{"paper":{"id":"matematyka-2009-operon-probna-rozszerzona","subject":"matematyka","category":"probna","year":2009,"month":"listopad","level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":11,"source_label":"Matematyka · Matura próbna · listopad 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},"questions":[{"id":"matematyka-2009-operon-probna-rozszerzona/zad/1","paper_id":"matematyka-2009-operon-probna-rozszerzona","number":"1","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2009,"month":"listopad","level":"rozszerzona","text":"Rozwiąż układ równań: cases 3 = |x| + |y| 3 = 2|x| + y cases . [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"uklady_rownan","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2009-operon-probna-rozszerzona/zad/2","paper_id":"matematyka-2009-operon-probna-rozszerzona","number":"2","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2009,"month":"listopad","level":"rozszerzona","text":"Określ, jaką liczbą - dodatnią czy ujemną, jest x + x , wiedząc, że x ( 2 , ) i (1 + x) ( 1 x - tg ,x ) + 1 3 = 0 . [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"trygonometria","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2009-operon-probna-rozszerzona/zad/3","paper_id":"matematyka-2009-operon-probna-rozszerzona","number":"3","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2009,"month":"listopad","level":"rozszerzona","text":"Obrazem odcinka AB , gdzie A = (1, 0) i B = (2, 1) w jednokładności o skali k > 1 i środku P jest odcinek CD , gdzie C = (4, 0) , D = (6, 2) . równanie okręgu o środku w punkcie P i promieniu |AB| . [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"analityczna","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2009-operon-probna-rozszerzona/zad/4","paper_id":"matematyka-2009-operon-probna-rozszerzona","number":"4","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2009,"month":"listopad","level":"rozszerzona","text":"Wykaż, że dla a > 1 i x > 1 zachodzi nierówność _a x + _x a 100 .","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"logarytmy","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2009-operon-probna-rozszerzona/zad/5","paper_id":"matematyka-2009-operon-probna-rozszerzona","number":"5","points":3,"ptype":"open","subject":"matematyka","category":"probna","year":2009,"month":"listopad","level":"rozszerzona","text":"Spiralę tworzymy następująco: kreślimy półokrąg o średnicy |AB| = 2r i środku O , do tego półokręgu dorysowujemy półokrąg o średnicy OB i środku C . Następnie kreślimy półokrąg o średnicy OC i środku D itd. Oblicz długość spirali złożonej z dziesięciu tak otrzymanych półokręgów. [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":"img/zz/matR_Operon_2009_listopad_zad5_rys1.png","solution_image":null,"topics":"ciagi","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2009-operon-probna-rozszerzona/zad/6","paper_id":"matematyka-2009-operon-probna-rozszerzona","number":"6","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2009,"month":"listopad","level":"rozszerzona","text":"Wykaż, że suma odwrotności pierwiastków wielomianu W(x) = x^4 + x^3 - 4x^2 - 2x + 4 jest liczbą wymierną.","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"wielomiany","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2009-operon-probna-rozszerzona/zad/7","paper_id":"matematyka-2009-operon-probna-rozszerzona","number":"7","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2009,"month":"listopad","level":"rozszerzona","text":"Punkty równoodległe od prostej o równaniu y = - 1 2 i punktu P = (0, 1 2 ) należą do wykresu funkcji f . ź wzór tej funkcji. [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"analityczna","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2009-operon-probna-rozszerzona/zad/8","paper_id":"matematyka-2009-operon-probna-rozszerzona","number":"8","points":4,"ptype":"open","subject":"matematyka","category":"probna","year":2009,"month":"listopad","level":"rozszerzona","text":"Boki trójkąta ABC są równe a, b, c . Oblicz długość środkowej poprowadzonej z wierzchołka A do boku a . [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"planimetria","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2009-operon-probna-rozszerzona/zad/9","paper_id":"matematyka-2009-operon-probna-rozszerzona","number":"9","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2009,"month":"listopad","level":"rozszerzona","text":"Liczby naturalne parzyste od 2 do 100 zapisujemy kolejno jedna za drugą, tworząc liczbę naturalną a . liczba a jest kwadratem pewnej liczby naturalnej? Wskazówka: zbadaj podzielność sumy cyfr. [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"liczby_naturalne","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2009-operon-probna-rozszerzona/zad/10","paper_id":"matematyka-2009-operon-probna-rozszerzona","number":"10","points":5,"ptype":"open","subject":"matematyka","category":"probna","year":2009,"month":"listopad","level":"rozszerzona","text":"Określ, dla jakich wartości parametru k równanie x^2 + (k + 1)x + 0 , 5(k + 5) = 0 ma dwa różne pierwiastki dodatnie. [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"parametr","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"},{"id":"matematyka-2009-operon-probna-rozszerzona/zad/11","paper_id":"matematyka-2009-operon-probna-rozszerzona","number":"11","points":6,"ptype":"open","subject":"matematyka","category":"probna","year":2009,"month":"listopad","level":"rozszerzona","text":"Trapez o ramionach długości 6 i 10 jest opisany na okręgu. Odcinek łączący środki ramion trapezu dzieli trapez na dwie części, których pola pozostają w stosunku 3 : 5 . Oblicz długości podstaw trapezu. [odp_otwarta1]","answer":null,"answer_text":null,"solution":null,"image":null,"solution_image":null,"topics":"planimetria","page_from":null,"source":"zadaniazmatur","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"zadaniazmatur","source_label":"Matematyka · Matura próbna · listopad 2009 (rozszerzona)","subject_label":"Matematyka","category_label":"Matura próbna"}]}