{"paper":{"id":"site-4324","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":18,"source_label":"Matematyka · Zbiory zadań · 2014 (rozszerzona)","subject_label":"Matematyka","category_label":"Zbiory zadań"},"questions":[{"id":"site-4324/zad/4325","paper_id":"site-4324","number":"4325","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Wielomian W(x) = 2x^3-bx^2-1 jest podzielny przez dwumian x+1. Wynika stąd, że A. b=-3, B. b=-1, C. b=1, D. b=3.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4325.svg","solution_image":"img/site/4324/rozw-4325.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-4324/zad/4326","paper_id":"site-4324","number":"4326","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Okrąg o równaniu (x+2)^2+(y-2)^2=4 ma dwa punkty wspólne z prostą o równaniu A. x=0, B. y=0, C. y=-x, D. y=x.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4326.svg","solution_image":"img/site/4324/rozw-4326.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-4324/zad/4327","paper_id":"site-4324","number":"4327","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Funkcja określona dla każdej liczby rzeczywistej x wzorem f(x) = x^5+5x-1 A. ma więcej niż dwa minima lokalne. B. ma dokładnie dwa minima lokalne. C. ma dokładnie jedno minimum lokalne. D. nie ma minimum lokalnego.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4327.svg","solution_image":"img/site/4324/rozw-4327.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-4324/zad/4328","paper_id":"site-4324","number":"4328","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Każda liczba x należąca do przedziału otwartego (pi/2, 3pi/4) spełnia nierówność A. tg x > sin x, B. cos x > sin x, C. cos x > tg x, D. tg x > cos x.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4328.svg","solution_image":"img/site/4324/rozw-4328.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-4324/zad/4329","paper_id":"site-4324","number":"4329","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Funkcja f jest określona dla wszystkich liczb rzeczywistych wzorem f(x) = 3^{x-2} + 3. Prosta l ma równanie y = 3,3. Ile punktów wspólnych mają wykres funkcji f i prosta l? A. Zero. B. Jeden. C. Dwa. D. Nieskończenie wiele.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4329.svg","solution_image":"img/site/4324/rozw-4329.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-4324/zad/4330","paper_id":"site-4324","number":"4330","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Dane są liczby a, b takie, że a-b=4 i ab=7. Oblicz a^3b+ab^3. Zakoduj w kratkach poniżej kolejno, od lewej do prawej, cyfry setek, dziesiątek i jedności otrzymanego wyniku.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4330.svg","solution_image":"img/site/4324/rozw-4330.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-4324/zad/4331","paper_id":"site-4324","number":"4331","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Długości boków prostokąta są równe 3 oraz 5. Oblicz sinus kąta ostrego, który tworzą przekątne tego prostokąta.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4331.svg","solution_image":"img/site/4324/rozw-4331.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-4324/zad/4332","paper_id":"site-4324","number":"4332","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Oblicz granicę lim_{n dąży do nieskończoności} n^2 / n+2 - (n+2)^2 / n+444","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4332.svg","solution_image":"img/site/4324/rozw-4332.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-4324/zad/4333","paper_id":"site-4324","number":"4333","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Funkcja f jest określona wzorem f(x) = x^2 / x-4 dla każdej liczby rzeczywistej x różnej od 4. Oblicz pochodną funkcji f w punkcie x=12.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4333.svg","solution_image":"img/site/4324/rozw-4333.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-4324/zad/4334","paper_id":"site-4324","number":"4334","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Funkcja f jest określona wzorem f(x)=x^4 dla każdej liczby rzeczywistej x. Wyznacz równanie prostej stycznej do wykresu funkcji f, która jest równoległa do prostej y=4x+7.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4334.svg","solution_image":"img/site/4324/rozw-4334.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-4324/zad/4335","paper_id":"site-4324","number":"4335","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Wyznacz wszystkie liczby rzeczywiste x, spełniające równanie sin 5x - sin x = 0.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4335.svg","solution_image":"img/site/4324/rozw-4335.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-4324/zad/4336","paper_id":"site-4324","number":"4336","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Niech P_n oznacza pole koła o promieniu 1 / 2^n, dla n >= 1. Oblicz sumę wszystkich wyrazów ciągu (P_n).","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4336.svg","solution_image":"img/site/4324/rozw-4336.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-4324/zad/4337","paper_id":"site-4324","number":"4337","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Wykaż, że jeżeli a > b >= 1, to a / 2+a^3 < b / 2+b^3.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4337.svg","solution_image":"img/site/4324/rozw-4337.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-4324/zad/4338","paper_id":"site-4324","number":"4338","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Wykaż, że jeżeli alfa, beta, gamma są kątami wewnętrznymi trójkąta i sin^2 alfa + \\sin^2 beta < \\sin^2\\ gamma, to cos gamma < 0.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4338.svg","solution_image":"img/site/4324/rozw-4338.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-4324/zad/4339","paper_id":"site-4324","number":"4339","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Punkt E jest środkiem boku BC prostokąta ABCD, w którym AB > BC. Punkt F leży na boku CD tego prostokąta oraz |kąt AEF| = 90 stopni. Udowodnij, że |kąt BAE| = |kąt EAF|.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4339.svg","solution_image":"img/site/4324/rozw-4339.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-4324/zad/4340","paper_id":"site-4324","number":"4340","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Oblicz prawdopodobieństwo warunkowe, że w trzykrotnym rzucie symetryczną sześcienną kostką do gry otrzymamy co najmniej jedną „jedynkę”, pod warunkiem że otrzymamy co najmniej jedną „szóstkę”.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4340.svg","solution_image":"img/site/4324/rozw-4340.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-4324/zad/4341","paper_id":"site-4324","number":"4341","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Dany jest okrąg o_0 o równaniu (x-3)^2+(y-1)^2=1. W pierwszej „ćwiartce” układu współrzędnych istnieją dwa okręgi o_1, o_2 styczne zewnętrznie do okręgu o_0 i jednocześnie styczne do obu osi układu współrzędnych. Oblicz odległość środków okręgów o_1 oraz o_2.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4341.svg","solution_image":"img/site/4324/rozw-4341.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-4324/zad/4342","paper_id":"site-4324","number":"4342","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":2014,"month":null,"level":"rozszerzona","text":"Okno na poddaszu ma mieć kształt trapezu równoramiennego, którego krótsza podstawa i ramiona mają długość po 4 dm. Oblicz, jaką długość powinna mieć dłuższa podstawa tego trapezu, aby do pomieszczenia wpadało przez to okno jak najwięcej światła, czyli aby pole powierzchni okna było największe. Oblicz to pole.","answer":null,"answer_text":null,"solution":null,"image":"img/site/4324/zad-4342.svg","solution_image":"img/site/4324/rozw-4342.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"}]}