Zadanie 9b, Paper 2, maj 2025, TZ3, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2025, TZ3, Paper 2 (zadania wzorowane), pytanie 9(b). 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 mapping aircraft P flies horizontally towards a straight survey strip AB of length 1.5 km. Its altitude is 4 km. The point G vertically below it lies on the extension of the strip before A, with GA = x km and x ≥ 0. Let .
Polecenie
At an instant when θ = 0.2 radians, the angle is increasing at 6 radians per hour. Find the aircraft’s speed.
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Odpowiedź z klucza
Solve on x≥0, obtaining x = 3.013167450…. Differentiate: dθ/dx = 4/[(x + 1.5)2 + 16] − 4/(x2 + 16) = −0.04951011631…. The chain rule gives 6 = (dθ/dx)(dx/dt), so dx/dt = −121.1873542… km/h. Speed is its magnitude, 121 km/h.
Schemat punktowania
M1: set the angle expression equal to 0.2. A1: x=3.01316745… . A1: correctly differentiate at least one inverse-tangent term. M1: use the chain rule relating θ, x and time. A1: correct full derivative relation. M1: substitute their x and the given angle rate. A1: positive speed 121 km/h. The middle three calculus marks do not depend on the initial root-finding marks. Technology derivative alternative: M1 substitute their x into a numerical derivative, M1 chain rule with θ,x,t, M1 substitute both rates, A2 correct speed, following the first two marks. Equivalent tangent route: tan θ=6/(x²+1.5x+16). In place of the three analytic calculus marks, award A1 for the correct derivative −6(2x+1.5)/(x²+1.5x+16)² of the right-hand side, M1 for the chain rule, and A1 for the complete relation sec²θ·dθ/dt=[−6(2x+1.5)/(x²+1.5x+16)²]·dx/dt. Retain the other four marks and the overall maximum of 7.
Komentarz. Paper 2: a graphic display calculator and the AA HL formula booklet may be used. Give exact values or three significant figures unless specified otherwise. Show mathematical or technology inputs and retain unrounded values between steps. Equivalent valid methods are accepted unless a method is specified.