zadania zamknięte 1-25 z obliczeniami. matura z mat pp

z.1 a = 2 - √3 a³ = a²*a = ( 2- √3)²(2 - √3) = (4 - 4√3 +3)(2 - √3) = = (7 - 4√3)(2 -√3) = 14 -7√3 - 8√3 +12 = 26 - 15√3 Odp. D z.2 3²⁰ + 3²⁰ + 3²⁰ = 3*3²⁰ = 3²⁰⁺¹ = 3²¹ Odp.B z.3 sinα*cos α = 0.5 (sinα + cosα)² = sin²α + 2sinα cosα + cos²α = = sin²α + cos²α + 2*0,5 = 1 + 1 = 2 W odpowiedziach nie ma takiego wyniku. z.4 log₁₂3 - 2log₁₂(1/2) = log₁₂ - log₁₂(1/2)² = log¹²3 - log₁₂(1/4) = = log₁₂ [3:(1/4)] = log₁₂ (3*4) = log₁₂12 = 1 Odp. A z.5 y - 3 = 0 oraz punkt (1; -1) Odp. x = 1 Odp. B z.6 W(x) = 2x⁴ - x² - 1 P(x) = 2x³ - x² + 1 W(x)*P(x) jest wielomianem stopnia 7, bo 2x⁴*2x³ = 4x⁷ z.7 f(x) = (2x = 3)/(1 - x²) 1 - x² = (1 -x)*(1 +x) D = R { -1; 1}, bo dla x = -1 lub x = 1 mianownik = 0. --------------------------------------------------------- z.8 210 - 150 = 60 60 : 150 = 0,40 = 40 % D) -------------------------------------------- z.9 AB = 3 oraz A1B1 = 5 k = AB : A1B1 = 3 : 5 = 3/5 P1 : P2 = k² = (3/5)² = 9/25 B) ------------------------------------------ z.10 B) (√2 + √8)² = 2 + 8 + 2*√2*√8 = 10 + 2*√16 = 10 +8 = 18 ---------------------------------------------------------------------- z.11 A = (1; -4) , B = (-2; 6) I AB I² = (-2 -1)² + (6 -(-4))² = (-3)² + 10² = 9 + 100 = 109 zatem AB = √109 A) ----------------------------------------------------------------- z.12 3*4 + 3*4 = 2*12 = 24 ------------------------------------------------------------------ z.13 n = 7 (n -2) *180⁰ = 5*180⁰ = 900⁰ A) ----------------------------------------------------------------- z.14 C) 23 : 5 = 4,6 ----------------------------------------------------------- z.15 (x -2)(1 -x) > 0 -(x-2)(x-1) > 0 (x-2)(x-1) < 0 x ∈ ( 1; 2) C) ------------------------------------- z.16 y = √2 x - 2 oraz y = - √0,5 x - 2 a*a1 = √2 *(-√0,5) = -√2*√0,5 = - 1 D) Proste są prostopadle ------------------------------------------- z.17 x + 2y = 7 ---> x = 7 - 2y -3x + 3y = 15 ------------------------------ -3*(7 -2y) + 3y = 15 -21 + 6y + 3y = 15 9y = 15 + 21 = 36 y = 4 x = 7 -2*4 = 7 - 8 = - 1 Odp. D ------------------------------------------------------ z.18 f(x) = 2x² + x - 5 < 1; 3 > p = -1/4 < 1 oraz a = 2 > 0 zatem funkcja f rośnie dla x > -1/4 czyli rośnie w całym <1 ; 3> f(3) = 2*3² +3 - 5 = 18 +3 - 5 = 16 D) --------------------------------------------------- z.19 x - 0,2x = 0,8 x 0,8x + y*0,8x = x y*0,8 x = x - 0,8 x = 0,2 x y = 0,2 x : 0,8 x = 2/8 = 1/4 = 25% B) --------------------------------------------------- z.20 d = √(3²+4²) = √25 = 5 p = √(5²+6²) = √61 B) ------------------------------------------- z.22 a = (-4; 6) , B = < -1; 7> A B = (-4; -1) B) ---------------------------------------------------- z.23 Pb = 2πrh = 24π ---> rh = 12 V = πr²h = 48π ---> r²h = 48 r²h : rh = r = 48 : 12 = 4 r = 4 B) --------------------------------------------- z.24 P(A) = 0,3 , P(B) = 0,4 , P(A u B) = 0,6 Korzystamy ze wzoru P(A u B ) = P(A) + P(B) - P(A n B) P(A n B) = P(A) + P(B) - P (A u B) = 0,3 + 0,4 - 0,6 = 0,1 A) ---------------------------------------------------------------- z.25 A) 22: 7 = 3,142 ≈ 3,14 =====================================================