Zadanie 2c, Paper 3, maj 2024, TZ2, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2024, TZ2, Paper 3 (zadania wzorowane), pytanie 2(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
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
Show that the probability of exactly one x-intercept is 3/64.
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Odpowiedź z klucza
Exactly one intercept requires b²=4ac, so b must be even: 2,4,6,8. For b=2, ac=1 has ordered pair (1,1). For b=4, ac=4 has (1,4),(2,2),(4,1). For b=6, ac=9 has (3,3). For b=8, ac=16 has (4,4). Thus there are 1+3+1+1=6 equally likely triples, giving 6/128=3/64.
Schemat punktowania
M1: discriminant zero. M1: use b²=4ac, evenness, or attempt to determine an admissible b/ac. A1: at least one correct candidate b∈{2,4,6,8} or ac∈{1,4,9,16}; a complete candidate list is not required for this mark. A1: all three ordered pairs at b=4. A1: any two correct single-pair cases among b=2,6,8. A1: the remaining single-pair case, completing six favorable outcomes. Combining these gives 6/128=3/64 (given result). Equivalent complete enumeration or probabilities earns the same marks.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Retain exact intermediate expressions; give three significant figures for numerical probabilities.