Zadanie 1a
For policy A with p=.12, find (i) the probability that the first inspection is on batch 4; (ii) the probability of at least one inspection among the first 8 batches.
Matematyka AA IB HL — maj 2024, TZ1, Paper 3 (zadania wzorowane)
Zadania wzorowane na pytaniach z tego arkusza: te same umiejętności, format i punktacja, inne dane i kontekst. Treści arkusza IB nie publikujemy.
W bazie 17 zadań z tego arkusza, razem 55 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
(i) Explain why P(X = x) = p(1 − p)x−1 for integer x≥1. (ii) Write E(X) in sigma notation.
Zadanie 1c
For |r|<1, a + ar + ar2 + ⋯ = a/(1 − r). (i) Differentiate both sides with respect to r and express in terms of a,r. (ii) Hence show that E(X)=1/p.
Zadanie 1d
For policy A with p=.12, find E(X) and Var(X).
Zadanie 1e
Show that under policy B the first inspection occurs on batch 3 with probability 405/1024.
Zadanie 1f
Explain why Y≤4.
Zadanie 1g
The distribution of Y is P(Y=1)=1/16,P(Y=2)=m,P(Y=3)=405/1024,P(Y=4)=n. (i) Find m,n. (ii) Show that E(Y)=3019/1024. (iii) Find Var(Y).
Zadanie 1h
(i) Using E(X)=1/p, find p giving E(X)=E(Y). (ii) Find Var(X) for that p. (iii) Which policy gives more consistent inspection intervals at this common mean? Give a reason.
Zadanie 2a
For G, find the second derivative.
Zadanie 2b
For G, find B and C.
Zadanie 2c
For G, show that line BC is y=2x−3/2.
Zadanie 2d
For G and its line BC, find x_D to three decimal places.
Zadanie 2e
For F, find the two inflection abscissae in terms of m.
Zadanie 2f
For F: (i) state x_B; (ii) find y_B and y_C in terms of m,n.
Zadanie 2g
For F, show that line BC is y = nx − 12m3/125.
Zadanie 2h
For F, show that .
Zadanie 2i
For F, find the exact value of (xB − xA)/(xC − xB).