{"paper":{"id":"matematyka-2019-operon-probna-rozszerzona","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":16,"source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},"questions":[{"id":"matematyka-2019-operon-probna-rozszerzona/zad/1","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"1","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Liczba 11 - 6 2 jest równa:","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"algebra","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/2","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"2","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Dziedziną funkcji f(x) = _ 2x - 3 x + 3 (x^3 - x^2 ) jest:","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"logarytmy","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/3","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"3","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Suma wszystkich współczynników wielomianu W(x) = (7x^3 - 5x^2 - 2x + 8 )^5 stojących przy nieparzystych potęgach zmiennej x wynosi:","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"wielomiany","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/4","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"4","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Ile maksymalnie rozwiązań może mieć równanie | |x| - 3 | - 2 = m , gdzie m R ?","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"bezwgledna","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/5","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"5","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Dany jest trapez równoramienny, w który wpisano okrąg. Odcinek łączący środki ramion trapezu ma długość 7 cm. Obwód tego trapezu jest równy:","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"planimetria","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/6","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"6","points":2,"ptype":"open","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Oblicz _ x -2 x^2 + 7x + 10 x^3 + 8 . cyfrę jedności i dwie kolejne cyfry po przecinku rozwinięcia dziesiętnego otrzymanego wyniku. [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"granice","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/7","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"7","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Wyznacz największą i najmniejszą wartość funkcji f(x) = x^2 + 8 x + 1 w przedziale 0, 3 . [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"funkcje","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/8","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"8","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Wykaż, że dla dowolnych liczb rzeczywistych x, y zachodzi nierówność 2x^2 + 5y^2 + 10 > 6xy + 4y .","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"algebra","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/9","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"9","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Dany jest trójkąt prostokątny o przyprostokątnych długości a i b , w którym kąt między środkową a wysokością wychodzącymi z wierzchołka kąta prostego ma miarę . Wykaż, że tg = |a^2 - b^2| 2ab .","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"dowody_geometryczne","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/10","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"10","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Rozwiąż równanie 3x + 7x = 0 w przedziale 0, . [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"trygonometria","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/11","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"11","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"W urnie umieszczono 4 kule białe i 8 kul czarnych. Losujemy jedną kulę. Jeżeli będzie biała, to wrzucamy ją z powrotem do urny i dorzucamy do niej jeszcze dwie białe kule. Jeżeli będzie czarna, to zatrzymujemy ją i dorzucamy dwie zielone kule do urny. Następnie losujemy z urny jednocześnie dwie kule. Oblicz prawdopodobieństwo zdarzenia, że obie z wylosowanych za drugim razem kul są białe. [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"prawdopodobienstwo","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/12","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"12","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Graniastosłup prawidłowy czworokątny o krawędzi podstawy a i dwa razy krótszej wysokości przecięto płaszczyzną przechodzącą przez przekątną podstawy i nachyloną do płaszczyzny podstawy pod kątem 60^ . Zaznacz ten kąt na rysunku oraz oblicz pole otrzymanego przekroju, wynik przedstaw w najprostszej postaci. [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"stereometria","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/13","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"13","points":6,"ptype":"open","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Wyznacz wartość parametru m , dla którego równanie (m^2 + m - 3)x^2 + (2m - 1)x + 2 = 0 ma dwa rozwiązania dodatnie takie, że jedno z nich jest dwa razy większe od drugiego. [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"parametr","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/14","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"14","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Wyznacz równanie okręgu opisanego na trójkącie, którego boki zawierają się w prostych o równaniach x + 6y - 12 = 0 ; x + y - 7 = 0 oraz x - 4y + 18 = 0 . [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"analityczna","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/15","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"15","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Rozwiąż nierówność 1 x - 3 + 1 (x - 3)^2 + 1 (x - 3)^3 + 2 - x , gdzie lewa strona nierówności jest szeregiem geometrycznym zbieżnym. Podaj odpowiednie założenia. [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"ciagi","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2019-operon-probna-rozszerzona/zad/16","paper_id":"matematyka-2019-operon-probna-rozszerzona","number":"16","points":7,"ptype":"open","subject":"matematyka","category":"probna","year":2019,"month":"listopad","level":"rozszerzona","text":"Powierzchnia całkowita graniastosłupa prawidłowego sześciokątnego jest równa S 3 . Wyznacz największą z możliwych objętość tego graniastosłupa, wynik zapisz w najprostszej postaci. [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"optymalizacja","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2019 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"}]}