Zadanie 2c, Paper 3, maj 2025, TZ2, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2025, TZ2, Paper 3 (zadania wzorowane), pytanie 2(c). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
Darmowe konto pozwoli wrócić do niego później.
Zadanie
Wspólne informacje do wiązki zadań
Three fictional research stations are called Westhaven (W), Meridian (M), and Relay (N). Compare a flat planning map with a spherical model. Bearings are measured clockwise from north.
Polecenie
In the spherical model the world has radius 5000 km. Coordinates are in thousands of kilometres, with origin at its centre. The north pole P and station N have vectors and . The positive x direction points west along the equator from N. An equatorial reference point E has vector .
For the spherical triangle EPN, define the angle at E as the angle between and .
(i) Calculate .
(ii) Prove that this spherical angle is 90°.
Korepetytor
Nie wiesz, jak ruszyć? Korepetytor zna to zadanie i poprowadzi Cię krok po kroku, bez gotowca.
3 rozmowy · bez karty · Zaloguj się
Sprawdź się
Jak poszło?
Pokaż rozwiązanieUkryj rozwiązanie
Odpowiedź z klucza
(i) (5, 0, 0) × (0, 0, 5) = (0, −25, 0). (ii) . Their dot product is zero and both vectors are nonzero, so the angle is 90°. Equivalently the magnitude of their cross product is 625, giving sine equal to 625/(25·25)=1.
Schemat punktowania
(i) M1: attempt the cross product; a sign error can retain this mark. A1: (0,-25,0). (ii) A1: (0,0,25). M1: use a scalar product between the normals, or a cross product together with the sine formula. A1: zero dot product or sine 1, establishing 90°. Allow consistent follow-through from (i), and internal follow-through if the second normal is explicitly identified.
Komentarz. Paper 3: calculators permitted. Give reproducible algebra or numerical work. Round final numerical answers to three significant figures unless stated otherwise.