Zadanie 1a
State u₁.
Matematyka AA IB HL — maj 2022, TZ2, Paper 1 (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 30 zadań z tego arkusza, razem 110 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
Given uₙ=−38, find n.
Zadanie 1c
Find the common difference.
Zadanie 2a
Prove that their sum is divisible by 3.
Zadanie 2b
Prove that the sum of their squares is never divisible by 3.
Zadanie 3a
State (i) the vertical asymptote and (ii) the horizontal asymptote.
Zadanie 3b
Sketch y=f(x), indicating its asymptotes and axis intercepts.
Zadanie 3c
Hence solve 0<f(x)<3.
Zadanie 3d
Solve 0 < (3∣x∣ − 2)/(∣x∣ + 2) < 3.
Zadanie 4
Find the least positive x satisfying .
Zadanie 5a
Show that b=40.
Zadanie 5b
The third term is the arithmetic mean of the second and fourth terms. Find all possible x.
Zadanie 6a
Show that f is odd.
Zadanie 6b
The range is a≤y≤b. Find a,b.
Zadanie 7
Using u=csc x or otherwise, evaluate in terms of the nonzero real parameter n.
Zadanie 8
A continuous random variable X has density for a≤x≤c, for c<x≤b, and 0 elsewhere. Assume a<c<b and c ≥ (2b + a)/3. Find its median in terms of a,b,c.
Zadanie 9
Prove by contradiction that 3x3 + 6x2 + 9x + 2 = 0 has no integer root.
Zadanie 10a
Show that p=0.40 and q=0.25.
Zadanie 10b
Find P(X>2).
Zadanie 10c
After three cycles the recorded total is 2 units. Find the probability the batch passes.
Zadanie 10d
Two independent uniform four-sector spinners each have sector labels 0,2,4,6. Their sum has the following distribution. A replacement pair of independent uniform four-sector spinners must produce this same sum distribution. Their labels are respectively…
- Tabela do uzupełnienia
Zadanie 10e
Two independent uniform four-sector spinners each have sector labels 0,2,4,6. Their sum has the following distribution. A replacement pair of independent uniform four-sector spinners must produce this same sum distribution. Their labels are respectively…
- Tabela do uzupełnienia
Zadanie 11a
Sketch y=f(x), labelling all asymptotes, local extrema and axis intercepts.
Zadanie 11b
(i) Show that . (ii) State its domain.
Zadanie 11c
Given (h∘g)(a) = π/4, find a in exact simplified form.
Zadanie 12a
Show that .
Zadanie 12b
Given , show that triangle Z₁OZ₂ is right-angled.
Zadanie 12c
For the stated isosceles 120° triangle, (i) express z₁ in terms of z₂; (ii) hence show that z12 + z22 = −z1z2.
Zadanie 12d
For this 120° configuration, suppose z₁,z₂ are the distinct roots of z2 + az + b = 0 with real a,b. Use z12 + z22 = −z1z2 to show that a²−b=0.
Zadanie 12e
Now consider z2 + az + 16 = 0. You may use a²=b for the 120° configuration. Deduce that the argument restriction 0<α−θ<π permits only one such triangle.