Zadanie 9c, Paper 1, maj 2026, TZ1, Matematyka AA IB SL
Wzorowane na: Matematyka AA IB SL — maj 2026, TZ1, Paper 1 (zadania wzorowane), pytanie 9(c). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
A calibration rig pulls a loaded platform horizontally using a cable at angle θ to the floor. Its required force is modelled by newtons, for 0 ≤ θ ≤ π/2. The constants A and μ are determined by measurements.
Polecenie
The required force is 240 N both for a horizontal pull and when θ = π/3.
The force is least when θ=α, with 0 < α < π/2.
(i) Show that .
(ii) Hence show that α = π/6, and find F(α).
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Odpowiedź z klucza
(i) Differentiate 240 times the reciprocal of cosθ+sinθ/√3. The inner derivative is −sinθ+cosθ/√3; the reciprocal derivative supplies a minus sign and the squared denominator, establishing the stated formula. (ii) At the interior minimum F′(α)=0. The denominator is positive, so sinα−cosα/√3=0 and tanα=1/√3. In the stated interval α=π/6. Substitution gives N. The derivative changes from negative to positive there, confirming the minimum.
Schemat punktowania
(i) M1: quotient rule or chain rule for a reciprocal. A1: correct inner derivative. A1: correct full derivative establishing the expression. (ii) M1: set F′(α)=0. A1: correct zero-numerator equation. M1: rearrange to a standard trig equation. A1: tanα=1/√3 or equivalent exact sine/cosine value, establishing α=π/6. A1: correct exact substitution into F. A1: 120√3 N. The last two marks are independent of the preceding four in (ii).
Komentarz. Paper 1: no calculator. Give exact answers and show reasoning. AA SL formula booklet allowed. Equivalent valid methods are accepted unless the question specifies one.