Zadanie 2h, Paper 3, maj 2024, TZ2, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2024, TZ2, Paper 3 (zadania wzorowane), pytanie 2(h). 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 random quadratic is generated as q(x) = ax2 + bx + c. Choose a and c independently and uniformly from {1,2,3,4}, and choose b independently and uniformly from {1,2,3,4,5,6,7,8}. Ordered coefficient choices are distinct and equally likely. Parts (a)–(f) concern this discrete generator; the separate continuous model is specified explicitly in (g) and (h).
Polecenie
In the continuous model q(x) = x2 + 2Zx + 4, with Z normal of mean 0.5 and standard deviation 1.5, suppose the graph has two distinct x-intercepts and . Find the conditional probability that both intercepts are greater than 1.
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Because X₁<X₂, require X₁>1. Both roots must be positive, so Z<−2. Then can be squared safely: (Z + 1)2 > Z2 − 4, hence 2Z+1>−4 and Z>−5/2. Thus the joint event is −2.5<Z<−2. Its probability is 0.02504022032…. The conditioning event has probability P(Z<−2)+P(Z>2)=0.20644560620…. Their ratio is 0.12129209618…≈0.121.
Schemat punktowania
M1: impose the smaller-root inequality or an equivalent boundary/root-product argument. A1: boundary Z=−2.5. A1: complete interval −2.5<Z<−2, including the sign/reality restriction; endpoint distinctions at −2 do not change the continuous probability. M1: evaluate the joint-event normal probability. A1: 0.0250402203…. M1: divide by the probability of two real distinct roots. A1: conditional probability 0.121.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Retain exact intermediate expressions; give three significant figures for numerical probabilities.