Zadanie 1a
Write ln 5 + ln 6 as ln k for a positive integer k.
Matematyka AA IB SL — maj 2025, TZ1, 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 23 zadania z tego arkusza, razem 80 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
Write 2ln 7 as ln k for a positive integer k.
Zadanie 1c
Write −ln(1/9) as ln k for a positive integer k.
Zadanie 2
The curve y = h(x) is defined for real x by . Its local minimum is M(a, b), with a > 1. Find a and b.
Zadanie 3a
Write down r.
Zadanie 3b
Expand (1 + t)4 and collect like powers of t.
Zadanie 3c
Hence or otherwise, find the balance after one year, rounded to the nearest dollar.
Zadanie 4
A region is bounded above by y = 1 + e2x, below by y = 1 − 2e2x, and by the vertical lines x = 0 and x = ln 2. Find its exact area.
Zadanie 5a
Find .
Zadanie 5b
Show that C and D are independent.
Zadanie 6a
(i) Show that . (ii) Hence find the mean of the eight areas in the form , where p is a positive rational number and a is a positive integer.
Zadanie 6b
Find the median area in the form cm2, with q a positive rational number.
Zadanie 7a
Determine (i) its maximum velocity and (ii) its maximum speed during the interval.
- Materiał graficzny
Zadanie 7b
(i) Write down T. (ii) Find the distance travelled in the first T seconds.
Zadanie 7c
Determine whether the trolley returns to its starting position at a time strictly after zero and no later than five seconds. Justify your answer.
Zadanie 8a
Write f(x) in the form a(x − h)2 + k, with integer constants a, h, k.
Zadanie 8b
Sketch y = f(x). Label the vertex and every intercept with a coordinate axis.
Zadanie 8c
Solve f(x) ≤ 24.
Zadanie 8d
(i) Write an expression for (f∘g)(x). (ii) Solve (f∘g)(x) ≤ 24.
Zadanie 9a
(i) Use triangle OCB to show that . (ii) Express h in terms of r and θ.
- Materiał graficzny
Zadanie 9b
Hence or otherwise show that .
Zadanie 9c
Show that .
Zadanie 9d
Find the cone’s volume in the form , where p is a positive rational number.