Zadanie 1a
Find the area of fabric in the sector.
Matematyka AA IB HL — maj 2025, 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 37 zadań z tego arkusza, razem 110 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
Find the radius of the open base.
Zadanie 2a
Write down the period of f.
Zadanie 2b
Find a and b.
Zadanie 3
At the beginning of 2021, a sealed reservoir holds 420000 litres. Its modelled rate of water loss is described by litres per year, where t is years after the beginning of 2021. Find the volume predicted for the beginning of…
Zadanie 4a
Use the appropriate regression equation to estimate height for a tree whose crown is 6 m wide.
Zadanie 4b
Find the mean height and the mean crown width.
Zadanie 4c
Find s.
Zadanie 5a
Find the gradient of Lk in terms of k.
Zadanie 5b
The smaller angle between Lk and the horizontal is π/10. Find all possible k.
Zadanie 6a
(i) Rewrite f(x) using only sine and cosine. (ii) Sketch its graph on the given domain and label its horizontal intercept.
Zadanie 6b
Find f−1(3).
Zadanie 6c
Given and 0 < α < π, find f(α).
Zadanie 7a
Find the horizontal flight speed.
Zadanie 7b
Find the angle made by the descending flight path with the horizontal, in degrees.
Zadanie 8a
Show that .
Zadanie 8b
Express e0.4i + e2.2i as reiθ, where r > 0 and −π < θ ≤ π.
Zadanie 8c
Hence or otherwise express h(t) as , with p > 0 and 0 < q < 2π.
Zadanie 9a
Use Euler’s method with step size 0.25 to estimate y at x = 3.
Zadanie 9b
The diagram shows two arcs of the associated solution relation. (i) Is the estimate in (a) above or below the upper arc at x = 3? Justify your answer using its curvature. (ii) Explain why forward Euler steps starting at (2, 0) cannot estimate the point on the…
- Materiał graficzny
Zadanie 10a
The first three assemblies are shown. Write down t3.
- Materiał graficzny
Zadanie 10b
Find t4.
Zadanie 10c
Show that .
Zadanie 10d
Rods are supplied in boxes of 50. Find the maximum number of tiers that can be built using 12 boxes.
Zadanie 10e
A complete assembly uses a whole number of boxes with no rods left over. Find its smallest possible positive number of tiers.
Zadanie 10f
Each rod is 12 cm long. Find the minimum number of rods needed for an assembly whose height exceeds 2.5 m. Ignore rod thickness.
Zadanie 11a
(i) Evaluate . (ii) Write down the mode of T. (iii) Determine whether the median is larger or smaller than the mode, with a reason.
Zadanie 11b
A cycle is classified as short if it lasts no more than 6 minutes. Find the probability that a randomly selected cycle is short.
Zadanie 11c
A cycle is classified as short when T ≤ 6. Given that a selected cycle is short, find the probability that it lasts no more than 5.5 minutes.
Zadanie 11d
Find the lower quartile of T.
Zadanie 11e
A performance score is S = a − bT, with a, b > 0. Its mean is 120 and its maximum possible value is 180. Using E(T) = 7.59472, find a and b, each rounded to the nearest integer.
Zadanie 11f
The calibration uses S = a − bT, with a, b > 0, mean score 120 and maximum possible score 180. Use E(T) = 7.59472 and round the resulting coefficients to the nearest integer before calculating the score variance. The score is S = a − bT, with the coefficients…
Zadanie 12a
Show that Fn is even for every allowed n.
Zadanie 12b
(i) Show that . (ii) Write F3(x) in ascending powers of x.
Zadanie 12c
(i) Find the largest K > 0 such that F exists throughout (−K, K). Give an exact answer. (ii) Express F(x) as 1/(a + bx2), determining a, b.
Zadanie 12d
Define g(x) = F(x) for 0 ≤ x < K. (i) Justify that g−1 exists. (ii) Find g−1(x) and its domain.
Zadanie 12e
Find the area of the shaded region bounded by portions of y = g(x), y = g−1(x) and the positive coordinate axes.
- Materiał graficzny