Zadanie 1a
Find QR.
- Materiał graficzny
Matematyka AA IB HL — maj 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 27 zadań z tego arkusza, razem 110 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
Find .
Zadanie 2
Let F(x) = 3x4 − 4x3 + px2 + qx + 1. The polynomial has factor x + 1, and division by x − 2 leaves remainder 45. Determine p and q.
Zadanie 3a
(i) Find a. (ii) State the mode of X.
Zadanie 3b
Find E(X).
Zadanie 3c
To gather feedback, the coordinator interviews the first 40 members who enter the workshop on Monday. Identify the sampling method.
Zadanie 4
A circle has centre O. Points A and B subtend a minor central angle of 2.2 radians. Triangle AOB has area 18 cm2. Point C is on the major arc from A to B. Find the length of arc ACB.
- Materiał graficzny
Zadanie 5a
Find f′(x).
Zadanie 5b
The tangent at x = 1 has inclination 35° to the positive horizontal axis. Determine all possible values of a.
Zadanie 6a
Find (i) ; (ii) .
Zadanie 6b
The angle between and equals the angle between and . Find p.
Zadanie 7
In the binomial expansion about x = 0 of , where q is a positive rational number, the coefficient of x6 is 320. Find q.
Zadanie 8
A calibration network contains n sensors. Each unordered pair is scheduled to exchange three test messages. One sensor is removed after participating in exactly eight messages; all pairs among the remaining sensors complete their three messages. In total 701…
Zadanie 9a
Show that .
- Materiał graficzny
Zadanie 9b
At an instant when θ = 0.2 radians, the angle is increasing at 6 radians per hour. Find the aircraft’s speed.
Zadanie 10a
(i) State u(1). (ii) Find the time when the first cart’s velocity is 4 m/s.
Zadanie 10b
Find the time when the first cart’s acceleration is 3 m/s2.
Zadanie 10c
(i) Find . (ii) Explain why this limit is not an attained trial velocity.
Zadanie 10d
The cart modelled by u finishes first. How far is the other cart from its endpoint at that instant?
Zadanie 11a
A processes 40 images. (i) Find the probability of at most five errors. (ii) Given at most five errors, find the probability of exactly three.
Zadanie 11b
For another batch of n images processed by A, P(E ≤ 5) = 0.508 to three significant figures. Find n.
Zadanie 11c
B and C have error probabilities 0.04 and 0.12. Of all images, 50% go to A, 30% to B and 20% to C. For an image selected at random, find (i) the probability of an error; (ii) the probability it was processed by A, given an error.
Zadanie 11d
Before the update, B and C have error probabilities 0.04 and 0.12. Of all images, 50% go to A, 30% to B and 20% to C. After an update, C’s error probability is x%, while the other error rates and routing shares remain unchanged. The overall…
Zadanie 12a
Find .
Zadanie 12b
By solving it, show that , where C is a constant.
Zadanie 12c
Sketch the particular solution through (−3, 2) for −4 ≤ x ≤ 4, labelling the coordinates of its local maximum and minimum.
Zadanie 12d
Consider two distinct solution curves passing through (−3, p) and (−3, q), where p ≠ q. Use their tangents at x = −3 to prove that every tangent at this x-coordinate, for this family of solutions, passes through a common point. State that point.