Zadanie 1d, Paper 3, listopad 2024, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — listopad 2024, 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
A digitization team begins year 1 with 4800 pages awaiting processing. During each year, incoming revisions increase the outstanding work by 12.5%, after which a fixed quota is processed at year-end. Treat page counts as continuous estimates until rounding is requested. With quota 400, T_n denotes the work at the start of year n, so T1 = 4800 and . With quota 750, use D_n instead. For either policy, a non-positive algebraic prediction means the queue has already cleared; actual work cannot be negative.
Polecenie
For quota 750, Dn = −1200(9/8)n−1 + 6000. (i) Show that Dn+1 − Dn = −150(9/8)n−1. (ii) Explain why the algebraic sequence is strictly decreasing.
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Odpowiedź z klucza
(i) Subtraction cancels the constants and gives −1200[(9/8)n − (9/8)n−1] = −1200(9/8)n−1(1/8) = −150(9/8)n−1. (ii) The power is positive, so every difference is negative. This is the algebraic sequence; the physical queue stays at zero after clearance.
Schemat punktowania
(i) M1: attempt to subtract consecutive explicit expressions or form the recurrence difference. A1: correct substituted difference, e.g.−1200[(9/8)^n−(9/8)^(n−1)]. A1: correct expansion/factorization, e.g.−1200(9/8)^(n−1)(1/8), establishing the target. Alternatively A1 for D_n/8−750 and A1 for substituting the explicit D_n. (ii) R1: the consecutive difference is negative (or the reverse difference positive), so the sequence strictly decreases. Merely citing negativity suffices; specific numerical examples alone do not.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Give exact forms unless an approximation is requested.