ETF MATF FON GRF TMF FORUM

Prijemni ispit na Matematičkom fakultetu u Beogradu

1. jul 2020.


Vreme za rad je 180 minuta.

1.Link zadatka Jednačina [inline]\max\{1+x,1-x\}=b[/inline] ima bar jedno rešenje ako i samo ako realan parametar [inline]b[/inline] zadovoljava uslov:
[inline]\text{A)}[/inline] [inline]b\in\mathbb{R}[/inline]      [inline]\text{B)}[/inline] [inline]b\ge1[/inline]      [inline]\text{C)}[/inline] [inline]b\le1[/inline]      [inline]\text{D)}[/inline] [inline]b\ge0[/inline]      [inline]\text{E)}[/inline] [inline]b>1[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]b\in\mathbb{R}[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]b\ge1[/inline]      [inline]\text{C)}[/inline] [inline]b\le1[/inline]      [inline]\text{D)}[/inline] [inline]b\ge0[/inline]      [inline]\text{E)}[/inline] [inline]b>1[/inline]              [inline]\text{N)}[/inline] ne znam

2.Link zadatka Ako za realne brojeve [inline]t[/inline] i [inline]k[/inline] prava [inline]y=kx+4[/inline] sadrži tačke [inline]\left(3,2^t\right)[/inline] i [inline]\left(2^t,4\right)[/inline], onda:
[inline]\text{A)}[/inline] [inline]t\in(-\infty,0][/inline]      [inline]\text{B)}[/inline] [inline]t\in(0,1][/inline]      [inline]\text{C)}[/inline] [inline]t\in(1,2][/inline]      [inline]\text{D)}[/inline] [inline]t\in(2,4][/inline]      [inline]\text{E)}[/inline] [inline]t\in(4,+\infty)[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]t\in(-\infty,0][/inline]      [inline]\text{B)}[/inline] [inline]t\in(0,1][/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline]t\in(1,2][/inline]      [inline]\text{D)}[/inline] [inline]t\in(2,4][/inline]      [inline]\text{E)}[/inline] [inline]t\in(4,+\infty)[/inline]              [inline]\text{N)}[/inline] ne znam

3.Link zadatka Najveća vrednost funkcije [inline]f(x)=16-\sqrt{2x^2+4x\sqrt2+8}[/inline] na njenom domenu je:
[inline]\text{A)}[/inline] [inline]16-2\sqrt2[/inline]      [inline]\text{B)}[/inline] [inline]12[/inline]      [inline]\text{C)}[/inline] [inline]16-4\sqrt2[/inline]      [inline]\text{D)}[/inline] [inline]14[/inline]      [inline]\text{E)}[/inline] [inline]16[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]16-2\sqrt2[/inline]      [inline]\text{B)}[/inline] [inline]12[/inline]      [inline]\text{C)}[/inline] [inline]16-4\sqrt2[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]14[/inline]      [inline]\text{E)}[/inline] [inline]16[/inline]              [inline]\text{N)}[/inline] ne znam

4.Link zadatka Najmanji prirodan broj [inline]a[/inline] za koji jednačina [inline]ax^2+bx+c=0[/inline] sa celobrojnim koeficijentima ima rešenja [inline]\displaystyle x_1=\frac{3+2\sqrt7}{5}[/inline] i [inline]\displaystyle x_2=\frac{3-2\sqrt7}{5}[/inline] jednak je:
[inline]\text{A)}[/inline] [inline]5[/inline]      [inline]\text{B)}[/inline] [inline]10[/inline]      [inline]\text{C)}[/inline] [inline]25[/inline]      [inline]\text{D)}[/inline] [inline]100[/inline]      [inline]\text{E)}[/inline] [inline]125[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]5[/inline]      [inline]\text{B)}[/inline] [inline]10[/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline]25[/inline]      [inline]\text{D)}[/inline] [inline]100[/inline]      [inline]\text{E)}[/inline] [inline]125[/inline]              [inline]\text{N)}[/inline] ne znam

5.Link zadatka Skup svih vrednosti realnog parametra [inline]c[/inline] za koje jednačina [inline]\sqrt{x+c}+\sqrt x=c[/inline] ima tačno jedno realno rešenje je:
[inline]\text{A)}[/inline] [inline]\{0\}[/inline]      [inline]\text{B)}[/inline] [inline][0,1][/inline]      [inline]\text{C)}[/inline] [inline][0,+\infty)[/inline]      [inline]\text{D)}[/inline] [inline][1,+\infty)[/inline]      [inline]\text{E)}[/inline] [inline]\{0\}\cup[1,+\infty)[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\{0\}[/inline]      [inline]\text{B)}[/inline] [inline][0,1][/inline]      [inline]\text{C)}[/inline] [inline][0,+\infty)[/inline]      [inline]\text{D)}[/inline] [inline][1,+\infty)[/inline]      [inline]\enclose{circle}{\text{E)}}[/inline] [inline]\{0\}\cup[1,+\infty)[/inline]              [inline]\text{N)}[/inline] ne znam

6.Link zadatka Skup rešenja nejednačine [inline]3\cdot4^x-7\cdot2^{x+1}\le5[/inline] je:
[inline]\text{A)}[/inline] [inline][\log_25,+\infty)[/inline]      [inline]\text{B)}[/inline] [inline][-\log_23,\log_25][/inline]      [inline]\text{C)}[/inline] [inline](-\infty,\log_25][/inline]      [inline]\text{D)}[/inline] [inline][\log_23,+\infty)[/inline]      [inline]\text{E)}[/inline] [inline](-\infty,\log_23][/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline][\log_25,+\infty)[/inline]      [inline]\text{B)}[/inline] [inline][-\log_23,\log_25][/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline](-\infty,\log_25][/inline]      [inline]\text{D)}[/inline] [inline][\log_23,+\infty)[/inline]      [inline]\text{E)}[/inline] [inline](-\infty,\log_23][/inline]              [inline]\text{N)}[/inline] ne znam

7.Link zadatka Skup rešenja nejednačine [inline]\displaystyle\frac{|x-3|-1}{x^2-4x+3}>0[/inline] je:
[inline]\text{A)}[/inline] [inline](-\infty,1)\cup(3,4)\cup(4,+\infty)[/inline]      [inline]\text{B)}[/inline] [inline](2,3)\cup(4,+\infty)[/inline]      [inline]\text{C)}[/inline] [inline](-\infty,2)\cup(4,+\infty)[/inline]      [inline]\text{D)}[/inline] [inline](-\infty,1)\cup(2,3)\cup(4,+\infty)[/inline]      [inline]\text{E)}[/inline] [inline](3,4)\cup(4,+\infty)[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline](-\infty,1)\cup(3,4)\cup(4,+\infty)[/inline]      [inline]\text{B)}[/inline] [inline](2,3)\cup(4,+\infty)[/inline]      [inline]\text{C)}[/inline] [inline](-\infty,2)\cup(4,+\infty)[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline](-\infty,1)\cup(2,3)\cup(4,+\infty)[/inline]      [inline]\text{E)}[/inline] [inline](3,4)\cup(4,+\infty)[/inline]              [inline]\text{N)}[/inline] ne znam

8.Link zadatka Ako je [inline]\log_2x+2\log_3y=0[/inline] i [inline]2\log_2x+\log_3y=5[/inline], onda je [inline]9\log_3x\cdot\log_2y[/inline] jednako:
[inline]\text{A)}[/inline] [inline]-50[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle-\frac{50}{9}[/inline]      [inline]\text{C)}[/inline] [inline]9[/inline]      [inline]\text{D)}[/inline] [inline]-9[/inline]      [inline]\text{E)}[/inline] [inline]50[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]-50[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle-\frac{50}{9}[/inline]      [inline]\text{C)}[/inline] [inline]9[/inline]      [inline]\text{D)}[/inline] [inline]-9[/inline]      [inline]\text{E)}[/inline] [inline]50[/inline]              [inline]\text{N)}[/inline] ne znam

9.Link zadatka Dužina duži koja je paralelna stranici trougla dužine [inline]a[/inline] i koja deli trougao na dva dela jednakih površina je:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{a}{2}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{a\sqrt2}{3}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{a\sqrt3}{2}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{a\sqrt2}{2}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{a\sqrt6}{4}[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle\frac{a}{2}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{a\sqrt2}{3}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{a\sqrt3}{2}[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]\displaystyle\frac{a\sqrt2}{2}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{a\sqrt6}{4}[/inline]              [inline]\text{N)}[/inline] ne znam

10.Link zadatka Osnova piramide je pravougli trougao s katetama [inline]6[/inline] i [inline]6\sqrt2[/inline], a svaka bočna ivica gradi s osnovom ugao od [inline]60^\circ[/inline]. Zapremina te piramide je:
[inline]\text{A)}[/inline] [inline]54\sqrt2[/inline]      [inline]\text{B)}[/inline] [inline]162\sqrt2[/inline]      [inline]\text{C)}[/inline] [inline]65[/inline]      [inline]\text{D)}[/inline] [inline]72[/inline]      [inline]\text{E)}[/inline] [inline]195[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]54\sqrt2[/inline]      [inline]\text{B)}[/inline] [inline]162\sqrt2[/inline]      [inline]\text{C)}[/inline] [inline]65[/inline]      [inline]\text{D)}[/inline] [inline]72[/inline]      [inline]\text{E)}[/inline] [inline]195[/inline]              [inline]\text{N)}[/inline] ne znam

11.Link zadatka Broj rešenja jednačine [inline]\displaystyle\frac{\text{tg }3x}{\text{tg }x}=0[/inline] na intervalu [inline][0,2\pi][/inline] je:
[inline]\text{A)}[/inline] [inline]2[/inline]      [inline]\text{B)}[/inline] [inline]4[/inline]      [inline]\text{C)}[/inline] [inline]5[/inline]      [inline]\text{D)}[/inline] [inline]6[/inline]      [inline]\text{E)}[/inline] [inline]7[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]2[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]4[/inline]      [inline]\text{C)}[/inline] [inline]5[/inline]      [inline]\text{D)}[/inline] [inline]6[/inline]      [inline]\text{E)}[/inline] [inline]7[/inline]              [inline]\text{N)}[/inline] ne znam

12.Link zadatka Dužine dveju stranica trougla su [inline]25[/inline] i [inline]30[/inline], a za njima naspramne uglove [inline]\alpha[/inline] i [inline]\beta[/inline] važi [inline]\beta=2\alpha[/inline]. Dužina treće stranice trougla je:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{1}{11}[/inline]      [inline]\text{B)}[/inline] [inline]11[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{44}{3}[/inline]      [inline]\text{D)}[/inline] [inline]13[/inline]      [inline]\text{E)}[/inline] [inline]25[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle\frac{1}{11}[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]11[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{44}{3}[/inline]      [inline]\text{D)}[/inline] [inline]13[/inline]      [inline]\text{E)}[/inline] [inline]25[/inline]              [inline]\text{N)}[/inline] ne znam

13.Link zadatka Zbir kvadrata svih vrednosti parametra [inline]n\in\mathbb{R}[/inline] za koje prava [inline]y=-2x+n[/inline] dodiruje krivu [inline]x^2+y^2-14x+29=0[/inline] jednak je:
[inline]\text{A)}[/inline] [inline]16[/inline]      [inline]\text{B)}[/inline] [inline]144[/inline]      [inline]\text{C)}[/inline] [inline]272[/inline]      [inline]\text{D)}[/inline] [inline]576[/inline]      [inline]\text{E)}[/inline] [inline]592[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]16[/inline]      [inline]\text{B)}[/inline] [inline]144[/inline]      [inline]\text{C)}[/inline] [inline]272[/inline]      [inline]\text{D)}[/inline] [inline]576[/inline]      [inline]\enclose{circle}{\text{E)}}[/inline] [inline]592[/inline]              [inline]\text{N)}[/inline] ne znam

14.Link zadatka Zbir prvih [inline]5[/inline] članova nekonstantnog aritmetičkog niza jednak je zbiru prvih [inline]8[/inline] njegovih članova, a proizvod prva [inline]3[/inline] člana tog niza jednak je proizvodu prvih [inline]6[/inline] njegovih članova. Proizvod prva tri člana niza je:
[inline]\text{A)}[/inline] [inline]0[/inline]      [inline]\text{B)}[/inline] [inline]6[/inline]      [inline]\text{C)}[/inline] [inline]12[/inline]      [inline]\text{D)}[/inline] [inline]20[/inline]      [inline]\text{E)}[/inline] [inline]30[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]0[/inline]      [inline]\text{B)}[/inline] [inline]6[/inline]      [inline]\text{C)}[/inline] [inline]12[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]20[/inline]      [inline]\text{E)}[/inline] [inline]30[/inline]              [inline]\text{N)}[/inline] ne znam

15.Link zadatka Imaginarni deo kompleksnog broja [inline]\displaystyle z=(-1+5i):\left(2-\frac{3+i}{2+i}\right)[/inline] je:
[inline]\text{A)}[/inline] [inline]8[/inline]      [inline]\text{B)}[/inline] [inline]1[/inline]      [inline]\text{C)}[/inline] [inline]-8[/inline]      [inline]\text{D)}[/inline] [inline]5[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{10}{3}[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]8[/inline]      [inline]\text{B)}[/inline] [inline]1[/inline]      [inline]\text{C)}[/inline] [inline]-8[/inline]      [inline]\text{D)}[/inline] [inline]5[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{10}{3}[/inline]              [inline]\text{N)}[/inline] ne znam

16.Link zadatka Koliko različitih realnih korena ima polinom [inline]\displaystyle p(x)=x^4-\left(x-\frac{1}{4}\right)^2[/inline]?
[inline]\text{A)}[/inline] [inline]0[/inline]      [inline]\text{B)}[/inline] [inline]1[/inline]      [inline]\text{C)}[/inline] [inline]2[/inline]      [inline]\text{D)}[/inline] [inline]3[/inline]      [inline]\text{E)}[/inline] [inline]4[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]0[/inline]      [inline]\text{B)}[/inline] [inline]1[/inline]      [inline]\text{C)}[/inline] [inline]2[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]3[/inline]      [inline]\text{E)}[/inline] [inline]4[/inline]              [inline]\text{N)}[/inline] ne znam

17.Link zadatka Neka su [inline]a[/inline], [inline]b[/inline] i [inline]c[/inline] proizvoljni prirodni brojevi. Koliko je od sledećih tvrđenja uvek tačno?
[inline](I)[/inline] ako [inline]ab[/inline] deli [inline]c[/inline], tada [inline]a[/inline] deli [inline]c[/inline] i [inline]b[/inline] deli [inline]c[/inline];
[inline](II)[/inline] ako [inline]a[/inline] deli [inline]c[/inline] i [inline]b[/inline] deli [inline]c[/inline], tada [inline]ab[/inline] deli [inline]c[/inline];
[inline](III)[/inline] ako [inline]a[/inline] deli [inline]b[/inline] i [inline]b[/inline] deli [inline]c[/inline], tada [inline]a[/inline] deli [inline]c[/inline];
[inline](IV)[/inline] [inline]\text{NZD }(a,b)[/inline] deli [inline]a+b[/inline].
[inline]\text{A)}[/inline] nijedno      [inline]\text{B)}[/inline] jedno      [inline]\text{C)}[/inline] dva      [inline]\text{D)}[/inline] tri      [inline]\text{E)}[/inline] četiri              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] nijedno      [inline]\text{B)}[/inline] jedno      [inline]\text{C)}[/inline] dva      [inline]\enclose{circle}{\text{D)}}[/inline] tri      [inline]\text{E)}[/inline] četiri              [inline]\text{N)}[/inline] ne znam

18.Link zadatka Domen funkcije [inline]f(x)=\sqrt{\sqrt{4x-3-x^2}\cdot\sin(\pi x)}[/inline] je:
[inline]\text{A)}[/inline] [inline]\{1\}\cup[2,3][/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\left[0,\frac{\pi}{2}\right]\cup[3,\pi][/inline]      [inline]\text{C)}[/inline] [inline](1,2)\cup(2,3)[/inline]      [inline]\text{D)}[/inline] [inline][0,1]\cup[2,3][/inline]      [inline]\text{E)}[/inline] [inline][2,3][/inline]              [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]\{1\}\cup[2,3][/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\left[0,\frac{\pi}{2}\right]\cup[3,\pi][/inline]      [inline]\text{C)}[/inline] [inline](1,2)\cup(2,3)[/inline]      [inline]\text{D)}[/inline] [inline][0,1]\cup[2,3][/inline]      [inline]\text{E)}[/inline] [inline][2,3][/inline]              [inline]\text{N)}[/inline] ne znam

19.Link zadatka U učionici se nalazi [inline]6[/inline] klupa sa po dva mesta (levo i desno), koje su poređane u red, jedna iza druge. Na koliko načina se na ovih [inline]12[/inline] mesta mogu rasporediti Pera, Mika i Laza, tako da ni u jednoj klupi ne sede dva učenika i da ne postoje dve uzastopne klupe u kojima se nalazi učenik?
[inline]\text{A)}[/inline] [inline]144[/inline]      [inline]\text{B)}[/inline] [inline]24[/inline]      [inline]\text{C)}[/inline] [inline]32[/inline]      [inline]\text{D)}[/inline] [inline]96[/inline]      [inline]\text{E)}[/inline] [inline]192[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]144[/inline]      [inline]\text{B)}[/inline] [inline]24[/inline]      [inline]\text{C)}[/inline] [inline]32[/inline]      [inline]\text{D)}[/inline] [inline]96[/inline]      [inline]\enclose{circle}{\text{E)}}[/inline] [inline]192[/inline]              [inline]\text{N)}[/inline] ne znam

Obrađeno u temi: LINK

20.Link zadatka Neka je [inline]D[/inline] tačka na stranici [inline]CA[/inline], a [inline]E[/inline] tačka na stranici [inline]BC[/inline] trougla [inline]ABC[/inline], tako da važi [inline]AB=BE[/inline] i [inline]AD=DE=EC[/inline]. Ako je [inline]\angle BCA=40^\circ[/inline], tada je razlika [inline]\angle CAB-\angle ABC[/inline] jednaka:
[inline]\text{A)}[/inline] [inline]10^\circ[/inline]      [inline]\text{B)}[/inline] [inline]20^\circ[/inline]      [inline]\text{C)}[/inline] [inline]30^\circ[/inline]      [inline]\text{D)}[/inline] [inline]40^\circ[/inline]      [inline]\text{E)}[/inline] [inline]50^\circ[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]10^\circ[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]20^\circ[/inline]      [inline]\text{C)}[/inline] [inline]30^\circ[/inline]      [inline]\text{D)}[/inline] [inline]40^\circ[/inline]      [inline]\text{E)}[/inline] [inline]50^\circ[/inline]              [inline]\text{N)}[/inline] ne znam


Izvor: LINK

Postupci: http://www.matf.bg.ac.rs/files/skice_resenja_jun2020.pdf


Napomena: Ukoliko vam treba pomoć oko rešavanja nekog od zadataka koji dosad nije obrađivan ni na jednoj temi, slobodno zatražite pomoć na forumu „Matemanija“, naravno uz poštovanje forumskih pravila.