{"paper":{"id":"site-1759","subject":"matematyka","category":"zbiory","year":2003,"month":null,"level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":11,"source_label":"Matematyka · Zbiory zadań · 2003 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},"questions":[{"id":"site-1759/zad/813","paper_id":"site-1759","number":"813","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2003,"month":null,"level":"rozszerzona","text":"Wyznacz najmniejszą i największą wartość funkcji, określonej wzorem f(x)=(x-1)/(5-x), w przedziale <0,7>.","answer":null,"answer_text":null,"solution":null,"image":"img/site/1759/zad-813.svg","solution_image":"img/site/1759/rozw-813.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 · 2003 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-1759/zad/815","paper_id":"site-1759","number":"815","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2003,"month":null,"level":"rozszerzona","text":"Dane jest równanie postaci a^2 * x - 1 = x+a, w którym niewiadomą jest x. Zbadaj liczbę rozwiązań tego równania, w zależności od parametru a.","answer":null,"answer_text":null,"solution":null,"image":"img/site/1759/zad-815.svg","solution_image":"img/site/1759/rozw-815.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 · 2003 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-1759/zad/817","paper_id":"site-1759","number":"817","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2003,"month":null,"level":"rozszerzona","text":"Wyznacz te wartości parametru a oraz b, przy których funkcja określona wzorem jest ciągła w punkcie x=2","answer":null,"answer_text":null,"solution":null,"image":"img/site/1759/zad-817.svg","solution_image":"img/site/1759/rozw-817.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 · 2003 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-1759/zad/818","paper_id":"site-1759","number":"818","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2003,"month":null,"level":"rozszerzona","text":"Suma n początkowych, kolejnych wyrazów ciągu (a_n), jest obliczana według wzoru S_n=n^2+3n. Wyznacz a_n. Wykaż, że ciąg (a_n) jest ciągiem arytmetycznym.","answer":null,"answer_text":null,"solution":null,"image":"img/site/1759/zad-818.svg","solution_image":"img/site/1759/rozw-818.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 · 2003 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-1759/zad/819","paper_id":"site-1759","number":"819","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2003,"month":null,"level":"rozszerzona","text":"Dziesiąty wyraz pewnego ciągu geometrycznego równa się 10. Oblicz iloczyn dziewiętnastu początkowych, kolejnych wyrazów tego ciągu.","answer":null,"answer_text":null,"solution":null,"image":"img/site/1759/zad-819.svg","solution_image":"img/site/1759/rozw-819.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 · 2003 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-1759/zad/820","paper_id":"site-1759","number":"820","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2003,"month":null,"level":"rozszerzona","text":"Rzucamy pięć razy symetryczną kostką sześcienną. Oblicz prawdopodobieństwo zdarzenia, polegającego na tym, że ,,jedynka'' wypadnie co najmniej cztery razy.","answer":null,"answer_text":null,"solution":null,"image":"img/site/1759/zad-820.svg","solution_image":"img/site/1759/rozw-820.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 · 2003 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-1759/zad/821","paper_id":"site-1759","number":"821","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2003,"month":null,"level":"rozszerzona","text":"W układzie współrzędnych są dane punkty: A(-9,-2) oraz B(4,2). Wyznacz współrzędne punktu C, leżącego na osi OY, tak że kąt ACB jest kątem prostym.","answer":null,"answer_text":null,"solution":null,"image":"img/site/1759/zad-821.svg","solution_image":"img/site/1759/rozw-821.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 · 2003 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-1759/zad/823","paper_id":"site-1759","number":"823","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2003,"month":null,"level":"rozszerzona","text":"Wybierz dwie dowolne przekątne sześcianu i oblicz cosinus kąta między nimi. Sporządź odpowiedni rysunek i zaznacz na nim kąt, którego cosinus obliczasz.","answer":null,"answer_text":null,"solution":null,"image":"img/site/1759/zad-823.svg","solution_image":"img/site/1759/rozw-823.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 · 2003 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-1759/zad/827","paper_id":"site-1759","number":"827","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2003,"month":null,"level":"rozszerzona","text":"Trapez równoramienny, o obwodzie równym 20 cm, jest opisany na okręgu. Wiedząc, że przekątna trapezu ma długość 41 cm, oblicz pole tego trapezu.","answer":null,"answer_text":null,"solution":null,"image":"img/site/1759/zad-827.svg","solution_image":"img/site/1759/rozw-827.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 · 2003 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-1759/zad/829","paper_id":"site-1759","number":"829","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2003,"month":null,"level":"rozszerzona","text":"Funkcja h jest określona wzorem h(x)=\\log_2(x^2-4)-\\log_2(x-5). Wyznacz wszystkie wartości parametru k, dla których równanie h(x)-\\log_2 k=0 ma dwa różne pierwiastki.","answer":null,"answer_text":null,"solution":null,"image":"img/site/1759/zad-829.svg","solution_image":"img/site/1759/rozw-829.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 · 2003 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-1759/zad/835","paper_id":"site-1759","number":"835","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2003,"month":null,"level":"rozszerzona","text":"Na kuli o promieniu R=4 cm opisujemy stożki o promieniu r i wysokości H. Spośród wszystkich takich stożków wyznacz ten, który ma najmniejszą objętość. Oblicz tę objętość. Oblicz promień i wysokość znalezionego stożka.","answer":null,"answer_text":null,"solution":null,"image":"img/site/1759/zad-835.svg","solution_image":"img/site/1759/rozw-835.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 · 2003 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"}]}