{"paper":{"id":"site-3064","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":5,"source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},"questions":[{"id":"site-3064/zad/6697","paper_id":"site-3064","number":"6697","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Dane są dwie liczby x i y, takie, że iloraz x/y jest równy 1+√5/2. Oblicz wartość wyrażenia x+y/x. Wynik podaj bez niewymierności w mianowniku.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3064/zad-6697.svg","solution_image":"img/site/3064/rozw-6697.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 (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3064/zad/6698","paper_id":"site-3064","number":"6698","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Dane są liczby a=√5-2 oraz b=√5+2. Oblicz wartość wyrażenia a⋅b/√a+√b : √a-√b/a-b dla podanych a i b.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3064/zad-6698.svg","solution_image":"img/site/3064/rozw-6698.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 (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3064/zad/6857","paper_id":"site-3064","number":"6857","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Liczbę a=(√2+√7)² można zapisać w postaci a=x+y√14, gdzie x∈Z oraz y∈Z. Uzupełnij poniższe równości. Wpisz właściwe liczby w wykropkowanych miejscach. x = .................. y = ..................","answer":null,"answer_text":null,"solution":null,"image":"img/site/3064/zad-6857.svg","solution_image":"img/site/3064/rozw-6857.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 (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3064/zad/6861","paper_id":"site-3064","number":"6861","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Okres T drgań wahadła w pewnym zegarze dany jest wzorem: T = 2π√l/g gdzie l oznacza długość wahadła, a g oznacza przyśpieszenie grawitacyjne. Przyjmij do obliczeń, że przyśpieszenie grawitacyjne na Ziemi wynosi g_Z = 9,81 m/s^2, a przyśpieszenie grawitacyjne na Księżycu wynosi g_K = 1,62 m/s^2. Oblicz T_K/T_Z - stosunek okresu drgań tego wahadła, gdyby znajdowało się ono na Księżycu, do okresu drgań tego samego wahadła znajdującego się na Ziemi. Wynik podaj z dokładnością do 0,01.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3064/zad-6861.svg","solution_image":"img/site/3064/rozw-6861.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 (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-3064/zad/6866","paper_id":"site-3064","number":"6866","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Dane jest wyrażenie (a/a+b - a^2/a^2-b^2) : a-b/a^2-b^2 gdzie a∈R, b∈R, a≠b, a≠-b. Przekształć dane wyrażenie do najprostszej postaci i oblicz jego wartość dla a=2/√3 oraz b=-1/√3.","answer":null,"answer_text":null,"solution":null,"image":"img/site/3064/zad-6866.svg","solution_image":"img/site/3064/rozw-6866.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 (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"}]}