{"paper":{"id":"matematyka-2011-operon-probna-rozszerzona","subject":"matematyka","category":"probna","year":2011,"month":"listopad","level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":11,"source_label":"Matematyka · Matura próbna · listopad 2011 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},"questions":[{"id":"matematyka-2011-operon-probna-rozszerzona/zad/1","paper_id":"matematyka-2011-operon-probna-rozszerzona","number":"1","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2011,"month":"listopad","level":"rozszerzona","text":"Znajdź ujemny pierwiastek równania ||2x - 1| - 2| = 4 . [odp_otwarta1]","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 2011 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2011-operon-probna-rozszerzona/zad/2","paper_id":"matematyka-2011-operon-probna-rozszerzona","number":"2","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2011,"month":"listopad","level":"rozszerzona","text":"Prostokąt o bokach długości a , b jest podobny do prostokąta o bokach długości a + 5 , b + 5 . Wykaż, że te prostokąty są kwadratami.","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 2011 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2011-operon-probna-rozszerzona/zad/3","paper_id":"matematyka-2011-operon-probna-rozszerzona","number":"3","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2011,"month":"listopad","level":"rozszerzona","text":"Dla jakich x liczby 1 2 tg x , x , x w podanej kolejności są kolejnymi wyrazami ciągu geometrycznego? [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 2011 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2011-operon-probna-rozszerzona/zad/4","paper_id":"matematyka-2011-operon-probna-rozszerzona","number":"4","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2011,"month":"listopad","level":"rozszerzona","text":"Wykaż, że dla dowolnej liczby a > 0 zachodzi nierówność ^2( a) + ^2( + a) 2 _ +a 10 - _ .","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 2011 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2011-operon-probna-rozszerzona/zad/5","paper_id":"matematyka-2011-operon-probna-rozszerzona","number":"5","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2011,"month":"listopad","level":"rozszerzona","text":"Wierzchołki trójkąta równobocznego ABC leżą na paraboli, będącej wykresem funkcji f(x) = x^2 - 6x . Punkt C leży w wierzchołku paraboli. Znajdź współrzędne jednego z pozostałych wierzchołków trójkąta. [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 2011 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2011-operon-probna-rozszerzona/zad/6","paper_id":"matematyka-2011-operon-probna-rozszerzona","number":"6","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2011,"month":"listopad","level":"rozszerzona","text":"W graniastosłupie prawidłowym czworokątnym długość krawędzi podstawy jest równa 2a . Miara kąta między przekątną podstawy a przekątną ściany bocznej wychodzącą z tego samego wierzchołka jest równa . Oblicz objętość graniastosłupa. [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 2011 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2011-operon-probna-rozszerzona/zad/7","paper_id":"matematyka-2011-operon-probna-rozszerzona","number":"7","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2011,"month":"listopad","level":"rozszerzona","text":"W konkursie Jaka to piosenka? uczestnik zna 12 spośród przygotowanych 20 piosenek. Prowadzący przedstawia mu 4 piosenki. Uczestnik musi odgadnąć tytuł co najmniej jednej piosenki, aby przejść do dalszego etapu konkursu. Oblicz prawdopodobieństwo, że uczestnik przejdzie do dalszego etapu konkursu. Wynik podaj z dokładnością do 0 , 01 . [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 2011 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2011-operon-probna-rozszerzona/zad/8","paper_id":"matematyka-2011-operon-probna-rozszerzona","number":"8","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2011,"month":"listopad","level":"rozszerzona","text":"Oblicz, dla jakich wartości parametru k punkt przecięcia prostych o równaniach y = -x , y = x + k należy do koła o nierówności (x + 1)^2 + (y + 1)^2 10 . [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 2011 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2011-operon-probna-rozszerzona/zad/9","paper_id":"matematyka-2011-operon-probna-rozszerzona","number":"9","points":6,"ptype":"open","subject":"matematyka","category":"probna","year":2011,"month":"listopad","level":"rozszerzona","text":"Wiadomo, że pierwiastkami wielomianu W(x) = x^3 + ax^2 + bx + 6 są liczby -1 i 2 . Rozwiąż nierówność W(x) > 0 . [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 · listopad 2011 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2011-operon-probna-rozszerzona/zad/10","paper_id":"matematyka-2011-operon-probna-rozszerzona","number":"10","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2011,"month":"listopad","level":"rozszerzona","text":"Wyznacz wszystkie wartości parametru m , dla których równanie (m - 1)x^2 + 2(m + 1)x + m + 4 = 0 ma jedno rozwiązanie. [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 2011 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2011-operon-probna-rozszerzona/zad/11","paper_id":"matematyka-2011-operon-probna-rozszerzona","number":"11","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2011,"month":"listopad","level":"rozszerzona","text":"W trójkącie o polu 1 4 ab dwa boki mają długości a i b . Znajdź długość trzeciego boku. [odp_otwarta1]","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 2011 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"}]}