Zadanie 1a
Show that .
Matematyka AA IB HL — listopad 2025, TZ3, 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 30 zadań z tego arkusza, razem 110 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
Hence or otherwise solve for 0° ≤ x ≤ 180°.
Zadanie 2a
Explain why the median of all 26 masses remains 18 g.
Zadanie 2b
Find L and H.
Zadanie 3a
Show that r = 24π/(2 + θ).
Zadanie 3b
Find θ.
Zadanie 4a
Given μ+Aσ=45, find A.
Zadanie 4b
Find μ and σ.
Zadanie 5
The curve y = 1 + ex/2 for x≥0, the line y=7 and the y-axis enclose a region. Rotate this region through 2π radians about the y-axis. Find the volume of the resulting solid.
Zadanie 6
Let P(x) = 3x2 + qx + r, with real coefficients and real roots α,β. Division by x+1 leaves remainder 14. Given α2 + β2 = 23/3, find q and r.
Zadanie 7a
Give the general solution of the first two equations in parametric form.
Zadanie 7b
For a=4,k=20, find the unique solution of all three equations.
Zadanie 7c
(i) State a for which the system has no unique solution. (ii) For this a, state k giving infinitely many solutions.
Zadanie 8a
How many run lists can be made with no restrictions?
Zadanie 8b
How many run lists keep all jobs of each test type consecutive?
Zadanie 8c
Three jobs of a single test type are chosen for a pilot; their order is irrelevant. How many selections are possible?
Zadanie 9
Prove by contradiction that u/v + v/u ≥ 2 for all positive real u,v.
Zadanie 10a
Find (i) the maximum displacement; (ii) the maximum velocity, over t≥0.
Zadanie 10b
Find (i) the minimum value of s(t) for t≥0; (ii) s(4.6).
Zadanie 10c
Hence find the total distance travelled during the first 4.6 seconds.
Zadanie 10d
State the first positive time T at which the stage crosses the reference mark.
Zadanie 10e
The stage crosses the reference mark at T,2T,3T,… . Let uₙ be its greatest distance from the mark during (n−1)T<t<nT. These distances form a geometric sequence. Find (i) its common ratio; (ii) the total distance travelled if the model continues indefinitely.
Zadanie 11a
Show that 625a/4 + 25b/2 = 1.
Zadanie 11b
(i) Show that 25a+b=0. (ii) Find a and b.
Zadanie 11c
Show that the median lifetime is smaller than the mode.
Zadanie 11d
A coating has already lasted more than 6 days. According to the model, what is the probability that it lasts at least 6.6 days?
Zadanie 12a
(i) Find y′ at x=1. (ii) Use two equal Euler steps to estimate y at x=1.2. (iii) Assuming concavity does not change on 1≤x≤1.2, state the concavity and justify it from your estimates.
Zadanie 12b
(i) Show that (x2 + xy)y′′ = 2x + y − x(y′)2 − (x + 17y)y′. (ii) Find y″ at x=1.
Zadanie 12c
Find constants A,B such that (1 + v)/(1 − 9v2) = A/(1 − 3v) + B/(1 + 3v), for v≠±1/3.
Zadanie 12d
Solve the given differential equation using y=vx and show that its initial-value solution satisfies x8(3y − x)2 = x + 3y.