{"paper":{"id":"matematyka-2016-operon-probna-rozszerzona","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":18,"source_label":"Matematyka · Matura próbna · listopad 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},"questions":[{"id":"matematyka-2016-operon-probna-rozszerzona/zad/1","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"1","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Zbiorem rozwiązań nierówności ||x + 3| - 5| < 2 jest:","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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/2","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"2","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Liczba tg 22,5^ + 1 tg 22,5^ jest równa:","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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/3","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"3","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Dany jest trójkąt o bokach 10 i 6 i kącie między nimi 120^ . Promień okręgu opisanego na tym trójkącie 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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/4","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"4","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Wielomian określony wzorem W(x) = 1 4 x^4 + 2 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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/5","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"5","points":1,"ptype":"closed","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Liczba _6 5 + 2 _ 36 3 jest równa:","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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/6","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"6","points":2,"ptype":"open","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Oblicz granicę _ n ( 3n^2 - 5n - 7 5n^2 + 3n + 2 - ( 2n - 1 3n + 1 )^3 ) . Zakoduj cyfrę jedności i dwie początkowe cyfry 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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/7","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"7","points":2,"ptype":"open","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Wyznacz największą liczbę spełniającą równanie x^3 + x^2 - 7x + 5 = 0 . 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":"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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/8","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"8","points":2,"ptype":"open","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Dany jest trapez prostokątny opisany na okręgu. Punkt styczności okręgu z dłuższym ramieniem trapezu dzieli to ramię na odcinki długości 8 i 11 . Oblicz obwód trapezu. Zakoduj cyfrę dziesiątek, jedności i jedną początkową cyfrę 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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/9","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"9","points":2,"ptype":"open","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Wyznacz dziedzinę wyrażenia W = x - 5 4 - x^2 . [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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/10","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"10","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Dana jest funkcja f określona wzorem f(x) = 3 x^4 + x^2 - 75 . Wyznacz równanie stycznej do wykresu funkcji f poprowadzonej w punkcie P = (-3, 1 5 ) . [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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/11","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"11","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Wykaż, że jeśli liczby a i b są dodatnie, to a^2 b^2 + b^2 a^2 + 3 ( a b + b a ) 8 .","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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/12","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"12","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Dany jest nieskończony ciąg geometryczny. Suma wszystkich wyrazów tego ciągu jest równa 40 , a suma wszystkich wyrazów o numerach nieparzystych jest równa 32 . Oblicz iloraz i pierwszy wyraz tego ciągu. [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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/13","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"13","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Wykaż, że jeśli i są kątami trójkąta takimi, że ^2 - ^2 = ( - ) , to trójkąt jest równoramienny lub 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 · listopad 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/14","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"14","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Na płaszczyźnie dany jest punkt A = (8, 4) . Prosta AB jest nachylona do osi OX pod kątem = 60^ . Wyznacz współrzędne punktu B , wiedząc, że |AB| = 22 . [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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/15","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"15","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Dany jest trapez ABCD . Punkt E jest punktem przecięcia się przekątnych trapezu. Ramiona trapezu przedłużono do przecięcia w punkcie F . Wykaż, że prosta EF dzieli dłuższą podstawę AB trapezu na połowy.","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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/16","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"16","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"W urnie jest 5 kul białych i 7 czarnych. Wyjmujemy losowo z tej urny dwie kule i odkładamy na bok. Następnie wyjmujemy z tej urny jedną kulę. Oblicz prawdopodobieństwo, że będzie to kula biała. [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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/17","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"17","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Dany jest trójmian kwadratowy f(x) = (m + 1)x^2 - (2m - 2)x - 2(m - 1) . Oblicz, dla jakich wartości parametru m suma odwrotności sześcianów dwóch różnych pierwiastków tego trójmianu jest mniejsza od 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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2016-operon-probna-rozszerzona/zad/18","paper_id":"matematyka-2016-operon-probna-rozszerzona","number":"18","points":7,"ptype":"open","subject":"matematyka","category":"probna","year":2016,"month":"listopad","level":"rozszerzona","text":"Punkt P o dodatnich współrzędnych należy do wykresu funkcji określonej wzorem f(x) = 2 x . Wyznacz odciętą punktu P tak, aby jego odległość od prostej o równaniu y = - 4 3 x - 2 była najmniejsza. Oblicz tę najmniejszą odległość. [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 2016 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"}]}