Državni izpitni center MATEMATIKA. Izpitna pola. Sobota, 3. junij 2017 / 120 minut
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1 Š i f r a k a n d i d a t a : Državni izpitni center *P171C10111* SPOMLADANSKI IZPITNI ROK MATEMATIKA Izpitna pola Sobota, 3. junij 017 / 10 minut Dovoljeno gradivo in pripomočki: Kandidat prinese nalivno pero ali kemični svinčnik, svinčnik, radirko, numerično žepno računalo brez grafičnega zaslona in možnosti simbolnega računanja, šestilo, trikotnik (geotrikotnik), ravnilo, kotomer in trigonir. Kandidat dobi dva konceptna lista in ocenjevalni obrazec. Priloga s formulami je na perforiranem listu, ki ga kandidat pazljivo iztrga. POKLICNA MATURA NAVODILA KANDIDATU Pazljivo preberite ta navodila. Ne odpirajte izpitne pole in ne začenjajte reševati nalog, dokler vam nadzorni učitelj tega ne dovoli. Prilepite oziroma vpišite svojo šifro v okvirček desno zgoraj na tej strani in na ocenjevalni obrazec ter na konceptna lista. Izpitna pola je sestavljena iz dveh delov. Prvi del vsebuje 11 nalog. Drugi del vsebuje 3 naloge, izmed katerih izberite in rešite dve. Število točk, ki jih lahko dosežete, je 70, od tega 50 v prvem delu in 0 v drugem delu. Za posamezno nalogo je število točk navedeno v izpitni poli. Pri reševanju si lahko pomagate s formulami na 3. in 4. strani. V preglednici z "x" zaznamujte, kateri dve nalogi v drugem delu naj ocenjevalec oceni. Če tega ne boste storili, bo ocenil prvi dve nalogi, ki ste ju reševali Rešitve pišite z nalivnim peresom ali s kemičnim svinčnikom in jih vpisujte v izpitno polo v za to predvideni prostor; grafe funkcij, geometrijske skice in risbe pa lahko rišete s svinčnikom. Če se zmotite, napisano prečrtajte in rešitev zapišite na novo. Nečitljivi zapisi in nejasni popravki bodo ocenjeni z 0 točkami. Osnutki rešitev, ki jih lahko naredite na konceptna lista, se pri ocenjevanju ne upoštevajo. Pri reševanju nalog mora biti jasno in korektno predstavljena pot do rezultata z vsemi vmesnimi računi in sklepi. Če ste nalogo reševali na več načinov, jasno označite, katero rešitev naj ocenjevalec oceni. Zaupajte vase in v svoje zmožnosti. Želimo vam veliko uspeha. Ta pola ima 4 strani, od tega 3 prazne. Državni izpitni center Vse pravice pridržane.
2 /4 *P171C101110*
3 *P171C * 3/4 FORMULE 1. Pravokotni koordinatni sistem v ravnini, linearna funkcija Razdalja dveh točk v ravnini: d( AB, ) ( x x ) + ( y y ) 1 1 Linearna funkcija: f ( x) kx + n Smerni koeficient: k y x y x 1 1 k k Naklonski kot premice: k tanϕ Kot med premicama: tanϕ 1 + kk 1 1. Ravninska geometrija (ploščine likov so označene s S) cv Trikotnik: S c 1 absinγ ss ( a)( s b)( s c), s a+ b+ c Polmera trikotniku očrtanega ( R) in včrtanega () r kroga: R abc, r 4S Enakostranični trikotnik: S a 3, v a 3, r a 3, R a ef Deltoid, romb: S Romb: Paralelogram: S absina Trapez: Dolžina krožnega loka: l π rα 180 Sinusni izrek: a b c R sina sin b sinγ Kosinusni izrek: a b + c bccosa S s S a sina S a+ c v, ( s a+ b+ c) Ploščina krožnega izseka: S π r α Površine in prostornine geometrijskih teles (S je ploščina osnovne ploskve) Prizma: P S + Spl, V Sv 1 Piramida: P S + Spl, V Sv 3 3 Krogla: P 4π r, V 4πr 3 Valj: Stožec: P π r + π rv, V π rv P π r +π rs, 1 V π rv 3 sin α + cos α 1 sin tana a cosa cos( α ± β) cosαcos β sinαsin β sin( α ± β) sinαcos β ± cosαsin β 4. Kotne funkcije 1+ tan a 1 cos a sinα sinαcosα cos α cos α sin α 5. Kvadratna funkcija, kvadratna enačba f ( x) ax + bx + c Teme: T( pq, ), p b, q D a 4a ax + bx + c 0 Ničli: x b± D 1,, D b 4ac a P perforiran list
4 4/4 *P171C * x 6. Logaritmi loga y x a y loga x nloga x loga x log a( xy) loga x + loga y logb x log b log x log x log y a a a y n a 7. Zaporedja Aritmetično zaporedje: an a1 + ( n 1) d, sn n ( a1 + ( n 1) d) n 1 Geometrijsko zaporedje: an aq n q 1 1, sn a1 q 1 G0np Navadno obrestovanje: Gn G0 + o, o 100 n p Obrestno obrestovanje: Gn Gr 0, r Obdelava podatkov (statistika) Aritmetična sredina: x1+ x xn x n fx 1 1+ fx fkx x f + f f 1 k k 9. Odvod Odvodi nekaterih elementarnih funkcij: n n 1 f ( x) x, f ( x) nx f( x) sin x, f ( x) cos x f( x) cos x, f ( x) sin x f( x) tan x, f ( x) 1 cos x f( x) ln x, f ( x) 1 x x x f( x) e, f ( x) e Pravila za odvajanje: f( x) + g( x) f ( x) + g ( x) ( ) f( xgx ) ( ) f ( xgx ) ( ) + f( xg ) ( x) ( ) kf ( x) kf ( x) ( ) f( x) f ( xgx ) ( ) f( xg ) ( x) gx ( ) g ( x) f gx ( ) f gx ( ) g ( x) ( ( )) ( ) Permutacije brez ponavljanja: Pn n! r Variacije brez ponavljanja: V! n n ( n r)! Variacije s ponavljanjem: ( p) V r n r 10. Kombinatorika in verjetnostni račun r r Vn Kombinacije brez ponavljanja: C! n n n ( r ) Verjetnost slučajnega dogodka A : ( ) n r! r!( n r)! P A m n število ugodnih izidov število vseh izidov
5 *P171C * 5/4 1. DEL Rešite vse naloge. 1. Za a 4 in b 3 z uporabo žepnega računala izračunajte vrednosti spodnjih izrazov: 3 a 3 ab 1 + a b log a (4 točke)
6 6/4 *P171C *. Dan je enakokraki trikotnik ABC z dolžino osnovnice c 8 cm in velikostjo kota α 30. Narišite skico in konstruirajte trikotnik ABC. Kot α konstruirajte s šestilom in ravnilom. o (4 točke)
7 *P171C * 7/4 3. Zapišite predpis za kvadratno funkcijo f, ki doseže največjo vrednost 4 pri x 1 in za katero velja f ( ) 3. (4 točke)
8 8/4 *P171C * 4. Rešite enačbo 1 5 x (4 točke)
9 *P171C * 9/4 5. Dan je splošni člen zaporedja an sin nπ, n. Natančno izračunajte prve tri člene zaporedja. 6 (4 točke)
10 10/4 *P171C * 6. Biatlonka Mojca je zadela 3 izmed 5 tarč (glej sliko, zadeta tarča je bela). Izračunajte verjetnost, da je zadela razporeditev zadetih tarč na sliki, če so vse razporeditve zadetih tarč enako verjetne. (4 točke)
11 *P171C * 11/4 7. Zapišite enačbo premice, ki gre skozi točko T (,3) in ima smerni koeficient k 1. Premico tudi narišite v dani koordinatni sistem. (4 točke) y x
12 1/4 *P171C101111* 8. Matej je v štirih dneh prebral knjigo s 10 stranmi besedila. Prvi dan je prebral 0 % celotnega besedila, naslednji dan 1 celotnega besedila, zadnja dva dni pa vsak dan enako število strani 4 besedila. Koliko strani besedila je Matej prebral zadnji dan? (5 točk)
13 *P171C * 13/4 9. Rešite sistem enačb: x + 5 y 7 in 3 x + 7 y 11. (5 točk)
14 14/4 *P171C * 10. V podjetju ima plačo 750 EUR en zaposleni, plačo 80 EUR dva zaposlena, plačo 1050 EUR osem zaposlenih, plačo 180 EUR en zaposleni in 400 EUR en zaposleni. Izračunajte mediano, modus in aritmetično sredino plač zaposlenih v podjetju. (6 točk)
15 *P171C * 15/4 11. Dan je pravokotni trikotnik ABC s pravim kotom pri oglišču C. Velikost kota pri oglišču A o je 73, dolžina hipotenuze c pa 6 cm. Narišite skico ter izračunajte dolžini katet in velikost kota pri oglišču B. (6 točk)
16 16/4 *P171C *. DEL Izberite dve nalogi, na naslovnici izpitne pole zaznamujte njuni zaporedni številki in ju rešite. 1. Dana je racionalna funkcija f ( x) 3x + 1. x Izračunajte manjkajoče vrednosti v preglednici. x 1 4 f ( x ) 1.. Zapišite enačbo tangente na graf funkcije f v točki 1 ( ) T 0,. 4 (5 točk) (5 točk)
17 *P171C * 17/4
18 18/4 *P171C *. Pločevinka ima obliko valja s prostornino π cm in višino 10 cm..1. Narišite skico mreže valja, ki je sestavljena iz dveh krogov in pravokotnika. Izračunajte polmer kroga in dolžini stranic pravokotnika... V prazno pločevinko smo do 3 4 nalili. (7 točk) višine nalili vodo. Izračunajte, koliko decilitrov vode smo (3 točke)
19 *P171C * 19/4
20 0/4 *P171C101110* 3. V 1. A razredu na neki šoli je 1 fantov in 16 deklet V 1. A razredu na tej šoli je dvakrat toliko deklet kolikor je deklet v. A razredu. Razmerje deklet in fantov v. A razredu je : 5. Koliko fantov in koliko deklet je v. A razredu? (5 točk) 3.. Učitelj matematike je v 1. A razredu za spraševanje naključno izbral 3 dijake. Kolikšna je verjetnost, da je izbral dva fanta in eno dekle? (5 točk)
21 *P171C101111* 1/4
22 /4 *P171C10111* Prazna stran
23 *P171C101113* 3/4 Prazna stran
24 4/4 *P171C101114* Prazna stran
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