2 tengsizlik qanday sohani ifodalaydi?

a)

Markazi O(1,0) nuqtada bo`lgan radiusi 2 ga teng bo`lgan yopiq doira

b)

Markazi O(1,0) nuqtada bo`lgan radiusi 2 ga teng bo`lgan aylana

c)

Markazi O(1,0) nuqtada bo`lgan radiusi 2 ga teng bo`lgan ochiq doira

d)

Markazi O(1,0) nuqtada bo`lgan radiusi 3 ga teng bo`lgan aylana

15.

w=f(z) aylanani aylanaga o’tkazuvchi akslantirishda f’(z₀)<1 bo’lsa aylana qanday o’zgaradi?

a)

siqiladi

b)

kengayadi

c)

o'zgarmaydi

d)

radius 1 dan kichik bo'ladi

16.

’’Agar f(z) funksiya E soxada golomorf bo’lib ,f(z)=const bo’lsa ,f(E)xam soxa bo’ladi’’ Teorema qanday nomlanadi?

a)

Koshi teoremasi

b)

Soxaning saqlanish prinspi

c)

Riman teoremasi

d)

1-tur akslantirish

17.

w=z² funksiya C/{0} da qanaqa akslantirish bo’ladi?

a)

1 tur

b)

2 tur

c)

3 tur

d)

ixtiyoriy

18.

f(z)=z² funksiyaning z₀=1/2 nuqtadagi burilish burchagini toping

a)

90

b)

180

c)

360

d)

Burilmaydi

19.

f(z)=6z+7 funksiyaning z₀=5 nuqtadagi cho’zilish koeffitsiyentini toping

a)

2

b)

4

c)

6

d)

36

20.

f(z)=z³ funksiyaning z₀=3 nuqtadagi cho’zilish koeffitsiyentini toping

a)

1

b)

2

c)

9

d)

27

21.

w=2z+3 akslantirish C_z tekislikdagi aylanani C_w tekislikka qanday akslantiradi?

a)

radiusi 2 barobar katta bo’lgan aylanaga

b)

3 barobar katta aylanaga

c)

To’g’ri chiziqqa

d)

Sharga

22.

Nuqtalar o’rniga kerakli so’zni qo’ying ”Agar w=f(z) akslantirish z₀ nuqtada cho’zilish va burchak saqlanish xossalariga ega bo’lsa,bunday akslantirishga z₀ nuqtada ….deyiladi.

a)

golomorf akslantirish

b)

konform akslantirish

c)

chiziqli akslantirish

d)

nochiziqli akslantirish

23.

Xosila argumentining geometrik ma’nosi nima ?

a)

ma’lum burchakka burish

b)

bir soxadan boshqa soxaga o’tish

c)

proporsionallik koeffitsiyenti

d)

cho’zilish koeffitsiyenti

24.

Xosila moduli geometric ma’nosi

a)

Burchak

b)

golomorf akslantirishni

c)

bir soxadan boshqa soxaga o’tish

d)

cho’zilish koeffitsiyenti

25.

$\int\ \frac{xdx}{x^2-4}$ integralni hisoblang

a)

$\frac{1}{2}\ln\left(x^2-4\right)+C$

b)

$\ln\left(x^2-4\right)^2+C$

c)

$x\ln\left(x^2-4\right)$

d)

$\ln\left(x^2-4\right)$

26.

$\lim_{x\rightarrow\left(-\frac{\pi}{2}\right)}\ \frac{\cos^2x}{1+\sin^3x}$ limit qiymatini toping

a)

$\frac{1}{3}$

b)

$\frac{1}{2}$

c)

$\frac{2}{3}$

d)

1

27.

$\lim_{x\rightarrow\infty}\left(\frac{x+1}{x-1}\right)^x\ \$ limit qiymatini toping

a)

$e^{-1}$

b)

$\sqrt[]{e}$

c)

e

d)

$e^2$

28.

$\lim_{x\rightarrow\infty}\left(\frac{x}{1+x}\right)^x$ limitni hisoblang

a)

$-e$

b)

$\frac{1}{e}$

c)

e

d)

$+\infty$

29.

$\lim_{x\rightarrow\infty}\sqrt[]{n}\left(\sqrt[]{n+2}-\sqrt[]{n-3}\right)\$ ni hisoblang.

a)

2/3

b)

4/3

c)

5/2

d)

3

30.

$x=C\left[a,b\right]$ da $p_1\left(\phi,\psi\right)=\max_{x\in\left[a,b\right]}\left|\phi\left(x\right)-\psi\left(x\right)\right|$ va $p_2\left(\phi,\psi\right)=\int_a^b\left|\phi\left(x\right)-\psi\left(x\right)\right|dx$ $\left(x,p_1\right)$ va $\left(x,p_2\right)$ metrik fazolar tula metrik fazolar bo’ladimi?

a)

$\left(x,p_1\right)$ tula emas $\left(x,p_2\right)$ tula

b)

$\left(x,p_1\right)$ tula $\left(x,p_2\right)$ tula emas

c)

$\left(x,p_1\right)$ tula emas

d)

$\left(x,p_1\right)$ tula emas $\left(x,p_2\right]$ tula emas

31.

$C\left[-1,1\right]$ fazoda $<x,f>=2\left[x\left(1\right)-x\left(0\right)\right]$ funktsionalning normasini toping.

a)

1

b)

2

c)

3

d)

4

32.

$C\left[-1,1\right]$ fazoda $<x,f>=\int_0^1x\left(t\right)dt$ funktsionalning normasini toping.

a)

1

b)

2

c)

3

d)

4

33.

$C\left[-1,1\right]$ fazoda $<x,f>=\int_0^1x\left(t\right)dt-x\left(0\right)$ funktsionalning normasini toping.

a)

1

b)

2

c)

3

d)

4

34.

$C\left[-1,1\right]$ fazoda $<x,f>=\int_{-1}^0x\left(t\right)dt-\int_0^1x\left(t\right)dt$ funktsionalning normasini toping.

a)

1

b)

2

c)

3

d)

4

35.

$C\left[-1,1\right]$ fazoda $<x,f>=\int_{-1}^1tx\left(t\right)dt$ funktsionalning normasini toping.

a)

1

b)

2

c)

3

d)

4

36.

$l_1$ fazoda $<x,f>=\sum_{k=1}^{\infty}\frac{x_k}{k}$ funktsionalning normasini toping.

a)

1

b)

2

c)

3

d)

4

37.

$A:L_2\ \left[0,1\right]\rightarrow L_1\left[0,1\right]$ operator $Ax\left(t\right)=tx\left(t\right)$ bo'lsa unga ko'shma operatorni toping.

a)

$ty\left(t\right)$

b)

$\int_t^1y\left(\tau\right)d\tau$

c)

$\int_0^1ty\left(t\right)dt$

d)

mavjud emas

38.

$A:L_2\ \left[0,1\right]\rightarrow L_1\left[0,1\right]$ operator $Ax\left(t\right)=\int_0^1tx\left(s\right)ds$ bo'lsa unga ko'shma operatorni toping.

a)

$ty\left(t\right)$

b)

$\int_0^1ty\left(t\right)dt$

c)

$\int_t^1y\left(\tau\right)d\tau$

d)

$\int_0^1ty\left(s\right)ds$

39.

$A:C\ \left[0;1\right]\ \rightarrow\ C\left[0;1\right]$ $Ax\left(t\right)=t^2x\left(0\right)$ operatorning normasini toping

a)

1

b)

2

c)

$\sqrt[]{2}$

d)

3

40.

Quyidagi to‘plamlardan qaysi biri $C\left[-1,1\right]$ fazoda qism fazo tashkil qilmaydi?

a)

Monoton funksiyalar to‘plami

b)

Uzluksiz differensiallanuvchi funksiyalar to‘plami

c)

$x\left(-1\right)=0$ startni qanoatlantiruvchi funksiyalar

d)

Barcha ko‘phadlar to‘plami

41.

Quyidagi to‘plamlardan qaysi biri $C\left[-1,1\right]$ fazoda qism fazo tashkil qilmaydi?

a)

$x\left(1\right)=1$ shartni qanoatlantiruvchi funksiyalar to‘plami

b)

Darajasi 100 dan oshmaydigan ko‘phadlar to‘plami

c)

Toq funksiyalar to‘plami

d)

Juft funksiyalar to‘plami

42.

Qanday fazo Banax fazosi deyiladi?

a)

To‘la normalangan fazo

b)

Skalyar ko‘paytma kiritilgan chiziqli fazo

c)

Har qanday normalangan fazo.

d)

Istalgan metrik fazo.

43.

Qanday fazo Evklid fazosi deyiladi?

a)

Skalyar ko‘paytma kiritilgan chiziqli fazo

b)

Har qanday normalangan fazo

c)

To‘la normalangan fazo

d)

Istalgan metrik fazo.

44.

To‘la bo‘lmagan separabel Evklid fazosini toping.

a)

$C_2\left[a,b\right]$

b)

$R^n$

c)

$C^{\ n}$

d)

$\iota_2$

45.

Noto‘g‘ri tasdiqni toping.

a)

Har qanday Evklid fazosida sanoqli ortonormal bazis mavjud.

b)

Evklid fazosida yig‘indi va songa ko‘paytirish amallari uzluksizdir.

c)

Evklid fazosida skalyar ko‘paytma amali uzluksizdir.

d)

Har qanday separabel Evklid fazosida sanoqli ortonormal bazis mavjud.

46.

Quyidagi tasdiqlarning qaysi biri to‘g‘ri?

a)

Separabel Evklid fazosida har qanday to‘la ortonormal sistema yopiq va aksincha.

b)

Separabel Evklid fazosida har qanday ortonormal sistema to‘ladir.

c)

Separabel Evklid fazosida har qanday ortonormal sistema yopiqdir.

d)

To‘la Evklid fazosida har qanday ortonormal sistema yopiqdir.

47.

Quyidagi tasdiqlardan qaysi biri o‘rinli?

a)

Har qanday ikki separabel Hilbert fazolari o‘zaro izomorfdir.

b)

Har qanday ikki Evklid fazolari o‘zaro izomorfdir.

c)

Har qanday ikki Hilbert fazolari o‘zaro izomorfdir.

d)

Har qanday ikki separabel Evklid fazolari o‘zaro izomorfdir.

48.

To‘g‘ri tasdiqni ajrating.

a)

Operatorlarni qo‘shish kommutativ, operatorlarni ko‘paytirish assotsiativ.

b)

Operatorlarni ko‘paytirish kommutativ.

c)

Operatorlarni ko‘paytirish assotsiativ va kommutativ.

d)

Operatorlarni qo‘shish assotsiativ

49.

$Dim\left(KerT\right)\ =\ \dim T^{\cdot}\ <\ \infty$ tasdiq Fredgolmning nechanchi teoremasiga mos keladi

a)

1-teoremasi

b)

2-teoremasi

c)

3-teoremasi

d)

4-teoremasi

50.

$\sin\pi\ \frac{z}{\left(z-1\right)^3}$ funksiyaning z = 1 nuqtadagi chegirmani toping

a)

0

b)

1

c)

-1

d)

2

51.

$\frac{1}{\sin z^2}$ funksiyaning z = 0 nuqtadagi chegirmani toping.

a)

0

b)

1/2

c)

1

d)

-1

52.

$f\left(z\right)=e^{\frac{1}{z}}$ ning maxsus nuqtalari va ularning turini aniqlang.

a)

$z=0$ muhim maxsus nuqta.

b)

z = 0 qutilib bo`ladigan maxsus nuqta.

c)

2-tartibli qutb

d)

z = 0 oddiy qutb.

53.

Agar butun kompleks tekislikda golomorf bo‘lgan $w=f\left(z\right)$ funksiya chegaralangan bo‘lsa, u holda ....

a)

$f\left(z\right)=const$

b)

$f\left(z\right)=0$

c)

$f\left(z\right)=\sin z$

d)

$f\left(z\right)=i$

54.

1/(1-z)² funksiyasining z = 0 nuqta atrofida Teylor qatoriga yoyilmasidagi a₂koeffitsientni toping.

a)

3

b)

-3

c)

1/2

d)

0

55.

$w=\frac{1}{z}$ funksiyani haqiqiy qismini toping

a)

$\frac{x}{x^2+y^2}$

b)

$\frac{1}{x}$

c)

$\frac{x}{x^2+y^2}$

d)

1

Funksional Analiz

Total questions: 55

Worksheet time: 9hrs 10mins

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