{"paper":{"id":"matematyka-2007-operon-probna-rozszerzona","subject":"matematyka","category":"probna","year":2007,"month":"probna","level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":12,"source_label":"Matematyka · Matura próbna · 2007 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},"questions":[{"id":"matematyka-2007-operon-probna-rozszerzona/zad/1","paper_id":"matematyka-2007-operon-probna-rozszerzona","number":"1","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2007,"month":"probna","level":"rozszerzona","text":"Dla jakich wartości parametru m funkcja f(x) = (m-1)x^2 + (m-1)x + m przyjmuje tylko wartości dodatnie? [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 · 2007 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2007-operon-probna-rozszerzona/zad/2","paper_id":"matematyka-2007-operon-probna-rozszerzona","number":"2","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2007,"month":"probna","level":"rozszerzona","text":"Oblicz _ 2 [3] 16 2 8 . [odp_otwarta1]","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 · 2007 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2007-operon-probna-rozszerzona/zad/3","paper_id":"matematyka-2007-operon-probna-rozszerzona","number":"3","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2007,"month":"probna","level":"rozszerzona","text":"Naszkicuj wykres funkcji y = 2^ x-|x| . Dla jakich wartości parametru m równanie 2^ x-|x| = m ma przynajmniej jedno rozwiązanie? [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"wykladnicza","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · 2007 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2007-operon-probna-rozszerzona/zad/4","paper_id":"matematyka-2007-operon-probna-rozszerzona","number":"4","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2007,"month":"probna","level":"rozszerzona","text":"Współczynniki wielomianu trzeciego stopnia W(x) tworzą rosnący ciąg geometryczny (licząc od współczynnika przy najwyższej potędze) o pierwszym wyrazie równym 1 i sumie równej 15 . Wyznacz wzór tego wielomianu. [odp_otwarta1]","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 · 2007 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2007-operon-probna-rozszerzona/zad/5","paper_id":"matematyka-2007-operon-probna-rozszerzona","number":"5","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2007,"month":"probna","level":"rozszerzona","text":"Wykaż, że liczba a = 29 - 12 5 - 2 5 jest całkowita.","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 · 2007 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2007-operon-probna-rozszerzona/zad/6","paper_id":"matematyka-2007-operon-probna-rozszerzona","number":"6","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2007,"month":"probna","level":"rozszerzona","text":"W urnie U_1 jest 5 kul białych i 7 czarnych, a w urnie U_2 są 4 białe i 8 czarnych. Rzucamy trzema monetami. Jeśli wyrzucimy dokładnie 2 orły - losujemy kulę z urny U_1 , w pozostałych przypadkach - z U_2 . Jakie jest prawdopodobieństwo, że w ten sposób wylosujemy kulę białą? [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 · 2007 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2007-operon-probna-rozszerzona/zad/7","paper_id":"matematyka-2007-operon-probna-rozszerzona","number":"7","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2007,"month":"probna","level":"rozszerzona","text":"Oblicz granicę ciągu: a_n = n^2 4+8+12+ +4n . [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 · 2007 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2007-operon-probna-rozszerzona/zad/8","paper_id":"matematyka-2007-operon-probna-rozszerzona","number":"8","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2007,"month":"probna","level":"rozszerzona","text":"Rozwiąż równanie: ax + 5x = b . [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 · 2007 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2007-operon-probna-rozszerzona/zad/9","paper_id":"matematyka-2007-operon-probna-rozszerzona","number":"9","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2007,"month":"probna","level":"rozszerzona","text":"Wykaż, że dwusieczna kąta wewnętrznego dowolnego trójkąta dzieli bok przeciwległy w stosunku pozostałych boków.","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 · 2007 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2007-operon-probna-rozszerzona/zad/10","paper_id":"matematyka-2007-operon-probna-rozszerzona","number":"10","points":6,"ptype":"open","subject":"matematyka","category":"probna","year":2007,"month":"probna","level":"rozszerzona","text":"Trójkąt o bokach długości 5, 8, 9 obraca się dokoła najdłuższego boku. Oblicz objętość powstałej bryły. [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 · 2007 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2007-operon-probna-rozszerzona/zad/11","paper_id":"matematyka-2007-operon-probna-rozszerzona","number":"11","points":7,"ptype":"open","subject":"matematyka","category":"probna","year":2007,"month":"probna","level":"rozszerzona","text":"W jakim punkcie x (0, 2) paraboli y = -x^2 + 4 należy poprowadzić styczną, aby trójkąt ograniczony tą styczną i dodatnimi półosiami osi współrzędnych miał najmniejsze pole? [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 · 2007 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2007-operon-probna-rozszerzona/zad/12","paper_id":"matematyka-2007-operon-probna-rozszerzona","number":"12","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2007,"month":"probna","level":"rozszerzona","text":"Wykaż, że jeśli , są różnymi kątami trójkąta spełniającymi warunek ( - ) = ^2 - ^2 , to ten trójkąt jest prostokątny.","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 · 2007 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"}]}