ETF MATF FON GRF TMF FORUM

Prijemni ispit na Matematičkom fakultetu u Beogradu

1. jul 2008.


Vreme za rad je 180 minuta.

1.Link zadatka Ako za realne brojeve [inline]a[/inline] i [inline]b[/inline] važi [inline]\left(2a-b-3\right)^2+\left(3a+b-7\right)^2=0[/inline], onda je [inline]3a-7b[/inline] jednako:
[inline]\text{A)}[/inline] [inline]-13[/inline]      [inline]\text{B)}[/inline] [inline]-8[/inline]      [inline]\text{C)}[/inline] [inline]-4[/inline]      [inline]\text{D)}[/inline] [inline]-1[/inline]      [inline]\text{E)}[/inline] [inline]2[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]-13[/inline]      [inline]\text{B)}[/inline] [inline]-8[/inline]      [inline]\text{C)}[/inline] [inline]-4[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]-1[/inline]      [inline]\text{E)}[/inline] [inline]2[/inline]              [inline]\text{N)}[/inline] ne znam

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2.Link zadatka Kružnica čiji se centar poklapa sa centrom kvadrata deli svaku od stranica tog kvadrata na tri jednaka dela. Odnos površina odgovarajućeg kruga i kvadrata je:
[inline]\text{A)}[/inline] [inline]5\pi:18[/inline]      [inline]\text{B)}[/inline] [inline]13\pi:36[/inline]      [inline]\text{C)}[/inline] [inline]\pi:6[/inline]      [inline]\text{D)}[/inline] [inline]\pi:4[/inline]      [inline]\text{E)}[/inline] [inline]2\pi:9[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]5\pi:18[/inline]      [inline]\text{B)}[/inline] [inline]13\pi:36[/inline]      [inline]\text{C)}[/inline] [inline]\pi:6[/inline]      [inline]\text{D)}[/inline] [inline]\pi:4[/inline]      [inline]\text{E)}[/inline] [inline]2\pi:9[/inline]              [inline]\text{N)}[/inline] ne znam

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3.Link zadatka Vrednost izraza [inline]\displaystyle\frac{1-5^{-1/2}}{1+5^{1/2}}-\frac{5^{1/2}-5^{-1/2}}{4}[/inline] je:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{\sqrt5+5}{2}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{\sqrt5-5}{10}[/inline]      [inline]\text{C)}[/inline] [inline]2\sqrt5[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle-\frac{2}{5-\sqrt5}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{\sqrt5+1}{4}[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle\frac{\sqrt5+5}{2}[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]\displaystyle\frac{\sqrt5-5}{10}[/inline]      [inline]\text{C)}[/inline] [inline]2\sqrt5[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle-\frac{2}{5-\sqrt5}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{\sqrt5+1}{4}[/inline]              [inline]\text{N)}[/inline] ne znam

4.Link zadatka Realan broj [inline]a[/inline] za koji važi [inline]\displaystyle\frac{1+2ai}{1-ai}=\frac{1}{4}+\frac{3\sqrt3}{4}i[/inline] jednak je:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{2\sqrt3}{3}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{\sqrt3}{2}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{\sqrt3}{3}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{\sqrt3}{4}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{\sqrt3}{6}[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle\frac{2\sqrt3}{3}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{\sqrt3}{2}[/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline]\displaystyle\frac{\sqrt3}{3}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{\sqrt3}{4}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{\sqrt3}{6}[/inline]              [inline]\text{N)}[/inline] ne znam

5.Link zadatka Vrednost izraza [inline]\displaystyle\left(\frac{1+i}{\sqrt2}\right)^{2008}+\left(\frac{1-i}{\sqrt2}\right)^{2008}[/inline] je:
[inline]\text{A)}[/inline] [inline]\sqrt2[/inline]      [inline]\text{B)}[/inline] [inline]i\sqrt2[/inline]      [inline]\text{C)}[/inline] [inline]0[/inline]      [inline]\text{D)}[/inline] [inline]2i[/inline]      [inline]\text{E)}[/inline] [inline]2[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\sqrt2[/inline]      [inline]\text{B)}[/inline] [inline]i\sqrt2[/inline]      [inline]\text{C)}[/inline] [inline]0[/inline]      [inline]\text{D)}[/inline] [inline]2i[/inline]      [inline]\enclose{circle}{\text{E)}}[/inline] [inline]2[/inline]              [inline]\text{N)}[/inline] ne znam

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6.Link zadatka Za rešenja [inline]x_1[/inline] i [inline]x_2[/inline] kvadratne jednačine važi [inline]\displaystyle x_1^2+x_2^2=-\frac{1}{4}[/inline] i [inline]\displaystyle\frac{1}{x_1}+\frac{1}{x_2}=2[/inline]. Ta jednačina glasi:
[inline]\text{A)}[/inline] [inline]4x^2-x+2=0[/inline]      [inline]\text{B)}[/inline] [inline]4x^2+x-2=0[/inline]      [inline]\text{C)}[/inline] [inline]4x^2-2x-1=0[/inline]      [inline]\text{D)}[/inline] [inline]4x^2+2x+1=0[/inline]      [inline]\text{E)}[/inline] [inline]4x^2-2x+1=0[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]4x^2-x+2=0[/inline]      [inline]\text{B)}[/inline] [inline]4x^2+x-2=0[/inline]      [inline]\text{C)}[/inline] [inline]4x^2-2x-1=0[/inline]      [inline]\text{D)}[/inline] [inline]4x^2+2x+1=0[/inline]      [inline]\enclose{circle}{\text{E)}}[/inline] [inline]4x^2-2x+1=0[/inline]              [inline]\text{N)}[/inline] ne znam

7.Link zadatka Koliko različitih realnih rešenja ima jednačina [inline]x^2+\sqrt{\left(x-1\right)^2}=1[/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] više od [inline]3[/inline]              [inline]\text{N)}[/inline] ne znam[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] više od [inline]3[/inline]              [inline]\text{N)}[/inline] ne znam

8.Link zadatka Vrednost izraza [inline]\displaystyle\left(\log_{1/2}^24\right)^{1/2}+\left(\log_2^2\frac{1}{4}\right)^{1/2}[/inline] je:
[inline]\text{A)}[/inline] [inline]0[/inline]      [inline]\text{B)}[/inline] [inline]-2[/inline]      [inline]\text{C)}[/inline] [inline]2[/inline]      [inline]\text{D)}[/inline] [inline]4[/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]-2[/inline]      [inline]\text{C)}[/inline] [inline]2[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]4[/inline]      [inline]\text{E)}[/inline] [inline]-4[/inline]              [inline]\text{N)}[/inline] ne znam

9.Link zadatka Koliko celih brojeva [inline]x[/inline] zadovoljava jednačinu [inline]x\cdot3^{\log_x5}=15[/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] više od [inline]3[/inline]              [inline]\text{N)}[/inline] ne znam[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] više od [inline]3[/inline]              [inline]\text{N)}[/inline] ne znam

10.Link zadatka Jednačina [inline]\left|x-1\right|-\left|x-2\right|+\left|x-3\right|=a[/inline] ima [inline]4[/inline] rešenja ako i samo ako parametar [inline]a[/inline] pripada intervalu:
[inline]\text{A)}[/inline] [inline]\left(0,1\right)[/inline]      [inline]\text{B)}[/inline] [inline]\left(0,1\right][/inline]      [inline]\text{C)}[/inline] [inline]\left[1,2\right][/inline]      [inline]\text{D)}[/inline] [inline]\left(1,2\right)[/inline]      [inline]\text{E)}[/inline] [inline]\left(2,+\infty\right)[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\left(0,1\right)[/inline]      [inline]\text{B)}[/inline] [inline]\left(0,1\right][/inline]      [inline]\text{C)}[/inline] [inline]\left[1,2\right][/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]\left(1,2\right)[/inline]      [inline]\text{E)}[/inline] [inline]\left(2,+\infty\right)[/inline]              [inline]\text{N)}[/inline] ne znam

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11.Link zadatka Vrednost izraza [inline]\displaystyle\cos\biggl(2\text{ arctg}\left(2\sin\frac{\pi}{3}\right)\biggr)[/inline] je:
[inline]\text{A)}[/inline] [inline]\displaystyle-\frac{\sqrt3}{2}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle-\frac{1}{2}[/inline]      [inline]\text{C)}[/inline] [inline]0[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{1}{2}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{\sqrt3}{2}[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle-\frac{\sqrt3}{2}[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]\displaystyle-\frac{1}{2}[/inline]      [inline]\text{C)}[/inline] [inline]0[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{1}{2}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{\sqrt3}{2}[/inline]              [inline]\text{N)}[/inline] ne znam

12.Link zadatka Izraz [inline]\displaystyle\sin\alpha+\sin\left(\alpha+\frac{2\pi}{3}\right)+\sin\left(\alpha+\frac{4\pi}{3}\right)[/inline] identički je jednak izrazu:
[inline]\text{A)}[/inline] [inline]3\sin\alpha[/inline]      [inline]\text{B)}[/inline] [inline]0[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{2\sqrt3}{3}[/inline]      [inline]\text{D)}[/inline] [inline]\sin\alpha[/inline]      [inline]\text{E)}[/inline] [inline]2\sin\alpha[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]3\sin\alpha[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]0[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{2\sqrt3}{3}[/inline]      [inline]\text{D)}[/inline] [inline]\sin\alpha[/inline]      [inline]\text{E)}[/inline] [inline]2\sin\alpha[/inline]              [inline]\text{N)}[/inline] ne znam

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13.Link zadatka Broj rešenja jednačine [inline]\sin2x=\cos x[/inline] na intervalu [inline]\left[-\pi,2\pi\right][/inline] je:
[inline]\text{A)}[/inline] [inline]2[/inline]      [inline]\text{B)}[/inline] [inline]3[/inline]      [inline]\text{C)}[/inline] [inline]4[/inline]      [inline]\text{D)}[/inline] [inline]5[/inline]      [inline]\text{E)}[/inline] veći od [inline]5[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]2[/inline]      [inline]\text{B)}[/inline] [inline]3[/inline]      [inline]\text{C)}[/inline] [inline]4[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]5[/inline]      [inline]\text{E)}[/inline] veći od [inline]5[/inline]              [inline]\text{N)}[/inline] ne znam

14.Link zadatka Dužine stranica jednog trougla su [inline]7\text{ cm}[/inline], [inline]8\text{ cm}[/inline] i [inline]13\text{ cm}[/inline]. Najveći ugao tog trougla jednak je:
[inline]\text{A)}[/inline] [inline]90^\circ[/inline]      [inline]\text{B)}[/inline] [inline]105^\circ[/inline]      [inline]\text{C)}[/inline] [inline]120^\circ[/inline]      [inline]\text{D)}[/inline] [inline]135^\circ[/inline]      [inline]\text{E)}[/inline] [inline]150^\circ[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]90^\circ[/inline]      [inline]\text{B)}[/inline] [inline]105^\circ[/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline]120^\circ[/inline]      [inline]\text{D)}[/inline] [inline]135^\circ[/inline]      [inline]\text{E)}[/inline] [inline]150^\circ[/inline]              [inline]\text{N)}[/inline] ne znam

15.Link zadatka Jednačina kruga simetričnog krugu [inline]\left(x+2\right)^2+\left(y-1\right)^2=4[/inline] u odnosu na tačku [inline]\left(1,2\right)[/inline] je:
[inline]\text{A)}[/inline] [inline]x^2-8x+y^2-6y+21=0[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle x^2-\frac{1}{4}x+y^2+\frac{1}{4}y-\frac{7}{2}=0[/inline]      [inline]\text{C)}[/inline] [inline]x^2+4x+y^2-2y+1=0[/inline]      [inline]\text{D)}[/inline] [inline]x^2-x+y^2+2y+1=0[/inline]      [inline]\text{E)}[/inline] [inline]x^2+8x+y^2+6y+21=0[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]x^2-8x+y^2-6y+21=0[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle x^2-\frac{1}{4}x+y^2+\frac{1}{4}y-\frac{7}{2}=0[/inline]      [inline]\text{C)}[/inline] [inline]x^2+4x+y^2-2y+1=0[/inline]      [inline]\text{D)}[/inline] [inline]x^2-x+y^2+2y+1=0[/inline]      [inline]\text{E)}[/inline] [inline]x^2+8x+y^2+6y+21=0[/inline]              [inline]\text{N)}[/inline] ne znam

16.Link zadatka Ako tačka [inline]M\left(x_0,y_0\right)[/inline] pripada pravoj [inline]8x+3y-15=0[/inline] i ako je jednako udaljena od tačaka [inline]A\left(8,2\right)[/inline] i [inline]B\left(2,4\right)[/inline], tada je proizvod [inline]x_0y_0[/inline] jednak:
[inline]\text{A)}[/inline] [inline]-9[/inline]      [inline]\text{B)}[/inline] [inline]0[/inline]      [inline]\text{C)}[/inline] [inline]6[/inline]      [inline]\text{D)}[/inline] [inline]9[/inline]      [inline]\text{E)}[/inline] [inline]12[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]-9[/inline]      [inline]\text{B)}[/inline] [inline]0[/inline]      [inline]\text{C)}[/inline] [inline]6[/inline]      [inline]\text{D)}[/inline] [inline]9[/inline]      [inline]\text{E)}[/inline] [inline]12[/inline]              [inline]\text{N)}[/inline] ne znam

17.Link zadatka Zbir svih trocifrenih brojeva deljivih sa [inline]13[/inline] je:
[inline]\text{A)}[/inline] [inline]37\,024[/inline]      [inline]\text{B)}[/inline] [inline]37\,128[/inline]      [inline]\text{C)}[/inline] [inline]37\,674[/inline]      [inline]\text{D)}[/inline] [inline]38\,220[/inline]      [inline]\text{E)}[/inline] [inline]38\,675[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]37\,024[/inline]      [inline]\text{B)}[/inline] [inline]37\,128[/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline]37\,674[/inline]      [inline]\text{D)}[/inline] [inline]38\,220[/inline]      [inline]\text{E)}[/inline] [inline]38\,675[/inline]              [inline]\text{N)}[/inline] ne znam

18.Link zadatka Koeficijent uz [inline]x^{10}[/inline] u sređenom obliku polinoma [inline]\left(3-2x^2\right)^7[/inline] jednak je:
[inline]\text{A)}[/inline] [inline]6048[/inline]      [inline]\text{B)}[/inline] [inline]-6048[/inline]      [inline]\text{C)}[/inline] [inline]2016[/inline]      [inline]\text{D)}[/inline] [inline]-20412[/inline]      [inline]\text{E)}[/inline] [inline]-2016[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]6048[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]-6048[/inline]      [inline]\text{C)}[/inline] [inline]2016[/inline]      [inline]\text{D)}[/inline] [inline]-20412[/inline]      [inline]\text{E)}[/inline] [inline]-2016[/inline]              [inline]\text{N)}[/inline] ne znam

19.Link zadatka Ako je zbir svih članova beskonačnog geometrijskog niza [inline]2a+a\sqrt2+a+\cdots[/inline] jednak [inline]8[/inline], onda broj [inline]a[/inline] pripada intervalu:
[inline]\text{A)}[/inline] [inline]\left[0,1\right)[/inline]      [inline]\text{B)}[/inline] [inline]\left[1,2\right)[/inline]      [inline]\text{C)}[/inline] [inline]\left[2,3\right)[/inline]      [inline]\text{D)}[/inline] [inline]\left[3,4\right)[/inline]      [inline]\text{E)}[/inline] [inline]\left[4,+\infty\right)[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\left[0,1\right)[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]\left[1,2\right)[/inline]      [inline]\text{C)}[/inline] [inline]\left[2,3\right)[/inline]      [inline]\text{D)}[/inline] [inline]\left[3,4\right)[/inline]      [inline]\text{E)}[/inline] [inline]\left[4,+\infty\right)[/inline]              [inline]\text{N)}[/inline] ne znam

20.Link zadatka Visina kupe maksimalne zapremine sa datom izvodnicom [inline]s[/inline] jednaka je:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{s\sqrt6}{6}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{s\sqrt2}{6}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{s\sqrt2}{2}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{s\sqrt3}{2}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{s\sqrt3}{3}[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle\frac{s\sqrt6}{6}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{s\sqrt2}{6}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{s\sqrt2}{2}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{s\sqrt3}{2}[/inline]      [inline]\enclose{circle}{\text{E)}}[/inline] [inline]\displaystyle\frac{s\sqrt3}{3}[/inline]              [inline]\text{N)}[/inline] ne znam

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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.