Zadanie 7c, Paper 2, listopad 2025, TZ3, Matematyka AA IB SL
Wzorowane na: Matematyka AA IB SL — listopad 2025, TZ3, Paper 2 (zadania wzorowane), pytanie 7(c). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
Two reserve funds each begin with USD 6000. Fund A receives a fixed monthly credit equal to 0.275% of its initial balance; these credits earn no further interest. Fund B earns 3% interest compounded annually. There are no deposits, withdrawals or fees.
Polecenie
(i) Write Fund B’s balance after n years in the form 6000qⁿ. (ii) Find the smallest number of complete years after which Fund B exceeds Fund A.
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Odpowiedź z klucza
(i) 6000(1.03)ⁿ. (ii) Solve 6000(1.03)ⁿ>6000+198n. A table or graph shows that at n=7 the balances are 7379.243193<7386, while at n=8 they are 7600.620488>7584. The first positive crossing is between 7 and 8, so the answer is 8 years. The difference has increasing first differences, so there is only one positive crossing after it starts negative.
Schemat punktowania
(i) M1: apply annual compound growth. A1:6000(1.03)ⁿ. (ii) M1: set the comparison in the correct direction; an equality used to locate the crossing is allowed. M1: solve/compare values around the positive crossing. A1:8 complete years.
Komentarz. Paper 2: calculator and AA SL formula booklet allowed. Give three significant figures unless directed otherwise, retaining unrounded values in calculations.