Zadanie 1a
Find a and b.
Matematyka AA IB HL — maj 2024, TZ2, 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 34 zadania z tego arkusza, razem 110 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
Find the area enclosed by the graphs.
Zadanie 2a
Verify that the mean y is 14.
Zadanie 2b
Find p and q.
Zadanie 3a
State the amplitude ratio for configuration D.
Zadanie 3b
Find the attenuation score for D.
Zadanie 3c
Another configuration has attenuation score 86. Find its amplitude ratio.
Zadanie 4a
Find v(2.3).
Zadanie 4b
Find the maximum velocity on the given time interval.
Zadanie 4c
Find the acceleration when the carriage changes direction.
Zadanie 5
Each independent attempt to receive a beacon has success probability 0.18. If X is the number of successes in n attempts, then . Find the least integer n for which P(X≥1)>0.995.
Zadanie 6
A soft hemispherical dome expands while retaining its hemispherical shape. Material enters at a constant rate of 8 cm³ s⁻¹. Its volume is V = (2/3)πr3. Find the rate at which its radius increases when V=36 cm³.
Zadanie 7
The region between the y-axis and the curve y = 3ln(x − 1), bounded by y=0 and y=6, is rotated about the y-axis. Find the volume of the resulting solid.
Zadanie 8a
Show that (i) ; (ii) .
Zadanie 8b
Hence or otherwise, solve for θ.
Zadanie 9
The curve y = (x + 2)/(ax2 + bx + c) has a local minimum at (1,2) and a vertical asymptote x=−1. The real constants a,b,c are nonzero. Find them.
Zadanie 10a
Find the mode of X.
Zadanie 10b
Find P(1≤X≤2).
Zadanie 10c
Find the median of X.
Zadanie 10d
Find the probability that a randomly selected customer spends at most USD 32.
Zadanie 10e
Find the expected amount spent per customer, to the nearest cent.
Zadanie 11a
Show that AB=37.47 m to four significant figures.
Zadanie 11b
Show that the angular speed is π/9 radians per second.
Zadanie 11c
Find T.
Zadanie 11d
Express α in terms of t.
Zadanie 11e
Use the sine rule in triangle ASD to show that , where .
Zadanie 11f
State how far south of A the robot is at t=0.
Zadanie 11g
For the first sweep, use . (i) Express the distance w between the robot and D in terms of t. (ii) Find when and where the scanner first reaches the robot.
Zadanie 12a
Use Euler’s method with step length π/12 to estimate y at x=π/3, giving three significant figures.
Zadanie 12b
Show that .
Zadanie 12c
Show that cot x is an integrating factor.
Zadanie 12d
Hence solve the initial-value problem and show that y=x tan x.
Zadanie 12e
The exact solution is y=x tan x. (i) Find y at x=π/3 to three significant figures. (ii) Considering the gradient, explain why the forward Euler estimate from π/6 using steps of π/12 is inaccurate here. (iii) Explain why that estimate is less than the exact…
Zadanie 12f
Using the differential equation and its solution y=x tan x, deduce that the curve has positive gradient throughout 0<x<π/2.