Prijemni ispit na Matematičkom fakultetu u Beogradu
29. jun 2015.
1.Link zadatka Podnožje visine deli hipotenuzu pravouglog trougla na odsečke čije su dužine [inline]16\text{ cm}[/inline] i [inline]9\text{ cm}[/inline]. Poluprečnik kruga upisanog u taj trougao je:
[inline]\text{A)}[/inline] [inline]4\text{ cm}[/inline] [inline]\text{B)}[/inline] [inline]5\text{ cm}[/inline] [inline]\text{C)}[/inline] [inline]6\text{ cm}[/inline] [inline]\text{D)}[/inline] [inline]8\text{ cm}[/inline] [inline]\text{E)}[/inline] [inline]10\text{ cm}[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]4\text{ cm}[/inline] [inline]\enclose{circle}{\text{B)}}[/inline] [inline]5\text{ cm}[/inline] [inline]\text{C)}[/inline] [inline]6\text{ cm}[/inline] [inline]\text{D)}[/inline] [inline]8\text{ cm}[/inline] [inline]\text{E)}[/inline] [inline]10\text{ cm}[/inline] [inline]\text{N)}[/inline] ne znam
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2.Link zadatka Ako je ivica kocke jednaka [inline]a[/inline], poluprečnik sfere koja dodiruje svih dvanaest ivica te kocke jednak je:
[inline]\text{A)}[/inline] [inline]a[/inline] [inline]\text{B)}[/inline] [inline]a\sqrt2[/inline] [inline]\text{C)}[/inline] [inline]\displaystyle\frac{a\sqrt2}{2}[/inline] [inline]\text{D)}[/inline] [inline]a\sqrt3[/inline] [inline]\text{E)}[/inline] [inline]\displaystyle\frac{a\sqrt3}{2}[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]a[/inline] [inline]\text{B)}[/inline] [inline]a\sqrt2[/inline] [inline]\enclose{circle}{\text{C)}}[/inline] [inline]\displaystyle\frac{a\sqrt2}{2}[/inline] [inline]\text{D)}[/inline] [inline]a\sqrt3[/inline] [inline]\text{E)}[/inline] [inline]\displaystyle\frac{a\sqrt3}{2}[/inline] [inline]\text{N)}[/inline] ne znam
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3.Link zadatka Date su kružna linija [inline]k[/inline] jednačinom [inline]x^2+y^2=a^2[/inline], [inline]a>0[/inline], i tačka [inline]P\left(a,a\right)[/inline]. Neka su [inline]A[/inline] i [inline]B[/inline] tačke u kojima prava [inline]PO[/inline] seče [inline]k[/inline], gde je [inline]O[/inline] koordinatni početak. Proizvod [inline]PA\cdot PB[/inline] jednak je:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{a^2}{2}[/inline] [inline]\text{B)}[/inline] [inline]a^2[/inline] [inline]\text{C)}[/inline] [inline]2a^2[/inline] [inline]\text{D)}[/inline] [inline]a^2\sqrt2[/inline] [inline]\text{E)}[/inline] [inline]\displaystyle\frac{a^2}{\sqrt2}[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle\frac{a^2}{2}[/inline] [inline]\enclose{circle}{\text{B)}}[/inline] [inline]a^2[/inline] [inline]\text{C)}[/inline] [inline]2a^2[/inline] [inline]\text{D)}[/inline] [inline]a^2\sqrt2[/inline] [inline]\text{E)}[/inline] [inline]\displaystyle\frac{a^2}{\sqrt2}[/inline] [inline]\text{N)}[/inline] ne znam
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4.Link zadatka Ako je odnos uglova nekog trougla jednak [inline]3:4:5[/inline], onda je odnos njegovih stranica jednak:
[inline]\text{A)}[/inline] [inline]2:\sqrt6:\left(1+\sqrt3\right)[/inline] [inline]\text{B)}[/inline] [inline]\sqrt3:\sqrt4:\sqrt5[/inline] [inline]\text{C)}[/inline] [inline]\left(1+\sqrt2\right):\sqrt6:\left(1+\sqrt3\right)[/inline] [inline]\text{D)}[/inline] [inline]3:4:5[/inline] [inline]\text{E)}[/inline] [inline]\displaystyle\frac{1}{3}:\frac{1}{4}:\frac{1}{5}[/inline] [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]2:\sqrt6:\left(1+\sqrt3\right)[/inline] [inline]\text{B)}[/inline] [inline]\sqrt3:\sqrt4:\sqrt5[/inline] [inline]\text{C)}[/inline] [inline]\left(1+\sqrt2\right):\sqrt6:\left(1+\sqrt3\right)[/inline] [inline]\text{D)}[/inline] [inline]3:4:5[/inline] [inline]\text{E)}[/inline] [inline]\displaystyle\frac{1}{3}:\frac{1}{4}:\frac{1}{5}[/inline] [inline]\text{N)}[/inline] ne znam
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5.Link zadatka Najmanji realan broj [inline]a[/inline] za koji jednačina [inline]\cos^4x+\sin^4x=a[/inline] ima rešenje jednak je:
[inline]\text{A)}[/inline] [inline]2[/inline] [inline]\text{B)}[/inline] [inline]\sqrt2[/inline] [inline]\text{C)}[/inline] [inline]1[/inline] [inline]\text{D)}[/inline] [inline]\displaystyle\frac{1}{\sqrt2}[/inline] [inline]\text{E)}[/inline] [inline]\displaystyle\frac{1}{2}[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]2[/inline] [inline]\text{B)}[/inline] [inline]\sqrt2[/inline] [inline]\text{C)}[/inline] [inline]1[/inline] [inline]\text{D)}[/inline] [inline]\displaystyle\frac{1}{\sqrt2}[/inline] [inline]\enclose{circle}{\text{E)}}[/inline] [inline]\displaystyle\frac{1}{2}[/inline] [inline]\text{N)}[/inline] ne znam
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6.Link zadatka Periodični decimalni broj [inline]2,727272\ldots[/inline], čiji se period sastoji od dve cifre, napisan je u obliku neskrativog razlomka. Zbir brojioca i imenioca tog razlomka jednak je:
[inline]\text{A)}[/inline] [inline]369[/inline] [inline]\text{B)}[/inline] [inline]299[/inline] [inline]\text{C)}[/inline] [inline]123[/inline] [inline]\text{D)}[/inline] [inline]41[/inline] [inline]\text{E)}[/inline] [inline]29[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]369[/inline] [inline]\text{B)}[/inline] [inline]299[/inline] [inline]\text{C)}[/inline] [inline]123[/inline] [inline]\enclose{circle}{\text{D)}}[/inline] [inline]41[/inline] [inline]\text{E)}[/inline] [inline]29[/inline] [inline]\text{N)}[/inline] ne znam
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7.Link zadatka Skup vrednosti parametra [inline]m[/inline] za koje jednačina [inline]\left(m-2\right)x^2-2mx+2m+2=0[/inline] ima dva realna rešenja suprotnog znaka je:
[inline]\text{A)}[/inline] [inline]\left(-1,2\right)[/inline] [inline]\text{B)}[/inline] [inline]\left(-2,1\right)[/inline] [inline]\text{C)}[/inline] [inline]\left(-2,0\right][/inline] [inline]\text{D)}[/inline] [inline]\left(0,2\right][/inline] [inline]\text{E)}[/inline] [inline]\left[-1,1\right][/inline] [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]\left(-1,2\right)[/inline] [inline]\text{B)}[/inline] [inline]\left(-2,1\right)[/inline] [inline]\text{C)}[/inline] [inline]\left(-2,0\right][/inline] [inline]\text{D)}[/inline] [inline]\left(0,2\right][/inline] [inline]\text{E)}[/inline] [inline]\left[-1,1\right][/inline] [inline]\text{N)}[/inline] ne znam
8.Link zadatka Zbir kvadrata rešenja jednačine [inline]\displaystyle x-\frac{2}{x}=i[/inline] jednak je:
[inline]\text{A)}[/inline] [inline]0[/inline] [inline]\text{B)}[/inline] [inline]2i[/inline] [inline]\text{C)}[/inline] [inline]5[/inline] [inline]\text{D)}[/inline] [inline]-i[/inline] [inline]\text{E)}[/inline] [inline]3[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]0[/inline] [inline]\text{B)}[/inline] [inline]2i[/inline] [inline]\text{C)}[/inline] [inline]5[/inline] [inline]\text{D)}[/inline] [inline]-i[/inline] [inline]\enclose{circle}{\text{E)}}[/inline] [inline]3[/inline] [inline]\text{N)}[/inline] ne znam
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9.Link zadatka Na slici je dat grafik funkcije [inline]f\left(x\right)=ax^2+bx+c[/inline].
Koliko ima pozitivnih među brojevima [inline]a[/inline], [inline]b[/inline], [inline]c[/inline], [inline]2ac-b^2[/inline] i [inline]a-b+c[/inline]?
[inline]\text{A)}[/inline] [inline]1[/inline] [inline]\text{B)}[/inline] [inline]2[/inline] [inline]\text{C)}[/inline] [inline]3[/inline] [inline]\text{D)}[/inline] [inline]4[/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]\enclose{circle}{\text{C)}}[/inline] [inline]3[/inline] [inline]\text{D)}[/inline] [inline]4[/inline] [inline]\text{E)}[/inline] [inline]5[/inline] [inline]\text{N)}[/inline] ne znam
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10.Link zadatka Polinom [inline]x^{2015}+x^{2014}+ax+b[/inline] je deljiv polinomom [inline]x^2-1[/inline]. Koliko je [inline]a^2+b^2[/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
11.Link zadatka Funkcija [inline]f\colon\mathbb{R}\to\mathbb{R}[/inline] zadata je sa [inline]f\left(x\right)=\left|x+1\right|+\left|2x+4\right|+4x[/inline]. Vrednost izraza [inline]f^{-1}\left(-7\right)+f^{-1}\left(-2\right)+f^{-1}\left(5\right)[/inline] je:
[inline]\text{A)}[/inline] [inline]-4[/inline] [inline]\text{B)}[/inline] [inline]-3[/inline] [inline]\text{C)}[/inline] [inline]0[/inline] [inline]\text{D)}[/inline] [inline]4[/inline] [inline]\text{E)}[/inline] [inline]14[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]-4[/inline] [inline]\enclose{circle}{\text{B)}}[/inline] [inline]-3[/inline] [inline]\text{C)}[/inline] [inline]0[/inline] [inline]\text{D)}[/inline] [inline]4[/inline] [inline]\text{E)}[/inline] [inline]14[/inline] [inline]\text{N)}[/inline] ne znam
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12.Link zadatka Operacija [inline]\ast[/inline] je definisana na skupu pozitivnih realnih brojeva sa [inline]x\ast y=\sqrt{xy}[/inline]. Koliko je [inline]\left(3\ast48\right)\ast9[/inline]?
[inline]\text{A)}[/inline] [inline]6\sqrt3[/inline] [inline]\text{B)}[/inline] [inline]6\sqrt[4]3[/inline] [inline]\text{C)}[/inline] [inline]6[/inline] [inline]\text{D)}[/inline] [inline]36[/inline] [inline]\text{E)}[/inline] [inline]108[/inline] [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]6\sqrt3[/inline] [inline]\text{B)}[/inline] [inline]6\sqrt[4]3[/inline] [inline]\text{C)}[/inline] [inline]6[/inline] [inline]\text{D)}[/inline] [inline]36[/inline] [inline]\text{E)}[/inline] [inline]108[/inline] [inline]\text{N)}[/inline] ne znam
13.Link zadatka Funkcija [inline]f\colon\left(0,+\infty\right)\to\mathbb{R}[/inline] definisana je sa [inline]f\left(x\right)=\log_6x+3\log_3\left(9x\right)[/inline]. Koliko je [inline]\displaystyle f\left(x\right)+f\left(\frac{1}{x}\right)[/inline]?
[inline]\text{A)}[/inline] [inline]-2[/inline] [inline]\text{B)}[/inline] [inline]9[/inline] [inline]\text{C)}[/inline] [inline]12[/inline] [inline]\text{D)}[/inline] [inline]15[/inline] [inline]\text{E)}[/inline] [inline]27[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]-2[/inline] [inline]\text{B)}[/inline] [inline]9[/inline] [inline]\enclose{circle}{\text{C)}}[/inline] [inline]12[/inline] [inline]\text{D)}[/inline] [inline]15[/inline] [inline]\text{E)}[/inline] [inline]27[/inline] [inline]\text{N)}[/inline] ne znam
14.Link zadatka Skup rešenja nejednačine [inline]3\cdot4^x-5\cdot2^{x+1}\le8[/inline] je:
[inline]\text{A)}[/inline] [inline]\left(-\infty,2\right][/inline] [inline]\text{B)}[/inline] [inline]\left[0,2\right][/inline] [inline]\text{C)}[/inline] [inline]\displaystyle\left[-\frac{2}{3},4\right][/inline] [inline]\text{D)}[/inline] [inline]\left[2,+\infty\right)[/inline] [inline]\text{E)}[/inline] [inline]\left(3,8\right)[/inline] [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]\left(-\infty,2\right][/inline] [inline]\text{B)}[/inline] [inline]\left[0,2\right][/inline] [inline]\text{C)}[/inline] [inline]\displaystyle\left[-\frac{2}{3},4\right][/inline] [inline]\text{D)}[/inline] [inline]\left[2,+\infty\right)[/inline] [inline]\text{E)}[/inline] [inline]\left(3,8\right)[/inline] [inline]\text{N)}[/inline] ne znam
15.Link zadatka Ako je [inline]-1<x<0[/inline], koji je broj najveći:
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{3}{x^2}[/inline] [inline]\text{B)}[/inline] [inline]3^{1/x}[/inline] [inline]\text{C)}[/inline] [inline]\displaystyle\frac{-3}{x}[/inline] [inline]\text{D)}[/inline] [inline]3^x[/inline] [inline]\text{E)}[/inline] [inline]\displaystyle\frac{3}{\sqrt{-x}}[/inline] [inline]\text{N)}[/inline] ne znam[inline]\enclose{circle}{\text{A)}}[/inline] [inline]\displaystyle\frac{3}{x^2}[/inline] [inline]\text{B)}[/inline] [inline]3^{1/x}[/inline] [inline]\text{C)}[/inline] [inline]\displaystyle\frac{-3}{x}[/inline] [inline]\text{D)}[/inline] [inline]3^x[/inline] [inline]\text{E)}[/inline] [inline]\displaystyle\frac{3}{\sqrt{-x}}[/inline] [inline]\text{N)}[/inline] ne znam
16.Link zadatka Niz [inline]\left(a_n\right)[/inline] je zadat sa [inline]a_1=1[/inline], [inline]a_{2n}=a_n-1[/inline], [inline]a_{2n+1}=a_n+1[/inline]. Koliko je [inline]a_{2015}[/inline]?
[inline]\text{A)}[/inline] [inline]5[/inline] [inline]\text{B)}[/inline] [inline]0[/inline] [inline]\text{C)}[/inline] [inline]8[/inline] [inline]\text{D)}[/inline] [inline]9[/inline] [inline]\text{E)}[/inline] [inline]10[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]5[/inline] [inline]\text{B)}[/inline] [inline]0[/inline] [inline]\text{C)}[/inline] [inline]8[/inline] [inline]\enclose{circle}{\text{D)}}[/inline] [inline]9[/inline] [inline]\text{E)}[/inline] [inline]10[/inline] [inline]\text{N)}[/inline] ne znam
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17.Link zadatka Data je aritmetička progresija [inline]a_1,a_2,a_3,\ldots[/inline]. Ako je [inline]a_1=3[/inline], [inline]a_n=136[/inline] i [inline]\displaystyle\sum_{k=1}^na_k=1390[/inline], koliko je [inline]a_2[/inline]?
[inline]\text{A)}[/inline] [inline]6[/inline] [inline]\text{B)}[/inline] [inline]13[/inline] [inline]\text{C)}[/inline] [inline]8[/inline] [inline]\text{D)}[/inline] [inline]20[/inline] [inline]\text{E)}[/inline] [inline]10[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]6[/inline] [inline]\text{B)}[/inline] [inline]13[/inline] [inline]\text{C)}[/inline] [inline]8[/inline] [inline]\text{D)}[/inline] [inline]20[/inline] [inline]\enclose{circle}{\text{E)}}[/inline] [inline]10[/inline] [inline]\text{N)}[/inline] ne znam
18.Link zadatka Koeficijent uz [inline]x^7[/inline] u razvoju [inline]\displaystyle\left(\frac{x^2}{2}-\frac{2}{x}\right)^8[/inline] jednak je
[inline]\text{A)}[/inline] [inline]\displaystyle\frac{7}{4}[/inline] [inline]\text{B)}[/inline] [inline]-14[/inline] [inline]\text{C)}[/inline] [inline]70[/inline] [inline]\text{D)}[/inline] [inline]-224[/inline] [inline]\text{E)}[/inline] [inline]448[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]\displaystyle\frac{7}{4}[/inline] [inline]\enclose{circle}{\text{B)}}[/inline] [inline]-14[/inline] [inline]\text{C)}[/inline] [inline]70[/inline] [inline]\text{D)}[/inline] [inline]-224[/inline] [inline]\text{E)}[/inline] [inline]448[/inline] [inline]\text{N)}[/inline] ne znam
19.Link zadatka Broj bijekcija skupa [inline]\left\{1,2,3,4,5,6,7\right\}[/inline] u sebe, koje preslikavaju parne brojeve u parne brojeve, jednak je:
[inline]\text{A)}[/inline] [inline]12[/inline] [inline]\text{B)}[/inline] [inline]24[/inline] [inline]\text{C)}[/inline] [inline]256[/inline] [inline]\text{D)}[/inline] [inline]144[/inline] [inline]\text{E)}[/inline] [inline]121[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]12[/inline] [inline]\text{B)}[/inline] [inline]24[/inline] [inline]\text{C)}[/inline] [inline]256[/inline] [inline]\enclose{circle}{\text{D)}}[/inline] [inline]144[/inline] [inline]\text{E)}[/inline] [inline]121[/inline] [inline]\text{N)}[/inline] ne znam
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20.Link zadatka Stranice knjige su numerisane prirodnim brojevima [inline]1,2,3,\ldots[/inline] u decimalnom zapisu. Koliko stranica ima knjiga, ako je za njihovu numeraciju upotrebljeno [inline]1500[/inline] cifara?
[inline]\text{A)}[/inline] [inline]500[/inline] [inline]\text{B)}[/inline] [inline]512[/inline] [inline]\text{C)}[/inline] [inline]536[/inline] [inline]\text{D)}[/inline] [inline]550[/inline] [inline]\text{E)}[/inline] [inline]555[/inline] [inline]\text{N)}[/inline] ne znam[inline]\text{A)}[/inline] [inline]500[/inline] [inline]\text{B)}[/inline] [inline]512[/inline] [inline]\enclose{circle}{\text{C)}}[/inline] [inline]536[/inline] [inline]\text{D)}[/inline] [inline]550[/inline] [inline]\text{E)}[/inline] [inline]555[/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.