ETF MATF FON GRF TMF FORUM

Prijemni ispit na Matematičkom fakultetu u Beogradu

29. jun 2011.



1.Link zadatka Jedna stranica paralelograma je [inline]4\text{ cm}[/inline], druga [inline]6\text{ cm}[/inline], a jedna dijagonala je [inline]4\sqrt2\text{ cm}[/inline]. Dužina druge dijagonale je:
[inline]\text{A)}[/inline] [inline]2\sqrt5\text{ cm}[/inline]      [inline]\text{B)}[/inline] [inline]4\sqrt{33}\text{ cm}[/inline]      [inline]\text{C)}[/inline] [inline]6\sqrt2\text{ cm}[/inline]      [inline]\text{D)}[/inline] [inline]6\sqrt3\text{ cm}[/inline]      [inline]\text{E)}[/inline] [inline]5\sqrt2\text{ cm}[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]2\sqrt5\text{ cm}[/inline]      [inline]\text{B)}[/inline] [inline]4\sqrt{33}\text{ cm}[/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline]6\sqrt2\text{ cm}[/inline]      [inline]\text{D)}[/inline] [inline]6\sqrt3\text{ cm}[/inline]      [inline]\text{E)}[/inline] [inline]5\sqrt2\text{ cm}[/inline]              [inline]\text{N)}[/inline] ne znam

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2.Link zadatka Najveća vrednost funkcije [inline]f\left(x\right)=\left|2x-1\right|-\left|3x+1\right|[/inline] je:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{5}{3}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{5}{2}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{1}{3}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{1}{2}[/inline]      [inline]\text{E)}[/inline] [inline]2[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]\displaystyle\frac{5}{3}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle\frac{5}{2}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{1}{3}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{1}{2}[/inline]      [inline]\text{E)}[/inline] [inline]2[/inline]              [inline]\text{N)}[/inline] ne znam

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3.Link zadatka Koliko celobrojnih rešenja ima nejednačina [inline]\displaystyle\frac{x}{x+4}\le\frac{1}{x+1}[/inline]?
[inline]\text{A)}[/inline] [inline]4[/inline]      [inline]\text{B)}[/inline] [inline]3[/inline]      [inline]\text{C)}[/inline] [inline]6[/inline]      [inline]\text{D)}[/inline] [inline]5[/inline]      [inline]\text{E)}[/inline] [inline]7[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]4[/inline]      [inline]\text{B)}[/inline] [inline]3[/inline]      [inline]\text{C)}[/inline] [inline]6[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]5[/inline]      [inline]\text{E)}[/inline] [inline]7[/inline]              [inline]\text{N)}[/inline] ne znam

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4.Link zadatka Dati su iskazi: [inline]1^\circ\;\log\bigl(\left(-2\right)\left(-3\right)\bigr)=\log\left(-2\right)+\log\left(-3\right)[/inline], [inline]2^\circ\;\log\left(-3\right)^2=2\log\left(-3\right)[/inline], [inline]3^\circ\;\log\left(-2\right)^4=2\log\left(-2\right)^2[/inline], [inline]\displaystyle4^\circ\;\log\frac{-2}{-3}=\log2-\log3[/inline]. Tačni su:
[inline]\text{A)}[/inline] svi      [inline]\text{B)}[/inline] nijedan      [inline]\text{C)}[/inline] [inline]1^\circ[/inline] i [inline]4^\circ[/inline]      [inline]\text{D)}[/inline] [inline]2^\circ[/inline] i [inline]3^\circ[/inline]      [inline]\text{E)}[/inline] [inline]3^\circ[/inline] i [inline]4^\circ[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] svi      [inline]\text{B)}[/inline] nijedan      [inline]\text{C)}[/inline] [inline]1^\circ[/inline] i [inline]4^\circ[/inline]      [inline]\text{D)}[/inline] [inline]2^\circ[/inline] i [inline]3^\circ[/inline]      [inline]\enclose{circle}{\text{E)}}[/inline] [inline]3^\circ[/inline] i [inline]4^\circ[/inline]              [inline]\text{N)}[/inline] ne znam

5.Link zadatka Vrednost realnog parametra [inline]m[/inline] za koju je zbir kvadrata rešenja jednačine [inline]x^2-mx+m-3=0[/inline] najmanji 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]\enclose{circle}{\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

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6.Link zadatka U jednakokrakom trapezu kraća osnovica i krak su dužine [inline]\sqrt{10}\text{ cm}[/inline], a duža osnovica je [inline]2\sqrt{10}\text{ cm}[/inline]. Površina kruga opisanog oko trapeza je:
[inline]\text{A)}[/inline] [inline]5\pi\text{ cm}^2[/inline]      [inline]\text{B)}[/inline] [inline]10\pi\text{ cm}^2[/inline]      [inline]\text{C)}[/inline] [inline]15\pi\text{ cm}^2[/inline]      [inline]\text{D)}[/inline] [inline]20\pi\text{ cm}^2[/inline]      [inline]\text{E)}[/inline] [inline]25\pi\text{ cm}^2[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]5\pi\text{ cm}^2[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]10\pi\text{ cm}^2[/inline]      [inline]\text{C)}[/inline] [inline]15\pi\text{ cm}^2[/inline]      [inline]\text{D)}[/inline] [inline]20\pi\text{ cm}^2[/inline]      [inline]\text{E)}[/inline] [inline]25\pi\text{ cm}^2[/inline]              [inline]\text{N)}[/inline] ne znam

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7.Link zadatka Zbir svih rešenja jednačine [inline]\displaystyle x+\sqrt{x^2+16}=\frac{40}{\sqrt{x^2+16}}[/inline] je:
[inline]\text{A)}[/inline] [inline]0[/inline]      [inline]\text{B)}[/inline] [inline]-3[/inline]      [inline]\text{C)}[/inline] [inline]3[/inline]      [inline]\text{D)}[/inline] [inline]5[/inline]      [inline]\text{E)}[/inline] [inline]8[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]0[/inline]      [inline]\text{B)}[/inline] [inline]-3[/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline]3[/inline]      [inline]\text{D)}[/inline] [inline]5[/inline]      [inline]\text{E)}[/inline] [inline]8[/inline]              [inline]\text{N)}[/inline] ne znam

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8.Link zadatka Ako je [inline]a=\sin35^\circ[/inline], [inline]b=\text{ctg }50^\circ[/inline] i [inline]c=\cos65^\circ[/inline], tada je:
[inline]\text{A)}[/inline] [inline]a<b<c[/inline]      [inline]\text{B)}[/inline] [inline]a<c<b[/inline]      [inline]\text{C)}[/inline] [inline]b<c<a[/inline]      [inline]\text{D)}[/inline] [inline]c<b<a[/inline]      [inline]\text{E)}[/inline] [inline]c<a<b[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]a<b<c[/inline]      [inline]\text{B)}[/inline] [inline]a<c<b[/inline]      [inline]\text{C)}[/inline] [inline]b<c<a[/inline]      [inline]\text{D)}[/inline] [inline]c<b<a[/inline]      [inline]\enclose{circle}{\text{E)}}[/inline] [inline]c<a<b[/inline]              [inline]\text{N)}[/inline] ne znam

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9.Link zadatka Broj rešenja sistema jednačina [inline]y+\log_{10}x=1[/inline], [inline]x^y=0,01[/inline] 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]4[/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] [inline]4[/inline]              [inline]\text{N)}[/inline] ne znam

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10.Link zadatka Ako je [inline]\displaystyle\cos\alpha=-\frac{40}{41}[/inline] i [inline]\displaystyle\pi<\alpha<\frac{3\pi}{2}[/inline], tada je [inline]\text{tg }\alpha[/inline] jednak:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{40}{9}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle-\frac{40}{9}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{9}{41}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{9}{40}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle-\frac{9}{40}[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle\frac{40}{9}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle-\frac{40}{9}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle\frac{9}{41}[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]\displaystyle\frac{9}{40}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle-\frac{9}{40}[/inline]              [inline]\text{N)}[/inline] ne znam

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11.Link zadatka Kriva koja je predstavljena na slici
Prijemni ispit MATF 2011. – slika
može biti grafik funkcije:
[inline]\text{A)}[/inline] [inline]\displaystyle y=1+\frac{2}{x+1}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle y=1-\frac{1}{x-1}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle y=-1+\frac{1}{x-1}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle y=-1-\frac{2}{x+1}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle y=1-\frac{1}{2\left(x+1\right)}[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle y=1+\frac{2}{x+1}[/inline]      [inline]\text{B)}[/inline] [inline]\displaystyle y=1-\frac{1}{x-1}[/inline]      [inline]\text{C)}[/inline] [inline]\displaystyle y=-1+\frac{1}{x-1}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle y=-1-\frac{2}{x+1}[/inline]      [inline]\enclose{circle}{\text{E)}}[/inline] [inline]\displaystyle y=1-\frac{1}{2\left(x+1\right)}[/inline]              [inline]\text{N)}[/inline] ne znam

12.Link zadatka Lopta je presečena dvema paralelnim ravnima koje se nalaze sa raznih strana centra. Jedna od njih seče loptu po krugu površine [inline]49\pi\text{ dm}^2[/inline], a druga po krugu površine [inline]400\pi\text{ dm}^2[/inline]. Ako je međusobna udaljenost tih ravni [inline]39\text{ dm}[/inline], onda je poluprečnik lopte jednak:
[inline]\text{A)}[/inline] [inline]25\text{ dm}[/inline]      [inline]\text{B)}[/inline] [inline]20\sqrt2\text{ dm}[/inline]      [inline]\text{C)}[/inline] [inline]15\sqrt3\text{ dm}[/inline]      [inline]\text{D)}[/inline] [inline]30\text{ dm}[/inline]      [inline]\text{E)}[/inline] [inline]24\text{ dm}[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]25\text{ dm}[/inline]      [inline]\text{B)}[/inline] [inline]20\sqrt2\text{ dm}[/inline]      [inline]\text{C)}[/inline] [inline]15\sqrt3\text{ dm}[/inline]      [inline]\text{D)}[/inline] [inline]30\text{ dm}[/inline]      [inline]\text{E)}[/inline] [inline]24\text{ dm}[/inline]              [inline]\text{N)}[/inline] ne znam

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13.Link zadatka Osnovna ivica pravilne četvorostrane piramide je [inline]8\text{ cm}[/inline], a središte osnove je od bočne strane na rastojanju [inline]2\text{ cm}[/inline]. Visina te piramide je:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{16}{3}\text{ cm}[/inline]      [inline]\text{B)}[/inline] [inline]4\sqrt2\text{ cm}[/inline]      [inline]\text{C)}[/inline] [inline]2\sqrt2\text{ cm}[/inline]      [inline]\text{D)}[/inline] [inline]\displaystyle\frac{4\sqrt3}{3}\text{ cm}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{2\sqrt3}{3}\text{ cm}[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle\frac{16}{3}\text{ cm}[/inline]      [inline]\text{B)}[/inline] [inline]4\sqrt2\text{ cm}[/inline]      [inline]\text{C)}[/inline] [inline]2\sqrt2\text{ cm}[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]\displaystyle\frac{4\sqrt3}{3}\text{ cm}[/inline]      [inline]\text{E)}[/inline] [inline]\displaystyle\frac{2\sqrt3}{3}\text{ cm}[/inline]              [inline]\text{N)}[/inline] ne znam

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14.Link zadatka Najmanja vrednost funkcije [inline]f\left(x\right)=\sin x+\cos x[/inline] za [inline]x\in\left[0,\pi\right][/inline] je:
[inline]\text{A)}[/inline] [inline]\sqrt2[/inline]      [inline]\text{B)}[/inline] [inline]-1[/inline]      [inline]\text{C)}[/inline] [inline]-\sqrt2[/inline]      [inline]\text{D)}[/inline] [inline]-2[/inline]      [inline]\text{E)}[/inline] [inline]1[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\sqrt2[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]-1[/inline]      [inline]\text{C)}[/inline] [inline]-\sqrt2[/inline]      [inline]\text{D)}[/inline] [inline]-2[/inline]      [inline]\text{E)}[/inline] [inline]1[/inline]              [inline]\text{N)}[/inline] ne znam

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15.Link zadatka Prava [inline]x-2y-2=0[/inline] seče kružnicu [inline]x^2-2x+y^2+6y=0[/inline] u tačkama [inline]A[/inline] i [inline]B[/inline]. Ako je [inline]O[/inline] centar kružnice, onda je [inline]\angle OAB[/inline] jednak:
[inline]\text{A)}[/inline] [inline]15^\circ[/inline]      [inline]\text{B)}[/inline] [inline]30^\circ[/inline]      [inline]\text{C)}[/inline] [inline]45^\circ[/inline]      [inline]\text{D)}[/inline] [inline]60^\circ[/inline]      [inline]\text{E)}[/inline] [inline]75^\circ[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]15^\circ[/inline]      [inline]\text{B)}[/inline] [inline]30^\circ[/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline]45^\circ[/inline]      [inline]\text{D)}[/inline] [inline]60^\circ[/inline]      [inline]\text{E)}[/inline] [inline]75^\circ[/inline]              [inline]\text{N)}[/inline] ne znam

16.Link zadatka Zbir drugog i jedanaestog člana aritmetičkog niza je [inline]28[/inline]. Zbir trećeg, petog, osmog i desetog člana tog niza je:
[inline]\text{A)}[/inline] [inline]14[/inline]      [inline]\text{B)}[/inline] [inline]28[/inline]      [inline]\text{C)}[/inline] [inline]56[/inline]      [inline]\text{D)}[/inline] [inline]84[/inline]      [inline]\text{E)}[/inline] Ne može se jednoznačno odrediti              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]14[/inline]      [inline]\text{B)}[/inline] [inline]28[/inline]      [inline]\enclose{circle}{\text{C)}}[/inline] [inline]56[/inline]      [inline]\text{D)}[/inline] [inline]84[/inline]      [inline]\text{E)}[/inline] Ne može se jednoznačno odrediti              [inline]\text{N)}[/inline] ne znam

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17.Link zadatka Broj racionalnih članova u razvoju stepena binoma [inline]\left(\sqrt[3]5+\sqrt3\right)^{2011}[/inline] je:
[inline]\text{A)}[/inline] [inline]1005[/inline]      [inline]\text{B)}[/inline] [inline]670[/inline]      [inline]\text{C)}[/inline] [inline]336[/inline]      [inline]\text{D)}[/inline] [inline]335[/inline]      [inline]\text{E)}[/inline] [inline]333[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]1005[/inline]      [inline]\text{B)}[/inline] [inline]670[/inline]      [inline]\text{C)}[/inline] [inline]336[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]335[/inline]      [inline]\text{E)}[/inline] [inline]333[/inline]              [inline]\text{N)}[/inline] ne znam

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18.Link zadatka Ako je broj [inline]x=2-i[/inline] rešenje jednačine [inline]x^3-2x^2-3x+a=0[/inline], onda je broj [inline]a[/inline] jednak:
[inline]\text{A)}[/inline] [inline]10[/inline]      [inline]\text{B)}[/inline] [inline]-10[/inline]      [inline]\text{C)}[/inline] [inline]0[/inline]      [inline]\text{D)}[/inline] [inline]20[/inline]      [inline]\text{E)}[/inline] [inline]-20[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]10[/inline]      [inline]\text{B)}[/inline] [inline]-10[/inline]      [inline]\text{C)}[/inline] [inline]0[/inline]      [inline]\text{D)}[/inline] [inline]20[/inline]      [inline]\text{E)}[/inline] [inline]-20[/inline]              [inline]\text{N)}[/inline] ne znam

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19.Link zadatka Vrednost izraza [inline]\displaystyle\left(\frac{1+i\sqrt3}{2}\right)^{2011}+\left(\frac{1-i\sqrt3}{2}\right)^{2011}[/inline] je:
[inline]\text{A)}[/inline] [inline]0[/inline]      [inline]\text{B)}[/inline] [inline]1[/inline]      [inline]\text{C)}[/inline] [inline]-1[/inline]      [inline]\text{D)}[/inline] [inline]i\sqrt3[/inline]      [inline]\text{E)}[/inline] [inline]-i\sqrt3[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]0[/inline]      [inline]\enclose{circle}{\text{B)}}[/inline] [inline]1[/inline]      [inline]\text{C)}[/inline] [inline]-1[/inline]      [inline]\text{D)}[/inline] [inline]i\sqrt3[/inline]      [inline]\text{E)}[/inline] [inline]-i\sqrt3[/inline]              [inline]\text{N)}[/inline] ne znam

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20.Link zadatka Date su tačke [inline]A\left(2,0\right)[/inline] i [inline]B\left(8,0\right)[/inline]. Ako je [inline]C\left(0,y\right)[/inline], [inline]y>0[/inline], tačka za koju je ugao [inline]\angle ACB[/inline] maksimalan, tada je:
[inline]\text{A)}[/inline] [inline]y=1[/inline]      [inline]\text{B)}[/inline] [inline]y=2[/inline]      [inline]\text{C)}[/inline] [inline]y=3[/inline]      [inline]\text{D)}[/inline] [inline]y=4[/inline]      [inline]\text{E)}[/inline] [inline]y=5[/inline]              [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]y=1[/inline]      [inline]\text{B)}[/inline] [inline]y=2[/inline]      [inline]\text{C)}[/inline] [inline]y=3[/inline]      [inline]\enclose{circle}{\text{D)}}[/inline] [inline]y=4[/inline]      [inline]\text{E)}[/inline] [inline]y=5[/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.