{"paper":{"id":"site-3917","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":12,"source_label":"Matematyka · Zbiory zadań · 2013 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},"questions":[{"id":"site-3917/zad/3918","paper_id":"site-3917","number":"3918","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Rozwiąż nierówność: pierwiastek z (x^2+4x+4) \\xge 11 - pierwiastek z (x^2-6x+9)","answer":null,"answer_text":null,"solution":null,"image":"img/site/3917/zad-3918.svg","solution_image":"img/site/3917/rozw-3918.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 · 2013 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3917/zad/3919","paper_id":"site-3917","number":"3919","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Wyznacz wszystkie wartości parametru m, dla których równanie (m+1)x^2 - 3mx + m + 1 = 0 ma dwa różne pierwiastki takie, że ich suma jest nie większa niż 2,5.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3917/zad-3919.svg","solution_image":"img/site/3917/rozw-3919.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 · 2013 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3917/zad/3920","paper_id":"site-3917","number":"3920","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Rozwiąż równanie 2 * tg x * cos x + 1 = 2 * cos x + tg x w przedziale <0, 2pi>.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3917/zad-3920.svg","solution_image":"img/site/3917/rozw-3920.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 · 2013 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3917/zad/3921","paper_id":"site-3917","number":"3921","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Wykaż, że prawdziwa jest równość pierwiastek 3 stopnia z (9 + pierwiastek z 80) + pierwiastek 3 z (9 - pierwiastek z 80) = 3.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3917/zad-3921.svg","solution_image":"img/site/3917/rozw-3921.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 · 2013 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3917/zad/3922","paper_id":"site-3917","number":"3922","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Uzasadnij, że jeżeli 2a + b >= 0, to 2a^3 + b^3 >= 3a^2b.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3917/zad-3922.svg","solution_image":"img/site/3917/rozw-3922.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 · 2013 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3917/zad/3923","paper_id":"site-3917","number":"3923","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"W równoległoboku ABCD miara kąta ostrego jest równa 30 stopni, a odległości punktu przecięcia się przekątnych od sąsiednich boków równoległoboku są równe 2 i pierwiastek z 3. Oblicz długość krótszej przekątnej tego równoległoboku.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3917/zad-3923.svg","solution_image":"img/site/3917/rozw-3923.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 · 2013 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3917/zad/3924","paper_id":"site-3917","number":"3924","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Punkty A=(2,0) i B=(4,2) leżą na okręgu o równaniu (x-1)^2 + (y-3)^2 = 10. Wyznacz na tym okręgu taki punkt C, aby trójkąt ABC był trójkątem równoramiennym o podstawie AB.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3917/zad-3924.svg","solution_image":"img/site/3917/rozw-3924.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 · 2013 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3917/zad/3925","paper_id":"site-3917","number":"3925","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Wykaż, że dla dowolnego kąta α prawdziwa jest tożsamość sin^4 α + cos^4 α = 1 + cos^2 2α / 2.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3917/zad-3925.svg","solution_image":"img/site/3917/rozw-3925.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 · 2013 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3917/zad/3926","paper_id":"site-3917","number":"3926","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Podstawą ostrosłupa prawidłowego trójkątnego ABCS jest trójkąt ABC. Kąt nachylenia krawędzi bocznej AS do płaszczyzny podstawy ostrosłupa jest równy kątowi między krawędziami bocznymi AS i BS zawartymi w ścianie bocznej ASB tego ostrosłupa (zob. rysunek). Oblicz kosinus tego kąta.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3917/zad-3926.svg","solution_image":"img/site/3917/rozw-3926.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 · 2013 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3917/zad/3927","paper_id":"site-3917","number":"3927","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Liczby a_1, a_2, ... , a_n są dodatnie i w podanej kolejności tworzą ciąg geometryczny. Uzasadnij, że prawdziwa jest równość pierwiastek n stopnia z {a_1 * a_2 * ... * a_n} = pierwiastek z {a_1\\c a_n}.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3917/zad-3927.svg","solution_image":"img/site/3917/rozw-3927.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 · 2013 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3917/zad/3928","paper_id":"site-3917","number":"3928","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Suma długości dwóch boków trójkąta równa się 4, a kąt między tymi bokami ma miarę 120 stopni. Oblicz najmniejszą wartość sumy kwadratów długości wszystkich boków tego trójkąta.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3917/zad-3928.svg","solution_image":"img/site/3917/rozw-3928.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 · 2013 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3917/zad/3929","paper_id":"site-3917","number":"3929","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Pierwiastkami wielomianu stopnia trzeciego są liczby 1, 3, 5. Współczynnik przy najwyższej potędze zmiennej tego wielomianu jest równy 1/2. Uzasadnij, że dla każdej liczby całkowitej nieparzystej wartość tego wielomianu jest liczbą podzielną przez 24.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3917/zad-3929.svg","solution_image":"img/site/3917/rozw-3929.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 · 2013 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"}]}