Zadanie 1a
Find a and b.
Matematyka AA IB HL — listopad 2022, Paper 2 (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 28 zadań z tego arkusza, razem 110 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
State the Pearson correlation coefficient.
Zadanie 1c
Predict the output at input 25, to the nearest integer.
Zadanie 2a
Find the coordinates of A.
Zadanie 2b
Find the x-coordinate of B.
Zadanie 2c
Find the total area between the graph and the x-axis from x=0 to x=2.
Zadanie 3
A geometric sequence begins with 80 and has fourth term 121.67. Let Sₙ be the sum of its first n terms. Find the least positive integer n for which Sₙ>12000.
Zadanie 4
A collection of unprocessed archival files shrinks according to N(t) = 24000ekt, k<0, where t is the number of years after 1 January 2017. By 1 January 2023,14% of the initial files had been processed. Estimate the number remaining on 1 January 2036.
Zadanie 5
For a≠0, the coefficient of x3 in (1 + ax)8/(14x2) is 12a4. Find a.
Zadanie 6a
Express a in terms of b.
Zadanie 6b
For b=2 and a = 1/(1 − 3e−2), find the median.
Zadanie 7a
Express cosθ in terms of n.
Zadanie 7b
Find the exact limit of θ as n tends to infinity.
Zadanie 8
A widening vessel is filled at a constant rate of 3 cm³/s. For liquid depth h cm, its volume is V = π(2h2 + h3/6), for 0≤h≤8. Find how fast the depth is increasing when the vessel contains 100 cm³.
Zadanie 9a
(i) Express AC in terms of α. (ii) Express AC in terms of β. (iii) Hence express α in terms of β.
Zadanie 9b
Find the maximum total area of fabric required, counting both layers of triangle ADC.
Zadanie 10a
Find P(T>48).
Zadanie 10b
Four cycles are checked in order. Find the probability that only the fourth lasts longer than 48 minutes.
Zadanie 10c
A large collection contains only cycles longer than 48 minutes. (i) Find the probability that a randomly chosen cycle from this collection lasts less than 62 minutes. (ii) Twelve cycles are independently selected from the collection. Find the probability that…
Zadanie 10d
Thresholds a<b satisfy P(a≤T≤b)=0.88 and P(T>b)=2P(T<a). Find the lower threshold a.
Zadanie 11a
Find f′(x).
Zadanie 11b
Find the point where the tangent to the graph of f is parallel to y=2x.
Zadanie 11c
The region bounded by the graph of f and the two coordinate axes is rotated once about the x-axis. Find the volume of the resulting solid.
Zadanie 11d
Find (i)(f∘g)(0); (ii)(f∘g)′(0).
Zadanie 12a
Write down and .
Zadanie 12b
If A,B,C are collinear, show that k=8.
Zadanie 12c
For k=8, L₁ passes through A,B,C. (i) Give a vector equation of L₁. (ii) L₂ is given by x = (y − 1)/2 = 2 − z. Show that L₁ and L₂ are skew.
Zadanie 12d
For k≠8, let Π be the plane through A,B,C. (i) Find its Cartesian equation. (ii) Find the point of Π nearest the origin.