Zadanie 1f, Paper 3, maj 2025, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2025, TZ1, Paper 3 (zadania wzorowane), pytanie 1(f). 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 polynomial can encode a finite probability distribution: for a nonnegative integer-valued random variable U, define . Two independent random selectors each return one of 1, 2, 3, 4 with equal probability. Let M be the smaller of the two outputs. Its probabilities for m = 1, 2, 3, 4 are respectively 7/16, 5/16, 3/16, 1/16, so GM(t) = (7t + 5t2 + 3t3 + t4)/16. In later parts, X counts experts in a two-person panel chosen uniformly from seven people, three experts and four trainees. Independently of that selection, two sensors attempt to register a pulse; their independent detection probabilities are 0.4 and p. Let Y count detections. The coefficient of t2 in GY(t) is 0.24. For Z = X + Y, you may use GZ(t) = GX(t)GY(t).
Polecenie
You may use E(Z) = GZ′(1). Use this derivative identity to find E(Z).
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Odpowiedź z klucza
. By the product rule, GZ′(1) = GX′(1)GY(1) + GX(1)GY′(1). Both polynomial values at 1 are 1, while GX′(1) = 6/7 and GY′(1) = 1. Therefore E(Z) = 13/7 = 1.857142… ≈ 1.86. Differentiating an expanded product or evaluating its derivative numerically at one is equivalent.
Schemat punktowania
A1: correct product of the two generating polynomials. M1: differentiate or apply the product-rule identity at one. M1: evaluate the component derivatives or the derivative of the expanded product. A1: correct expectation. For direct calculator derivative evaluation, award A1 for the product, M1 for evaluation at one and A2 for the correct result. An incorrect quadratic from (e) can earn at most three marks through consistent method; do not award the product accuracy mark for a polynomial of an inappropriate form.
Komentarz. Paper 3: calculator use is permitted, but reasoning must be shown. Exact equivalent forms and appropriate numerical answers to three significant figures are accepted unless stated otherwise.