{"paper":{"id":"site-4427","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":16,"source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},"questions":[{"id":"site-4427/zad/4441","paper_id":"site-4427","number":"4441","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Na rysunku przedstawiony jest zbiór wszystkich liczb rzeczywistych spełniających nierówność |2x-8| <= 10. Stąd wynika, że A. k=2, B. k=4, C. k=5, D. k=9.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4441.svg","solution_image":"img/site/4427/rozw-4441.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/4442","paper_id":"site-4427","number":"4442","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Dana jest funkcja f określona wzorem f(x) = { x-2 dla x <= 0; ||x+3|-4| dla x>0 }. Równanie f(x) = 1 ma dokładnie A. jedno rozwiązanie, B. dwa rozwiązania, C. cztery rozwiązania, D. pięć rozwiązań.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4442.svg","solution_image":"img/site/4427/rozw-4442.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/4443","paper_id":"site-4427","number":"4443","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Liczba (3 - 2 * pierwiastek z 3)^3 jest równa A. 27 - 24 * pierwiastek z 3, B. 27 - 30 * pierwiastek z 3, C. 135 - 78 * pierwiastek z 3, D. 135 - 30 * pierwiastek z 3.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4443.svg","solution_image":"img/site/4427/rozw-4443.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/4444","paper_id":"site-4427","number":"4444","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Równanie 2 * sin x + 3 * cos x = 6 w przedziale (0, 2*pi). A. nie ma rozwiązań rzeczywistych. B. ma dokładnie jedno rozwiązanie rzeczywiste. C. ma dokładnie dwa rozwiązania rzeczywiste. D. ma więcej niż dwa rozwiązania rzeczywiste.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4444.svg","solution_image":"img/site/4427/rozw-4444.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/4445","paper_id":"site-4427","number":"4445","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Odległość początku układu współrzędnych od prostej o równaniu y = 2x+4 jest równa A. pierwiastek z 5 / 5, B. 4 * pierwiastek z 5 / 5, C. 4/5, D. 4.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4445.svg","solution_image":"img/site/4427/rozw-4445.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/4446","paper_id":"site-4427","number":"4446","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Oblicz granicę lim {n -> infty} ( {11n^3 + 6n + 5 / 6n^3 + 1} - {2n^2 + 2n + 1 / 5n^2 - 4} ) W poniższe kratki wpisz kolejno cyfrę jedności i pierwsze dwie cyfry po przecinku rozwinięcia dziesiętnego otrzymanego wyniku.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4446.svg","solution_image":"img/site/4427/rozw-4446.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/4447","paper_id":"site-4427","number":"4447","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Liczby (-1) i 3 są miejscami zerowymi funkcji kwadratowej f. Oblicz f(6) / f(12).","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4447.svg","solution_image":"img/site/4427/rozw-4447.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/4449","paper_id":"site-4427","number":"4449","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Dwusieczne czworokąta ABCD wpisanego w okrąg przecinają się w czterech różnych punktach: P, Q, R, S (zobacz rysunek). Wykaż, że na czworokącie PQRS można opisać okrąg.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4449.svg","solution_image":"img/site/4427/rozw-4449.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/4450","paper_id":"site-4427","number":"4450","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Długości boków czworokąta ABCD są równe: |AB|=2, |BC|=3, |CD|=4, |DA|=5. Na czworokącie ABCD opisano okrąg. Oblicz długość przekątnej AC tego czworokąta.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4450.svg","solution_image":"img/site/4427/rozw-4450.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/4451","paper_id":"site-4427","number":"4451","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"W pierwszej urnie umieszczono 3 kule białe i 5 kul czarnych, a w drugiej urnie 7 kul białych i 2 kule czarne. Losujemy jedną kulę z pierwszej urny, przekładamy ją do urny drugiej i dodatkowo dokładamy do urny drugiej jeszcze dwie kule tego samego koloru, co wylosowana kula. Następnie losujemy dwie kule z urny drugiej. Oblicz prawdopodobieństwo zdarzenia polegającego na tym, że obie kule wylosowane z drugiej urny będą białe.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4451.svg","solution_image":"img/site/4427/rozw-4451.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/4473","paper_id":"site-4427","number":"4473","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Funkcja f określona jest wzorem f(x) = x^3 - 2x^2 + 1 dla każdej liczby rzeczywistej x. Wyznacz równania tych stycznych do wykresu funkcji f, które są równoległe do prostej o równaniu y = 4x.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4473.svg","solution_image":"img/site/4427/rozw-4473.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/4474","paper_id":"site-4427","number":"4474","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Dany jest trójmian kwadratowy f(x) = (m+1)x^2 + 2(m-2)x - m+4. Wyznacz wszystkie wartości parametru m, dla których trójmian f ma dwa różne pierwiastki rzeczywiste x_1, x_2 spełniające warunek x_1^2 - x_2^2 = x_1^4 - x_2^4.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4474.svg","solution_image":"img/site/4427/rozw-4474.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/4480","paper_id":"site-4427","number":"4480","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Podstawą ostrosłupa ABCDS jest kwadrat ABCD. Krawędź boczna SD jest wysokością ostrosłupa, a jej długość jest dwa razy większa od długości krawędzi podstawy. Oblicz sinus kąta między ścianami bocznymi ABS i CBS tego ostrosłupa.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4480.svg","solution_image":"img/site/4427/rozw-4480.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/4481","paper_id":"site-4427","number":"4481","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Suma wszystkich czterech współczynników wielomianu W(x) = x^3+ax^2+bx+c jest równa 0. Trzy pierwiastki tego wielomianu tworzą ciąg arytmetyczny o różnicy równej $3$. Oblicz współczynniki a, b i c. Rozważ wszystkie możliwe przypadki.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4481.svg","solution_image":"img/site/4427/rozw-4481.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/4500","paper_id":"site-4427","number":"4500","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Rozpatrujemy wszystkie stożki, których przekrojem osiowym jest trójkąt o obwodzie 20.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-4500.svg","solution_image":"img/site/4427/rozw-4500.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4427/zad/7685","paper_id":"site-4427","number":"7685","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2015,"month":null,"level":"rozszerzona","text":"Udowodnij, że dla każdej liczby rzeczywistej x prawdziwa jest nierówność x^4 - x^2 - 2x + 3 > 0","answer":null,"answer_text":null,"solution":null,"image":"img/site/4427/zad-7685.svg","solution_image":"img/site/4427/rozw-7685.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań · 2015 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"}]}