{"paper":{"id":"site-4005","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":11,"source_label":"Matematyka · Zbiory zadań · 2013 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},"questions":[{"id":"site-4005/zad/4006","paper_id":"site-4005","number":"4006","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Liczba 2 jest jednym z pierwiastków wielomianu W(x)=x^4+x^3+ax^2+x+b. Reszta z dzielenia wielomianu W(x) przez dwumian x+1 jest równa -12. Oblicz a i b oraz wyznacz pozostałe pierwiastki rzeczywiste wielomianu W(x).","answer":null,"answer_text":null,"solution":null,"image":"img/site/4005/zad-4006.svg","solution_image":"img/site/4005/rozw-4006.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-4005/zad/4007","paper_id":"site-4005","number":"4007","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Rozwiąż równanie cos x + cos2x = 0 dla x w <-pi,pi>.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4005/zad-4007.svg","solution_image":"img/site/4005/rozw-4007.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-4005/zad/4008","paper_id":"site-4005","number":"4008","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Ciąg arytmetyczny (a_n), określony dla n>=1 jest malejący i a_1=6. Siódmy, dziesiąty i szesnasty wyraz tego ciągu w podanej kolejności są równe trzem początkowym wyrazom ciągu geometrycznego (b_n), określonego dla n >= 1. Oblicz sumę dziesięciu początkowych wyrazów ciągu (b_n).","answer":null,"answer_text":null,"solution":null,"image":"img/site/4005/zad-4008.svg","solution_image":"img/site/4005/rozw-4008.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-4005/zad/4009","paper_id":"site-4005","number":"4009","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Wyznacz wszystkie wartości parametru m, dla których równanie x^2 - (m-4)x + m^2-7m+12 = 0 ma dwa różne pierwiastki x_1, x_2 spełniające warunek (x_1+x_2)(x_1+x_2-1)=x_1^2+x_2^2.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4005/zad-4009.svg","solution_image":"img/site/4005/rozw-4009.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-4005/zad/4010","paper_id":"site-4005","number":"4010","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Dane są dwa okręgi o promieniach R i r, styczne zewnętrznie i styczne do tej samej prostej l w punktach A i B (tak jak na rysunku). Udowodnij, że |AB| = 2*pierwiastek(R*r).","answer":null,"answer_text":null,"solution":null,"image":"img/site/4005/zad-4010.svg","solution_image":"img/site/4005/rozw-4010.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-4005/zad/4011","paper_id":"site-4005","number":"4011","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Ze zbioru {1, 2, 3, ..., 341} losujemy dwie różne liczby. Oblicz prawdopodobieństwo zdarzenia polegającego na tym, że iloczyn wylosowanych liczb będzie podzielny przez 11 i 31.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4005/zad-4011.svg","solution_image":"img/site/4005/rozw-4011.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-4005/zad/4012","paper_id":"site-4005","number":"4012","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Punkty A=(1,3) i D=(2,1) są wierzchołkami trapezu ABCD o podstawach AB i CD. Wierzchołki B i C leżą na osi Ox, a ramię BC tego trapezu ma długość 2. Wyznacz równania prostych AB i CD.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4005/zad-4012.svg","solution_image":"img/site/4005/rozw-4012.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-4005/zad/4013","paper_id":"site-4005","number":"4013","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Wykres funkcji logarytmicznej f(x) = log_p x, którego fragment przedstawiony jest na rysunku, przechodzi przez punkt A = (9/4, 2). a) Oblicz wartość p. b) Narysuj wykres funkcji g określonej wzorem g(x) = |f(x+2)|-3. c) Podaj wszystkie wartości parametru m, dla których równanie g(x)=m ma dwa rozwiązania ujemne.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4005/zad-4013.svg","solution_image":"img/site/4005/rozw-4013.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-4005/zad/4014","paper_id":"site-4005","number":"4014","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Oblicz, ile jest liczb pięciocyfrowych, w zapisie których występuje dokładnie jedna cyfra 9 oraz dokładnie dwie cyfry 0.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4005/zad-4014.svg","solution_image":"img/site/4005/rozw-4014.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-4005/zad/4015","paper_id":"site-4005","number":"4015","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Dany jest ostrosłup prawidłowy trójkątny ABCS o podstawie ABC. Krawędź podstawy tego ostrosłupa ma długość a. Punkt P jest środkiem wysokości SO ostrosłupa. Odcinki AP i BP są prostopadłe. Oblicz pole powierzchni całkowitej tego ostrosłupa.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4005/zad-4015.svg","solution_image":"img/site/4005/rozw-4015.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-4005/zad/4016","paper_id":"site-4005","number":"4016","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2013,"month":null,"level":"rozszerzona","text":"Udowodnij, że jeżeli ciąg (a_n) określony dla n>=1 jest geometryczny, to dla dowolnych liczb całkowitych dodatnich n takich, że n>=11 prawdziwa jest równość (a_{n+1})^2 = a_{n-10} * a_{n+12}.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4005/zad-4016.svg","solution_image":"img/site/4005/rozw-4016.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"}]}