{"paper":{"id":"site-4077","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":11,"source_label":"Matematyka · Zbiory zadań · 2014 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},"questions":[{"id":"site-4077/zad/4078","paper_id":"site-4077","number":"4078","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Rozwiąż nierówność z dwiema wartościami bezwzględnymi |x+6| - 2|x-4| <= 2x-3.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4077/zad-4078.svg","solution_image":"img/site/4077/rozw-4078.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 · 2014 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4077/zad/4079","paper_id":"site-4077","number":"4079","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"W czworokąt ABCD, w którym |AD| = 5*pierwiastków z 3 i |CD| = 6, można wpisać okrąg. Przekątna BD tworzy z bokiem AB czworokąta kąt o mierze 60, natomiast z bokiem AD tworzy kąt, którego sinus jest równy 3/4. Wyznacz długości boków AB i BC oraz długość przekątnej BD tego czworokąta.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4077/zad-4079.svg","solution_image":"img/site/4077/rozw-4079.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 · 2014 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4077/zad/4080","paper_id":"site-4077","number":"4080","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Udowodnij, że dla każdej liczby rzeczywistej x i każdej liczby rzeczywistej y prawdziwa jest nierówność x(x-1) + y(y-1) >= xy - 1.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4077/zad-4080.svg","solution_image":"img/site/4077/rozw-4080.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 · 2014 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4077/zad/4081","paper_id":"site-4077","number":"4081","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Rozwiąż nierówność -2*sin3x >= 1 w przedziale <0,2pi>.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4077/zad-4081.svg","solution_image":"img/site/4077/rozw-4081.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 · 2014 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4077/zad/4087","paper_id":"site-4077","number":"4087","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Na przyprostokątnych AC i BC trójkąta prostokątnego ABC zbudowano, na zewnątrz trójkąta, kwadraty ACDE i BFGC. Odcinek AF przecina przyprostokątną BC w punkcie L, a odcinek BE przecina przyprostokątną AC w punkcie K (zobacz rysunek). Udowodnij, że |KC|=|LC|.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4077/zad-4087.svg","solution_image":"img/site/4077/rozw-4087.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 · 2014 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4077/zad/4123","paper_id":"site-4077","number":"4123","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Wyznacz wszystkie wartości parametru m, dla których równanie x^2+(2m-5)x+2m+3=0 ma dwa różne pierwiastki rzeczywiste x_1, x_2 takie, że (x_1+x_2)^2 >= x_1^2*x_2^2 >= x_1^2+x_2^2.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4077/zad-4123.svg","solution_image":"img/site/4077/rozw-4123.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 · 2014 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4077/zad/4124","paper_id":"site-4077","number":"4124","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Odcinek AB o długości 4 jest zawarty w prostej o równaniu y=3/4x-3/2. Symetralna odcinka AB przecina oś Oy w punkcie P=(0,6). Oblicz współrzędne końców odcinka AB.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4077/zad-4124.svg","solution_image":"img/site/4077/rozw-4124.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 · 2014 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4077/zad/4125","paper_id":"site-4077","number":"4125","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Trzy liczby są kolejnymi wyrazami ciągu geometrycznego, którego iloraz jest różny od 1. Jeżeli weźmiemy kolejno drugą z nich, pierwszą i trzecią, to otrzymamy trzy kolejne wyrazy ciągu arytmetycznego. Jeżeli pierwszy wyraz tego ciągu arytmetycznego zmniejszymy o 7, drugi pozostawimy bez zmian, a trzeci zwiększymy o 3, to otrzymamy trzy kolejne wyrazy ciągu geometrycznego. Oblicz te liczby.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4077/zad-4125.svg","solution_image":"img/site/4077/rozw-4125.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 · 2014 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4077/zad/4126","paper_id":"site-4077","number":"4126","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Reszta z dzielenia wielomianu W(x) = 6x^3+(m+4)x^2-2x-1 przez dwumian x-m jest równa 8. Oblicz wartość m oraz pierwiastki tego wielomianu.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4077/zad-4126.svg","solution_image":"img/site/4077/rozw-4126.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 · 2014 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4077/zad/4127","paper_id":"site-4077","number":"4127","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Podstawą ostrosłupa jest kwadrat ABCD o boku długości 25. Ściany boczne ABS i BCS mają takie same pola, każde równe 250. Ściany boczne ADS i CDS też mają jednakowe pola, każde równe 187,5. Krawędzie boczne AS i CS mają równe długości. Oblicz objętość tego ostrosłupa.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4077/zad-4127.svg","solution_image":"img/site/4077/rozw-4127.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 · 2014 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-4077/zad/4128","paper_id":"site-4077","number":"4128","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"W urnie jest dziesięć kul: 4 białe, 3 czarne, 2 zielone i 1 niebieska. Losujemy jednocześnie trzy kule z urny. Oblicz prawdopodobieństwo zdarzenia polegającego na tym, że wśród wylosowanych kul nie ma kul w tym samym kolorze. Wynik przedstaw w postaci ułamka zwykłego nieskracalnego.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4077/zad-4128.svg","solution_image":"img/site/4077/rozw-4128.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 · 2014 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"}]}