Zadanie 1a
Find x.
Matematyka AA IB HL — maj 2022, TZ1, 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 33 zadania z tego arkusza, razem 110 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
Find the population standard deviation.
Zadanie 2a
Express the area of one hole in terms of θ.
Zadanie 2b
The remaining plate has area 25π − 9 cm². Find θ.
Zadanie 3a
Show that 2k²−k+0.105=0.
Zadanie 3b
Find k and justify your choice.
Zadanie 3c
Hence find E(X).
Zadanie 4a
Find the time when the particle is at rest.
Zadanie 4b
Find its acceleration when it changes direction.
Zadanie 4c
Find its total distance travelled over the given interval.
Zadanie 5a
Find x.
Zadanie 5b
Find P(A′|B′).
Zadanie 6a
Show that f is odd.
Zadanie 6b
Solve f(x)≥g(x).
Zadanie 7a
Find the possible range of |a+b|.
Zadanie 7b
Find p when |a+b| is as small as possible.
Zadanie 7c
Find q=(x,y), with x,y>0, such that |q|=13 and q is perpendicular to a.
Zadanie 8a
Express the product of its roots in terms of k.
Zadanie 8b
Determine k for which one root is positive and the other negative.
Zadanie 9a
Initially all five containers must occupy consecutive slots. Count the arrangements.
Zadanie 9b
They are rearranged with at least one empty slot between F and every other container. The other four need not be consecutive or separated from one another. Count the arrangements.
Zadanie 10a
Sketch y=f(x), labelling both endpoints.
Zadanie 10b
(i) Show that . (ii) State its domain and range.
Zadanie 10c
(i) Show that water at depth h has volume V = π(h3/3 + 3h). (ii) Find the maximum water volume.
Zadanie 10d
Find the time needed to fill the vessel.
Zadanie 10e
Find dh/dt when half the water capacity has been filled.
Zadanie 11a
Find the probability that a line-A piece has thickness below 117 μm.
Zadanie 11b
Among 15 independently selected line-A pieces, find the probability that exactly 6 have thickness below 117 μm.
Zadanie 11c
A piece is selected from the combined output. (i) Find the probability its thickness is below 117 μm. (ii) Given that its thickness is below 117 μm, find the probability it came from line A.
Zadanie 11d
Line A is adjusted: its mean stays 120 μm but its standard deviation becomes σ>0. Line B and the output proportions are unchanged. The combined probability of thickness below 117 μm is now 0.18. Find σ.
Zadanie 12a
Use Euler’s method with step 0.08 to estimate y(1.4).
Zadanie 12b
Use y=vx to show that x dv/dx = v2 − 2v − 3.
Zadanie 12c
(i) Solve the differential equation and show that y = (15x + x5)/(5 − x4) on its initial-value branch. (ii) Find the actual value y(1.4). (iii) Use the graph of this solution to explain why five Euler steps of 0.08 from (1,4) give a poor estimate at 1.4.