Zadanie 1e, Paper 3, listopad 2025, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — listopad 2025, TZ1, Paper 3 (zadania wzorowane), pytanie 1(e). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
Darmowe konto pozwoli wrócić do niego później.
Zadanie
Wspólne informacje do wiązki zadań
In parts 1a–1d consider H : y = −9/x, x≠0. Its normal at x=t, t≠0, is written y=mx+c. We study normal lines that pass through a fixed point (0,c), rather than normal lines with a prescribed slope. Count geometrically distinct lines: one line may be normal at two different points. In parts 1e–1f use F : y = −k2/x, where k>0. Define A=(kt,−k/t), with t≠0,±1. Its normal meets F again at B. The line OA, where O=(0,0), meets F again at C.
Polecenie
The normal to F at A is y = −t2x + kt3 − k/t. (i) Show that its intersections with F satisfy x2 − k(t − t−3)x − k2/t2 = 0. (ii) Hence, using a sum/product of roots or another valid method, find B.
Korepetytor
Nie wiesz, jak ruszyć? Korepetytor zna to zadanie i poprowadzi Cię krok po kroku, bez gotowca.
3 rozmowy · bez karty · Zaloguj się
Sprawdź się
Jak poszło?
Pokaż rozwiązanieUkryj rozwiązanie
Odpowiedź z klucza
(i) Equate −k²/x=−t²x+kt³−k/t and multiply by x. Rearrangement gives t²x²−k(t³−1/t)x−k²=0; division by t² gives the target. (ii) One root is x_A=kt. The product is −k²/t², so x_B=−k/t³. On F, y_B=−k²/(−k/t³)=kt³. Thus B=(−k/t³,kt³). The two x-roots have opposite signs and are distinct for real nonzerot,k. Factoring (x−kt)(x+k/t³) or using the quadratic formula is equivalent.
Schemat punktowania
(i) A1:correct curve-normal equality. M1:clear the denominator and form a quadratic. A1:correct rearrangement to the target. Merely verifying the known root can earn at most the method mark. (ii) M1:use root sum/product, factorization or quadratic formula. A1:x_B=−k/t³. A1:y_B=kt³. For the root-sum/product M1, recognize the relevant coefficient even with a sign error, as in the source; subsequent accuracy marks still require the correct result.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Show complete reasoning and reproducible numerical work. Give exact results unless a stated approximation is requested.