Ispitni Zadaci Iz Matematike

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Zadaci iz matematike

Text of Ispitni Zadaci Iz Matematike

Via tehnika kola

Via tehnika kola

Doboj

Vesna Mii

Doboj 2003

I ispitni rokGrupa I1. Izraunati korjen .

2.Ako je u razvoju koeficijenti drugog lana za 44 je manji od koeficijenata treeg lana, odrediti koji lan ne sadri x.

3. Sastaviti jednainu ravni koja prolazi kroz presjek ravni 2x-3y-z+5=0 i x+3y+2z-2=0 i taku M(2,1,0).4. Rijeit i diskutovati sistem jednaina:

5. Nacrtati grafik funkcije i ispitati tok: .

6.

7. Rijeiti diferencijalnu jednainu:

1.

2.

je koeficijent etvrtog lana, a etvrti lan ne sadri x.

3.

4.

Sistem je neodreen

Sistem je neodreen

5.

3.) Funkcija nije periodina

5.) Znak funkcije :

6.)

7.

8.

9.Grafik 6.

(*)

-

-

,

(1)+(3)

, , ,

Uvrstimo u (*)

,

7.

Grupa II

1. Pokazati da predstavlja krunicu sa centrom u taki i .2. Rijeiti matrinu jednainu E jedinina matrica i .

3. Odrediti jednainu prave koja prolazi kroz taku M(1,1,0) i paralelna je sa ravnima i .

4. Nacrtati grafik i ispitati tok funkcije : .

5. Rijeiti integral .

6. Rijeiti diferencijalnu jednainu : .7. Dokazati da je svaki prirodan broj n :

djeljivo sa 7.

1.

2.

,

3.

prava p : (Uslov paralelnosti prave p i ravni , je )

Uzmimo n=-4 , l=5 , m=-7

4.

funkcija nije periodina

, nema nula

5.) Znak funkcije

6.)

7.) Znak prvog izvoda

5.

6.

7.

djeljivo sa 7

I korak : n=1

II korak : n=k (pretpostavimo da je funkcija tana, za n=k)

III korak : n=k+1

a to znai da je formula tana i za m=k+1.II ispitni rokGrupa I

1. Izraunati x i y iz jednaina :

2. Rijeiti i diskutovati sistem :

3. Odrediti taku B simetrinu taki u odnosu na ravan

4. Ispitati i nacrtati grafik funkcije : .

5.

6.

1.

/

uvrstimo u prvu jednainu

2.

Sistem je neodreen

Sistem je neodreen

3.

vektor normale ravni

zadovoljava

jednainu ravni

(1)

(*)

uvrstimo u (1)

uvrstimo u (*)

je simetrina taka

je traena taka4.

1.) D:

2.) Nije ni parna ni neparna

3.) Nije periodina

4.)

5.)Znak funkcije:

Znak

5.

6.

Grupa II1. Ako je nai emu je jednako .

2. Rijeiti matrinu jednainu AX=B , X=?.

3. Sastaviti jednainu ravni koja prolazi kroz presjek ravni i i kroz taku M (2,1,0).

4.

5. Nacrtati grafik funkcije i ispitati tok funkcije : .6.

1.

Dakle,

2.

3.

Traena ravan

4.

5.)

1.)

2.) funkcija nije ni parna ni neparna.

3.) Nije periodina.

4.) presjek s x-osom5.) Znak funkcije

6.) Prvi izvod funkcije

7.) Znak prvog izvoda

8.) Asimptote

kosa asimptota funkcije .

9.) Grafik

6.

III ispitni rokGrupa I

1.Nai redni broj onog lana razvoja binoma koja ne sadri a i b na isti eksponent.

2.

. X=?

3.Sastaviti jednainu ravni koja prolazi kroz taku A (3,2,1) i normalna je na ravnima i .4. Nacrtati grafik funkcije i ispitati tok .5. .

6. Rijeiti diferencijalnu jednainu :

1.

Ako je redni broj traenog lana k+1 , onda je :

Poto po uslovu zadatka a i b imaju iste eksponente to je :

Znai, deseti lan po redu sadravae i a i b sa jednakim eksponentima.

2.

Za

Za a=1 , ima beskonano rjeenja, a za a=-2, nema rjeenja.

3.

4.

1.) .

2.) Nije ni parna ni neparna funkcija.

3.) Nije periodina.

4.)

5.) Znak funkcije

6.) Prvi izvod funkcije

7.) Znak prvog izvoda

8.)Grafik5.

6.

Grupa II

1.Izraunati:

2.

3.Kako glasi jednaina ravni koja prolazi takama M(3,-5,1) i N(4,1,2),a normalna je na ravan .4. Ispitati i nacrtati grafik funkcije .

5. .

6.

1.

2.

Za sistem je neodreen, ima beskonano mnogo rjeenja.

3.

4.

1.)

2.) funkcija je neparna.3.) Nije periodina4.) presjek sa x-osom5.)Znak funkcije

prevojna takaKosa asimptota

kosa asimptota

6.Grafik

5.

6.

IV Ispitni rokGrupa I

1. Rijeiti jednainu : .2. Odrediti koeficijente lanova koji sadre u razvoju .3. Odrediti projekciju take A(2,1,2) na pravu .

4. Nacrtati grafik funkcije .

5.

6. Rijeiti diferencijalnu jednainu .

1.Rijeiti

2.

I lan je

II lan je

je etvrti lan i on sadri .

3.

Uvrstimo u (*)

Taka .4.

1.)

2.)

funkcija nije parna ni neparna

3.)Funkcija nije periodina4.)

5.)Znak funkcije:

Funkcija pozitivna

(grafik iznad x-ose)

6.

Znak

5.

Uvrstimo

Dakle,

6.

Grupa II

1. Rijeiti jednainu : .

2. Odrediti za koje vrijednosti parametara a i b sistem jednaina:

a.) ima samo jedno rjeenje,

b.) nema rjeenja,

c.) ima beskonano mnogo rjeenja.

3. Sastaviti jednainu prave koja prolazi kroz taku M(1,1,0) i paralelna je sa ravnima i .

4. Nacrtati i ispitati funkcije : .

5. Rijeiti integral .

6. Rijeiti diferencijalnu jednainu .1.

2.

Diskusija

Za sistem ima jedinstveno rjeenje

Za , imamo sistem je neodreen

(ima beskonano mnogo rjeenja)

Za sistem je nemogu, nema rjeenja.

3.

4.

2.Nije ni parna ni neparna

3.Nije periodina Znak funkcije :

5.Izvod :

Znak

y=2 je asimptota

Grafik

5.

(1) , ,

(2)

(3)

Uvrstimo A i B u (2):

uvrstimo u (3)

6.

smjena:

Grupa III

1.Izraunati

2.Diskutovati rjeenja sistema jednaina u zavisnosti od parametra a:

3.Kako glasi jednaina ravni koja prolazi takama a normalna je na ravan .4. Grafik funkcije nacrtati i ispitati tok:

5.

6. Rijeiti diferencijalnu jednainu uz poetni uslov .

1.

2.

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3.

4.

1.

2.Nije ni parna ni neparna3.Nije periodina

4.

Znak y

, y=0 asimptota

5.

6.

Kako je y=1 za x=0 imamo tj. .

Traeno partikularno rjeenje je .

V Ispitni rokGrupa I

1. Rijeiti jednainu : .

2. Dokazati da je matrica :

Nula polinoma .

3. Odrediti presjenu taku pravih :

i

4. Nacrtati grafik funkcije : .5. Izraunati integral : .

6. Rijeiti jednainu : .

1.

2.

3.

Uslov presjeka pravih je ispunjen.

Uvrstimo x, y, z u (p2):

Uvrstimo u (*):

4.

3.) Funkcija nije periodina

4.) y=0 presjek s x osom

5.) Znak funkcije

Znak

5.

6.

Grupa II

1. U zavisnosti od realnog parametra m rijeiti i diskutovati sistem :

2. Nai vrijednost x u izrazu iji je trei lan razvoja 1000000.

3. Data su tjemena paralelograma ABCD.Sastaviti jednainu

dijagonale BD ovog paralelograma.

4. Nacrtati grafik funkcije .

5. .

6. Rijeiti diferencijalnu jednainu : .

1.

neodreen

sistem je nemogu.2.

3.

Jednaina prave kroz dvije take

4.

1.) .

2.) Nije ni parna ni neparna funkcija.

3.) Nije periodina.

4.) 5.) Znak funkcije

6.) Prvi izvod funkcije

7.) Znak prvog izvoda

8.)Grafik

5.

6.

VI Ispitni rok

Grupa I

1. Odrediti x u izrazu u kome je odnos sedmog lana od poetka razvoja binoma prema sedmom lanu od kraja .

2. Izraunati .

3. Sastaviti jednainu normale povuene iz take (2,3,1) na pravu :

.

4. Nacrtati grafik funkcije .

5. Odrediti integral .

6. Rijeiti diferencijalnu jednainu : .

1.

Sedmi lan razvoja binoma je:

a sedmi lan od kraja je :

Binomni koeficijenti su im jednaki, a to proizilazi iz njihove osobine simetrinosti.

Prema uslovu zadatka je:

2.

3.

zadovoljava jednainu prave

uvrstimo u

normala :

Traena jednaina normale : .

4.

3.Nije periodina

Znak y

Prvi izvod :

Znak

5.

6.

Grupa II1. Rijeiti matrinu jednainu :

.

2. Dokazati jednakost : .

3. Nai presjeke ravni sa koordinatnim osama i koordinatnim ravnima.

4. Nacrtati grafik funkcije .

5. .

6. .

1.

2.

1.)

2.) , Pretpostavimo da je (F) tano za .

tj. (T)

3.) Treba dokazati da je (F) tano i za

(T)3.

Datu ravan svedemo na segmentni oblik:

Vidi se da su odsjeci na koordinatnim osama:

Presjek ravni sa xy ravni

4.

1.) .

2.) parna funkcija

3.)

Funkcija je periodina sa periodom

4.)

5.) Znak funkcije

6.) Prvi izvod funkcije

(keZ)

7.) Znak prvog izvoda

8.) Grafik5.

6.

VII Ispitni rok

Grupa I

1. Dato je , gdje je . Pokazati da je .

2. Rijeiti matrinu jednainu :

3. Odrediti rastojanje paralelnih pravih : .4. Nacrtati grafik funkcije .

5. Odrediti integral .

6. Odrediti opte rjeenje diferencijalne jednaine : .

1.

Dakle, vrijedi :

2.

3.

4.

Domen

Parnost funkcija nije ni parna ni neparna.

Nije periodina Nule funkcije

Znak funkcije

Prvi izvod funkcije

Asimptote

kosa asimptota funkcije .

Grafik

5.

6.

Grupa II

1. Nai vrijednost x u izrazu iji je etvrti lan razvoja binoma 200.

2. Rijeiti matrini jednainu AX=B.

3.Napisati jednainu ravni koja prolazi kroz taku A(3,-5,1) i paralelna je ravni

.4. Nacrtati grafik funkcije .

5.

6.

1.

IV lan je

2.

3.

Jednaina traene ravni

4.

1.)

2.) Funkcija nije parna ni neparna.3.) Period .

4.)

5.) Znak funkcije

8.) Grafik

5.

6.

Sadraj Stranica

I Ispitni rok ........................................................................................

II Ispitni rok ......................................................................................

III Ispitni rok ....................................................................................

IV Ispitni rok ....................................................................................

V Ispitni rok .....................................................................................

VI Ispitni rok ....................................................................................

VII Ispitni rok ..................................................................................

Sadraj .............................................................................................

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