{"paper":{"id":"matematyka-2015-operon-probna-rozszerzona","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":18,"source_label":"Matematyka · Matura próbna · listopad 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},"questions":[{"id":"matematyka-2015-operon-probna-rozszerzona/zad/1","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"1","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Funkcja określona wzorem f(x) = |x + 3| + 5 :","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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/2","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"2","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Dokładna wartość liczby 15^ to:","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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/3","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"3","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Funkcja określona wzorem f(x) = 1 5 x^5 - 1 2 x^4 + 1 3 x^3 :","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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/4","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"4","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Dany jest ostrosłup prawidłowy czworokątny, którego wszystkie krawędzie mają długość a . Ostrosłup przecięto płaszczyzną przechodzącą przez przekątną podstawy i środek przeciwległej do niej krawędzi bocznej. Pole otrzymanego przekroju jest równe:","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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/5","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"5","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Dany jest ciąg określony wzorem rekurencyjnym cases a_1 = 4 a_ n+1 = 3a_n - n 2 cases . Czwarty wyraz tego ciągu jest równy:","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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/6","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"6","points":2,"ptype":"open","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Dane są punkty A = (5, 2) , B = (1, -3) , C = (-2, -8) . Oblicz odległość punktu A od prostej l przechodzącej przez punkty B, C . Zakoduj trzy początkowe cyfry po przecinku rozwinięcia dziesiętnego otrzymanego wyniku. [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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/7","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"7","points":2,"ptype":"open","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Oblicz sinus najmniejszego kąta trójkąta o bokach a = 8, b = 10, c = 12 . Zakoduj trzy początkowe cyfry po przecinku rozwinięcia dziesiętnego otrzymanego wyniku. [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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/8","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"8","points":2,"ptype":"open","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Dana jest funkcja określona wzorem f(x) = x^2 - 1 x^2 - 4 . Oblicz wartość pochodnej tej funkcji dla x = - 7 . Zakoduj cyfrę jedności i dwie początkowe cyfry po przecinku rozwinięcia dziesiętnego otrzymanego wyniku. [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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/9","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"9","points":2,"ptype":"open","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Oblicz granicę _ n 2 + 4 + 6 + + 2n 11n^2 - 1 . Zakoduj trzy początkowe 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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/10","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"10","points":2,"ptype":"open","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Pierwiastkami równania x^2 + 7x + 4 = 0 są liczby x_1, x_2 . Oblicz wartość sumy sześcianów liczb x_1, x_2 . Zakoduj cyfrę setek, dziesiątek i jedności wartości bezwzględnej otrzymanego wyniku. [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"rownania_i_nierownosci","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/11","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"11","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Wykaż, że jeśli _ 24 6 = a , to _6 256 = 4(1 - a) a .","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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/12","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"12","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Wyznacz równanie stycznej do okręgu o równaniu x^2 - 6x + y^2 + 10y = 0 prostopadłej do prostej 3x - 4y + 5 = 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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/13","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"13","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Dany jest trójmian f(x) = x^2 + (m + 2)x + 4 . Wyznacz parametr m , jeśli wiadomo, że ciąg (x_1, (m + 5), x_2) , gdzie x_1, x_2 są różnymi miejscami zerowymi tego trójmianu, jest geometryczny. [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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/14","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"14","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Dany jest trójkąt równoboczny ABC , w którym punkt D jest środkiem boku AB . Przez punkt D poprowadzono prostą pod kątem do boku AB , która przecięła bok BC w punkcie E takim, że pole trójkąta BDE jest równe 1 8 pola trójkąta ABC . Wykaż, że = 30^ .","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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/15","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"15","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Rozwiąż równanie 2x + 4x = 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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/16","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"16","points":7,"ptype":"open","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Puszka ma kształt walca o objętości dm ^3 . Wyznacz promień podstawy i wysokość walca, aby pole powierzchni całkowitej puszki było najmniejsze. Oblicz to 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 · listopad 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/17","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"17","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"W urnie U_1 są 3 kule białe i 7 czarnych, a w urnie U_2 jest 5 kul białych i 4 czarne. Wybieramy losowo kulę z urny U_1 i wkładamy do urny U_2 . Następnie z urny U_2 losujemy 2 kule. Oblicz prawdopodobieństwo, że w ten sposób wylosujemy 2 kule 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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2015-operon-probna-rozszerzona/zad/18","paper_id":"matematyka-2015-operon-probna-rozszerzona","number":"18","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2015,"month":"listopad","level":"rozszerzona","text":"Trzy liczby tworzą ciąg arytmetyczny. Jeśli pierwszą liczbę zmniejszymy o 1 , drugą liczbę zwiększymy o 15 , a trzecią zwiększymy o 37 , to otrzymamy ciąg geometryczny. Wyznacz te liczby, jeśli wiadomo, że ich suma jest równa 63 . [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 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"}]}