ETF MATF FON GRF FORUM

Prijemni ispit na Matematičkom fakultetu u Beogradu

30. jun 2010.



1.Link zadatka Koliko elemenata ima skup [inline]A[/inline] ako je: [inline]A\cap\left\{3,5,8,11\right\}=\left\{5,8\right\}[/inline], [inline]A\cup\left\{4,5,11,13\right\}=\left\{4,5,7,8,11,13\right\}[/inline], [inline]\left\{8,13\right\}\subset A[/inline] i [inline]A\subset\left\{5,7,8,9,11,13\right\}[/inline]?
[inline]\text{A)}[/inline] [inline]2[/inline]      [inline]\text{B)}[/inline] [inline]3[/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline]4[/inline]      [inline]\text{D)}[/inline] [inline]5[/inline]      [inline]\text{E)}[/inline] [inline]6[/inline]              [inline]\text{N)}[/inline] ne znam

2.Link zadatka Sistem jednačina [equation]\begin{array}{rcl} 3x+y & = & 13,\\ \left(a-2\right)x-5y & = & 10,\\ x-y & = & 7, \end{array}[/equation] ima rešenje ako i samo ako parametar [inline]a[/inline] ima vrednost:
[inline]\text{A)}[/inline] [inline]-4[/inline]      [inline]\text{B)}[/inline] [inline]-2[/inline]      [inline]\text{C)}[/inline] [inline]0[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]2[/inline]      [inline]\text{E)}[/inline] [inline]4[/inline]              [inline]\text{N)}[/inline] ne znam

3.Link zadatka Koja je [inline]2010[/inline]-ta cifra posle zapete u decimalnom zapisu broja [inline]\displaystyle\frac{2010}{7}[/inline]?
[inline]\text{A)}[/inline] [inline]1[/inline]      [inline]\text{B)}[/inline] [inline]2[/inline]      [inline]\text{C)}[/inline] [inline]4[/inline]      [inline]\text{D)}[/inline] [inline]5[/inline]      [inline]\enclose{circle}{\text{E)}}[/inline] [inline]7[/inline]              [inline]\text{N)}[/inline] ne znam

4.Link zadatka Ako je [inline]\displaystyle\frac{b-2a}{4a+3b}=2[/inline], [inline]a,b\ne0[/inline], [inline]4a+3b\ne0[/inline], onda je [inline]\displaystyle\frac{2a^2-3ab+5b^2}{4a^2+3b^2}[/inline] jednako:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{4}{7}[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]\displaystyle\frac{7}{4}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{2}{5}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{5}{2}[/inline]      [inline]\text{E)}[/inline] [inline]1[/inline]              [inline]\text{N)}[/inline] ne znam

5.Link zadatka Vrednost izraza [inline]\displaystyle\frac{25^{0,3}\cdot5^{0,4}}{125^{-\frac{1}{3}}}[/inline] pripada intervalu:
[inline]\text{A)}[/inline] [inline]\left(-\infty,0\right][/inline]      [inline]\text{B)}[/inline] [inline]\left(0,1\right][/inline]      [inline]\text{C)}[/inline] [inline]\left(1,5\right][/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]\left(5,25\right][/inline]      [inline]\text{E)}[/inline] [inline]\left(25,+\infty\right)[/inline]              [inline]\text{N)}[/inline] ne znam

6.Link zadatka Ako su [inline]x[/inline] i [inline]y[/inline] realni brojevi, takvi da je [inline]\left(2+i\right)\left(x+iy\right)=5-5i[/inline], tada je zbir [inline]x+y[/inline] jednak:
[inline]\text{A)}[/inline] [inline]2[/inline]      [inline]\text{B)}[/inline] [inline]3[/inline]      [inline]\text{C)}[/inline] [inline]1[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]-2[/inline]      [inline]\text{E)}[/inline] [inline]-3[/inline]              [inline]\text{N)}[/inline] ne znam

7.Link zadatka Skup rešenja nejednačine [inline]x+3\left|x-1\right|<5[/inline] je:
[inline]\text{A)}[/inline] [inline]\left(-\infty,2\right)[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]\left(-1,2\right)[/inline]      [inline]\text{C)}[/inline] [inline]\left(-1,1\right)[/inline]      [inline]\text{D)}[/inline] [inline]\left(1,2\right)[/inline]      [inline]\text{E)}[/inline] [inline]\left(-1,+\infty\right)[/inline]              [inline]\text{N)}[/inline] ne znam

8.Link zadatka Zbir površina svih kvadrata u koordinatnoj ravni čija su temena tačke [inline]O\left(0,0\right)[/inline] i [inline]P\left(1,3\right)[/inline] iznosi:
[inline]\text{A)}[/inline] [inline]40[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]25[/inline]      [inline]\text{C)}[/inline] [inline]20[/inline]      [inline]\text{D)}[/inline] [inline]15[/inline]      [inline]\text{E)}[/inline] [inline]10[/inline]              [inline]\text{N)}[/inline] ne znam

9.Link zadatka Dati su iskazi: [inline]\text{(I) }\log\left(ab\right)=\log a+\log b[/inline] za sve [inline]a,b\in\mathbb{R}\setminus\left\{0\right\}[/inline]; [inline]\text{(II) }\displaystyle\log\frac{a}{b}=\log a-\log b[/inline] za sve [inline]a,b\in\mathbb{R}\setminus\left\{0\right\}[/inline]; [inline]\text{(III) }\log a^2=2\log a[/inline] za sve [inline]a\in\mathbb{R}\setminus\left\{0\right\}[/inline]; [inline]\text{(IV) }\log\left(-a\right)\left(-b\right)=\log\left(-a\right)+\log\left(-b\right)[/inline] za sve [inline]a<0[/inline], [inline]b<0[/inline]. Tačni su iskazi:
[inline]\text{A)}[/inline] svi      [inline]\text{B)}[/inline] nijedan      [inline]\enclose{circle}{\text{C)}}[/inline] samo [inline]\text{(IV)}[/inline]      [inline]\text{D)}[/inline] [inline]\text{(III)}[/inline] i [inline]\text{(IV)}[/inline]      [inline]\text{E)}[/inline] [inline]\text{(I)}[/inline] i [inline]\text{(II)}[/inline]              [inline]\text{N)}[/inline] ne znam

10.Link zadatka Vrednost izraza [inline]\text{tg }40^\circ\text{tg }45^\circ\text{tg }50^\circ[/inline] je:
[inline]\text{A)}[/inline] [inline]0[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{\sqrt3}{3}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{1}{3}[/inline]      [inline]\text{D)}[/inline] [inline]\sqrt3[/inline]      [inline]\enclose{circle}{\text{E)}}[/inline] [inline]1[/inline]              [inline]\text{N)}[/inline] ne znam

11.Link zadatka Grafik funkcije [inline]f\left(x\right)=ax^2+bx+c[/inline] prikazan je na slici.
Prijemni ispit MATF 2010. – slika
Tačan je iskaz:
[inline]\text{A)}[/inline] [inline]a>0,\;b>0,\;c<0[/inline]      [inline]\text{B)}[/inline] [inline]a>0,\;b>0,\;c>0[/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline]a>0,\;b<0,\;c<0[/inline]      [inline]\text{D)}[/inline] [inline]a>0,\;b<0,\;c>0[/inline]      [inline]\text{E)}[/inline] [inline]a<0,\;b<0,\;c<0[/inline]              [inline]\text{N)}[/inline] ne znam

12.Link zadatka Rešenje jednačine [inline]\displaystyle\frac{1}{1-\sqrt{1-x}}+\frac{1}{1+\sqrt{1-x}}=\frac{4\sqrt3}{\sqrt{1-x}}[/inline] pripada intervalu:
[inline]\text{A)}[/inline] [inline]\left(-\infty,-2\right][/inline]      [inline]\text{B)}[/inline] [inline]\left(-2,-1\right][/inline]      [inline]\text{C)}[/inline] [inline]\left(-1,0\right][/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]\left(0,1\right][/inline]      [inline]\text{E)}[/inline] [inline]\left(1,+\infty\right)[/inline]              [inline]\text{N)}[/inline] ne znam

13.Link zadatka Dužine stranica trougla [inline]ABC[/inline] su [inline]BC=4\sqrt3\text{ cm}[/inline] i [inline]CA=4\text{ cm}[/inline], a [inline]\angle A=120^\circ[/inline]. Dužina stranice [inline]AB[/inline] je:
[inline]\text{A)}[/inline] [inline]2\sqrt3\text{ cm}[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]4\text{ cm}[/inline]      [inline]\text{C)}[/inline] [inline]3\sqrt2\text{ cm}[/inline]      [inline]\text{D)}[/inline] [inline]3\text{ cm}[/inline]      [inline]\text{E)}[/inline] [inline]5\text{ cm}[/inline]              [inline]\text{N)}[/inline] ne znam

14.Link zadatka Brojevi [inline]a_1,a_2,\ldots,a_{20}[/inline] obrazuju aritmetički niz. Ako je zbir svih članova sa neparnim indeksima jednak [inline]320[/inline], a zbir svih članova sa parnim indeksima jednak [inline]350[/inline], onda je [inline]a_{11}[/inline] jednako:
[inline]\text{A)}[/inline] [inline]32[/inline]      [inline]\text{B)}[/inline] [inline]34[/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline]35[/inline]      [inline]\text{D)}[/inline] [inline]36[/inline]      [inline]\text{E)}[/inline] [inline]38[/inline]              [inline]\text{N)}[/inline] ne znam

15.Link zadatka Broj rešenja jednačine [inline]\displaystyle\sin x\cos\frac{\pi}{5}+\cos x\sin\frac{\pi}{5}=\frac{\sqrt3}{2}[/inline] koja pripadaju intervalu [inline]\displaystyle\left[0,\frac{\pi}{2}\right][/inline] je:
[inline]\text{A)}[/inline] [inline]0[/inline]      [inline]\text{B)}[/inline] [inline]1[/inline]      [inline]\enclose{circle}{\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

16.Link zadatka Prava [inline]p[/inline] sadrži centar kružnice [inline]k[/inline] i tačku [inline]P[/inline] van te kružnice, i seče kružnicu u tačkama [inline]A[/inline] i [inline]B[/inline] tako da je [inline]PA=8\text{ cm}[/inline] i [inline]PB=18\text{ cm}[/inline]. Ako je [inline]T[/inline] tačka te kružnice takva da je prava [inline]PT[/inline] njena tangenta, onda je dužina duži [inline]PT[/inline] jednaka:
[inline]\enclose{circle}{\text{A)}}[/inline] [inline]12\text{ cm}[/inline]      [inline]\text{B)}[/inline] [inline]6\sqrt3\text{ cm}[/inline]      [inline]\text{C)}[/inline] [inline]8\sqrt2\text{ cm}[/inline]      [inline]\text{D)}[/inline] [inline]9\sqrt2\text{ cm}[/inline]      [inline]\text{E)}[/inline] [inline]10\text{ cm}[/inline]              [inline]\text{N)}[/inline] ne znam

17.Link zadatka Ako je [inline]\log_83=a[/inline] i [inline]\log_35=b[/inline], onda je [inline]\log_{10}6[/inline] jednak:
[inline]\enclose{circle}{\text{A)}}[/inline] [inline]\displaystyle\frac{3a+1}{3ab+1}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{3a}{3ab+1}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{1}{3ab+1}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{a+ab+3}{ab+3}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{a+3}{ab+3}[/inline]              [inline]\text{N)}[/inline] ne znam

18.Link zadatka Ako je [inline]\alpha[/inline] ugao koji dijagonala kocke obrazuje sa ravni njene osnove, onda je:
[inline]\text{A)}[/inline] [inline]0<\alpha\le15^\circ[/inline]      [inline]\text{B)}[/inline] [inline]15^\circ<\alpha\le30^\circ[/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline]30^\circ<\alpha\le45^\circ[/inline]      [inline]\text{D)}[/inline] [inline]45^\circ<\alpha\le60^\circ[/inline]      [inline]\text{E)}[/inline] [inline]60^\circ<\alpha\le90^\circ[/inline]              [inline]\text{N)}[/inline] ne znam

19.Link zadatka Najveća vrednost funkcije [inline]f\left(x\right)=e^x+e^{-x}[/inline] na segmentu [inline]\left[-1,2\right][/inline] je:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{1}{e}[/inline]      [inline]\text{B)}[/inline] [inline]1[/inline]      [inline]\text{C)}[/inline] [inline]e^2[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle e+\frac{1}{e}[/inline]      [inline]\enclose{circle}{\text{E)}}[/inline] [inline]\displaystyle e^2+\frac{1}{e^2}[/inline]              [inline]\text{N)}[/inline] ne znam

20.Link zadatka Od svih tačaka kružnice [inline]x^2+y^2=4[/inline] tačka [inline]\left(x_0,y_0\right)[/inline] je najdalje od prave [inline]x-2y-1=0[/inline]. Zbir [inline]x_0+y_0[/inline] je jednak:
[inline]\text{A)}[/inline] [inline]\displaystyle-\frac{2\sqrt5}{5}[/inline]      [inline]\text{B)}[/inline] [inline]0[/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline]\displaystyle\frac{2\sqrt5}{5}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{4\sqrt5}{5}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{6\sqrt5}{5}[/inline]              [inline]\text{N)}[/inline] ne znam


Izvor: http://www.matf.bg.ac.rs/files/reseni_zadaci_juni2010.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.