{"paper":{"id":"site-2038","subject":"matematyka","category":"zbiory","year":2009,"month":null,"level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":11,"source_label":"Matematyka · Zbiory zadań · 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},"questions":[{"id":"site-2038/zad/2034","paper_id":"site-2038","number":"2034","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2009,"month":null,"level":"rozszerzona","text":"Rozwiązanie zadania z matury próbnej 2009: Na rysunku narysowano fragment wykresu funkcji f(x)=2^{x-3}-b określonej dla x w zbiorze liczb rzeczywistych. a) Podaj wartość b. b) Naszkicuj wykres funkcji g(x)=|f(x)|. c) Podaj wszystkie wartości parametru p, dla których równanie g(x)=p ma dokładnie jedno rozwiązanie.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2038/zad-2034.svg","solution_image":"img/site/2038/rozw-2034.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 · 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2038/zad/2035","paper_id":"site-2038","number":"2035","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2009,"month":null,"level":"rozszerzona","text":"Rozwiązanie zadania z matury próbne 2009: Rozwiąż nierówność |x+3|+|3x+9|<|x+5|.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2038/zad-2035.svg","solution_image":"img/site/2038/rozw-2035.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 · 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2038/zad/2036","paper_id":"site-2038","number":"2036","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2009,"month":null,"level":"rozszerzona","text":"Rozwiązanie zadania z matury próbne 2009: Jeden z końców odcinka leży na paraboli o równaniu y=x^2, a drugi na prostej o równaniu y=2x-6. Wykaż, że długość tego odcinka jest nie mniejsza od pierwiastek z 5. Sporządź odpowiedni rysunek.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2038/zad-2036.svg","solution_image":"img/site/2038/rozw-2036.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 · 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2038/zad/2037","paper_id":"site-2038","number":"2037","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2009,"month":null,"level":"rozszerzona","text":"Rozwiązanie zadania z matury próbne 2009: Oblicz prawdopodobieństwo P(A'nB')$, jeśli P(A')={1/3}, P(B')={1/4} i P(AnB)={1/2}.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2038/zad-2037.svg","solution_image":"img/site/2038/rozw-2037.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 · 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2038/zad/2039","paper_id":"site-2038","number":"2039","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2009,"month":null,"level":"rozszerzona","text":"Na rysunku przedstawiono fragment wykresu funkcji h otrzymanego przez przesunięcie o wektor [2,1] wykresu funkcji f określonej wzorem f(x) = a/x dla x w zbiorze liczb rzeczywistych i x różnego od 0.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2038/zad-2039.svg","solution_image":"img/site/2038/rozw-2039.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 · 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2038/zad/2040","paper_id":"site-2038","number":"2040","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2009,"month":null,"level":"rozszerzona","text":"Porównaj liczby a^b oraz b^a, gdzie a=[(2-\\sqrt3)^{1\\o2}+(2+\\sqrt3)^{1\\o2}]^2, b={81^{-1}\\c\\prw3\\o27^{-2}\\c\\prw{4}{9}}","answer":null,"answer_text":null,"solution":null,"image":"img/site/2038/zad-2040.svg","solution_image":"img/site/2038/rozw-2040.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 · 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2038/zad/2041","paper_id":"site-2038","number":"2041","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2009,"month":null,"level":"rozszerzona","text":"Rozwiązanie zadanie z matury próbnej 2009: Dane jest równanie (x+3)[x^2+(p+4)x+(p+1)^2]=0$ z niewiadomą x. a) Rozwiąż to równanie dla p=1. b) Wyznacz wszystkie wartości parametru p, dla których równanie ma tylko jedno rozwiązanie.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2038/zad-2041.svg","solution_image":"img/site/2038/rozw-2041.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 · 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2038/zad/2042","paper_id":"site-2038","number":"2042","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2009,"month":null,"level":"rozszerzona","text":"Rozwiązanie zadania z matury próbnej 2009: Trapez równoramienny jest opisany na okręgu. Suma długości krótszej podstawy i ramienia trapezu jest równa 30. Wyraź pole tego trapezu jako funkcję długości jego ramiena. Wyznacz dziedzinę tej funkcji.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2038/zad-2042.svg","solution_image":"img/site/2038/rozw-2042.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 · 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2038/zad/2043","paper_id":"site-2038","number":"2043","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2009,"month":null,"level":"rozszerzona","text":"Rozwiązanie zadania z matury próbne 2009: Środek okręgu przechodzącego przez punkty A=(1,4) i B=(-6,3) leży na osi Ox. a) Wyznacz równanie tego okręgu. b) Wyznacz równanie prostej prostopadłej do prostej AB i oddalonej od początku układu współrzędnych o pierwiastek z 2.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2038/zad-2043.svg","solution_image":"img/site/2038/rozw-2043.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 · 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2038/zad/2044","paper_id":"site-2038","number":"2044","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2009,"month":null,"level":"rozszerzona","text":"Sinusy kątów ostrych trójkąta prostokątnego oraz liczba 1 tworzą ciąg geometryczny. Oblicz sinus najmniejszego kąta tego trójkąta.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2038/zad-2044.svg","solution_image":"img/site/2038/rozw-2044.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 · 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2038/zad/2045","paper_id":"site-2038","number":"2045","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2009,"month":null,"level":"rozszerzona","text":"Rozwiązanie zadania z matury próbnej 2009: Dany jest ostrosłup prawidłowy czworokątny, w którym wszystkie krawędzie mają równą długość. Zaznacz na rysunku kąt utworzony przez dwie sąsiednie ściany boczne tego ostrosłupa i oblicz kosinus tego kąta.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2038/zad-2045.svg","solution_image":"img/site/2038/rozw-2045.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 · 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"}]}