ETF MATF FON GRF TMF FORUM

Prijemni ispit na Građevinskom fakultetu u Beogradu

9. jul 2014.


Test ima [inline]20[/inline] zadataka na dve stranice. Zadaci [inline]1–3[/inline] vrede po [inline]4[/inline] poena, zadaci [inline]4–17[/inline] vrede po [inline]5[/inline] poena i zadaci [inline]18–20[/inline] vrede po [inline]6[/inline] poena. Pogrešan odgovor donosi [inline]−10\%[/inline] poena od broja poena predviđenih za tačan odgovor. Zaokruživanje [inline]N[/inline] ne donosi ni pozitivne, ni negativne poene. U slučaju zaokruživanja više od jednog, kao i u slučaju nezaokruživanja nijednog odgovora, dobija se [inline]−1[/inline] poen.

1.Link zadatka Vrednost izraza [inline]\displaystyle\left(\sqrt6-\frac{6}{\sqrt6+2}\right):\bigl(\left(\sqrt[4]3-\sqrt[4]2\right)\left(\sqrt[4]3+\sqrt[4]2\right)\bigr)[/inline] jednaka je:
[inline]\text{A)}[/inline] [inline]2\sqrt3[/inline]      [inline]\text{B)}[/inline] [inline]\sqrt3[/inline]      [inline]\text{C)}[/inline] [inline]\sqrt2[/inline]      [inline]\text{D)}[/inline] [inline]2\sqrt2[/inline]      [inline]\text{E)}[/inline] [inline]2\sqrt6[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\enclose{box}{\text{A)}}[/inline] [inline]2\sqrt3[/inline]      [inline]\text{B)}[/inline] [inline]\sqrt3[/inline]      [inline]\text{C)}[/inline] [inline]\sqrt2[/inline]      [inline]\text{D)}[/inline] [inline]2\sqrt2[/inline]      [inline]\text{E)}[/inline] [inline]2\sqrt6[/inline]              [inline]\text{N)}[/inline] Ne znam

2.Link zadatka Ako je [inline]x>0[/inline] i [inline]f(x)=\log_2x^2+5\log_24x[/inline], onda je [inline]\displaystyle f(x)+f\left(\frac{1}{x}\right)[/inline] jednako:
[inline]\text{A)}[/inline] [inline]10\log_2x[/inline]      [inline]\text{B)}[/inline] [inline]20\log_2x[/inline]      [inline]\text{C)}[/inline] [inline]0[/inline]      [inline]\text{D)}[/inline] [inline]10[/inline]      [inline]\text{E)}[/inline] [inline]20[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\text{A)}[/inline] [inline]10\log_2x[/inline]      [inline]\text{B)}[/inline] [inline]20\log_2x[/inline]      [inline]\text{C)}[/inline] [inline]0[/inline]      [inline]\text{D)}[/inline] [inline]10[/inline]      [inline]\enclose{box}{\text{E)}}[/inline] [inline]20[/inline]              [inline]\text{N)}[/inline] Ne znam

3.Link zadatka Rešenje nejednačine [inline]\displaystyle\frac{1}{x}\lt x[/inline] je skup oblika:
[inline]\text{A)}[/inline] [inline][a,+\infty)[/inline]      [inline]\text{B)}[/inline] [inline][a,b)[/inline]      [inline]\text{C)}[/inline] [inline](-\infty,a]\cup[b,+\infty)[/inline]      [inline]\text{D)}[/inline] [inline](a,+\infty)[/inline]      [inline]\text{E)}[/inline] [inline][a,b)\cup[c,+\infty)[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\text{A)}[/inline] [inline][a,+\infty)[/inline]      [inline]\text{B)}[/inline] [inline][a,b)[/inline]      [inline]\text{C)}[/inline] [inline](-\infty,a]\cup[b,+\infty)[/inline]      [inline]\text{D)}[/inline] [inline](a,+\infty)[/inline]      [inline]\enclose{box}{\text{E)}}[/inline] [inline][a,b)\cup[c,+\infty)[/inline]              [inline]\text{N)}[/inline] Ne znam

4.Link zadatka Zbir rešenja jednačine [inline]\left|x^2+3x+2\right|-3|x+2|=0[/inline] je:
[inline]\text{A)}[/inline] [inline]-6[/inline]      [inline]\text{B)}[/inline] [inline]-4[/inline]      [inline]\text{C)}[/inline] [inline]-2[/inline]      [inline]\text{D)}[/inline] [inline]0[/inline]      [inline]\text{E)}[/inline] [inline]2[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\text{A)}[/inline] [inline]-6[/inline]      [inline]\enclose{box}{\text{B)}}[/inline] [inline]-4[/inline]      [inline]\text{C)}[/inline] [inline]-2[/inline]      [inline]\text{D)}[/inline] [inline]0[/inline]      [inline]\text{E)}[/inline] [inline]2[/inline]              [inline]\text{N)}[/inline] Ne znam

5.Link zadatka Ako je [inline](a_n)[/inline] rastući aritmetički niz, [inline]a_1+a_3+a_5=-12[/inline] i [inline]a_1a_3a_5=80[/inline], onda je [inline]a_1[/inline] jednako:
[inline]\text{A)}[/inline] [inline]10[/inline]      [inline]\text{B)}[/inline] [inline]4[/inline]      [inline]\text{C)}[/inline] [inline]2[/inline]      [inline]\text{D)}[/inline] [inline]-4[/inline]      [inline]\text{E)}[/inline] [inline]-10[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\text{A)}[/inline] [inline]10[/inline]      [inline]\text{B)}[/inline] [inline]4[/inline]      [inline]\text{C)}[/inline] [inline]2[/inline]      [inline]\text{D)}[/inline] [inline]-4[/inline]      [inline]\enclose{box}{\text{E)}}[/inline] [inline]-10[/inline]              [inline]\text{N)}[/inline] Ne znam

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6.Link zadatka Skup rešenja nejednačine [inline]2\cdot25^x-10^x\lt10\cdot4^x[/inline] sadržan je u skupu:
[inline]\text{A)}[/inline] [inline](-\infty,0)[/inline]      [inline]\text{B)}[/inline] [inline](0,\infty)[/inline]      [inline]\text{C)}[/inline] [inline](2,\infty)[/inline]      [inline]\text{D)}[/inline] [inline](-\infty,2)[/inline]      [inline]\text{E)}[/inline] [inline](1,\infty)[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\text{A)}[/inline] [inline](-\infty,0)[/inline]      [inline]\text{B)}[/inline] [inline](0,\infty)[/inline]      [inline]\text{C)}[/inline] [inline](2,\infty)[/inline]      [inline]\enclose{box}{\text{D)}}[/inline] [inline](-\infty,2)[/inline]      [inline]\text{E)}[/inline] [inline](1,\infty)[/inline]              [inline]\text{N)}[/inline] Ne znam

7.Link zadatka Koliko različitih delilaca ima broj [inline]1200[/inline] (uključujući broj [inline]1[/inline] i sam broj [inline]1200[/inline])?
[inline]\text{A)}[/inline] [inline]26[/inline]      [inline]\text{B)}[/inline] [inline]28[/inline]      [inline]\text{C)}[/inline] [inline]30[/inline]      [inline]\text{D)}[/inline] [inline]32[/inline]      [inline]\text{E)}[/inline] [inline]34[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\text{A)}[/inline] [inline]26[/inline]      [inline]\text{B)}[/inline] [inline]28[/inline]      [inline]\enclose{box}{\text{C)}}[/inline] [inline]30[/inline]      [inline]\text{D)}[/inline] [inline]32[/inline]      [inline]\text{E)}[/inline] [inline]34[/inline]              [inline]\text{N)}[/inline] Ne znam

8.Link zadatka Polinom [inline]P(x)=x^4+ax^3+bx[/inline] deljiv je polinomom [inline]Q(x)=x^2+4x+4[/inline]. Ostatak pri deljenju polinoma [inline]P(x)[/inline] polinomom [inline]x-2[/inline] jednak je:
[inline]\text{A)}[/inline] [inline]33[/inline]      [inline]\text{B)}[/inline] [inline]23[/inline]      [inline]\text{C)}[/inline] [inline]32[/inline]      [inline]\text{D)}[/inline] [inline]-23[/inline]      [inline]\text{E)}[/inline] [inline]-32[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\text{A)}[/inline] [inline]33[/inline]      [inline]\text{B)}[/inline] [inline]23[/inline]      [inline]\enclose{box}{\text{C)}}[/inline] [inline]32[/inline]      [inline]\text{D)}[/inline] [inline]-23[/inline]      [inline]\text{E)}[/inline] [inline]-32[/inline]              [inline]\text{N)}[/inline] Ne znam

9.Link zadatka Zbir svih kompleksnih brojeva [inline]z=x+iy[/inline] ([inline]x,y\in\mathbb{R}[/inline], [inline]i^2=-1[/inline]), takvih da je [inline]\overline z+|2z+i|=3+i[/inline] je:
[inline]\text{A)}[/inline] [inline]-2[/inline]      [inline]\text{B)}[/inline] [inline]-2-2i[/inline]      [inline]\text{C)}[/inline] [inline]-2i[/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]-2[/inline]      [inline]\enclose{box}{\text{B)}}[/inline] [inline]-2-2i[/inline]      [inline]\text{C)}[/inline] [inline]-2i[/inline]      [inline]\text{D)}[/inline] [inline]2i[/inline]      [inline]\text{E)}[/inline] [inline]2[/inline]              [inline]\text{N)}[/inline] Ne znam

10.Link zadatka Prave [inline]y-x=a[/inline] i [inline]x-y=b[/inline], gde su [inline]a>0[/inline] i [inline]b>0[/inline] pozitivni realni parametri, seku koordinatne ose redom u tačkama [inline]A[/inline], [inline]B[/inline], [inline]C[/inline] i [inline]D[/inline]. Ako je površina četvorougla [inline]ABCD[/inline] jednaka [inline]200[/inline], onda je [inline]a+b[/inline] jednako:
[inline]\text{A)}[/inline] [inline]10[/inline]      [inline]\text{B)}[/inline] [inline]-20[/inline]      [inline]\text{C)}[/inline] [inline]20[/inline]      [inline]\text{D)}[/inline] [inline]40[/inline]      [inline]\text{E)}[/inline] [inline]-40[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\text{A)}[/inline] [inline]10[/inline]      [inline]\text{B)}[/inline] [inline]-20[/inline]      [inline]\enclose{box}{\text{C)}}[/inline] [inline]20[/inline]      [inline]\text{D)}[/inline] [inline]40[/inline]      [inline]\text{E)}[/inline] [inline]-40[/inline]              [inline]\text{N)}[/inline] Ne znam

11.Link zadatka Ako je [inline]\text{tg }1007^\circ=m[/inline], onda je [inline]\sin2014^\circ[/inline] jednak:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{2m}{1+m^2}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{1-m^2}{1+m^2}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{1-m^2+m}{1+m^2}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{m^2-1}{1+m^2}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle-\frac{2m}{1+m^2}[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\enclose{box}{\text{A)}}[/inline] [inline]\displaystyle\frac{2m}{1+m^2}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{1-m^2}{1+m^2}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{1-m^2+m}{1+m^2}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{m^2-1}{1+m^2}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle-\frac{2m}{1+m^2}[/inline]              [inline]\text{N)}[/inline] Ne znam

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12.Link zadatka Broj realnih rešenja jednačine [inline]\sqrt{2x-3}-\sqrt{x+2}=\sqrt{3-x}[/inline] jednak je:
[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]\infty[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\enclose{box}{\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]\infty[/inline]              [inline]\text{N)}[/inline] Ne znam

13.Link zadatka Vrednost izraza [inline]1+i+i^2+\cdots+i^{2014}[/inline], gde je [inline]i^2=-1[/inline], jednaka je:
[inline]\text{A)}[/inline] [inline]1[/inline]      [inline]\text{B)}[/inline] [inline]i[/inline]      [inline]\text{C)}[/inline] [inline]-1[/inline]      [inline]\text{D)}[/inline] [inline]-i[/inline]      [inline]\text{E)}[/inline] [inline]1+i[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\text{A)}[/inline] [inline]1[/inline]      [inline]\enclose{box}{\text{B)}}[/inline] [inline]i[/inline]      [inline]\text{C)}[/inline] [inline]-1[/inline]      [inline]\text{D)}[/inline] [inline]-i[/inline]      [inline]\text{E)}[/inline] [inline]1+i[/inline]              [inline]\text{N)}[/inline] Ne znam

14.Link zadatka Zbir kvadrata rešenja jednačine [inline]\displaystyle\log_{x^2}5+\log_{x^4}5=\frac{3}{2}[/inline] je:
[inline]\text{A)}[/inline] [inline]10[/inline]      [inline]\text{B)}[/inline] [inline]5[/inline]      [inline]\text{C)}[/inline] [inline]0[/inline]      [inline]\text{D)}[/inline] [inline]25[/inline]      [inline]\text{E)}[/inline] [inline]100[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\enclose{box}{\text{A)}}[/inline] [inline]10[/inline]      [inline]\text{B)}[/inline] [inline]5[/inline]      [inline]\text{C)}[/inline] [inline]0[/inline]      [inline]\text{D)}[/inline] [inline]25[/inline]      [inline]\text{E)}[/inline] [inline]100[/inline]              [inline]\text{N)}[/inline] Ne znam

15.Link zadatka Pravilni šestougao stranice [inline]a[/inline] rotira oko svoje veće dijagonale. Zapremina tako nastalog rotacionog tela jednaka je:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{a^3\sqrt3\pi}{3}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{4a^3\sqrt3\pi}{3}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{4a^3\pi}{3}[/inline]      [inline]\text{D)}[/inline] [inline]a^3\pi[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{a^3\pi}{3}[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\text{A)}[/inline] [inline]\displaystyle\frac{a^3\sqrt3\pi}{3}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{4a^3\sqrt3\pi}{3}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{4a^3\pi}{3}[/inline]      [inline]\enclose{box}{\text{D)}}[/inline] [inline]a^3\pi[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{a^3\pi}{3}[/inline]              [inline]\text{N)}[/inline] Ne znam

16.Link zadatka Zbir najmanje i najveće vrednosti funkcije [inline]f(x)=\left|x^2-2x\right|+\left|-x^2+5x-6\right|[/inline] na [inline]\displaystyle\left[\frac{3}{2},\frac{5}{2}\right][/inline] je:
[inline]\text{A)}[/inline] [inline]3[/inline]      [inline]\text{B)}[/inline] [inline]2[/inline]      [inline]\text{C)}[/inline] [inline]0[/inline]      [inline]\text{D)}[/inline] [inline]1[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{3}{2}[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\text{A)}[/inline] [inline]3[/inline]      [inline]\text{B)}[/inline] [inline]2[/inline]      [inline]\text{C)}[/inline] [inline]0[/inline]      [inline]\text{D)}[/inline] [inline]1[/inline]      [inline]\enclose{box}{\text{E)}}[/inline] [inline]\displaystyle\frac{3}{2}[/inline]              [inline]\text{N)}[/inline] Ne znam

17.Link zadatka Rešenje nejednačine [inline]\sin4x>\cos2x[/inline] na intervalu [inline](0,\pi)[/inline] je:
[inline]\text{A)}[/inline] [inline]\displaystyle\left(\frac{\pi}{12},\frac{\pi}{4}\right)\cup\left(\frac{5\pi}{12},\frac{3\pi}{4}\right)[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\left(\frac{\pi}{8},\frac{5\pi}{12}\right)[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\left(\frac{\pi}{8},\frac{5\pi}{8}\right)[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\left(\frac{\pi}{12},\frac{\pi}{8}\right)\cup\left(\frac{5\pi}{12},\frac{3\pi}{4}\right)[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\left(\frac{\pi}{12},\frac{\pi}{8}\right)[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\enclose{box}{\text{A)}}[/inline] [inline]\displaystyle\left(\frac{\pi}{12},\frac{\pi}{4}\right)\cup\left(\frac{5\pi}{12},\frac{3\pi}{4}\right)[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\left(\frac{\pi}{8},\frac{5\pi}{12}\right)[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\left(\frac{\pi}{8},\frac{5\pi}{8}\right)[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\left(\frac{\pi}{12},\frac{\pi}{8}\right)\cup\left(\frac{5\pi}{12},\frac{3\pi}{4}\right)[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\left(\frac{\pi}{12},\frac{\pi}{8}\right)[/inline]              [inline]\text{N)}[/inline] Ne znam

18.Link zadatka Tačka [inline]M(x,y)[/inline] na pravoj [inline]p\colon2x+y+2=0[/inline] najbliža je hiperboli [inline]7x^2-4y^2=28[/inline]. Tada je [inline]5y-5x[/inline] jednako:
[inline]\text{A)}[/inline] [inline]20[/inline]      [inline]\text{B)}[/inline] [inline]25[/inline]      [inline]\text{C)}[/inline] [inline]24[/inline]      [inline]\text{D)}[/inline] [inline]-25[/inline]      [inline]\text{E)}[/inline] [inline]-20[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\text{A)}[/inline] [inline]20[/inline]      [inline]\text{B)}[/inline] [inline]25[/inline]      [inline]\enclose{box}{\text{C)}}[/inline] [inline]24[/inline]      [inline]\text{D)}[/inline] [inline]-25[/inline]      [inline]\text{E)}[/inline] [inline]-20[/inline]              [inline]\text{N)}[/inline] Ne znam

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19.Link zadatka Broj rešenja jednačine [inline]\displaystyle\cos x+|\cos x|=2-\frac{2}{\pi}x[/inline] jednak je:
[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]5[/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{box}{\text{D)}}[/inline] [inline]3[/inline]      [inline]\text{E)}[/inline] [inline]5[/inline]              [inline]\text{N)}[/inline] Ne znam

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20.Link zadatka Dat je trougao [inline]ABC[/inline] sa temenima [inline]A(0,0)[/inline], [inline]B(4,0)[/inline] i [inline]C(3,2)[/inline]. U trougao [inline]ABC[/inline] upisan je pravougaonik [inline]MNPQ[/inline] maksimalne površine tako da temena [inline]M[/inline] i [inline]N[/inline] leže na osi [inline]Ox[/inline]. Dužina dijagonale ovog pravougaonika jednaka je:
[inline]\text{A)}[/inline] [inline]\sqrt3[/inline]      [inline]\text{B)}[/inline] [inline]\sqrt2[/inline]      [inline]\text{C)}[/inline] [inline]\sqrt6[/inline]      [inline]\text{D)}[/inline] [inline]\sqrt5[/inline]      [inline]\text{E)}[/inline] [inline]2[/inline]              [inline]\text{N)}[/inline] Ne znam[inline]\text{A)}[/inline] [inline]\sqrt3[/inline]      [inline]\text{B)}[/inline] [inline]\sqrt2[/inline]      [inline]\text{C)}[/inline] [inline]\sqrt6[/inline]      [inline]\enclose{box}{\text{D)}}[/inline] [inline]\sqrt5[/inline]      [inline]\text{E)}[/inline] [inline]2[/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.