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Words 814

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Words 814

Pages 4

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a 2n +1 + b 2n +1 = (a + b ) (a 2n − a 2n −1b + a 2n −2b 2 − .... + a 2b 2n −2 − ab 2n −1 + b 2n )

2 Evklidov in višinski izrek v pravokotnem trikotniku: a 2 = ca1 , b 2 = cb1 , vc = a1b1 Polmera trikotniku očrtanega in včrtanega kroga: R = abc , r = S , s = a + b + c s 2 4S Kotne funkcije polovičnih kotov: sin x = ± 1 − cos x ; cos x = ± 1 + cos x ; tan x = sin x 2 2 2 1 + cos x 2 2 Kotne funkcije trojnih kotov: sin 3x = 3 sin x − 4 sin 3 x , cos 3x = 4 cos3 x − 3 cos x Adicijski izrek: sin (x + y ) = sin x cos y + cos x sin y

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cos (x + y ) = cos x cos y − sin x sin y tan x + tan y tan (x + y ) = 1 − tan x tan y Faktorizacija: x +y x −y x +y x −y sin x + sin y = 2 sin cos , sin x − sin y = 2 cos sin 2 2 2 2 x +y x −y x +y x −y cos x + cos y = 2 cos cos , cos x − cos y = −2 sin sin 2 2 2 2 sin (x ± y ) sin (y ± x ) tan x ± tan y = , cot x ± cot y = cos x cos y sin x sin y Razčlenitev produkta kotnih funkcij: sin x sin y = − 1 [ cos (x + y ) − cos (x − y )] 2 1 [ cos (x + y ) + cos (x − y )] cos x cos y = 2 1 [ sin (x + y ) + sin (x − y )] sin x cos y = 2

Razdalja točke T0 x 0 , y 0 od premice ax + by − c = 0 :

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(

)

d T0 , p =

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ax 0 + by 0 − c a 2 + b2

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Ploščina trikotnika z oglišči A (x 1, y1 ) , B (x 2 , y2 ) , C (x 3 , y 3 ) : S = 1 (x 2 − x 1 )(y 3 − y1 ) − (x 3 − x 1 )(y2 − y1 ) 2 e Elipsa: e 2 = a 2 − b 2 , ε = a ; a > b e Hiperbola: e 2 = a 2 + b 2 , ε = a , a je realna polos p Parabola: y 2 = 2px , gorišče G , 0 2 Integrala: dx 1 x dx x ∫ x 2 + a 2 = a arc tan a + C , ∫ a 2 − x 2 = arc sin a + C

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01. Dana je funkcija s predpisom f (x ) = −3x + 5 . Izračunajte f − 1 . Izračunajte, za kateri x je 2 11 . Za katere x so vrednosti funkcije negativne? vrednost te funkcije 2

(8 točk)

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02. Prvi člen aritmetičnega zaporedja je −4 , peti člen pa 8 . Izračunajte diferenco (razliko) in stoti člen tega zaporedja.

(5 točk)

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03. Poenostavite izraz ((−a )4 ) ⋅ (−a )−3 : a 9 .

(5 točk)

3

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04. V ostrokotnem trikotniku, v katerem je stranica b daljša od stranice a , merijo: stranica a = 17 cm , višina v c = 4 cm in težiščnica tc = 5 cm . Izračunajte stranico c in ploščino trikotnika. Narišite skico.

(8 točk)

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05. Kompleksno število (5 − 10i)2 (2 + i)−1 zapišite v obliki a + bi, a, b ∈ R .

(6 točk)

8

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06. Zapišite polinom tretje stopnje, katerega graf je narisan v koordinatnem sistemu.

(6 točk)

y

4 3 2 1

–2

–1

1

2

x

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07. Elipsa s središčem v izhodišču koordinatnega sistema ima dve temeni T1 (2, 0) in T2 (−2, 0) ter poteka skozi točko A

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3, 2 . Zapišite njeno enačbo in drugi dve temeni. 2

(7 točk)

)

10

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08. Vektorja a in b na spodnji sliki sta dolga 4 enote, kot med njima pa je 120° .

b

120°

a

Skicirajte vektor c = −2a + 1 b ter izračunajte skalarna produkta a ⋅ b in a ⋅ c . 2

(8 točk)

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09. Naj bo f (x ) = a ⋅ 3x −1 + b, a, b ∈ R . Določite števili a in b tako, da bo f (1) = −1 in f (3) = −17 . Zapišite še definicijsko območje Df in zalogo vrednosti Z f tako dobljene funkcije.

(7 točk)

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10. Izračunajte ničle funkcij f (x ) = sin x in g (x ) = 2 sin x + 1 . 3 3

(7 točk)

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11. Marjetica ima 21 prijateljic in 11 prijateljev (le enemu prijatelju je ime Andrej in le enemu Borut). Na zabavo bo povabila 3 svoje prijateljice in 4 prijatelje. Na koliko načinov lahko to stori? Kolikšna je verjetnost, da bosta med temi povabljenci Andrej in Borut, če bo Marjetica izbirala povabljence naključno?

(6 točk)

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12. Na sliki je graf funkcije f (x ) = a . Na dve decimalki izračunajte število a , če je ploščina x osenčenega lika na sliki enaka 4 .

(7 točk)

y

1

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