Zadanie 2c, Paper 3, maj 2026, TZ3, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2026, TZ3, Paper 3 (zadania wzorowane), pytanie 2(c). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
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A scanning head moves on a straight guide, with displacement measured from its central position O. One test run has metres for 0 ≤ t ≤ 2π/3 seconds. The general model is , where t≥0, A,n>0 and −π≤β≤π. Write v=ds/dt and a=d²s/dt². Restoring acceleration is modelled by a=−n²s. For an observation taken at a uniformly random phase, the displacement random variable S has density for −A<s<A and zero otherwise. This density concerns random observation times, not a change to the deterministic motion law.
Polecenie
(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.
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Odpowiedź z klucza
(i) a=dv/dt=(dv/ds)(ds/dt)=v(dv/ds). (ii) Separate and integrate v dv=−n²s ds: v²/2=−n²s²/2+C. At a turning point s=±A and v=0, giving C=n²A²/2. Therefore v²=n²(A²−s²). Alternatively use initial data s(0)=Acosβ, v(0)=−nAsinβ to determine the same constant. The separated branch identity extends to turning points by continuity. (iii) Speed |v| is largest when s=0, giving nA. Equivalently maximize |−nAsin(nt+β)|.
Schemat punktowania
(i) A1: a valid chain-rule equality establishing the relation. (ii) M1: separate variables. A1: correct integrated quadratic relation. M1: use an initial or turning-point condition to determine the constant. A1: correct condition/substitution. A1: correct constant and resulting identity. (iii) M1: identify the maximum at s=0 or maximize the trigonometric speed factor. A1: nA. The constant may also be fixed at s=0 with v=±nA, obtained from the given trigonometric motion; award the same condition method/accuracy marks. Using a general time and sin²+cos²=1 is equally valid.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Supply complete proofs and sufficient reasoning; numerical evidence alone does not establish a general identity.