Zadanie 1d, Paper 3, listopad 2025, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — listopad 2025, TZ1, Paper 3 (zadania wzorowane), pytanie 1(d). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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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
On one pair of axes sketch (i) H; (ii) its two distinct normals passing through (0,1). Mark each point of normal contact. Exact contact coordinates and line equations are not required.
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H lies in quadrantsII andIV with coordinate-axis asymptotes. The two normals have negative but different slopes and intersect at (0,1). Numerically the contact parameters solve t³/9−9/t=1: t≈3.23968822 and t≈−2.73973642. The points are (t,−9/t); draw each corresponding perpendicular line through (0,1), with the contact points visibly marked.
Schemat punktowania
(i) A1:correct two-branch hyperbola in quadrantsII andIV. (ii) A1:a negative-slope normal through (0,1) with its correct branch contact marked. A1:the second distinct normal through the same point, with the other contact marked. Two parallel normals do not meet this construction.
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.