Zadanie 1b, Paper 3, listopad 2025, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — listopad 2025, TZ1, Paper 3 (zadania wzorowane), pytanie 1(b). 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
(i) Eliminate t to show that every such normal has m<0 and (−m)c2 = 9(m2 − 1)2. (ii) State the set of possible intercepts c.
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From m=−t²/9, t=±3√(−m), with m<0. Substitute into c=t³/9−9/t to get c=±3(m²−1)/√(−m). Squaring gives (−m)c²=9(m²−1)². For (ii), c(t)=t³/9−9/t is continuous and strictly increasing on each of (−∞,0),(0,∞), since c′=t²/3+9/t²>0. On each half-axis its limits run from −∞ to+∞, so every real c is possible. A calculator plot of c(t) also supports the required range statement.
Schemat punktowania
(i) A1:relate m and t from the normal slope. A1:t=±3√(−m), with m<0. M1:substitute into the intercept. A1:valid simplification/squaring establishing the relation. (ii) A1:c∈R. A proof of the range is not separately required for this one mark.
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.