Zadanie 1a
Take b=2,d=3. (i) Sketch y=f(x), labelling its axis intercepts. (ii) Hence explain why f(x)=0 has no real root.
Matematyka AA IB HL — maj 2026, TZ3, Paper 3 (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 15 zadań z tego arkusza, razem 55 punktów, każde z kluczem i schematem punktowania.
Zadanie 1b
Take b=−2,d=1. Sketch y=f(x), labelling its axis intercepts.
Zadanie 1c
For the general family, (i) find f′(x); (ii) prove that f′ is odd.
Zadanie 1d
For a real double root α, describe the identifying feature of y=f(x) at x=α.
Zadanie 1e
Prove that −α is also a double root.
Zadanie 1f
(i) Using f′(α)=0 or otherwise, show that α4 = −b/2. (ii) Hence show that d = b2/4.
Zadanie 1g
Under the double-root assumption, state the b-values for which f(x)=0 has (i) at least one real root; (ii) at least one purely imaginary root.
Zadanie 1h
For b<0, show that the real zeros of f″ are 0 and .
Zadanie 1i
Let . At the inflection point x=t, the tangent is y=mx+c, where m = (16b/7)t3. Show that c = d + 9b2/28.
Zadanie 2a
For the test run, find (i) the amplitude; (ii) the initial displacement; (iii) the first positive time at which the head passes O.
Zadanie 2b
Find ds/dt and d²s/dt² and thereby verify the restoring-acceleration equation.
Zadanie 2c
(i) Use the chain rule to establish a = v dv/ds on a monotone motion branch. (ii) Solve the resulting differential equation to prove v2 = n2(A2 − s2). (iii) Hence or otherwise find the maximum speed.
Zadanie 2d
Show that .
Zadanie 2e
On the branch moving in the positive direction, write the positive velocity as v₊(s). For −A<s<A, the density has form fS(s) = k/v+(s) with k>0. Find k.
Zadanie 2f
(i) Find E(S), with justification. (ii) Interpret this value in the scanning-head model.