Zadanie 2c, Paper 3, maj 2026, TZ2, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2026, TZ2, 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 robot makes three consecutive mutually perpendicular moves of positive integer lengths u,v,w. Its straight-line distance from start to finish is a positive integer t, so u2 + v2 + w2 = t2. Such a route is recorded as (u,v,w;t). One way to construct routes uses z1 = a + bi, z2 = c + di, where a,b,c,d are positive integers: u = ∣z1∣2 − ∣z2∣2, , , and t = ∣z1∣2 + ∣z2∣2. Only choices with u,v,w>0 are allowed.
Polecenie
Take z1 = 4 + 8i, z2 = c + 7i. (i) Show that w=16c−56. (ii) Show that the only possible positive integer c are 4 and 5. (iii) For c=5, find u,v,w,t.
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Odpowiedź z klucza
(i) w=2(8c−4·7)=16c−56. (ii) w>0 gives c>7/2, hence c≥4. Also u=80−(c²+49)=31−c²>0 gives c<√31, hence c≤5. The real-product component v=2(4c+56)>0 for these values, so both c=4 and 5 work. (iii) For c=5, z₁conj(z₂)=76+12i, |z₁|²=80, |z₂|²=74. Therefore u=6,v=152,w=24,t=154. Check: 6²+152²+24²=154².
Schemat punktowania
(i) A1: correct substitution into 2(bc−ad), establishing the given expression. (ii) M1: impose w>0. A1: c>=4, c>3, or the exact real bound c>7/2. M1: impose u>0. A1: c<=5, c<6, or the exact real bound c<sqrt31, establishing the possibilities. (iii) M1: substitute into the product or v,w expressions. A1: both v=152 and w=24. M1: substitute into the norm-sum/difference formulas. A1: u=6. A1: t=154. The norm marks are independently available from the product marks.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Give exact values or three significant figures unless requested otherwise; proofs and numerical solutions require supporting work.