{"paper":{"id":"site-2866","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"variant":null,"exam_pdf":null,"key_pdf":null,"question_count":37,"source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},"questions":[{"id":"site-2866/zad/2867","paper_id":"site-2866","number":"2867","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Obliczam wyrażenie z pierwiastkami korzystając ze wzorów skróconego mnożenia.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2867.svg","solution_image":"img/site/2866/rozw-2867.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2868","paper_id":"site-2866","number":"2868","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Miary dwóch kątów trójkąta wynoszą pi/6 i pi/5. Oblicz miarę trzeciego kąta. Odpowiedź podaj w stopniach.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2868.svg","solution_image":"img/site/2866/rozw-2868.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2869","paper_id":"site-2866","number":"2869","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Dane jest równanie sin x = a^2+1, z niewiadomą x. Wyznacz wszystkie wartości parametru a, dla których dane równanie nie ma rozwiązań. Zadanie z informatora maturalnego.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2869.svg","solution_image":"img/site/2866/rozw-2869.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2871","paper_id":"site-2866","number":"2871","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Funkcja f jest określona wzorem f(x) = { x+5 dla x<-5; -x+2 dla -5<=x<5; x-6 dla x<=5 }. Miejscami zerowymi tej funkcji są liczby a) -5,2,6 b) 2,6 c) -5,2 d) -5,-2,6","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2871.svg","solution_image":"img/site/2866/rozw-2871.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2888","paper_id":"site-2866","number":"2888","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"W kolejce do kasy biletowej ustawiły się cztery dziewczynki i pięciu chłopców. Liczba wszystkich możliwych ustawień osób w tej kolejce wynosi a) 4!+5!, b) 9!, c) 4\\c5, d) 4!\\c5!.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2888.svg","solution_image":"img/site/2866/rozw-2888.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2889","paper_id":"site-2866","number":"2889","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Rozwiąż równanie \\log_5(\\log_4(\\log_2 x))=0.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2889.svg","solution_image":"img/site/2866/rozw-2889.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2890","paper_id":"site-2866","number":"2890","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Funkcja f jest określona wzorem f(x)={1 / x+1}-1 dla wszystkich liczb rzeczywistych x\\not=1. Rozwiąż nierówność f(x)>f(2-x).","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2890.svg","solution_image":"img/site/2866/rozw-2890.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2891","paper_id":"site-2866","number":"2891","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Narysuj wykres funkcji f określonej w przedziale <-2,2> wzorem f(x)=2^x-1.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2891.svg","solution_image":"img/site/2866/rozw-2891.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2892","paper_id":"site-2866","number":"2892","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Narysuj wykres funkcji f określonej w przedziale <-2,2> wzorem f(x)=2^{x-1}.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2892.svg","solution_image":"img/site/2866/rozw-2892.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2896","paper_id":"site-2866","number":"2896","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Pole wycinka koła o promieniu 3 cm jest równe 2 cm^2. Oblicz miarę łukową kąta środkowego tego wycinka.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2896.svg","solution_image":"img/site/2866/rozw-2896.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2897","paper_id":"site-2866","number":"2897","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Punkty A=(1,1), B=(5,5), C=(3,5) są wierzchołkami trapezu równoramiennego ABCD niebędącego równoległobokiem, w którym AB||CD. a) Wyznacz równanie osi symetrii tego trapezu. b) Oblicz pole tego trapezu.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2897.svg","solution_image":"img/site/2866/rozw-2897.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2898","paper_id":"site-2866","number":"2898","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Na okręgu zaznaczono sześć różnych punktów. Ile różnych wielokątów wypukłych o wszystkich wierzchołkach w tych punktach można narysować?","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2898.svg","solution_image":"img/site/2866/rozw-2898.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2902","paper_id":"site-2866","number":"2902","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Dla jakich wartości parametru m reszta z dzielenia wielomianu x^{17}-mx^{15}+(m-2)x^{10}+2x+m^2-2 przez dwumian x-1 jest równa 3?","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2902.svg","solution_image":"img/site/2866/rozw-2902.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2903","paper_id":"site-2866","number":"2903","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Wyznacz równanie okręgu o środku A=(2,3), stycznego do prostej o równaniu x-2y+1=0.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2903.svg","solution_image":"img/site/2866/rozw-2903.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2926","paper_id":"site-2866","number":"2926","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Niech A będzie zbiorem wszystkich liczb x, które spełniają równość |x-1|+|x-3|=2. Niech B będzie zbiorem wszystkich punktów na osi liczbowej, których suma odległości od punktów 4 i 6 jest nie większa niż 4. Zaznacz na osi liczbowej zbiory A i B oraz wszystkie punkty, które należą jednocześnie do A i do B.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2926.svg","solution_image":"img/site/2866/rozw-2926.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2942","paper_id":"site-2866","number":"2942","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Przedział (-{3/2},0) jest zbiorem wszystkich rozwiązań nierówności 2/x < m z niewiadomą x. Oblicz m.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2942.svg","solution_image":"img/site/2866/rozw-2942.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2943","paper_id":"site-2866","number":"2943","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Rozpatrujemy wszystkie prostokąty o polu równym 6, których dwa sąsiednie boki zawarte są w osiach Ox i Oy układu współrzędnych. Wyznacz równanie krzywej będącej zbiorem tych wierzchołków rozpatrywanych prostokątów, które nie leżą na żadnej z osi układu współrzędnych. Narysuj tę krzywą.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2943.svg","solution_image":"img/site/2866/rozw-2943.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2945","paper_id":"site-2866","number":"2945","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Miary pięciu kątów tworzą ciąg arytmetyczny. Drugim wyrazem tego ciągu jest 150 stopni, a czwartym 270 stopni. Oblicz sumę sinusów tych pięciu kątów.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2945.svg","solution_image":"img/site/2866/rozw-2945.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2948","paper_id":"site-2866","number":"2948","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Dane jest równanie x^2+(3m-2)x=-m-2 z niewiadomą x. Sformułuj warunki, jakie powinien spełniać parametr m, by to równanie miało dwa różne pierwiastki, których suma odwrotności jest dodatnia.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2948.svg","solution_image":"img/site/2866/rozw-2948.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2953","paper_id":"site-2866","number":"2953","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Wyznacz pierwsze trzy wyrazy ciągu geometrycznego wiedząc, że są one dodatnie, ich suma jest równa 21 oraz suma ich odwrotności jest równa 7/12.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2953.svg","solution_image":"img/site/2866/rozw-2953.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/2954","paper_id":"site-2866","number":"2954","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Z szuflady, w której znajduje się 10 różnych par rękawiczek wybieramy losowo cztery rękawiczki. Opisz zbiór wszystkich zdarzeń elementarnych, a następnie oblicz prawdopodobieństwo zdarzeń: A wśród wylosowanych rękawiczek nie będzie pary, B wśród wylosowanych rękawiczek będzie dokładnie jedna para.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-2954.svg","solution_image":"img/site/2866/rozw-2954.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3034","paper_id":"site-2866","number":"3034","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Wyznacz wartości parametru p, dla których równanie |x-2|+|x+3|=p ma dokładnie dwa rozwiązania.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3034.svg","solution_image":"img/site/2866/rozw-3034.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3035","paper_id":"site-2866","number":"3035","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Wykaż, że dla a w (2,3) zachodzi równość pierwiastek z {a^2-6a+9} / 3-a + pierwiastek z {a^2-4a+4} / a-2 = 2.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3035.svg","solution_image":"img/site/2866/rozw-3035.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3036","paper_id":"site-2866","number":"3036","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Dane jest równanie x^2+bx+c=0 z niewiadomą x. Wyznacz wartości b oraz c tak, by były one rozwiązaniami danego równania.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3036.svg","solution_image":"img/site/2866/rozw-3036.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3037","paper_id":"site-2866","number":"3037","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Dane są funkcje liniowe g i h określone wzorami: g(x)=ax+b i h(x)=bx+a. Wiadomo, że funkcja g jest rosnąca, a funkcja h malejąca. a) Wyznacz pierwszą współrzędną punktu przecięcia wykresów tych funkcji b) Oblicz liczby a i b wiedząc, że wykresy funkcji g i h są prostymi prostopadłymi, a punkt ich przecięcia leży na osi Ox","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3037.svg","solution_image":"img/site/2866/rozw-3037.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3039","paper_id":"site-2866","number":"3039","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Dany jest ciąg (a_n) mający tę własność, że dla każdej liczby naturalnej n suma n początkowych wyrazów tego ciągu jest równa {1/2}(7n^2-n). Oblicz dwudziesty wyraz tego ciągu. Wykaż, że (a_n) jest ciągiem arytmetycznym.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3039.svg","solution_image":"img/site/2866/rozw-3039.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3040","paper_id":"site-2866","number":"3040","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Proste zawierające ramiona BC i DA trapezu ABCD przecinają się w punkcie S. Dane są: |AB|=6, |CD|=2 oraz obwód trójkąta SCD równy pierwiestek z 18. Oblicz obwód trójkąta SAB.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3040.svg","solution_image":"img/site/2866/rozw-3040.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3041","paper_id":"site-2866","number":"3041","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"W pewnym trapezie kąty przy dwóch przeciwległych wierzchołkach mają miary alfa oraz 90+alfa Jedno z ramion tego trapezu ma długość t. Wyznacz różnicę długości podstaw tego trapezu.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3041.svg","solution_image":"img/site/2866/rozw-3041.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3042","paper_id":"site-2866","number":"3042","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Czworokąt ABCD jest wpisany w okrąg. Dane są |BC|=a, |CD|=b, |kąt DAB|=alfa. Wyznacz długość przekątnej BD.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3042.svg","solution_image":"img/site/2866/rozw-3042.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3043","paper_id":"site-2866","number":"3043","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Podstawą ostrosłupa ABCDS jest kwadrat ABCD o boku długości 4. Odcinek DS jest wysokością ostrosłupa i ma długość 6. Punkt M jest środkiem odcinka DS. Oblicz pole przekroju ostrosłupa płaszczyzną BCM.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3043.svg","solution_image":"img/site/2866/rozw-3043.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3044","paper_id":"site-2866","number":"3044","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Ze zbioru {1,2,...,2n+5} wybieramy jednocześnie dwie liczby. Na ile sposobów możemy to zrobić, tak aby otrzymać dwie liczby takie, że: a) ich różnica będzie liczbą parzystą, b) suma ich kwadratów będzie liczbą podzielną przez cztery?","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3044.svg","solution_image":"img/site/2866/rozw-3044.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3045","paper_id":"site-2866","number":"3045","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Narysuj przekrój równoległościanu płaszczyzną PQR","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3045.svg","solution_image":"img/site/2866/rozw-3045.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3046","paper_id":"site-2866","number":"3046","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Wiedząc, że dla pewnego ciągu geometrycznego (a_n) o wyrazach dodatnich prawdziwa jest równość S_{14}=5*S_7, oblicz iloraz tego ciągu. Symbol S_n oznacza sumę n początkowych wyrazów ciągu (a_n).","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3046.svg","solution_image":"img/site/2866/rozw-3046.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3054","paper_id":"site-2866","number":"3054","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Wielomian f jest określony wzorem f(x)=ax^4-9x^3+3x^2+7x+b dla pewnych liczb pierwszych a i b. Wiadomo, że liczba 3/2 jest pierwiastkiem tego wielomianu. Oblicz a i b.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3054.svg","solution_image":"img/site/2866/rozw-3054.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3056","paper_id":"site-2866","number":"3056","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Dane jest równanie x^2+mx+m-1=0 z niewiadomą x. Uzasadnij, że dla każdej liczby całkowitej m wszystkie rozwiązania tego równania są liczbami całkowitymi.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3056.svg","solution_image":"img/site/2866/rozw-3056.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3194","paper_id":"site-2866","number":"3194","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Wielomiany W(x)=ax(x+b)^2 i V(x)=x^3+2x^2+x są równe. Oblicz a i b.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3194.svg","solution_image":"img/site/2866/rozw-3194.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2866/zad/3195","paper_id":"site-2866","number":"3195","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":null,"text":"Wyrażenie {3 / x-3} - {x / x+1} zapisz w postaci ilorazu dwóch wielomianów.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2866/zad-3195.svg","solution_image":"img/site/2866/rozw-3195.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań","subject_label":"Matematyka","category_label":"Zbiory zadań"}]}