Prijemni ispit na Matematičkom fakultetu u Beogradu
27. jun 2012.
1.Link zadatka Data je tačka [inline]P\left(1,1\right)[/inline] i elipsa [inline]9x^2+16y^2=144[/inline]. Jednačina sečice elipse kojoj je [inline]P[/inline] središte je:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{x}{4}+\frac{y}{3}=\frac{7}{12}[/inline] [inline]\text{B)}[/inline] [inline]\displaystyle\frac{x}{16}+\frac{y}{9}=\frac{25}{144}[/inline] [inline]\text{C)}[/inline] [inline]\displaystyle\frac{x}{4}+\frac{y}{3}=\frac{1}{5}[/inline] [inline]\text{D)}[/inline] [inline]16x+9y=25[/inline] [inline]\text{E)}[/inline] [inline]3x+4y=7[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle\frac{x}{4}+\frac{y}{3}=\frac{7}{12}[/inline] [inline]\enclose{circle}{\text{B)}}[/inline] [inline]\displaystyle\frac{x}{16}+\frac{y}{9}=\frac{25}{144}[/inline] [inline]\text{C)}[/inline] [inline]\displaystyle\frac{x}{4}+\frac{y}{3}=\frac{1}{5}[/inline] [inline]\text{D)}[/inline] [inline]16x+9y=25[/inline] [inline]\text{E)}[/inline] [inline]3x+4y=7[/inline] [inline]\text{N)}[/inline] ne znam
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2.Link zadatka U zarubljenu kupu poluprečnika veće osnove [inline]4[/inline] upisana je lopta zapremine [inline]36\pi[/inline]. Zapremina zarubljene kupe je:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{239\pi}{4}[/inline] [inline]\text{B)}[/inline] [inline]\displaystyle\frac{481\pi}{8}[/inline] [inline]\text{C)}[/inline] [inline]\displaystyle\frac{359\pi}{6}[/inline] [inline]\text{D)}[/inline] [inline]\displaystyle\frac{219\pi}{17}[/inline] [inline]\text{E)}[/inline] [inline]\displaystyle\frac{298\pi}{5}[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle\frac{239\pi}{4}[/inline] [inline]\enclose{circle}{\text{B)}}[/inline] [inline]\displaystyle\frac{481\pi}{8}[/inline] [inline]\text{C)}[/inline] [inline]\displaystyle\frac{359\pi}{6}[/inline] [inline]\text{D)}[/inline] [inline]\displaystyle\frac{219\pi}{17}[/inline] [inline]\text{E)}[/inline] [inline]\displaystyle\frac{298\pi}{5}[/inline] [inline]\text{N)}[/inline] ne znam
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3.Link zadatka Neka su dužine stranica trougla [inline]4[/inline], [inline]5[/inline] i [inline]7[/inline]. Trougao je:
[inline]\text{A)}[/inline] pravougli [inline]\text{B)}[/inline] oštrougli [inline]\text{C)}[/inline] tupougli [inline]\text{D)}[/inline] ne postoji [inline]\text{E)}[/inline] ne može se odrediti [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] pravougli [inline]\text{B)}[/inline] oštrougli [inline]\enclose{circle}{\text{C)}}[/inline] tupougli [inline]\text{D)}[/inline] ne postoji [inline]\text{E)}[/inline] ne može se odrediti [inline]\text{N)}[/inline] ne znam
4.Link zadatka U trouglu [inline]ABC[/inline] je ugao kod temena [inline]A[/inline] dva puta veći od ugla kod temena [inline]B[/inline]. Ako su naspram temena [inline]A[/inline], [inline]B[/inline], [inline]C[/inline] redom stranice [inline]a[/inline], [inline]b[/inline], [inline]c[/inline], onda je:
[inline]\text{A)}[/inline] [inline]a^2=b^2+c^2[/inline] [inline]\text{B)}[/inline] [inline]a^2=bc[/inline] [inline]\text{C)}[/inline] [inline]a^2=b\left(b+c\right)[/inline] [inline]\text{D)}[/inline] [inline]2c=a+b[/inline] [inline]\text{E)}[/inline] [inline]b^2=a\left(a+c\right)[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]a^2=b^2+c^2[/inline] [inline]\text{B)}[/inline] [inline]a^2=bc[/inline] [inline]\enclose{circle}{\text{C)}}[/inline] [inline]a^2=b\left(b+c\right)[/inline] [inline]\text{D)}[/inline] [inline]2c=a+b[/inline] [inline]\text{E)}[/inline] [inline]b^2=a\left(a+c\right)[/inline] [inline]\text{N)}[/inline] ne znam
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5.Link zadatka Jednačina [inline]4\sin x+3\cos x=a[/inline] ima realna rešenja ako i samo ako parametar [inline]a[/inline] pripada intervalu:
[inline]\text{A)}[/inline] [inline]\displaystyle\left[-\frac{7}{2},\frac{7}{2}\right][/inline] [inline]\text{B)}[/inline] [inline]\left[-5,5\right][/inline] [inline]\text{C)}[/inline] [inline]\left[-7,7\right][/inline] [inline]\text{D)}[/inline] [inline]\left[-12,12\right][/inline] [inline]\text{E)}[/inline] [inline]\left[-25,25\right][/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle\left[-\frac{7}{2},\frac{7}{2}\right][/inline] [inline]\enclose{circle}{\text{B)}}[/inline] [inline]\left[-5,5\right][/inline] [inline]\text{C)}[/inline] [inline]\left[-7,7\right][/inline] [inline]\text{D)}[/inline] [inline]\left[-12,12\right][/inline] [inline]\text{E)}[/inline] [inline]\left[-25,25\right][/inline] [inline]\text{N)}[/inline] ne znam
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6.Link zadatka Broj rešenja jednačine [inline]\left(4\cos^2x+4\cos x-3\right)\sqrt{5\sin x}=0[/inline] u intervalu [inline]\left[0,2\pi\right)[/inline] je:
[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]3[/inline] [inline]\text{E)}[/inline] [inline]5[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]1[/inline] [inline]\text{B)}[/inline] [inline]2[/inline] [inline]\text{C)}[/inline] [inline]4[/inline] [inline]\enclose{circle}{\text{D)}}[/inline] [inline]3[/inline] [inline]\text{E)}[/inline] [inline]5[/inline] [inline]\text{N)}[/inline] ne znam
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7.Link zadatka Broj rešenja jednačine [inline]4^{\cos2x}+4^{\cos^2x}=3[/inline] u intervalu [inline]\left[0,2\pi\right)[/inline] je:
[inline]\text{A)}[/inline] [inline]6[/inline] [inline]\text{B)}[/inline] [inline]2[/inline] [inline]\text{C)}[/inline] [inline]1[/inline] [inline]\text{D)}[/inline] [inline]4[/inline] [inline]\text{E)}[/inline] [inline]8[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]6[/inline] [inline]\text{B)}[/inline] [inline]2[/inline] [inline]\text{C)}[/inline] [inline]1[/inline] [inline]\enclose{circle}{\text{D)}}[/inline] [inline]4[/inline] [inline]\text{E)}[/inline] [inline]8[/inline] [inline]\text{N)}[/inline] ne znam
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8.Link zadatka Jednačina [inline]\left|-x^2+5x-4\right|=ax[/inline] ima četiri realna 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(-1,+\infty\right)[/inline] [inline]\text{C)}[/inline] [inline]\left(0,2\right)[/inline] [inline]\text{D)}[/inline] [inline]\left(0,+\infty\right)[/inline] [inline]\text{E)}[/inline] [inline]\left(-1,0\right)[/inline] [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]\left(0,1\right)[/inline] [inline]\text{B)}[/inline] [inline]\left(-1,+\infty\right)[/inline] [inline]\text{C)}[/inline] [inline]\left(0,2\right)[/inline] [inline]\text{D)}[/inline] [inline]\left(0,+\infty\right)[/inline] [inline]\text{E)}[/inline] [inline]\left(-1,0\right)[/inline] [inline]\text{N)}[/inline] ne znam
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9.Link zadatka Skup rešenja nejednačine [inline]\log_{1/2}\left(3x^2+7x+4\right)<\log_{1/2}\left(x^2+2x+7\right)[/inline] je:
[inline]\text{A)}[/inline] [inline]\displaystyle\left(-\infty,-3\right)\cup\left(\frac{1}{2},+\infty\right)[/inline] [inline]\text{B)}[/inline] [inline]\displaystyle\left(-3,\frac{1}{2}\right)[/inline] [inline]\text{C)}[/inline] [inline]\displaystyle\left(-3,-\frac{4}{3}\right)\cup\left(-1,\frac{1}{2}\right)[/inline] [inline]\text{D)}[/inline] [inline]\displaystyle\left(-\infty,-3\right)\cup\left(-\frac{4}{3},-1\right)\cup\left(\frac{1}{2},+\infty\right)[/inline] [inline]\text{E)}[/inline] [inline]\left(-\infty,+\infty\right)[/inline] [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]\displaystyle\left(-\infty,-3\right)\cup\left(\frac{1}{2},+\infty\right)[/inline] [inline]\text{B)}[/inline] [inline]\displaystyle\left(-3,\frac{1}{2}\right)[/inline] [inline]\text{C)}[/inline] [inline]\displaystyle\left(-3,-\frac{4}{3}\right)\cup\left(-1,\frac{1}{2}\right)[/inline] [inline]\text{D)}[/inline] [inline]\displaystyle\left(-\infty,-3\right)\cup\left(-\frac{4}{3},-1\right)\cup\left(\frac{1}{2},+\infty\right)[/inline] [inline]\text{E)}[/inline] [inline]\left(-\infty,+\infty\right)[/inline] [inline]\text{N)}[/inline] ne znam
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10.Link zadatka Zbir rešenja jednačine [inline]\bigl|\left|x-1\right|-5\bigr|=6[/inline] je:
[inline]\text{A)}[/inline] [inline]12[/inline] [inline]\text{B)}[/inline] [inline]-10[/inline] [inline]\text{C)}[/inline] [inline]2[/inline] [inline]\text{D)}[/inline] [inline]3[/inline] [inline]\text{E)}[/inline] [inline]0[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]12[/inline] [inline]\text{B)}[/inline] [inline]-10[/inline] [inline]\enclose{circle}{\text{C)}}[/inline] [inline]2[/inline] [inline]\text{D)}[/inline] [inline]3[/inline] [inline]\text{E)}[/inline] [inline]0[/inline] [inline]\text{N)}[/inline] ne znam
11.Link zadatka Koliko ima realnih brojeva [inline]a[/inline] takvih da funkcije [inline]f\left(x\right)=ax[/inline] i [inline]g\left(x\right)=x+a[/inline] zadovoljavaju jednakost [inline]f\bigl(g\left(x\right)\bigr)=g\bigl(f\left(x\right)\bigr)[/inline] za sve [inline]x\in\mathbb{R}[/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]\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
12.Link zadatka Broj rešenja jednačine [inline]\sqrt{x+2}=x-1[/inline]:
[inline]\text{A)}[/inline] nema rešenja [inline]\text{B)}[/inline] ima jedno rešenje i ono je pozitivno [inline]\text{C)}[/inline] ima jedno rešenje i ono je negativno [inline]\text{D)}[/inline] ima jedno pozitivno i jedno negativno rešenje [inline]\text{E)}[/inline] ima dva rešenja i oba su pozitivna [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] nema rešenja [inline]\enclose{circle}{\text{B)}}[/inline] ima jedno rešenje i ono je pozitivno [inline]\text{C)}[/inline] ima jedno rešenje i ono je negativno [inline]\text{D)}[/inline] ima jedno pozitivno i jedno negativno rešenje [inline]\text{E)}[/inline] ima dva rešenja i oba su pozitivna [inline]\text{N)}[/inline] ne znam
13.Link zadatka Neka su [inline]x_1[/inline], [inline]x_2[/inline] rešenja jednačine [inline]x^2+\left(2-m\right)x-m-3=0[/inline]. Vrednost realnog parametra [inline]m[/inline] za koju [inline]x_1^2+x_2^2[/inline] ima najmanju vrednost je:
[inline]\text{A)}[/inline] [inline]2[/inline] [inline]\text{B)}[/inline] [inline]-3[/inline] [inline]\text{C)}[/inline] [inline]0[/inline] [inline]\text{D)}[/inline] [inline]1[/inline] [inline]\text{E)}[/inline] [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]0[/inline] [inline]\enclose{circle}{\text{D)}}[/inline] [inline]1[/inline] [inline]\text{E)}[/inline] [inline]5[/inline] [inline]\text{N)}[/inline] ne znam
14.Link zadatka Skup svih prirodnih brojeva razbijen je na grupe na sledeći način: [inline]\left\{1\right\}[/inline], [inline]\left\{2,3\right\}[/inline], [inline]\left\{4,5,6\right\}[/inline], [inline]\left\{7,8,9,10\right\}[/inline]... Zbir svih brojeva [inline]99.[/inline] grupe je:
[inline]\text{A)}[/inline] [inline]511932[/inline] [inline]\text{B)}[/inline] [inline]490901[/inline] [inline]\text{C)}[/inline] [inline]501509[/inline] [inline]\text{D)}[/inline] [inline]471981[/inline] [inline]\text{E)}[/inline] [inline]485199[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]511932[/inline] [inline]\text{B)}[/inline] [inline]490901[/inline] [inline]\text{C)}[/inline] [inline]501509[/inline] [inline]\text{D)}[/inline] [inline]471981[/inline] [inline]\enclose{circle}{\text{E)}}[/inline] [inline]485199[/inline] [inline]\text{N)}[/inline] ne znam
15.Link zadatka Vrednost izraza [inline]\left(1+i\right)^{2012}+\left(1-i\right)^{2012}[/inline] je:
[inline]\text{A)}[/inline] [inline]2^{1007}[/inline] [inline]\text{B)}[/inline] [inline]-2^{1007}[/inline] [inline]\text{C)}[/inline] [inline]2^{2012}i[/inline] [inline]\text{D)}[/inline] [inline]-2^{1007}i[/inline] [inline]\text{E)}[/inline] [inline]2^{2012}[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]2^{1007}[/inline] [inline]\enclose{circle}{\text{B)}}[/inline] [inline]-2^{1007}[/inline] [inline]\text{C)}[/inline] [inline]2^{2012}i[/inline] [inline]\text{D)}[/inline] [inline]-2^{1007}i[/inline] [inline]\text{E)}[/inline] [inline]2^{2012}[/inline] [inline]\text{N)}[/inline] ne znam
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16.Link zadatka Pravilni [inline]n[/inline]-tougao ima [inline]5[/inline] puta više dijagonala nego stranica ako je [inline]n[/inline] jednako:
[inline]\text{A)}[/inline] [inline]8[/inline] [inline]\text{B)}[/inline] [inline]12[/inline] [inline]\text{C)}[/inline] [inline]16[/inline] [inline]\text{D)}[/inline] [inline]13[/inline] [inline]\text{E)}[/inline] [inline]5[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]8[/inline] [inline]\text{B)}[/inline] [inline]12[/inline] [inline]\text{C)}[/inline] [inline]16[/inline] [inline]\enclose{circle}{\text{D)}}[/inline] [inline]13[/inline] [inline]\text{E)}[/inline] [inline]5[/inline] [inline]\text{N)}[/inline] ne znam
17.Link zadatka Ako je [inline]\displaystyle a={2012\choose1004}[/inline], [inline]\displaystyle b={2012\choose1006}[/inline] i [inline]\displaystyle c={2012\choose1007}[/inline], onda je:
[inline]\text{A)}[/inline] [inline]c<a<b[/inline] [inline]\text{B)}[/inline] [inline]a<c<b[/inline] [inline]\text{C)}[/inline] [inline]a<b<c[/inline] [inline]\text{D)}[/inline] [inline]c<b<a[/inline] [inline]\text{E)}[/inline] [inline]b<c<a[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]c<a<b[/inline] [inline]\enclose{circle}{\text{B)}}[/inline] [inline]a<c<b[/inline] [inline]\text{C)}[/inline] [inline]a<b<c[/inline] [inline]\text{D)}[/inline] [inline]c<b<a[/inline] [inline]\text{E)}[/inline] [inline]b<c<a[/inline] [inline]\text{N)}[/inline] ne znam
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18.Link zadatka Jednačina krive prikazane na slici je:
[inline]\text{A)}[/inline] [inline]\displaystyle y=1+\sin\left(x-\frac{\pi}{4}\right)[/inline] [inline]\text{B)}[/inline] [inline]\displaystyle y=1+\sin\left(x+\frac{\pi}{4}\right)[/inline] [inline]\text{C)}[/inline] [inline]\displaystyle y=1-\cos\left(x-\frac{\pi}{4}\right)[/inline] [inline]\text{D)}[/inline] [inline]\displaystyle y=1-\sin\left(x-\frac{\pi}{4}\right)[/inline] [inline]\text{E)}[/inline] [inline]\displaystyle y=1-\cos\left(x+\frac{\pi}{4}\right)[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle y=1+\sin\left(x-\frac{\pi}{4}\right)[/inline] [inline]\text{B)}[/inline] [inline]\displaystyle y=1+\sin\left(x+\frac{\pi}{4}\right)[/inline] [inline]\text{C)}[/inline] [inline]\displaystyle y=1-\cos\left(x-\frac{\pi}{4}\right)[/inline] [inline]\enclose{circle}{\text{D)}}[/inline] [inline]\displaystyle y=1-\sin\left(x-\frac{\pi}{4}\right)[/inline] [inline]\text{E)}[/inline] [inline]\displaystyle y=1-\cos\left(x+\frac{\pi}{4}\right)[/inline] [inline]\text{N)}[/inline] ne znam
19.Link zadatka Razlika između najveće i najmanje vrednosti funkcije [inline]f\left(x\right)=4x-6-x^2[/inline] na intervalu [inline]\left[-3,3\right][/inline] je:
[inline]\text{A)}[/inline] [inline]29[/inline] [inline]\text{B)}[/inline] [inline]1[/inline] [inline]\text{C)}[/inline] [inline]24[/inline] [inline]\text{D)}[/inline] [inline]25[/inline] [inline]\text{E)}[/inline] [inline]6[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]29[/inline] [inline]\text{B)}[/inline] [inline]1[/inline] [inline]\text{C)}[/inline] [inline]24[/inline] [inline]\enclose{circle}{\text{D)}}[/inline] [inline]25[/inline] [inline]\text{E)}[/inline] [inline]6[/inline] [inline]\text{N)}[/inline] ne znam
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20.Link zadatka Posle sniženja cene ulaznica broj posetilaca utakmica porastao je za [inline]50\%[/inline], a prihod je porastao za [inline]26\%[/inline]. Za koliko procenata su snižene cene ulaznica?
[inline]\text{A)}[/inline] [inline]8\%[/inline] [inline]\text{B)}[/inline] [inline]16\%[/inline] [inline]\text{C)}[/inline] [inline]24\%[/inline] [inline]\text{D)}[/inline] [inline]36\%[/inline] [inline]\text{E)}[/inline] [inline]38\%[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]8\%[/inline] [inline]\enclose{circle}{\text{B)}}[/inline] [inline]16\%[/inline] [inline]\text{C)}[/inline] [inline]24\%[/inline] [inline]\text{D)}[/inline] [inline]36\%[/inline] [inline]\text{E)}[/inline] [inline]38\%[/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.