{"paper":{"id":"matematyka-2018-operon-probna-rozszerzona","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":16,"source_label":"Matematyka · Matura próbna · listopad 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},"questions":[{"id":"matematyka-2018-operon-probna-rozszerzona/zad/1","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"1","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"Jeżeli (a_n) jest nieskończonym i niemonotonicznym ciągiem geometrycznym, w którym a_1 = 16 i a_3 = 1 , to suma wszystkich jego wyrazów wynosi:","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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/2","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"2","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"Dziedziną funkcji f(x) = _ x+1 (4 - 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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/3","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"3","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"Równanie |3 - 1 x | = m ma dwa różne rozwiązania dodatnie wtedy i tylko wtedy, gdy:","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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/4","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"4","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"Funkcja f(x) = x+3 (x-2)^2 :","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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/5","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"5","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"Czworokąt ABCD przedstawiony na rysunku jest wpisany w okrąg. Miara kąta jest równa:","answer":null,"answer_text":null,"solution":null,"image":"img/zz/matR_OPERON_2018_11_zad5_rys1.png","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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/6","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"6","points":2,"ptype":"open","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"Oblicz granicę _ x -2 ( x-3 x+2 - x^3-52 x^3+8 ) . Zakoduj kolejno, od lewej do prawej, cyfrę jedności i dwie pierwsze 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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/7","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"7","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"Rozwiąż nierówność 3x - |2x - 7| < 11 . [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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/8","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"8","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"Rozwiąż równanie (x + 6 ) + x = 3 2 w przedziale 0; 2 . [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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/9","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"9","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"Trapez prostokątny jest opisany na okręgu o promieniu 5 . Kąt ostry trapezu ma miarę 45^ . Oblicz długości odcinków, na które punkt styczności okręgu podzielił ramię pochyłe trapezu. [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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/10","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"10","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"W trójkącie ABC : |AB| = c , |AC| = b , |BC| = a oraz | BAC| = i | ABC| = (zobacz rysunek).\nWykaż, że jeżeli = 2 , to a^2 - b^2 = bc .","answer":null,"answer_text":null,"solution":null,"image":"img/zz/matR_OPERON_2018_11_zad10_rys1.png","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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/11","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"11","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"Wielomian W(x) = 2x^3 + ax^2 + bx + c jest podzielny przez trójmian x^2 + x - 6 , a przy dzieleniu przez dwumian x + 1 daje resztę 6 . Wyznacz wartości współczynników a, b i c . [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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/12","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"12","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"Ze zbioru wszystkich liczb naturalnych dodatnich nie większych od 30 losujemy kolejno 2 razy po jednej liczbie bez zwracania. Oblicz prawdopodobieństwo tego, że otrzymamy w ten sposób parę liczb, których iloczyn jest mniejszy od 30 pod warunkiem, że pierwsza wylosowana liczba jest mniejsza od drugiej wylosowanej liczby. [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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/13","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"13","points":6,"ptype":"open","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"Wyznacz wszystkie wartości parametru m , dla których dwa różne rozwiązania x_1 i x_2 równania (m + 1)x^2 + 2 2 x - m^2 + 2 = 0 spełniają warunek x_1^2 + x_2^2 m - x_1 x_2 . [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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/14","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"14","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"Liczbę 272 przedstaw w postaci sumy czterech całkowitych składników tworzących ciąg geometryczny i takich, że trzeci składnik jest o 48 większy od pierwszego. [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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/15","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"15","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"Napisz równania wszystkich prostych, które są jednocześnie styczne do paraboli o równaniu y = 1 4 x^2 - 1 i do okręgu o równaniu x^2 + (y + 6)^2 = 8 . [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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2018-operon-probna-rozszerzona/zad/16","paper_id":"matematyka-2018-operon-probna-rozszerzona","number":"16","points":7,"ptype":"open","subject":"matematyka","category":"probna","year":2018,"month":"listopad","level":"rozszerzona","text":"W graniastosłupie prawidłowym trójkątnym suma długości trzech różnych krawędzi wychodzących z jednego wierzchołka wynosi S . Wyznacz objętość tego graniastosłupa jako funkcję długości jednej z jego krawędzi i podaj dziedzinę tej funkcji. Oblicz wymiary graniastosłupa, którego objętość jest największa. Oblicz tę objętość. [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 2018 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"}]}